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Biases from omitted risk effects in standard gamble utilities
Journal of Health Economics 23 (2004) 695–735
Biases from omitted risk effects in standard gamble utilities Robin Pope∗ Center for European Integration Studies (ZEI), University of Bonn, Walter Flex Strasse 3, D-53113 Bonn, Germany Received 9 May 2003; received in revised form 11 July 2003; accepted 30 September 2003
Abstract Utilities elicited under the “probability equivalence” version of the standard gamble procedure systematically exaggerate the utility increments of expensive health interventions that bring the recipient towards full health. Switching to the alternative “certainty equivalence” version would introduce the reverse bias. The biases arise from anticipated pre- and post-decision risk effects that expected utility theory omits. To include these risk effects, partition the future by changes in knowledge into: (1) pre-choice periods; (2) the pre-outcome future before the chosen act’s outcome becomes known; and (3) the more distant post-outcome future, when that outcome will have become known. © 2003 Elsevier B.V. All rights reserved. JEL classification: D81; I10 Keywords: Standard gamble; Certainty and risk effects; Commitment; “As if certain” evaluation of outcomes; Changing stages of knowledge; Pre-choice; Pre- and post-outcome periods
1. Utilities and health policy Should an individual or health provider invest in preventive procedures? For example Alexander Technique (AT) education can improve movements, reducing the chance of educees damaging their hips later. What sort of restorative procedures should be chosen in the case of those with already damaged hips? The answers to these and other health decisions depend on issues such as costs, equity, efficacy/efficiency, the chooser’s prior health care commitments—and on the improvement in the health care recipient’s utility from each intervention. ∗ Tel.: +49-228-73-18-88/73-92-18/91-40-361; fax: +49-228-73-18-09. E-mail address: [email protected] (R. Pope). URL: http://www.zei.de/.
0167-6296/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jhealeco.2003.09.006
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1.1. Risky versus sure acts We cannot, other things equal, necessarily choose the intervention guaranteeing the highest utility. This is because only some interventions have guaranteed health outcomes at the time they are chosen—are what we term sure acts. Many have more than one possible health outcome—are what we term risky acts. Decisions therefore cannot be formulated with the goal of maximising utility in a certain world. They need to be formulated with a goal that recognises the existence of risk and uncertainty. We shall not distinguish acts with risky outcomes from acts with uncertain outcomes. Rather we shall assume that all uncertainties about the health outcome Y generated by an intervention are known at the time of choosing that act in the form of probabilities p0 , p1 , . . . for each of the risky act’s finite number of possible health outcomes, Y0 , Y1 , . . . , and an exact date when Y will be learned—i.e. when all probabilities become 0 or 1.1 More formally then, in this paper an act is such a probability distribution over outcomes. At the time of choice, each sure act’s guaranteed outcome has a probability of one and all the other outcomes have a probability of zero, while each risky act has at least two outcomes with non-degenerate probabilities. 1.2. Utilities of outcomes versus the value of a risky act The evaluation of an outcome we term a utility. It is central to understanding this paper to appreciate that axiomatised expected utility theory requires choosers to evaluate outcomes excluding consideration of their actual probabilities. Under this theory, actual probabilities enter only the value of a risky act—enter as the aggregation weights for summing its mutually exclusive utilities of outcomes to form this value. 1.3. Total versus per annum utilities of outcomes We focus on the health dimension of each outcome, and term its utility the quality adjusted life years saved (QALY). Decomposing a QALY introduces complications (Richardson et al., 1996). TNevertheless, tthe QALY of a health outcome is often decomposed into: QALY = t=1 Ut /(1 + d) where T is the number of life years remaining; Ut utility in year t and d the time preference discount rate. 1.4. The events sequence matters When using per annum utilities, keep salient the sequence of events, since what follows what affects the total utility derived from a health outcome (Drummond et al., 1997, pp. 146–147). As this paper will show, it is especially important to keep salient any sequence 1 We thus ignore situations in which the chooser lacks: (1) numerical probabilities; or even (2) the full specification of the event space itself (in which case the chooser may be later surprised by a completely unanticipated health outcome). Situations with a lack of (1) were emphasised by Keynes (1921). Situations with a lack of (2) were emphasised by Knight (1933) according to Langlois and Cosgel’s (1993) re-interpretation of Knightian uncertainty. Such situations must be ignored to use standard gambles to elicit utilities.
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involving a change in knowledge from risk endured followed by risk resolved. This is because the health care recipient may experience a radically different level of utility during the two phases of this sequence. She may for instance while facing risk about her health outcome, have her utility depressed by fear below what it would otherwise have been. Then after that risk is resolved—if it is resolved in the form of the risky health intervention act having turned out to be successful—she may have her utility elevated by relief above what it would otherwise have been. 1.5. Utilities independent of knowledge ahead Let utility be measured on the vertical axis with ‘0’ for the worst quality of life derivable from a health outcome during each year. Let this utility of 0 be derived from a health outcome of death. Let the utility scale end with a maximum value of ‘1’ for the best quality of life health outcome during each year. Let this utility of 1 be reaped from perfect health. Let a set of health outcomes from the worst to the best be denoted along the horizontal axis (a big enough finite set that visually the dots denoting each distinct outcome appear as a continuous line). For those with already damaged hips, let us in Fig. 1 denote four of possible health intervention acts with their associated health outcomes: utility best 1
0.9
0.05
worst 0
Y0
death (possible) (iv) operation
. .
Y1
low mobility much pain (i) nothing
.
A
B
.
.
Y2 low mobility limited pain
Y3 moderate mobility limited pain
(ii) pain killers
(iii) Al'r Technique
health
YF outcomes full mobility (possible or certain) no pain (possible) (iv) operation
interventions
Fig. 1. Discovering utilities of outcomes—is A true or B true? Or does this knowledge-ahead-independent x-axis preclude true utilities? Note: (1) There is only a one-to-one correspondence between outcomes and sure acts. Risky act (iv) of an operation has two possible outcomes Y0 and YF . (2) If B is true, the utility increment from a successful (iv) (operation) over (i) (do nothing) is large 1 − 0.05 = 0.95—maybe large enough to prefer the operation despite its cost and risk of death. But if A is true, that increment is a modest 1 − 0.9 = 0.1—maybe too small to prefer the operation, particularly if the risk of death is sizable. Also (iii) (Alexander Technique education with no risk of death) is nearly as good as a successful operation if B not A is true. (3) The x-axis of Fig. 1 limits probabilities to being aggregation weights—weights for aggregating the utilities of alternative possible outcomes into an overall valuation of a risky act. The x-axis excludes utility function C of Table 1 wherein knowledge ahead (about which outcome will occur) affects the utilities of outcomes.
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(i) nothing with the health outcome Y1 of low mobility and much pain; (ii) pain killers with the health outcome Y2 of low mobility but limited pain; (iii) Alexander Technique education to enhance mobility with the health outcome Y3 of moderate mobility and limited pain; (iv) risky hip replacement operation with the outcome YF of full mobility and no pain (if successful, or outcome Y0 of death (if unsuccessful). The recommendation of (iv) (an expensive risky operation) depends partly on the increment to utility of a successful health outcome of operating over (i) (do nothing). The importance of discovering the true utilities of the individual health outcomes is illustrated in Fig. 1. The utility increment from the expensive operation over doing nothing could be a modest 0.1 units. This would be the case if utility is already at 0.9 in the case of doing nothing because the true curve is A. But the utility increment from the expensive operation could be a hefty 0.95 units. This would be the case if utility is a miserable 0.05 in the case of doing nothing because the true curve is B. 1.6. Utilities based on knowledge ahead It could be that the utility increment from an operation cannot be depicted by any curve on Fig. 1 because situation C of Table 1 holds. In Fig. 1, utilities of outcomes are independent of knowledge ahead. This means that each outcome causes a unique utility, allowing a curve to denote how the set of outcomes map into utility. In situation C by contrast utilities of outcomes are knowledge ahead based. In situation C, the utility of an outcome differs depending on how much the health recipient knows ahead about her health outcome—about which acts are available to her, which act will be chosen to replace the status-quo; her likelihood of a successful outcome from this intervention, and what is the case if the intervention is unsuccessful. That is, in situation C a recipient’s utility from an outcome is not based exclusively on the health outcome after it becomes known. Table 1 Utility increment from outcome YF over outcome Y1
A B C
Utility function
Increment in utility
The concave–convex curve A in Fig. 1 is the true utility shape The convex–concave curve B in Fig. 1 is the true utility shape Factors like fear mean that the utility of each individual outcome Yi is not caused by that outcome alone. It is caused partly by what the health care recipient knows ahead about: (i) available acts; (ii) which act will be chosen to replace the status-quo; and (iii) the other possible outcomes of the chosen and rejected acts
This is a mighty 0.95 This is a modest 0.10 Factors like fear mean that the utility increment is not depictable in Fig. 1. It is based also on what the health care recipient knows ahead about the set of available acts and associated other possible outcomes replacing the status quo
Note: The information in Table 1 concerns the utilities of the individual outcomes. The value of an act is constructed by aggregating (e.g. by probability weights) the utilities of individual outcomes.
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An example of situation C is Betty’s utility from the best health outcome YF . For Betty there is no unique utility caused by YF . Betty’s utility from YF is the maximum utility of 1 as depicted in Fig. 1 if YF is the outcome of a guaranteed-to-be successful hip replacement operation. But she gets a much lower utility from YF if it is the outcome of a risky operation under which she has a chance of dying. Her lower utility from YF (should it eventuate under a risky operation) is partly because of her fear of dying during the time in between choosing the operation and learning that the outcome was YF . There is an almost universal belief that utilities for constructing QALYs can be depicted with an outcomes axis as in Fig. 1 wherein outcomes determine utilities independently of what the health care recipient knows ahead definitely and probabilistically about the outcome and the entire decision situation. Situation C is rarely contemplated. Prominent minority dissenting views from the health economics perspective are in Richardson (1994, 2001) and Sen (2002). 1.7. Axiomatised generalised expected utility and Fig. 1 Axiomatised expected utility theory is not any formula that values acts as a probability weighted sum of their component utilities of outcomes. It is the formula that obeys all the restrictions that an axiom set implying this formula imposes. These are restrictions on how acts and outcomes are evaluated and given values (utilities). These restrictions include the requirement that utilities from the five health outcomes Y0 to YF can be depicted on Fig. 1—a variant of the figure in Friedman and Savage (1948). This 1948 article expounds how under axiomatised expected utility theory, outcomes are evaluated independently of the chooser’s degree of knowledge ahead of the outcome. It explains how an outcome is evaluated as having exactly the same utility if the chooser knows in advance it is guaranteed to occur, as if the chooser has in advance merely probabilistic knowledge of learning later that it has occurred. This precludes a chooser’s risk attitude affecting the utility of an outcome. From full knowledge that she has the guaranteed outcome YF of full mobility and no pain, Betty derives no more utility than if she only learns later that outcome YF has occurred. Learning later occurs whenever that outcome YF was not guaranteed, but merely probabilistic knowledge ahead at the time of choosing the operation. At that time of choosing the operation it might be that Betty only learns later whether the outcome is full mobility and no pain, YF , or death, Y0 . According to axiomatised expected utility, living through a post-choice period in which she might die in no way reduces her total utility! Expected utility theory’s constraint that the utilities of outcomes be identical under certainty and under risk recurs with most generalisations of expected utility theory. It recurs for instance in the Quiggin and cumulative prospect versions of rank dependent theories that are applied to health economics issues (Bleichrodt and Quiggin, 1999; Bleichrodt et al., 2001). This identity treatment of the utilities of risky and riskless outcomes enters such rank dependent theories via their imposition of a preference for acts with stochastically dominant (de)cumulative distributions of outcomes. Imposition of such a preference precludes choosers from considering utility derived from anything other than the outcome alone. It precludes the utility of an outcome being affected by whether a chooser’s knowledge ahead of that outcome is merely probabilistic (risky) or certain.
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1.8. The use of standard gambles to elicit utilities Utilities of outcomes that are “knowledge-ahead independent”, in conjunction with the other features of axiomatised expected utility theory, enable standard gambles to be used to deduce utilities. The numbers so elicited are the chooser’s true utilities only if the participant’s choices obey all the restrictions of axiomatised expected utility theory—including its restriction that the utilities of outcomes are “knowledge-ahead independent”. Use of standard gambles to deduce the utilities used in QALYs is widespread, Torrance (1986). The typical way health economists use expected utility theory’s standard gamble procedure is in the probability equivalent version, as follows. Describe to a participant the two possible health outcomes Y0 of death and YF of full health. Next ask that participant to select a probability number p (between 0 and 1) that would make choosing this risky operation as attractive as choosing a guaranteed outcome Y1 of low mobility + much pain. This number p1 is the utility for Y1 . Repeat the exercise for guaranteed outcomes Y2 and Y3 . The resulting probability numbers p2 and p3 are the utilities of the health outcomes Y2 and Y3 , respectively. 1.9. Systematic bias in standard gamble utilities If the standard gamble procedure for eliciting utilities is valid, the three utility numbers p1 , p2 and p3 lie on an identical utility curve under the reverse “certainty equivalent” version of the standard gamble procedure. Under this version participants select from the continuum of health outcomes intermediate safe (i.e. certain) health outcomes that they find as attractive as having the probability numbers p1 , p2 and p3 of full health and thus a probability 1 − pi , i = 1, 2, 3 of death. If choosers obey all the restrictions of expected utility theory so that the utilities elicited are their true utilities, their certainty equivalent choices are respectively Y1 , Y2 and Y3 . But this check on the appropriateness of utilities so elicited has not been made by health economists. It would have been a check to alert them to the procedure’s inappropriateness. Hershey et al. (1982) showed that the utility curve of money nearly always twists from concave to convex like A in Fig. 1 when scientists use the “probability equivalence” version, from convex to concave like B when scientists use the reverse “certainty equivalence” version. Thus, the two versions of the standard gamble: 1, by probability equivalences, and 2, by certainty equivalences, systematically and predictably yield opposite shapes of utilities in the lower outcome range, and then both inflect to yield opposite shapes for the higher outcome range. To check, put on the board a figure scaled 0 to 1 on the vertical utility axis, US$ 0–1 billion on the horizontal money axis. Then ask one colleague for three probability equivalences and insert them in this figure. Draw another figure scaled 0–50 on the vertical utility axis, £0–50 on the horizontal money axis. Ask another colleague for three certainty equivalences and insert them in the second figure.2 This normally suffices to identify a marked twist from concave to convex under probability equivalence, and from convex to concave under 2
The new scale reduces the likelihood that the second asked seeks to proffer answers consistent with the first.
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certainty equivalence, with the twists in each case occurring at the point of the respondent’s starting intermediate outcome (under probability equivalence) and respondent’s starting probability (under certainty equivalence). Subsequent work reinforces the seriousness of the discrepancies generated by these reverse twists in the utility curve under the two versions, Hershey and Schoemaker (1985). In health applications, this twisting would mean the utility of some good health outcomes are arbitrarily stretched or compressed depending on which way the standard gamble is used. In Fig. 1, curve B illustrates such stretching (curve A such compressing) for the utilities of health outcomes Y2 and Y3 relative to those of health outcomes Y0 and Y1 . Health economists typically use the probability equivalence version of the standard gamble procedure and thus attribute huge utility improvements to some expensive interventions that improve health outcomes toward full health. Were they instead to use the certainty equivalence version, they would, as in curve A of Fig. 1, tend to attribute minor utility improvements to some expensive interventions that improve health outcomes toward full health.3 Thus, any user of the standard gamble procedure in health economics can predictably twist the policy conclusions to advocate dearer interventions with better health outcomes, or the converse. Standard gamble utilities are manipulable. Major predictable changes in estimates are a hall-mark of major omitted variables bias. Progress has already been made in identifying the omitted variables in standard gamble utilities (e.g. Krzysztofowicz, 1994), and in other expected utility theory techniques for eliciting utilities (e.g. Inder and O’Brien, 2003). A decision theory is usable as an approximation if it omits minor causal chains that rarely have a discernible impact on choice so that gigantic data bases are required to detect their minute effects out of background noise. To omit causal factors that generate such major errors that their effects are transparent above random noise—even in small samples—is to virtually guarantee large scale misdirection of health budgets. The omitted variables bias in standard gamble utilities stems from expected utility theory’s omission of the knowledge-ahead-based cause–effect chains yielding pre and post-choice risk effects. 1.10. Paper format Section 2 introduces the stages of knowledge periodisation necessary to delineate the sources of the risk effects excluded under expected utility theory and many of its generali3 Hershey and Schoemaker (1985) propose that the certainty equivalent curve A is the true shape. Their erroneous belief that the standard gamble procedure includes risk attitude (Schoemaker, 1982), causes them to begin their search for an explanation with the premiss that the discrepancies between the two versions must be due to irrationality on the part of the participants (not an error in the elicitation procedure itself in irrationally excluding risk attitude). The only form of irrationality they find that fits the facts is if participants reframe the probability equivalence questions in a way that, in conjunction with some irrational loss aversion, results in their true concave convex curve being twisted into a convex–concave curve. Such an account fails to explain the twist in their perceived true curve at the point of the first answer elicited. The example they give is that if participants are asked to choose between US$ 100 for sure and a 50–50 chance of either US$ 200 or zero, participants might reframe it as a choice between $0 for sure and a 50–50 chance of either gaining or losing US$ 100, p1224. Had Hershey and Schoemaker understood that the standard gamble procedure excludes risk attitude, they would have appreciated that neither shape can be true in the sense of representing utilities of a reasonable rational person who, as they themselves state, should take risk attitude into account. Their hunt for the explanation would not have confined them to seeking some irrationality account.
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sations. Section 3 illustrates with a non-emotional risk effect, inability to commit. Section 4 examines proposals to include stages of knowledge risk effects within axiomatised expected utility theory and shows how each proposal fails. Section 5 explains how the omitted risk effects cause the systematic biases inherent in expected utility theory’s standard gamble procedure. Section 6 concludes. 2. Choice allowing for changing knowledge and risk effects 2.1. Von Neumann and Morgenstern omit risk attitude because of a contradiction For expected utility theory, Ramsey’s 1926 lecture presented a partial axiomatisation (1950), and von Neumann and Morgenstern’s 1944 book, Theory of Games and Economic Behaviour presented a full axiomatisation. Von Neumann and Morgenstern (1944, 1947, 1953, 1972, p. 28) state their regret that they have “practically defined numerical utility” . . . as excluding knowledge ahead risk effects. Morgenstern reported in 1974 shortly before his death on the deep conviction of himself and von Neumann that such knowledge ahead risk effects exist and should be included (Morgenstern, 1979). Von Neumann and Morgenstern (1947, 1953, 1972, pp. 628–632) had added an appendix to the 1947 edition of their book to explain to their critics why they had had to leave to future researchers the task of including these knowledge ahead risk effects. They reported that these “cannot be formulated free of contradiction on this level”. The contradiction encountered and described in this appendix is their puzzlement that mutually exclusive outcomes (e.g. the mutually exclusive outcomes of death and full health) interact with each other (as they do in the case of fear and inability to commit). The contradiction is solved by recognising that risky acts have at least a two period future (Pope, 1985). There is a pre-outcome period during which the mutually exclusive outcomes interact with each other since both are possibilities in the mind of the chooser. There is a sequel post-outcome period, beginning when the outcome is learned. During the pre-outcome period, people can reap a utility of gambling from juggling these mutually exclusive possible outcomes in their minds. During the post-outcome period, by memory and physical heritage that there used to be other possible outcomes, there remain effects of the prior lack of certainty. For risk to be contrasted with the certainty of sure acts at the time of choice, the risky acts require this epistemic two-period future. This two period future is one demarcated by the change in knowledge between the two periods, from a risky future, to full knowledge (in simple cases). Periodising by when in the future risk is resolved, reveals a timing contradiction in von Neumann and Morgenstern’s (1944, 1947, 1953, 1972, p. 19) assumption that all outcomes of all acts are known at the same time. In a choice set comprising a sure and risky act, the timing contradiction is as follows. Let Betty choose between these two acts at t = 0. If Betty chooses the sure act, she knows the outcome at time t = 0. But if instead she chooses the risky act, she knows the outcome at a different later time, t > 0. At time t = 0, she merely probabilistically knows the outcome of the risky act. It is only at the later time t > 0 that the risk is resolved, and she will know that one of the previously possible outcomes is the actual outcome, and that the other previously possible outcomes are now impossible.
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In order to (1) avoid the timing contradiction of assuming that choosers learn the outcomes of sure and risky acts simultaneously, and (2) reach the higher level von Neumann and Morgenstern sought (wherein mutually exclusive outcomes can interact), we need to move to a new framework. We need to move from expected utility theory’s atemporal framework (backwards decomposable into an atemporal set of axioms) to a dynamic framework in which changes in knowledge demarcate the future from the point of choice. We do this in the next sections. 2.2. Anticipated changes in knowledge and risk effects At the time of deciding under risk, choosers are ignorant of which of various possible outcomes will ensue. Choosers can however anticipate changes in what they know, even though they cannot be confident whether the new knowledge will be good or bad. When Betty encounters the problem of a mushy hip, she can anticipate that after not knowing what are her available choices, she will later ascertain these, and after not knowing what to choose, that she will later have chosen. If she chooses a risky hip replacement operation, she can anticipate knowing later what she does not know now, that it is successful, and also can anticipate knowing nothing later, because her risky hip operation is unsuccessful and she dies. See Table 2. Table 2 Three anticipated changes in knowledge Ki , i = 1, 2, 3 after encountering a problem Time
Knowledge with respect to a future or past event
K1 From
Pre-choice set period
To
Choice set identified
Only probabilistic knowledge of what the sub-acts of search/negotiation in ascertaining and creating available acts will identify as the choice set, e.g. unclear which interventions are feasible for Betty’s damaged hip Knowing with a probability of 1 the choice set, e.g. which intervention acts are feasible
K2 From
Pre-act choice period
To
Act chosen
K3 From
Pre-outcome period
To
Post-outcome period
Only probabilistic knowledge about what the sub-acts of evaluating the available acts will identify as the act to choose, e.g. uncertain which intervention act will turn out to be the best Knowing with a probability of 1 the act chosen, e.g. chose a hip replacement operation Only probabilistic knowledge of which will be learned to be the chosen act’s actual outcome, e.g. the outcome could be full mobility and no pain, or it could be death Knowing with a probability of 1 the chosen act’s actual outcome, e.g. knowing that the risky operation was successful, or dead
Risk effects (such as those arising out of Betty’s fear) arise out of anticipated possible changes in probabilistic knowledge from being uncertain whether the operation will be successful to knowing whether the operation was successful. If there could never be this change in knowledge, Betty has no fear she will die. Without any anticipated future changes in knowledge, Betty and all of us have chosen a future perceived as certain with regard to
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its effects on us—chosen a sure act. Without any anticipated future changes in knowledge, we experience no risk effects from unknown outcomes. In the simplest risky choices illustrated in Table 2, anticipated changes in knowledge causing risk effects are changes from merely probabilistic knowledge to full certainty. In this table, each future change in knowledge creates at that future date a disparity between: 1, what was previously a set of possible mutually exclusive events and associated utilities (of which the chooser had a non-degenerate probabilistic knowledge); and 2, the chooser’s new knowledge at that future date, namely, (a) full certainty about the actual occurrence of one previously possible event, and (b) full certainty of the non-occurrence of all the other previously possible mutually exclusive events. Anticipated future changes in knowledge generate anticipated risk effects such as fear. Risk effects influence choices if and only if anticipated effects of changes in the probabilistic knowledge impact on the chooser’s evaluation of acts and outcomes. As shown in Section 1, in axiomatised expected utility theory (and in its generalisations to the rank dependent models of Quiggin and cumulative prospect theory) acts and outcomes are evaluated independent of Ki , i = 1, 2, 3—i.e. independent of knowing ahead the choice set, the act, and which outcome will occur—and thus excluding all pre- and post-decision risk effects. Theories that exclude anticipated pre- and post-decision risk effects are simpler. But as in the well-known comment of Einstein, a theory should be simple, but not too simple. By today, we know that expected utility theory’s standard gamble procedure to elicit utilities has a hallmark of being too simple, a hallmark of major omitted risk effects. 2.3. Old misleading and new neutral terminology Misleading terminology has hindered scientists discerning the seriousness of omitting the anticipated effects of pre- and post-decision risk effects. The name most often used for the post-decision risk effects of K3 excluded under expected utility theory is the “(dis)utility of gambling”. This term has the disadvantage that Friedman and Savage (1948) created confusion by altering it to mean a (concave) convex “as if certain” utility of monetary outcomes function. This name has the further disadvantage of frivolous (even derogatory, unethical, irrational) connotations. These misleadingly create the impression that the (positive and negative) satisfactions omitted under expected utility theory are minor—or even desirable ones to omit. There are disadvantages in using terms “risk attitude” and “risk aversion” (preference) for the downgrading (upgrading) of the value of assets that have higher riskiness as measured by the dispersion of the assets returns. The disadvantages are that Marschak (1950) confusingly changed risk aversion (preference) to mean a concave (convex) “as if certain” utility function of asset outcomes. Marschak’s terminological changes became ensconced in the construction and naming of the Arrow–Pratt measures of risk aversion (e.g. Arrow, 1971; Pratt, 1964). Pope (1996/1997, 2001) lists other names used for the anticipated post-decision risk effects of K3 and confusing changes in terminology or misleading connotations associated with each, then introduces a new terminology to avoid the confusions/misleading connotations. Under this new terminology, satisfactions derived from the outcomes alone
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are termed primary. Primary satisfactions are knowledge ahead independent in the sense of the utilities depicted in Fig. 1. Such utilities are satisfactions from learning that a particular outcome has occurred (or will occur) which are independent of whether that outcome was at the point of choice: certain having a probability of 1; very unlikely having only a probability of 0.05; and better or worse than other possible outcomes that instead might have occurred (yet occur). Secondary satisfactions by contrast are based on such degrees and aspects of knowledge (Table 3). Table 3 Secondary satisfactions anticipated at the time a problem requires action Event
1 Pre-act search 2 Choice set identified 3 Pre-act evaluation
4 Act chosen 5 Background conditions
6 Learn outcome 7 Background conditions
When satisfaction reaped
Source of secondary satisfaction, probabilistic knowledge of an event or its wisdom
Pre-choice set period: before stop Uncertainty about which acts are search for available acts available and what will ensue Time point following event 1 Pre-choice period: before end Uncertainty about which act to valuation of available acts choose and certainty or uncertainty on wisdom of 1–2 and what will ensue Time point following event 3 Pre-outcome period: before Uncertainty or certainty about outcome becomes known outcomes and wisdom of 1–4 and what will ensue Time point following event 5 Post-outcome period: after Prior uncertainty or certainty about outcome has become known outcomes and wisdom of 1–4
Primary satisfactions from the events 1–7 themselves, i.e. independent of the knowledge ahead based sources listed in the last column. Secondary satisfactions from the events are those dependent on chooser’s prior or current probabilistic knowledge of events 1–7 or their wisdom, i.e. dependent on the knowledge ahead based sources listed in the last column. Note: (1) Satisfactions include both positive ones and negative ones (i.e. dissatisfactions). (2) Satisfactions denote levels of attainment of the chooser’s goals. These need not be solely or at all related to personal degrees of pleasure or pain, but could be entirely satisfactions with respect to attainments or features of another individual or of society or the planet, etc. (3) Making a choice involves a series of prior choices, here termed pre-acts. (4) It is only in the simplest risky choice situation that there is a single pre- and post-outcome period. In more complex cases, there are multiple pre-outcome periods. Multiple pre-outcome periods occur for instance whenever aspects of the outcome get learned at different times in the future. (5) For a sure act, the pre-outcome uncertain period is degenerate, of zero duration. (6) Axiomatic derivations of expected utility theory contain no entities denoting background conditions. Hence, a pre-requisite for its axiomatic justification is that all relevant background conditions must be subsumable into an elaborated specification of individual outcomes of an act. Background conditions which are sources of primary satisfactions can be so subsumed. By contrast background conditions which are sources of secondary satisfactions cannot be so subsumed: elaborating outcomes to include secondary satisfactions destroys that theory’s axiomatic basis, as also its applicability to multi-stage acts (Pope, 1983, 1984, 1989, 2000).
In Table 3, the positive and negative satisfactions from the risk effects that “dominance preserving” theories exclude are no longer derogatorily termed utilities of gambling. Rather they are given the more neutral name of (positive and negative) secondary satisfactions.
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The pair of names is not intended to give the impression that secondary satisfactions are unimportant relative to primary satisfactions, ones that as a first approximation can be ignored. To the contrary in many decision contexts, secondary satisfactions are quantitatively more important. Table 4 Changing knowledge demarcating six sources of emotional secondary satisfactions Problem at time t = −1 s ↑ Pre-decision period (act unknown) ↓
Anticipated future 1
Bad stress, e.g. time lost choosing and Good stress, e.g. brain growth from challenge of choosing
Act choice at time t = 0 ↑
2
Pre-outcome period (act’s outcome unknown)
3
Fear that one of the act’s possible bad outcomes may come and Hope that one of the act’s possible good outcomes may come Depression at guaranteed bad aspects of the future and Joys in guaranteed good aspects of the future
↓ Learn outcome at time t = 1 ↑
4
Disappointment that an act’s good outcome did not occur or Thankfulness that an act’s bad outcome did not occur
5
Regret and blame since rejected acts had better outcomes or Rejoicing and praise since rejected acts had worse outcomes
Post-outcome period (act’s outcome known) 6
Bad shock at act’s completely unforeseen bad outcome or Good shock at acts’ completely unforeseen good outcome
↓ New problem at time t = 2 1–6 help in this next pre-decision period, in making the next choice at t = 3. We reason, choose partly through the emotions. Brain damage removing fear removes good decision making (Demasio, 1984) See on emotional sources of secondary satisfactions in the pre-decision period: Janis and Mann (1977), Lazarus (2000), Somerfield and McCrae (2000) and Hardie (2002); pre-outcome period: Plato (1967) and Smith (1778), Darwin (1874), Canaan (1926), Bernard (1963), Hart (1930), Pope (1983, 1984), Wu (1991), Rothenberg (1992, 1994), Schade and Kunreuther (2000) and Inder and O’Brien (2003); post-outcome period: von Neumann and Morgenstern (1944), Savage (1954), Markowitz (1959), Loomes and Sugden (1982), Bell (1983), Hagen (1985) and Berns et al. (2001).
Table 4 instances emotional sources of secondary satisfactions and locates these in the evolving stages of knowledge anticipated on encountering a problem. In Table 4, and henceforth, we have collapsed the two pre-choice periods (that of selecting a choice set and choosing within it) into one period. This collapse is for brevity, not to downplay the significance of pre-decision risk effects. Consider for instance the evidence that some health insurance is taken out to avoid the stress of making financial decisions after getting ill and needing health help.
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There is extensive medical evidence of the impact of emotional states on health, including those pertaining to the emotional sources of secondary satisfactions in Table 4. But since some believe that emotional effects should not enter public health decisions, it is important that health economists appreciate that secondary satisfactions stem also from non-emotional sources. One such source is an (in)ability to commit. To delineate its role, let us look at an amalgam of the decisions that recently faced the author’s mother Betty (who had a mushy hip) and a brother-in-law helping administer AIDS projects in Africa.
3. A non-emotional source of secondary satisfaction in a health choice Betty’s problem of a “mushy” hip joint yields her QALY (quality adjusted life years) with the two features depictable on the knowledge ahead independent health outcomes axis of Fig. 1, namely: (a) pain and (b) low mobility, plus (c), a knowledge ahead based commitment effect of her QALY not depictable on the x-axis of Fig. 1. A person committing at time t = 0—as distinct from a person lying and merely saying that they will commit—must know at t = 0 with probability 1 that nothing ahead will preclude that commitment. (We shall deal later in Section 4.1 with the secondary satisfactions associated with partial, conditional commitment effects.) In Betty’s case, let the commitment concern the project described in (c) of Table 5. It encapsulates a vivid commitment effect problem in communities where the AIDS inroads are severe: all the qualified care givers are infected. Their considerable risk of being too ill to serve for long precludes them from being able to elicit the level of co-operation from others in the community to introduce some sorts of AIDS amelioration projects. Correspondingly, inability to commit creates problems for organisations fostering community health, such as the World Bank, the World Health Organisation, local governments and non-government organisations. Table 5 Betty’s decision situation (a) (b) (c)
(d)
Her hip hurts, reducing her QALY below that gained from no pain She has restricted mobility, reducing her QALY below that gained from full mobility Her other potential source of QALY is a community job. To do it, she must tomorrow commit unconditionally to work on it for the next 5 years (apart from 1 month per annum of holiday/sick leave), something she has already decided to do if her hip decision does not preclude her committing She must decide tomorrow on her health intervention act
In reading (c) bear in mind commitment’s knowledge ahead prerequisites. In particular, note that a necessary condition for Betty tomorrow to have her desired ability to commit to the project (i.e. guarantee that she will do it), is that tomorrow Betty knows with probability 1 that for these next 5 years, nothing will prevent her working full time on it. Let us exclude four common features of actual decisions that expected utility theory excludes: (i) inherently multi-dimensional goals, (ii) erroneous anticipations about an act’s possible outcomes and the satisfactions to be reaped from these outcomes and associated acts and related costs of decision making,
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(iii) satisfactions indices that cannot be mapped into real numbers, (iv) choosers ignorant of how to rank acts in terms of the satisfaction yielded them. Let us for simplicity also exclude choosers who have considerable merit with respect to being practicable and rational, namely choosers who satisfice—i.e. take any act that reaches a satisfactory level of utility (see, e.g. Simon, 1983). Also let us exclude another common feature of many actual decisions, namely choosers whose satisfactions we deem irrational. With these exclusions, we can offer an intuitive decisive gold standard, a sufficient condition for identifying when a choice theory selects an irrational act (Table 6). Table 6 Intuitive decisive gold standard: sufficient condition for an irrational choice theory If Then
For any choice set, we know with certainty at time of choice that another available act than that chosen by our theory would yield Betty a better QALY Our theory is irrational
What makes Table 6 uncontroversial is that it avoids the contentious issue of: 1. whether the only rational summation rule for forming an overall valuation of a risky act is to aggregate its mutually exclusive utilities by their probabilities (as happens with a number of decision procedures including expected utility theory) or 2. whether alternative aggregation rules may be as reasonable or even more reasonable in some circumstances as Keynes (1921), Allais (1952, 1979a,b, 1986) and others have argued. Other aggregation rules are used in many generalisations of expected utility theory and in some non-nested alternatives to expected utility theory. We shall show that axiomatised expected utility theory fails this intuitive gold standard under both versions that are internally consistent with its own implication that utilities be a function of outcomes alone as in Fig. 1. These are the versions of Ramsey (1950), that outcomes be evaluated independently of knowledge ahead, and of Friedman and Savage (1948), that outcomes be evaluated “as if certain”. We shall show that the failure to meet this intuitive gold standard stems from the omission of actual secondary satisfactions. We do this in stages. First, we specify Betty’s decision procedure for choice sets comprising only sure acts and make simplifying assumptions to allow an easy partition of Betty’s satisfactions into primary and secondary ones. We present a scenario 1, a choice set under certainty, and show that, whereas the Ramsey version of expected utility theory fails it, both Betty’s decision procedure and the Friedman–Savage version of expected utility theory survive it. But under certainty there is no need for expected utility theory. The Friedman–Savage version survives the gold standard since it does not impute illusory secondary satisfactions under its “as if certain” rule. All outcomes really are certain. We next introduce a scenario 2, a new choice set which includes a risky act, something realistic in health choices, and also essential for the application of the standard gamble procedure. We elaborate on Betty’s decision procedure to include the evaluation of risky acts. We show that for the particular scenario 2 choice set examined, both the Friedman–Savage version and the Ramsey version fail the intuitive gold standard whereas Betty’s decision procedure survives it. The intuitive gold standard has then been used for two particular choice sets, with a particular constellation of satisfactions. We then explain the general implications of this finding of the irrationality of axiomatised expected utility theory. The
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sequel section (Section 4) examines proposals to overcome the irrationality by means of opportunity cost, elaborated outcomes, and different valuing of outcomes. It shows that each proposal for including fear or commitment destroys expected utility theory’s axiomatic base—destroys its unique mapping from outcomes into utilities that is required in order to use standard gamble procedures to elicit utilities. First, let us introduce Betty’s way of choosing among sure acts, Table 7. Table 7 Betty’s choice procedure under certainty 1
Betty decides on her objective, to maximise U = U1 + U2 + U3 + U4 + U5
2 3
4
(1)
where U is an index of the QALY caused by her act choice, Ut is that act’s satisfaction in year t, t = 1, . . . , 5, and Ut is normed with 0 her worst conceivable satisfaction, and 1 her highest Betty discovers which acts are available to her Betty, who has no constraints, costs or benefits from her pre-decision analysis, anticipates today precisely her U caused by each available act. The fact that she anticipates satisfactions from all the actual cause–effect chains initiated if she were to choose each act and traces these through to ascertain each act’s U is indicated with the script typeface. (An ordinary typeface U will denote what satisfactions Fig. 1 and expected utility theory permits/requires Betty to include.) At her time of choice, t = 0, Betty chooses an act with highest U
For simplicity in depicting the partitioning of Betty’s total QALY index U between its component primary and secondary satisfactions, let these be additively separable, i.e. let: Ut = Pt + St (2) where Pt is a cardinal index of anticipated primary satisfactions in year t, St is a cardinal index of anticipated secondary satisfactions in year t. We can now decompose U—the sum of Ut over 5 years and index of Betty’s QALY caused by her chosen act. We can decompose it into her primary satisfaction over the 5 years plus her secondary satisfaction over the 5 years. That is, ifs P = P1 + P2 + P3 + P4 + P5 (3) and S = S1 + S2 + S3 + S4 + S5
(4)
U=P +S
(5)
then
Let Betty discover that her available acts are those of Table 8. Table 8 Betty’s choice set: scenario 1 under certainty HR+ Hip replacement operation plus intensive physiotherapy: This yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation followed by two consecutive months of sick leave doing supervised exercises, then full health HR Hip replacement operation with minor physiotherapy: This yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation requiring only 2 weeks of sick leave, then nearly full health AT Alexander Technique education to reduce her asymmetric posture and movement: This yields her an outcomes flow of small steady increments in mobility and decreases in pain from her after work education in this technique
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Table 9 Betty’s anticipations of her primary and secondary satisfactions when choosing tomorrow Available sure acts
Outcomes, knowledge effects, satisfactions flows
QALY
Hip replacement operation plus intensive physiotherapy Outcome flow, YHR+ Year 1, queue Year 2, new hip, long leave Years 3–5, full health P1 = 0.1 P2 = 0.3 P3 = 0.5 P4 = 0.5 Primary satisfactions, Pt Effect flow of YHR+ known Years 1–5, no job since knows she will take long sick leave in year 2 S1 = 0 Secondary satisfactions, St S2 = 0 S3 = 0 S4 = 0 = P t + St U1 = 0.1 U2 = 0.3 U3 = 0.5 U4 = 0.5 Satisfaction in year t, Ut QALY, U = Ut = P + S
P5 = 0.5
P = 1.9
S5 = 0 U5 = 0.5
S=0 P + S = Ut U = 1.9
P = 1.48
HR Hip replacement operation with minor physiotherapy Outcome flow, YHR Year 1, queue Primary satisfactions, Pt P1 = 0.1 Effect flow of YHR known Secondary satisfactions, St S1 = 0.5 = P + S U1 = 0.6 Satisfaction in year t, U t t t QALY, U = Ut = P + S
Year 2, new hip, short leave Years 3–5, nearly full health P2 = 0.33 P3 = 0.35 P4 = 0.35 Years 1–4, progress on full time 5 year job S2 = 0.5 S3 = 0.5 S4 = 0.5 U2 = 0.83 U3 = 0.85 U4 = 0.85
P5 = 0.35 Year 5, job done S5 = 0.5 U5 = 0.85
AT Alexander Technique Outcome flow, YAT Primary satisfactions, Pt Effect flow of YAT known Secondary satisfactions, St Satisfaction in year t, Ut = Pt + St QALY, U = Ut = P + S
Years 1–5, annual, ↑ in mobility↓ in pain as better gait, no leave P2 = 0.15 P3 = 0.2 P4 = 0.25 Years 1–4, progress on full time 5-year job S2 = 0.5 S3 = 0.5 S4 = 0.5 U2 = 0.65 U3 = 0.7 U4 = 0.75
P5 = 0.3 Year 5, job done S5 = 0.5 U5 = 0.8
P1 = 0.1 S1 = 0.5 U1 = 0.6
S = 2.5 P + S = Ut U = 3.98
P=1 S = 2.5 P + S = Ut U = 3.5
HR > AT HR+ : HR better than AT which is much better than HR+ . Betty’s rational choice is HR. In calculating her QALY (quality of life U), the Ramsey version of axiomatised expected utility theory limits Betty to P, primary satisfactions, i.e. has her irrationally choose her worst act HR+ .
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Let the cause–effect chains initiated by each available act, the contributions to Betty’s satisfaction of each of these effects and her QALY index U from each act be as in Table 9. Betty’s utility from primary satisfactions is what she derives each year from her amount of pain, mobility and sick leave independently of knowing beforehand what these amounts would be. The amount of such utility is recorded in the second row of Table 9 for each of her three available acts, HR+ , HR and AT. Any utility from positive secondary satisfactions is what she derives from already knowing at her point of choice that her future hip outcome flow involves sufficiently little or no sick leave and so is able to commit and take on her project job. The amount of such utility is recorded in the fourth row for each of the three interventions, HR+ , HR and AT. Her total utility from the combined effects of her primary plus secondary satisfactions is recorded in the last row for each act. Comparing rows 2 and 4 for each act, it can be seen that Betty’s secondary (knowledge ahead based) satisfactions are quantitatively bigger than her primary satisfactions. Readers may realise from introspection that there are many occasions and choosers for whom this will be the case. Step 3 of Betty’s procedure of Table 7 enjoins her to include in her evaluation of each act all cause–effect chains initiated by the act that enter her QALY. Thereby she includes in her calculation of her QALY both primary and secondary satisfactions. Betty makes her best choice of HR and obtains her highest possible QALY of 3.98. Consider now Betty’s choice if in step 3 of Table 7 she excludes her secondary satisfactions. This is Ramsey’s version of expected utility theory. His version allows her to have the requisite feature of Fig. 1, namely that (for a given origin and scale), her mapping from outcomes into utilities is unique. She would choose HR+ since it is the act with the highest amount of primary satisfactions. But due to its associated extended sick leave, she would be unable to commit and undertake the project job. She would be choosing the act with the guaranteed lowest total QALY. The Ramsey version thus fails the intuitive gold standard. Consider next the Friedman–Savage version of expected utility theory. This is that outcomes be evaluated “as if certain”. It meets the intuitive gold standard and chooses exactly as does Betty. This is because the choice set comprises three sure acts, rendering irrelevant the “as if certain” means of constraining utilities of outcomes to be identical under risk and certainty. But once we alter the choice set to include a risky act, “as if certain” evaluations can impute satisfactions that are illusory under risk. If the illusorily imputed positive satisfactions are large enough, these constrain Betty to make a choice guaranteed to yield her a lower number of utils than had she made a different choice. To illustrate, let Betty be faced with a choice between the risky act RHR and the sure act AT of scenario 2, Table 10. Table 10 Betty’s choice set: scenario 2 under risk RHR
AT
Risky hip replacement operation with minor physiotherapy: This yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation in which she has a 5% chance of dying, and a 95% chance of success in which case she will require only 2 weeks of sick leave, then nearly full health Alexander Technique education to reduce her asymmetric posture and movement: This yields her an outcomes flow of small steady increments in mobility and decreases in pain from her after work education in this technique
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Betty now needs a more elaborate decision procedure than that of Table 7 wherein, step 4, she chooses the act with the highest U, her index of QALY (sum of Ut over 5 years). This is because for risky act RHR, her U lacks a single guaranteed value: there are two mutually exclusive Us that may occur. In order to compare her overall valuation of risky RHR with her guaranteed level of QALY from sure act AT, Betty modifies step 4 of Table 7 to incorporate a rule for aggregating RHR’s two mutually exclusive Us, Table 11. Table 11 Betty’s choice procedure under risk 1
Betty decides on her objective, to maximise U = U1 + U2 + U3 + U4 + U5
2 3
4
(1)
where U is an index of the QALY caused by her act choice, Ut is that act’s satisfaction in year t, t = 1, . . . , 5, and Ut is normed with 0 her worst conceivable satisfaction, and 1 her highest Betty discovers which acts are available to her Betty, who has no constraints, costs or benefits from her pre-decision analysis, anticipates today precisely her U caused by each available act. The fact that she anticipates satisfactions from all the actual cause–effect chains initiated if she were to choose each act and traces these through to ascertain each act’s U, is indicated with the script typeface. (An ordinary typeface U will denote what satisfactions Fig. 1 and expected utility theory permits/requires Betty to include.) At her time of choice, t = 0, Betty chooses an act with highest expected U
In step 4 of Table 11, it can be seen that, like axiomatised expected utility theory, Betty uses probability weights to aggregate mutually exclusive utilities (QALYs) of a risky act to form her overall valuation of a risky act. But do not be deceived into thinking that she has adopted that theory. In understanding how axiomatised expected utility theory places a value on risky act RHR, it is critical that health economists bear in mind the difference between the value of a risky act and the utilities of its component utilities of outcomes. Step 3 in Table 11 is what distinguishes Betty’s evaluation of a risky act from that of axiomatised expected utility theory. In step 3, she rationally anticipates and includes (in her evaluation of their impact on her QALY index) all the actual cause effect chains of each possible mutually exclusive outcome. By contrast axiomatised expected utility theory’s constraint that utilities of outcomes to be the same under risk and certainty, requires Betty to omit some actual cause–effect chains of outcomes (the Ramsey version) and in addition to impute some illusory cause–effect chains of outcomes (the Friedman–Savage version). Because the physiotherapy is minor, not needing a long absence from work, under sure act HR, Betty could commit. Under risky act RHR by contrast, these positive secondary satisfactions of being able to commit are non-existent. Instead Betty has negative secondary satisfactions in the pre-outcome year from fear of dying. Let Betty’s primary satisfactions outcomes flow under RHR be identical to that under sure act HR in year 1 (the pre-outcome period), and thereafter: zero if she is dead; those of HR for years 2 to 5 if the operation is successful. Then from Table 8, we can see that even if Betty learns afterwards that the outcome of the hip replacement operation is success, RHR will net her 1.48 utils in primary satisfactions, and that her QALY will be less than this by the amount of her negative secondary satisfactions from fear. Betty’s decision procedure, Table 11, avoids adding in certainty effects when these are absent, and ensures that risk effects are added in when present. Under risky RHR, her Table 11 decision procedure recognises that her utils are reduced (compared to sure HR)
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each year by her inability to commit, and in addition further reduced by her fears in her pre-operation year (her pre-outcome period). Her Table 11 decision procedure computes her utility as it actually is, namely less than 0.1 in her pre-operation year (due to her fear), and an expected 0.95 × (0.33 + 3[0.35]) ≈ 1.31 utils out of her three post-operation years. This amount of less than 0.1 + 1.31 ≈ 1.41 utils, is below her guaranteed 3.4 utils under the safe Alexander Technique, sure act AT. Betty’s decision procedure for valuing outcomes allows her to reasonably choose AT. But the Friedman–Savage “as if certain” evaluation of outcomes would force Betty to ignore risk in evaluating the outcome of a successful operation under RHR. From Table 9, it would force Betty to impute to RHR the “as if certain” utility of 3.98 that she would reap under sure act HR. The Friedman–Savage “as if certain” evaluation of RHR has two sources of error. One is that its imputed utility to risky act RHR excludes actual negative satisfactions, Betty’s fear. It ignores how fear reduces Betty’s pre-outcome utils. The other source of error is that it includes non-existent positive secondary satisfactions from Betty’s ability to commit to her full time 5-year job. This commitment does not exist. The combination of excluding actual negative secondary satisfactions from a risk effect (fear) plus adding in illusory positive secondary satisfactions from a non-existent certainty effect (ability to commit) causes Betty to inflate the value of RHR to 3.81 and choose it, even though RHR is in reality guaranteed to yield her a lower sum of utils than AT, to in reality yield less than 1.48 utils even if successful, as against AT’s guaranteed 3.4 utils. The Friedman and Savage version of expected utility theory thus fails the intuitive gold standard. On analogous illusory effects of certainty in Savage’s sure-thing principle, see Pope (1991a). Under a choice between RHR and AT, the Ramsey version of expected utility theory, similarly ranks RHR higher and thus fails the intuitive gold standard. It would consider only primary satisfactions. From Table 9, these would be 1.48 utils under RHR, only 1 util under AT. It might be thought that there could be other ways of imposing expected utility theory’s constraint that utilities of outcomes are identical when outcomes are risky and certain. It might be thought that at the time of evaluation and choice of act, there are intermediate probability values above zero—which excludes valuing any outcomes—and less than 1—the probability number used in Friedman and Savage (1948)—that meet this constraint. For instance it might be thought that the constraint can be met by having each outcome, no matter what its actual probability of occurrence, evaluated as if its probability of occurring were 0.5. To see that this cannot be the case, note that to be afraid of dying, death and better outcomes must be within the act’s set of possible outcomes. To be unable to commit to a job, extensive time off work must be within the set of outcomes. Thus, an outcome’s associated secondary satisfactions depend not only on itself, but partly also on the other possible outcomes—on whether the alternative possible outcomes (whose probabilities sum 0.5) lie above or below, or on both sides of this outcome, and on their range. There are no other alternative unspecified outcomes to consider in the Friedman–Savage version (each outcome is evaluated as if it is certain). But evaluating each outcome as if it has a probability of 0.5 means that there are alternative unspecified outcomes. Each different set of alternative outcomes will yield a different set of associated secondary satisfactions. The mappings from outcomes into utilities will be non-unique (for a given origin and scale). Their being
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non-unique precludes such utility mappings being versions of axiomatised expected utility theory. Expected utility theory thus fails the intuitive gold standard on the only two versions that preserve its required unique mapping from outcomes into utilities (for a given origin and scale). It failed in the particular decision situations considered. These were engineered to demonstrate that theory’s irrationality in general. The examples focussed on positive secondary satisfactions from certainty effects since these will be central to the biases generated in the particular choice sets used in standard gamble elicitations. This does not mean that secondary satisfactions are primarily negative in the case of risk effects (i.e. positive in the case of certainty effects). To the contrary, as expounded and empirically demonstrated elsewhere (Scitovsky, 1976; Albers et al., 2000; Pope, 2001, 2002), in some decision situations the predominance of secondary satisfactions are positive for risk effects. In the spectrum of positive and negative satisfactions generated by risk effects and illustrated in Table 4 above, it is an empirical question whether the positive or the negative dominate in particular a decision situation. Human physiology and associated financial and other resource issues aid in predicting in which decision situations the positive secondary satisfactions from risk effects dominate. The fact that both versions of expected utility theory here failed the intuitive gold standard for rationality does not mean that both invariably fail it. Whether they fail it depends on the relative magnitude of primary and secondary satisfactions under different acts in the choice set. There are situations in which, relative to primary satisfactions, the secondary ones (actual and illusory) are sufficiently minor that both versions pass the intuitive gold standard. What the above failures of expected utility theory prove is that it is an accident of the choice set and the chooser’s level of secondary satisfactions whenever that theory passes the intuitive gold standard, not a consequence of expected utility theory being rational. In a similar way, the accident of a dice throw can sometimes select an act that survives this intuitive gold standard. But we do not suggest that reasonable choosers make their decisions by throws of a dice.
4. Proposals for expected utility theory to meet rationality standards 4.1. Opportunity cost included by elaborating acts, outcomes in the given situation Some may feel that sources of secondary satisfactions like commitment effects can be incorporated into expected utility theory by means of treating them as opportunity costs. An opportunity cost of an act is the best alternative. To determine the best alternative to each act in the choice set, it is first necessary to rank the acts, and thus evaluate (determine the utility of) each act separately. The value of an act minus its opportunity cost is its utility minus that of its best alternative. This value is positive in the case of a uniquely first ranked act, zero for equally first ranked acts and negative for acts ranked below the first.4 Under 4 The concept of opportunity cost is thus redundant if, as in this paper, the alternative acts are specified explicitly. The concept of opportunity cost is widely used in economics texts and cost benefit analyses when instead the
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certainty (scenario 1 and Table 8), the Ramsey version of expected utility theory failed the intuitive gold standard, and thus failed to correctly evaluate the best alternative act. Under risk (scenario 2 and Table 10), both versions of expected utility theory failed to identify the best alternative act. Now to specify an act involves specifying its outcome flow(s). Are there other ways of specifying Betty’s outcome flow(s)—and hence acts—than as in Tables 8 and 10? If we specify some acts as involving a commitment outcome flow, can expected utility correctly incorporate the opportunity cost and reach a rational choice? An answer of yes is one interpretation of Luce and Raiffa’s claim that under axiomatised expected utility theory “the given situation” is taken into account in the utilities, 1957, p. 15. This claim is correct if interpreted as pertaining to primary satisfactions. These can be included in how we define (or re-define or elaborate) acts and hence outcomes (sources of primary satisfaction) without destroying the axiomatic base of expected utility theory, e.g. the amount of sick leave. In Fig. 1, there is no mention of the amount of time unavailable for work. Table 12 gives elaborated outcomes, elaborated to include this amount. Table 12 Betty’s conceivable sure acts and associated outcome flows over her 5-year time horizon specified independently of her knowing whether they are to occur when she chooses tomorrow Sure act
Outcome flow
Suicide Operation dirty hospital Do nothing Pain killers Alexander Technique Hip replacement operation with minor physiotherapy Hip replacement operation + intensive physiotherapy Hip replacement operation now
Y0 Y0 + YDN YPK YAT YHR YHR+ YF
Death the day after tomorrow Year’s wait, elective surgery, death within year of surgery Annual mobility decrements, pain increments, ever able to work Annual mobility decrements, limited pain, ever able to work Annual mobility increments, pain decrements, ever able to work Year’s wait for elective surgery as public patient, short period unable to work, thereafter nearly full health Year’s wait for elective surgery as public patient, 2 months unable to work, thereafter full health Surgery today, 1 day unable to work, thereafter full health
Time unavailable for work is a source of primary satisfaction as is a hip mobility and pain level. Any configuration of a degree of hip mobility, pain and time unavailable for work can be combined with any set of probabilities. None creates contradictions between the probabilities denoting knowledge of outcomes and the outcomes themselves. Table 12’s specification of sure acts and associated outcome flows does list some aspects of Betty’s decision situation that affect her satisfaction. But the list defining these sure acts in terms of their outcome flows makes no mention of those sources of satisfaction for Betty that depend on her knowing which act or outcome she will choose tomorrow. Thereby the choice set remains unspecified. These are situations in which it is assumed implicitly that the alternative act to a project under consideration can be decomposed into independent decisions about multiple sub-acts, e.g. three sub-acts concerning the use of labour, capital, land if these three resources were not employed on the project under investigation. It may be deemed that each would otherwise be used in competitive markets in which case their respective opportunity costs can be separately assessed as their market values. This saves the analyst the trouble of identifying the particular alternative usage to which each resource would be put and constructing explicitly the other acts (and associated sub-acts) in the choice set.
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list excludes aspects of the situation which depend on Betty’s probabilistic knowledge at her time of choice tomorrow of whether: (a) she will either die over the next 5 years, (b) be unable to work for any extended time interval. But a necessary condition for Betty having her desired ability tomorrow to commit to a project full time over the next 5 years (i.e. guarantee that she will do it) over this entire interval, is that she knows tomorrow with probability 1 that she has avoided all acts which imply a positive probability of (a) or (b). Table 12’s specifications of sure acts and associated outcome flows includes whether (a) and (b) will ensue, but does not include the necessary condition for Betty to commit, namely her knowledge ahead with probability 1 that (a) and (b) are excluded. By excluding information about this knowledge ahead, Table 12 excludes reference to Betty’s commitment source of secondary satisfaction. Axiomatised expected utility theory requires the specifications of acts (and hence outcomes) to be knowledge ahead independent in precisely the manner of Table 12. See e.g. Savage (1954) and Tables 14 and 16 below. Knowledge ahead independence precludes including commitment as an opportunity cost by inserting it in the specification of the act (i.e. specifying it as part of the act’s outcomes flow). To see that expected utility theory’s standard gamble procedure requires this independence, suppose we augment outcomes with knowledge ahead based sources of secondary satisfaction such as commitment considerations. Commitment in Betty’s case stems from knowing that she has a health outcome sufficiently good to guarantee her starting work within a week and taking no more than a month off each year for 5 years. Suppose we include such knowledge in the definition of her acts and associated outcomes, as in Table 13. Table 13 Sure acts and health outcome flows conflated with ability/inability to commit Y0C
.. . YFC
(i) Death immediately after tomorrow + (ii) + inability to commit, no job Implication of (ii): Betty tomorrow at time t = 0 will lack knowledge with probability 1 that under her chosen an act she will be alive for 5 years without an extended period out of work .. . (i) Immediate elective surgery, 1 day unable to work, thereafter full health + (ii) Ability to commit, from tomorrow at time t = 0 committed to and doing a 5-year full time job Implication of (ii): Betty at time t = 0 knows with probability 1 that under her chosen act she will be alive for 5 years without an extended period out of work
Note: For brevity, we list here only the conflations from Table 12 for the extreme outcome flows which yield the best and worst utilities.
Precede the best outcome in Table 13, YFC , with a probability number less than one with the complementary probability for the worst outcome in Table 13, Y0C . We must do this under the standard gamble procedure. But in so doing we have the internal contradiction of simultaneously stating that tomorrow at her time of choice, time t = 0, she knows with complete certainty she can commit (Table 13’s definition of YFC ) and knows with complete certainty she cannot commit (the death outcome has a positive probability).
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It might be thought that some forms of partial, conditional commitment effects (less than Betty’s case where the commitment has to be 100%) might avoid the contradiction. This however is never the case. To see this, take another decision situation in which Betty could execute the community project job if she can 90% commit tomorrow at time t = 0. Then Betty would undertake the job if she had a 90% commit tomorrow at time t = 0. In this decision situation: (a) the extreme elaborated worst outcome has an implication (ii)
that Betty knows with probability more than 10% that she will not be alive for 5 years; and (b) the extreme elaborated best outcome has an implication (ii)
that Betty knows with probability 90% or more that she will live be alive for 5 years. We now seek to apply a common probability number to these two extreme outcomes of Table 13, as required under the standard gamble procedure. Our problem is that no common probability number can be applied without encountering a contradiction. Any probability of being alive for 5 years big enough to enable the partial commitment and its secondary satisfactions of being committed and having a job, is too big a probability for the other extreme elaborated outcome of an inability to commit, thus being uncommitted and having no job. We would also be lacking in general a cardinal utility scale unique apart from scale and origin as is required for axiomatised expected utility theory to uniquely rank acts (Pope, 1983, 1984, 1989, 2000, 2002a). 4.2. Fear included by elaborating acts, outcomes in the “gestalt” It is widely believed that outcomes can be elaborated to include emotional sources of secondary satisfactions. For instance Samuelson (1952a) claimed that acts and outcomes could be elaborated within axiomatised expected utility theory to include emotional sources of secondary satisfactions. His example was a source of positive secondary satisfaction, suspense (1952a, p. 676). This belief is evident in Luce and Raiffa’s claim that the outcomes of axiomatised expected utility theory can be elaborated to include the “gestalt” of the situation, where gestalt refers to secondary satisfactions (1957, p. 26). Their example of such a “gestalt” was a positive secondary satisfaction from thrills. The claim recurs in Markowitz who asserted that elaborated outcomes to include two positive sources of secondary satisfaction, wonder and surprise, were within axiomatised expected utility theory (1959, pp. 225–226). But the claim is incorrect. To insert emotional sources of secondary satisfactions into the specifications of outcomes is to destroy axiomatised expected utility theory and its standard gamble procedure just as completely as does the insertion of non-emotional sources of secondary satisfactions. To see this, take the case of Betty’s secondary (dis)satisfaction from an emotional source, fear. Elaborate the outcomes of Fig. 1 to include this fear. Add to the worst and best outcomes of death and full health the elaboration that for Betty this is preceded by an amount of fear, setting this amount of fear at what Betty says she would experience if the chance is 50/50 of these two outcomes. So elaborating, we have set the probabilities and thus have to adopt the certainty equivalent version to ask Betty a question. We ask her which intermediate guaranteed outcome is equivalent. We cannot however keep our elaborated outcomes axis and ask her any more questions. Why? Because if we alter
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the probability of death, this will alter her amount of her fear. If for instance, we reduce the probability of death, we now have an even better outcome at the top, full health with a smaller amount of fear (now that the risk of death is lower), and at the bottom, the worst outcome is now better than before, for the same reason. The standard gamble procedure of eliciting multiple certainty equivalents is thus unavailable. It depends on the worst and best outcomes remaining the same as we ask successive questions. That is, it depends on being able to identify the worst and the best outcomes independently of knowledge ahead of whether they will occur. The probability equivalent method faces the same problems. Suppose we start with a sure health outcome, say limited mobility, elaborated to include its certainty effects. For elaborated outcomes, we need to ask Betty to state two things: (1) the risky act’s probability of death (and conversely full health) that would make it as good for her as this sure outcome, and (2) the amount of fear associated with this probability. Her answers are then our elaborated worst and best outcomes. But to proceed, we must ask Betty about a second probability number for death and full health that would leave her indifferent between another sure health outcome and another risky act. This second risky act will have a different amount of fear, and so our worst and best outcomes have shifted. This renders the standard gamble procedure of eliciting multiple probability equivalents unavailable. It depends on the worst and best outcomes remaining the same as we ask successive questions. That is, it depends on being able to identify the worst and the best outcomes independently of knowing ahead whether they will occur. Elaborated outcomes to include emotional sources of secondary satisfaction like fear, it can be seen, render the standard gamble procedure unavailable. Some readers might nevertheless think that under expected utility theory the effects of commitment and fear and other sources of secondary satisfactions do not need to enter these lists of how acts and outcomes are specified (defined) to get included. They might think that these can enter a decision theory through how Betty values the acts and their associated outcomes. They might think that, by giving their outcome flows higher total utilities (QALYs), Betty can take into account the fact that some sure acts (e.g. YDN , YPK , YAT , YHR and YF of Table 12) furnish her scope for commitment. They might think that, by giving their outcome flows lower total utilities (QALYs) Betty can take into account the fact that some risky acts cause her fear. This is correct—and precisely what Betty’s decision procedure of Table 11 does—but axiomatised expected utility theory irrationally precludes this. Below are five ways (set forth in Sections 4.3–4.7) of seeing that expected utility theory excludes Betty’s rational inclusion of secondary satisfactions in how she values her knowledge ahead independently specified acts and outcomes flows. 4.3. The lack of distinction between a sure act and its corresponding outcome flow One way is to observe that axiomatised expected utility theory requires identical utilities for: (a) a sure act (which embeds knowing at the point of choice that the outcome is certain) and (b) the corresponding outcome flow (which has to be specified in a knowledge ahead independent way to avoid contradictions when preceded by a non-degenerate probability).
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The identity of (a) and (b) means that many axiomatisations lack distinct symbols for acts and outcomes (e.g. that of von Neumann and Morgenstern). It means that the same probability weighted sum of utilities duals for valuing acts and outcomes, as discussed further in Section 4.6 and Table 16. It means that Luce and Raiffa (1957) can and do shift almost page to page in whether an expression refers to a sure act or to an individual outcome. The context has to be read (often several times) to ascertain which. The difference between sure acts and individual outcomes is deemed irrelevant because the chooser’s probabilistic knowledge of whether an act will happen in the future is deemed irrelevant to how the chooser should value that act or outcome. Knowledge of this probability is the essence of commitment. 4.4. Inconsistency in the standard gamble QALYs from secondary satisfactions A second way of discerning that a secondary satisfaction from a certainty effect like ability to commit cannot be taken into account under axiomatised expected utility theory by how outcomes are valued is to look carefully at the standard gamble elicitation procedure. We have shown in Sections 4.1 and 4.2, that standard gamble procedures fail and are inconsistent if we attempt to include secondary satisfactions by elaborating the outcomes and hence acts along the x-axis of Fig. 1. We now show that analogous inconsistencies arise in standard gamble QALYs if we do not elaborate the x-axis, but instead attempt to include secondary satisfactions in how Fig. 1’s knowledge ahead independent acts and outcomes are valued. In Fig. 1 and Table 12, Betty has already ranked the set of outcomes independently of knowing ahead whether they will occur. Such a ranking is by definition with respect to primary sources of satisfactions alone. In this knowledge-ahead independent ranking of outcomes, Y0 , death 2 days’ hence, is Betty’s worst outcome and her best, YF , full health over her entire 5 year time horizon from 2 days’ hence. In Section 1, we have normalised these worst and best utilities from primary satisfactions as 0 and 1 respectively. Now we attempt to change the vertical axis to include secondary satisfactions in the utilities. Let us term this Fig. 2, even though we shall not depict it in the paper, merely visualise it in our minds. Let us normalise Fig. 2’s utilities origin and scale to have 0 the worst and 1 the best from primary plus secondary satisfactions, and use the information in Tables 7, 9 and 12 to compute these QALYs (total utilities). From Table 7, we note that Betty’s minimum primary plus secondary satisfaction is 0 utils per year, and her maximum primary plus secondary satisfactions for Betty is 1 utils per year, and thus over her 5 year horizon, 5. From Table 9, she would get half of the maximum of 5 utils from her primary satisfactions if in full health, and the other half from her secondary satisfaction if able to commit. From the last act listed in Table 12, we can see that she would reap this maximum of 5 utils under the conceivable but unavailable sure act of a safe hip replacement operation now. So as to have the maximum 1 not 5 utils in our new Fig. 2, all the QALYs in Table 9 must be rescaled, divided by 5. We then choose the probability equivalence version of eliciting utilities of outcomes. We ask Betty to nominate a probability between 0 and 1 that will give her as much QALY from a chance of instant death or full health as to make her indifferent between that lottery and getting the guaranteed outcomes flow YAT afforded by AT, the sure act of Alexander Technique of education. But immediately we encounter a problem. Once the full health
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outcome is risky, it no longer yields Betty her maximum conceivable utility. This outcome now yields her primary satisfactions (which as rescaled are 1/2 a util) and negative secondary satisfactions from fear, with no positive secondary satisfactions from an ability to commit. Our extreme best outcome has slipped from being at the top to being somewhere in the middle. We cannot apply the standard gamble procedure. It needs a stable best outcome. We try again. We construct a Fig. 3 that we shall not depict in the paper, merely visualise it in our minds. This time we set the maximum on Fig. 3’s vertical utility axis as Betty’s maximum satisfactions from primary satisfactions alone. These are from full health each year. From Table 9 this would sum to 2.5 over her 5 year horizon. When we rescale to have a maximum of 1 util on our vertical axis, this means dividing all the other numbers in Table 9 by 2.5. We again choose the probability equivalence version of eliciting outcomes. We ask Betty to nominate a probability number between 0 and 1 that will give her as much QALY from a chance of instant death or full health that would make her indifferent between that lottery and getting the guaranteed outcomes flow YAT afforded by AT, the sure act of Alexander Technique of education. We look in Table 9 to form the answer. It should be the QALY for AT divided by 2.5, namely 3.5/2.5 = 1.75. But 1.75 exceeds her maximum QALY from primary satisfactions of 1. An answer of 1.75 would exceed the upper bound of our vertical utility axis due to Betty valuing her positive secondary satisfactions (from ability to commit enabled under this act) sufficiently highly relative to her primary satisfactions from its intermediate health outcome. This also violates another requirement for using the standard gamble procedure, namely that the extreme outcome has the maximum utility. Her answer of 1.75 is in addition an improper fraction when we asked Betty to scale her probabilities to lie between 0 and 1. No probability number between 0 and 1 exists to render Betty indifferent to a guaranteed YAT with its beneficial commitment effects, and the lottery of either almost instant death or almost instant full health. Under the standard gamble probability equivalence method of eliciting utilities, the utility of outcome YAT is elicited by choosing at t = 0 a positive probability of death (and complementary probability of full health) that leaves the chooser indifferent between this lottery and a guaranteed outcome YAT . The lottery’s positive probability of death robs her of the probability of 1 of staying alive at her time of making this standard gamble choice, t = 0. This probability of 1 at t = 0 for not dying is a pre-requisite of her committing at t = 0, and reaping her satisfactions from commitment effects. Remember, to commit to her job, Betty must know ahead with probability 1 that nothing will preclude fulfilling her job commitment at t = 0. Since commitment is the dominating source of her total satisfaction and hence QALY, she lacks any answer consistent with expected utility theory. The analogous conclusion follows from attempting to implement the certainty equivalent version retaining unelaborated outcomes on the horizontal axis and including secondary satisfactions on the vertical utility axis. Betty’s answers would be thrown out as “inconsistent”, those of a recalcitrant irrational chooser. After devoting our time to filling out this or a related questionnaire for a colleague, some of us turn in answers that the researcher tells us disappointedly they had to discard since our answers were “inconsistent”. The researcher sees the alternatives as either (1) an expensive, time-consuming and not always successful procedure of “educating” the participant to answer “consistently”, or (2) to correct for the participant’s perceived “biases”, e.g. with use of prospect theory, Bleichrodt, Pinto and Wakker (2001).
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Not many participants are as recalcitrantly “inconsistent” as Betty. This is because the researchers employing the standard gamble technique (and all others that focus on primary satisfactions) fail to alert their participants that secondary satisfactions exist and should be taken into account. Such researchers fail to lay out all risks and future changes in knowledge involved in risky acts, and how these generate risk and certainty effects. Unaware of the existence of these effects themselves, such researchers misinterpret the imposition of a preference for stochastically dominating acts and sets of outcomes as a mere “consistency” requirement. But Betty (and any other participant perceptive enough to discern the secondary satisfactions that they reap out of risky decision situations), takes into account “the given situation” in valuing outcomes with respect to secondary satisfactions. Betty disobeys the knowledge ahead independent evaluation rules of expected utility theory. 4.5. No room for commitment in expected utility theory’s act-value equation A third way of seeing that knowledge ahead based sources cannot be included in how an outcome is defined and still preserve expected utility theory is to look at its equation for valuing an act, Eq. (6) in Table 14 for the case where there are seven conceivable outcomes, Yi , i = 1, 2, 3, AT, HR, HR+ , F. Table 14 Axiomatised expected utility theory’s rule for valuing individual outcomes EU implies that the value of any act regardless of aspects of its given situation is V, V = E[U(Y)] = p0 U(Y0 ) + p1 U(Y1 ) + p2 U(Y2 ) + pAT U(YAT ) + pHR U(YHR ) + pHR+ U(YHR+ ) + pF U(YF )
(6)
where pi is the probability of outcome Yi , i = 1, 2, 3, AT, HR, HR+ and F, V must hold for all sets of pi that sum to 1. That is, the utility attached to an outcome has to be the same regardless of whether its probability number is 1 (enabling commitment in the case of the intermediate sure acts/outcomes) or less than 1 (precluding commitment). This precludes modifying the valuation of outcomes to incorporate any aspect of the actual “given situation” to reflect commitment and any other post-decision risk or certainty effect in the valuation of acts and outcomes which are themselves defined in a knowledge-independent way
In this table, it can be seen that under axiomatised expected utility theory the outcome YF of almost instant full health + only 1 day unable to work, has the same utility number regardless of whether it is: 1, guaranteed to occur, as in the case of the sure act of the unavailable instant operation; or 2, merely possible since the outcome of almost instant death, Y0 , is also possible. But if death is a possibility, this precludes Betty from giving her 5 year job commitment, and undertaking her project job. This identity of the utilities into which outcomes map regardless of whether the outcomes are ultra risky, fairly safe or absolutely certain, precludes commitment, fear, or any other knowledge ahead based aspect of the actual decision situation being included in the valuation of the outcome. 4.6. Commitment effects and the utility of multi-stage acts Any (non-degenerate) multi-stage act is risky. Let Betty’s choice be between multi-stage act MHR and the sure act AT of Table 15.
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Table 15 Betty with a choice set including a multi-stage act Betty conceives choosing tomorrow between the two acts, a multi-stage act MHR in which the risk will only be resolved just over a year later, and the sure act AT MHR A 20% chance of learning next year that she must do HR+ Hip replacement operation plus intensive follow-up physiotherapy which yields her an outcomes flow of a year’s wait in the surgery queue then the operation followed by two consecutive months doing supervised exercises most of her waking hours, then full health An 80% chance of learning next year that she must do HR Hip replacement operation with minor physiotherapy which yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation requiring only 2 weeks of sick leave, then nearly full health AT Alexander Technique education to reduce her asymmetric posture and movements that caused the “mushy” hip joint, which yields her an outcomes flow of small steady increments in mobility and decreases in pain from her post-work once weekly solo session with a teacher trained in this technique
In MHR, we conceive the situation in which Betty has no choice about whether to follow a hip replacement operation with intensive physiotherapy. The health providers will dictate to her after her hip replacement operation a year hence whether her life takes the HR+ branch of two consecutive months doing supervised exercises most of her waking hours, or her life takes the HR branch, leaving her free to resume work with less than a month’s sick leave. Risky MHR destroys Betty’s opportunity to combine HR with her ardently desired full-time 5-year job. As Table 16 shows axiomatised expected utility theory requires that the value of a multi-stage act be a probability weighted aggregation of its constituent utilities of sub-acts, outcomes, and that it can be decomposed back into these constituents. Table 16 Axiomatised expected utility theory’s knowledge-independence feature which excludes secondary satisfactions Axiomatised EU implies that V, the rule for aggregating mutually exclusive possibilities by their associated probability weights, p1 , p2 , . . . into an overall value, (7) V = E[U(H)] = p0 U(h0 ) + p1 U(h1 ) + · · · holds when H refers to Either (a) An outcome space, in which case its possible values hi are the possible outcomes of an act, i.e. the Yi ’s of Eq. (5) and Table 9, the possible outcomes of an act Or (b) An act space, in which case its possible values h0 , h1 , . . . , are themselves possible acts and hence in Table 15, (8) V(MHR) = 0.2 U(HR+ ) + 0.8 U(HR) This identity of formulae in the valuation of outcomes and acts over all probability distributions of acts and outcomes means that a multi-stage (compound) act decomposes into a weighted sum of the utilities of its component acts. In turn such decomposition requires knowledge ahead independence of which act the chooser will select and of the chooser’s array of different conceivable “given situations” for each probability distribution over acts and outcomes in the (i) specification of acts and outcomes, and (ii) U(·), the utility transform for evaluating satisfaction from acts and outcomes
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This requirement affords a bird’s eye view of why axiomatised expected utility theory precludes commitment or any other pre- or post-risk or certainty effect entering either how acts and outcomes are valued and defined (Table 16). The knowledge ahead independence features (a) and (b) of Table 16 preclude Betty taking her job commitment ability into account in evaluating her risky multi-stage act MHR and comparing it with simple act AT. From Eq. (7), Betty evaluates and defines MHR as the probability weighted sum of the utilities of two component “acts” HR+ and HR. She gives each of these sub-acts the same value (utility number) as they would have if at the point of choice they were sure acts, definitely coming up in year 2 (the case in Table 8), or merely probabilistically going to occur (the case when these are embedded in the multi-stage act MHR of Table 15). There are only two means of obtaining axiomatised expected utility theory’s required identity in an act’s value when it is guaranteed, and when it will merely be probabilistically undertaken, because a sub-act in another multi-stage act. One is to apply the Friedman–Savage “as if certain” evaluation rule to acts (as distinct from outcomes). Under it Betty would value the risky component acts HR+ and HR “as if sure” acts. From Table 9, she evaluates them respectively as 1.9 and 3.98. Under this “as if certain” approach, Betty would then proceed to compute MHR’s value as the probability weighted sum is 0.2 × 1.9 + 0.8 × 3.98. This gives MHR a value of about 3.57. This value exceeds that of her available sure act AT whose value is 3.4, and so causes Betty to choose MHR. Choosing MHR is irrational. It is not merely irrational in the weak sense that its actual expected value (0.2 × 1.9 + 0.8 × 1.48 ∼ 1.57) is below that of AT. It is irrational in the strong sense that it fails the intuitive gold standard. The best possible outcome under MHR is only 1.9 utils, i.e. MHR is guaranteed to yield less utils than sure act AT. This has happened because expected utility theory’s “as if certain” imputation for MHR irrationally includes positive secondary satisfactions from HR’s ability to commit that she lacks. Let us now consider the alternative way of achieving axiomatised expected utility theory’s requirement that acts have identical values when risky and certain at the time of choice—i.e. when it is known and when merely it is merely probabilistically known at the time of choice that the act will be undertaken. This is the Ramsey-style version of obeying axiomatised expected utility theory’s Eq. (7) in Table 16. This version excludes all secondary satisfactions, all knowledge ahead based effects such as ability to commit, fear.5 Excluding all secondary satisfactions overstates the value of MHR. It overstates the value of MHR because it ignores Betty’s negative secondary satisfaction from fear of dying (something absent when the operations were guaranteed to be successful and thus absent from Table 9). Conversely, it understates the value of AT by ignoring the 2.5 utils she obtains from the secondary satisfaction of being able to commit to her job. This understatement alone, even apart from its overstatement of the value of MHR, is enough to make MHR seem better when in fact it is guaranteed to yield her a lower expected QALY. Thus, the Ramsey-style version also fails the intuitive gold standard. It would not lead Betty to the rational choice of AT any more than would Friedman and Savage (1948) “as if certain” evaluations. 5 The qualifier style is added since Ramsey’s pre-axiomatisation does not consider multi-stage acts, and thus this is an extension of his interpretation that choosers must value outcomes independently of knowledge ahead, independently of the probabilities of the outcomes occurring.
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In summary then, neither by how acts and outcomes are valued, nor by elaborating, redefining acts and outcomes, can axiomatised expected utility theory include the risk effects of Betty’s inability to commit, and her fear when these effects exist. It cannot do this in a choice set including only a simple risky act (as is required to apply the standard gamble procedure) or in a choice set including a multi-stage act. 4.7. Job and commitment inseparable Some might feel that Betty’s need for total commitment to a full-time 5-year job is esoteric, that expected utility theory as used in health economics should not cater for unusual sources of satisfaction. They may grant that jobs and feeling useful give choosers finance and improve their health, but feel that the commitment aspect can be side-stepped, that what Betty wants is the job (which it might seem that expected utility theory can readily include), not commitment to it. This is to fail to unpack the meaning of a job. A job is a causal word. The person doing a job deliberately causes something to change—the job to get done. Causation takes time and the job only gets done by that doer if that doer stays committed for the requisite duration. Commitment is thus an integral part of specifying and defining any job, and thus in determining QALYs.6 The suggestion that we might replace the commitment factor with a job factor in elaborating outcomes, in no way eliminates the knowledge ahead based factor of commitment and the destruction of expected utility theory’s standard gamble process for eliciting QALYs. 5. Post-decision risk effects yield the standard gamble utilities bias 5.1. Pre-choice planning problems and risky choice investment compromises Our next task is to see how one particular class of secondary satisfactions yields predictable concave-convex twists when expected utility theory’s standard gamble procedure is presumed to map out utility curves. The answer relates to why individuals and organisations typically check their own ability to commit (and also seek commitment from relevant parties before deciding on a job (whether a community project or an operation on a patient or . . . ) a partnership (whether personal or business), having babies, hiring workers, and numerous other activities that directly and indirectly impinge on QALYs. Our answer to these questions should recognise that two major factors are: 1, the difficulty of foreseeing the multiple possible future choices and outcome flows entailed in the presence of risk, and determining what satisfaction might be gained from future unanticipatable choices in the chaos of no commitment, no guaranteed continuity in anything; and 2, the investment compromises entailed in the face of multiple alternative outcome flows. 6 By contrast some outputs (outcomes) of jobs can be specified and defined without reference to commitment. We can define the outcome that Betty receives an artificial hip in a year’s time as a result of a job done by a surgery team, independently of Betty knowing now at her time of choice that this will happen. The surgery team’s job, though, of putting in the artificial hip requires its commitment for that period of time.
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5.2. Pre-choice planning problems Central to both versions of the standard gamble procedure is a risky act in which the extreme outcomes of death and full health, each have a positive probability. Every alternative includes a comparison with such an extreme risky act. The situation of RHR, Betty’s risky hip operation with death a possible outcome after a lengthy pre-outcome period (a year’s wait for elective surgery) is appropriately close to the extreme risk situation used in the standard gamble procedure. Indeed it was selected to allow us to imaginatively understand how the standard gamble procedure functions. Under the standard gamble procedure Betty must, for each utility obtained, ascertain a sure act that is equivalent to an extreme risky act. To choose among acts that are far apart might require little planning and analysis.7 But the task is to select equivalent ones. In making such close comparisons, Betty would, as a standard gamble participant, have to assess fairly accurately her satisfactions over her pre-outcome year of uncertainty before knowing whether she has avoided dying on the operating table or soon after. Determining these involves determining the appropriate way to spend the pre-outcome year of uncertainty under the extreme risky act. Deciding this involves her in numerous planning problems. How much time should she spend doing those activities one would do if one knew the outcome will be death? How much should she spend doing those completely different activities she would do if she knew the outcome will be virtually full health? Ascertaining which such sub-acts are available, and choosing among them the best compromise course of action for the uncertain pre-outcome year may involve assessing long futures of branched possibilities, and extensive negotiations with family, doctors, hospitals. Betty needs to repeat this exercise for every different probability mix (or certainty equivalent) of the extreme outcomes. These pre-choice planning problems of arriving at QALYs, the overall values of the different extreme risky acts, are ignored under axiomatised expected utility theory and its standard gamble procedure. We did above also. We assumed Betty could ascertain these with perfect accuracy within her decision deadline of a day hence. Even that day is rather longer than we normally offer participants in a standard gamble procedure before they have to proffer their QALY answers. Savage (1954) recognised the long chains of branched possibilities inherent in using axiomatised expected utility theory to give a utility to any risky act (let alone one so risky as to have extreme outcomes). Savage defended expected utility theory requiring this with the argument that to do otherwise than “look before you leap” is to be irrationally myopic. But as Simon responded, it cannot be rational to have a decision theory that irrationally ignores the extensive time successful business men devote to negotiating and creating alternatives, 7 This was the case for the real life Betty who unlike the one in this paper, had such pain and immobility that she did not need detailed planning and analysis to choose an operation, even though her risk of death from that operation was 30%. Her problem was not that of assessing whether its possible primary and secondary satisfactions would render it her best alternative. Rather for her the difficulty lay in the pre-choice set problem of negotiating with family, doctors and hospitals that this act be made available to her, something she only succeeded in achieving after years of lobbying, and the added negotiation power of having been awarded a Queens Medal for her community service. In the case of eliciting QALYs under standard gamble procedures, participants wearing the hat of the health provider, have the complication of branched futures on the alternative uses of funds, especially when the recipient has a good chance of dying on the operating table.
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and irrationally ignores the complexity of an integrated analysis of all future implications of acts (e.g. Simon, 1991a,b). Commitment reduces the number of branched possibilities facing a chooser. A commitment-free world involves ultra risky futures with enormous numbers of alternatives of what others will later decide to do. It is a veritable impossibility at the physiological, emotional and financial levels of doing the Savage (1954) pre-choice “look before you leap” analysis of such futures. As Savage (1954) owned, not even choice over a family picnic could really be analysed in this way. Mere mortal Betty has very limited anticipatory powers for seeing the future out of even the small number of potential alternative acts we identify. To solve many of the dynamic programming problems her whole life would not be long enough to ascertain all the ramifications of a single sure act. She doubles what she has to evaluate to obtain a precise answer if the risky act has two possible outcomes, trebles it if there are three possible outcomes, and so forth.
5.3. Risky choice and investment compromise Keynes’ General Theory (1936) generated our modern macro-economics investmentconsumption distinction. Consumption yields its satisfaction now and thus is independent of risk, of what may or may not happen in the future. Investment by contrast yields satisfaction in the future, and thus is subject to risk, to what may or may not happen in the future. Keynes objected to expected utility theory which excludes pre and post-decision risk and certainty effects. Wold (1952a,b) provides an investment example showing that in the presence of investment expected utility theory yields arbitrary implausible choices as admitted in the Econometrica responses (Samuelson, 1952b; Savage, 1952a,b). By the later 1950s (e.g. Markowitz, 1959) it had been recognised that expected utility can address only a single choice in a life, and cannot deal with investment. It was thought that expected utility theory can deal with consumption flows such as QALYs. This however was to fail to realise the role of investment in all consumption activity, whether physiological or emotional. Betty’s autonomous nervous system invests (prepares) when it anticipates she will eat, drink, go into the cold or heat, have sex, sleep and so forth. Beyond a certain range, the more unpredictable the future, the more inefficient her bodily preparatory investment, and the lower her physiological consumption of available items. If the alternative outcome flows are extreme with death too likely—she can perform none of these functions. Her body freezes. Conversely, there is less compromise and more efficiency in Betty’s emotional investments (and thereby subsequent emotional consumption) if the risk in her future stays within a given range. Her desired emotional and financial investments, and the split between investment and consumption is very different if she is guaranteed to die within a year as against she is guaranteed full health for 5 years. This is likewise the case for health providers, governments, business, spouses, parents and so forth. The investment compromises for catering for huge risk (vastly different possible futures) greatly detract from the satisfaction flows that can be reaped. This is so even when such problems are tackled with the best dynamic programming algorithms run on the fastest super computers in the world. On how to im-
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prove such compromise investments by building in flexibility to reduce the risk of ruin or extremely low returns (e.g. De Neufville and Munéra, 1983; De Neufville and McCord, 1983; De Neufville and Odoni, 2002). 5.4. Commitment, sure acts to alleviate planning and investment costs For simplicity in Table 5, Betty’s project was impossible without full commitment tomorrow—without this the project was guaranteed to bear no fruit. In other cases, with a worker (and other resources) less than 100% committed, there may be a possibility of some useful project output. It is costly and difficult however for a Betty to do the planning to determine whether it is worthwhile proceeding with such partially committed workers, costly too for the other side of the coin, the funding agency.8 Take the World Bank as the potential funder of the Bettys doing community aid amelioration projects, and considering whether to allocate all its funds to Bettys who cannot fully commit, or to allocate some funds to different sorts of projects that have some chance of bearing some fruits when run by Bettys who cannot fully commit. The World Bank faces financial and time constraints. It is not necessarily desirable that it allocate resources to assess what else it will do if it later discovers it has lost two fifths of the money that would have gone on this project’s results. Money and time are required to compute what else to do to help the community if Betty has a high chance of dying during the project. This requires assessing whether hiring someone likely to die and implementing the back-up plan is better than allocating the funds only if Betty can fully commit having a negligible chance of dying. There is nothing necessarily irrational about the World Bank making commitment a pre-requisite for funding some community projects. It cannot use its entire budget up on feasibility studies for dealing with inabilities to commit. The World Bank must use its judgement on how much of its budget to spend on feasibility studies to deal with hires who cannot commit. It might deem it worthwhile to do a feasibility study on whether it is worthwhile to hire those with only a third ability to commit, as is the practice of a major car manufacturer in South Africa, which, recognising that AIDS precludes about one in three workers continuing in production for long, trains three for each assembly line position with two paid to be standbys for those on the line who have to drop out due to AIDS. But the World Bank might also deem it better to save the funds on such a feasibility study, and instead devote those funds to other health providers who can offer the full commitment. 8 If in the paper’s example Betty had not cared intrinsically about the project, but only getting paid, and if the World Bank required full commitment of her, Betty’s ability to commit would be likewise a major source of her secondary satisfactions from her salary obtained via her ability to commit. The analysis would be identical to that above if the World Bank shared the same probabilistic knowledge as Betty on her ability to commit (risk of dying or being out of work for an extended time), or if Betty did not lie. If Betty were ready to lie and had a capacity to deceive the World Bank, then the general conclusion still holds, but the analysis requires inserting for the job secondary satisfactions, a transform from the actual probability distribution over outcomes from different health interventions to the distribution that Betty through lies has the capacity to convince the World Bank is the case. Betty’s own perception of her probability distribution over outcomes from her alternative acts is then the pertinent one only for her ascertaining her fear of dying (and if pertinent, fear of being discovered later to have lied to the World Bank).
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Nor is there anything irrational about the commitments the government imposes in allocating a garbage contract. Nor is there anything irrational in a host seeking a commitment that the guest will arrive tonight to have the dinner being prepared. The optimal investments in organising the community, the garbage, the dinner are generally radically different under alternative outcomes. Discovering and evaluating compromise investments for coping with risky acts is a costly time consuming process. It would be an endless regress were a chooser so irrational as to attempt cognitively to optimise this complex process. It is an endless regress that expected utility theory simply ignores and assumes implicitly that chooser’s have the answers to all these calculations. On the disasters introduced by attempting to optimise in complex decision situations (e.g. MacKinnon and Wearing, 1983). These planning and compromise investment costs inherent in evaluating and choosing risky acts make their inroads into financial, time and physiological stress resources. They result in most people making many choices that seem likely to avoid large scale post-choice risk effects. Genes, as well as environmental factors make for individual differences in the reluctance for novelty inherent in risky acts. Two thirds of the population are averse to novelty and post-decision risk effects, one third extremely so. On the work in discovering the ten sequences on DNA pertaining to the genetic component in this dislike of post-decision risk effects, see Hamer and Copeland (1998). On adverse decisions made by those lacking enough aversion to post-decision risk effects through damage to pertinent segments of the brain, see Demasio (1984). 5.5. Post-decision risk effects yield the standard gamble utilities bias Axiomatised expected utility theory excludes the chooser’s pre-decision planning costs and those of relevant other parties who contribute to the chooser’s environment of penalties for failing to commit and imposing risk and thereby planning costs on others. On Ramsey’s version, axiomatised expected utility theory similarly excludes all other pre- and post-decision risk effects, and their mirror image, certainty effects. It limits satisfactions to post-decision primary satisfactions. Under the alternative “as if certain” version, it again excludes all risk effects, though not the converse certainty effects. Expected utility theory’s standard gamble procedures elicit what we erroneously infer are utilities. They elicit them with pre- and post-decision risk effects at their maximum. They always involve the extreme outcomes, the best and the worst. They furthermore typically begin with risk effects at their maximum for most people, i.e. a 50% chance of the extreme values (in a QALY context) either death or full health. (The risk effects would be by comparison more modest for many choosers, were the probabilities 99.99 and 0.01%, since then one outcome is approaching one of the extreme outcomes.) For these omitted risk effects to yield unbiased utilities, the ignored pre- and post-decision risk effects would need to be orthogonal to the included primary satisfactions derived from the outcomes alone. Then the effect of omitting these risk effects would be to generate non-systematic variation in the utility curve elicited under the certainty equivalence and the probability equivalence methods. Then omitting these risk effects would yield merely random differences making it impossible to retrodict from the shape of the utility curve whether the certainty equivalence or the probability equivalence method of elicitation had been used.
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But we know from Hershey et al. (1982) that choosing the probability equivalence version of the standard gamble systematically tends to a concave curve for low monetary outcomes, twisting into a convex shape for high monetary outcomes, and vice-versa for the certainty equivalence version. To describe how these reverse twists relate to reverse bunchings in the answers elicited, we can use Fig. 1 as we do not need outcomes that lie some objectively determined distance from each other, merely that their positions on the x-axis are pre-specified. Let us draw a line between 0,0 (denoting the zero utility of the worst outcome) and 1,1 (denoting the maximum utility from the best outcome), and call its slope S. Let us split the outcome space into high and low sub-sets at a middle health outcome Ym (between Y2 and Y3 ). The certainty equivalent’s convexity below, concavity above, means that relative to S, the elicited utility curve’s slope is steeper for intermediate outcomes, flatter near the extreme outcomes—i.e. relative to certainty equivalents elicited for low probabilities of either extreme outcome, there is a bunching together of the certainty equivalents elicited for intermediate probabilities (for when the probabilities of both extreme outcomes are sizable). Conversely, the probability equivalent procedure’s concavity below, convexity above, means that relative to S, the elicited utility curve’s slope is flatter for intermediate outcomes, steeper near the extreme outcomes—i.e. relative to those for very low or very high outcomes, there is a bunching together of the probability equivalents elicited for intermediate outcomes. Thus under the two procedures there are: A systematic (albeit opposite) reversals in what is bunched at exceedingly low outcomes relative to intermediate ones, and B systematic (albeit opposite) reversals in what is bunched at intermediate relative to exceedingly high outcomes. First for the certainty equivalent, and then for the probability equivalent, procedure we explain below how the pre- and post-decision risk effects (that are assumed to be non-existent under these elicitation procedures) in fact contaminate the answers and cause A and B. Under the certainty equivalent procedure, let us begin by putting participants in the situation of a 50/50 chance of death or full health, and ask for their certainty equivalent. Risk’s planning problems and some other pre- and post-decision risk effects give the typical participant a desire to switch out of their current situation of a 50/50 chance of death or full health, and say that a low health outcome is worth accepting to avoid the risk of death. But the causal chain underlying this answer is not representable on Fig. 1 or any other mapping from primary sources of satisfaction into utilities. Fig. 1 says nowhere there is a 50/50 chance of death or full health. Mis-representing the answer by plotting it on Fig. 1 is spurious, misleading, since the answer represents an escape from the risk of death as well as that very low health outcome. That is, the utility plot suffers from omitted variables bias. The utility plot is a conflation of a primary satisfaction (from the low health outcome) and omitted causes, the pre- and post-decision risk effects. The set of three utilities 0, this fraction and 1, do not, as implied by Fig. 1 and axiomatised expected utility theory, represent utilities that participants would place on death independent of knowing they are about to die, of low health independent of knowing this will be their fate, and full health independent of knowing whether this will really be their fate (with their alternative fate not some slightly low health outcome, but the other extreme, death).
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Let us next put participants in the situation of 75/25 chance of death or full health and identify its certainty equivalent. Since the adverse pre- and post-decision risk effects are fairly similar to the initial situation since both extreme outcomes still have substantial probabilities, the typical answer is fairly close to the earlier one. Bunching is only absent when we ask for certainty equivalents to gambles in which the probabilities of one of the extreme outcomes is sufficiently tiny that the typical chooser might neglect pre- and post-decision risk effects. The converse conflation and omitted variable bias occurs under the probability equivalence method. Let it begin by putting participants in the situation of being guaranteed the middle health outcome Ym . We ask them to nominate a probability of death or full health that would be on the borderline of warranting switching out of security and undertaking this sure act. Risk’s planning problems and some other pre- and post-decision risk effects give many a desire to avoid switching out of security into a chance of death or full health. The typical answer p is accordingly a high chance of full health. They require this high chance to warrant loss of security and taking on board all the pre- and post-decision risk effects. But these causal chains are not representable on Fig. 1 or any other mapping from primary sources of satisfaction into utilities. Fig. 1 says nowhere there is a 1-p chance of death, and a p chance of full health. Mis-representing p by plotting it on Fig. 1 is spurious and misleading since p represents the cost of incurring the risk as well as satisfactions from that middle health outcome Ym . That is, the utility plot suffers from omitted variables bias. The utility plot is a conflation of a primary satisfaction (from the middle health outcome Ym ) and omitted causes in the form of pre- and post-decision risk effects. The set of three utilities 0, p and 1, do not—as implied by Fig. 1 and axiomatised expected utility theory—represent the three utilities that participants would place on: 1 death independent of knowing they are about to die; 2 low health independent of knowing this will be their fate, and 3 full health independent of knowing whether this or death will be their fate. Let us next put participants in the situation of being guaranteed another intermediate health outcome, Y2 . We ask them to nominate a probability of death or full health that would be on the borderline of warranting switching out of security and undertaking this sure act. Concern about pre- and post-decision risk effects cause most participants to demand a high price of giving up even this level of security, and thus to nominate a probability fairly close to p, their original answer. This bunching is only absent when they are asked to nominate probability equivalences sufficiently close to the extreme outcomes, that one of the probabilities is sufficiently tiny for the typical chooser to neglect pre- and post-decision risk effects. 5.6. Bias from other omitted risk effects including fear For simplicity the analysis of the biases in standard gamble utilities in Sections 5.1–5.5 has focussed on a single source of omitted secondary satisfaction, that of the certainty effect of ability to commit. In their omission of secondary satisfactions, these elicitations omit all other secondary satisfactions as well. These include a range of financial risk cost effects (i.e. certainty benefit effects) explored elsewhere (Pope, 1983, 1991b). They also encompass a range of emotional risk effects, some of which are negative (e.g. Betty’s fear)
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and others positive (Table 4) explored in Pope (1983, 1989, 1995, 1998). If the post-choice negative emotional secondary satisfactions under risky acts outweigh the corresponding positive emotional secondary satisfactions, these exacerbate the bias in standard gamble utility elicitations.
6. Conclusions There are pronounced biases in standard gamble utilities for use in QALYs and other decision making purposes. These arise out of omitted risk and certainty effects. Such risk and certainty effects reflect planning difficulties at the neurophysiological level. The autonomous nervous system requires a degree of predictability for its planning and investment processes. This is manifest in and paralleled by emotional, financial and other physical planning difficulties and investment compromises necessitated when there is risk. The standard gamble procedure for eliciting utilities involves contrasting extreme risk (both extreme outcomes possible) with the minimum risk (certainty). The procedure is valid only under the implausible constraint that outcomes yield identical utilities under extreme risk and certainty. This implausible constraint is imposed by axiomatised expected utility theory. This paper has offered steps toward restoring the chooser’s anticipated satisfaction (including that from health outcomes) to its necessarily central place in any descriptive or rational choice procedure. It has taken the next steps of demonstrating what expected utility theory includes, what it omits, and of appraising what it omits and includes, showing that it excludes factors critical to health. To indicate their criticality, it has introduced a new more neutral terminology for what expected utility theory includes and excludes. The new term for what this theory includes is primary satisfactions. The new term for what it entirely excludes under the Ramsey version of axiomatised expected utility theory, is secondary satisfactions, namely pre- and post-decision risk and certainty effects. Both primary and secondary satisfactions are critical to health. The empirical magnitude of the two will vary across decision situations and choosers. The paper has further shown that the Friedman and Savage “as if certain” version of the utilities of outcomes under axiomatised expected utility theory differs from that of Ramsey. It differs by including a subset of secondary satisfactions, namely certainty effects, even when these effects are non-existent! Like the Ramsey version, the Friedman and Savage version excludes secondary satisfactions arising from risk effects. Central to health economists including risk effects in QALYs is to set out the changing stages of knowledge inherent in any situation of risk and uncertainty. Changing knowledge is the hall mark and defining characteristic of risk. But axiomatised expected utility theory and most of its rank dependent generalisations ignore this and model risky choice as an atemporal procedure. This has resulted in many health economists treating probabilities exclusively as if they are atemporal shares of simultaneously existing objects or events, like slices of a cake. They play this atemporal role in the aggregation of outcomes to form an overall valuation of an act. The other role of probabilities, namely as denoters of our knowledge ahead, has a fundamental temporal dimension In this regard, a probability of for instance 0.5 for two outcomes
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denotes that the chooser now has merely probabilistic knowledge ahead of the future outcome. The risk of which of the two outcomes is the case may later be resolved. When this risk is fully resolved, the probability of one of the two outcomes becomes 1 and the other 0. This paper has shown how to divide up the future by stages of knowledge to identify the changing risk and certainty effects caused by our anticipated changing stages of knowledge about the future. Such risk and certainty effects include fear and ability to commit. Meaningful QALY estimates under risky futures entail laying out these changing stages of knowledge facing choosers. It further entails alerting choosers to these effects in the same manner that utility elictations alert choosers to knowledge ahead independent features of health states. This paper has shown that to include anticipated consequences of future changes in knowledge, we must move beyond the knowledge-wise atemporal standard decision framework adopted by expected utility theory and most of its generalisations (e.g. the Quiggin and cumulative prospect rank dependent theories). In moving to the new stages of knowledge framework we can consistently (without contradictions) include: 1, QALY consequences independent of knowing ahead and thus independent of risk effects (primary satisfactions) and that are included in the standard decision framework; plus 2, QALY consequences based on knowing ahead about risks, certainties (secondary satisfactions) and that are excluded in the standard framework. Thus, this new framework, unlike that of axiomatised expected utility theory, includes all consequences—all cause–effect chains affecting satisfactions—and thereby provides the solid comprehensive consequentialism required for QALY applications.
Acknowledgements I thank for valued comments Jeff Richardson and Lillian Bulfone of CHPE, Monash University; Reinhard Selten of the Experimental Economics Laboratory, Bonn University, John Quiggin, Department of Economics, University of Queensland, Tony Culyer, Department of Economics, York University, Duncan Foley, New School University, Vivian Walsh, Muhlenberg College, two anonymous referees of this journal; and for earlier background discussions hugely assisting my understanding, Maurice Allais, Ken Arrow, Will Baumol, the late Ole Hagen, Hans Jensen, Duncan Luce, Roy Radner, Howard Raiffa, Paul Schoemaker, the late Herb Simon and Paul Samuelson. Remaining errors are my own.
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Biases from omitted risk effects in standard gamble utilities Robin Pope∗ Center for European Integration Studies (ZEI), University of Bonn, Walter Flex Strasse 3, D-53113 Bonn, Germany Received 9 May 2003; received in revised form 11 July 2003; accepted 30 September 2003
Abstract Utilities elicited under the “probability equivalence” version of the standard gamble procedure systematically exaggerate the utility increments of expensive health interventions that bring the recipient towards full health. Switching to the alternative “certainty equivalence” version would introduce the reverse bias. The biases arise from anticipated pre- and post-decision risk effects that expected utility theory omits. To include these risk effects, partition the future by changes in knowledge into: (1) pre-choice periods; (2) the pre-outcome future before the chosen act’s outcome becomes known; and (3) the more distant post-outcome future, when that outcome will have become known. © 2003 Elsevier B.V. All rights reserved. JEL classification: D81; I10 Keywords: Standard gamble; Certainty and risk effects; Commitment; “As if certain” evaluation of outcomes; Changing stages of knowledge; Pre-choice; Pre- and post-outcome periods
1. Utilities and health policy Should an individual or health provider invest in preventive procedures? For example Alexander Technique (AT) education can improve movements, reducing the chance of educees damaging their hips later. What sort of restorative procedures should be chosen in the case of those with already damaged hips? The answers to these and other health decisions depend on issues such as costs, equity, efficacy/efficiency, the chooser’s prior health care commitments—and on the improvement in the health care recipient’s utility from each intervention. ∗ Tel.: +49-228-73-18-88/73-92-18/91-40-361; fax: +49-228-73-18-09. E-mail address: [email protected] (R. Pope). URL: http://www.zei.de/.
0167-6296/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jhealeco.2003.09.006
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1.1. Risky versus sure acts We cannot, other things equal, necessarily choose the intervention guaranteeing the highest utility. This is because only some interventions have guaranteed health outcomes at the time they are chosen—are what we term sure acts. Many have more than one possible health outcome—are what we term risky acts. Decisions therefore cannot be formulated with the goal of maximising utility in a certain world. They need to be formulated with a goal that recognises the existence of risk and uncertainty. We shall not distinguish acts with risky outcomes from acts with uncertain outcomes. Rather we shall assume that all uncertainties about the health outcome Y generated by an intervention are known at the time of choosing that act in the form of probabilities p0 , p1 , . . . for each of the risky act’s finite number of possible health outcomes, Y0 , Y1 , . . . , and an exact date when Y will be learned—i.e. when all probabilities become 0 or 1.1 More formally then, in this paper an act is such a probability distribution over outcomes. At the time of choice, each sure act’s guaranteed outcome has a probability of one and all the other outcomes have a probability of zero, while each risky act has at least two outcomes with non-degenerate probabilities. 1.2. Utilities of outcomes versus the value of a risky act The evaluation of an outcome we term a utility. It is central to understanding this paper to appreciate that axiomatised expected utility theory requires choosers to evaluate outcomes excluding consideration of their actual probabilities. Under this theory, actual probabilities enter only the value of a risky act—enter as the aggregation weights for summing its mutually exclusive utilities of outcomes to form this value. 1.3. Total versus per annum utilities of outcomes We focus on the health dimension of each outcome, and term its utility the quality adjusted life years saved (QALY). Decomposing a QALY introduces complications (Richardson et al., 1996). TNevertheless, tthe QALY of a health outcome is often decomposed into: QALY = t=1 Ut /(1 + d) where T is the number of life years remaining; Ut utility in year t and d the time preference discount rate. 1.4. The events sequence matters When using per annum utilities, keep salient the sequence of events, since what follows what affects the total utility derived from a health outcome (Drummond et al., 1997, pp. 146–147). As this paper will show, it is especially important to keep salient any sequence 1 We thus ignore situations in which the chooser lacks: (1) numerical probabilities; or even (2) the full specification of the event space itself (in which case the chooser may be later surprised by a completely unanticipated health outcome). Situations with a lack of (1) were emphasised by Keynes (1921). Situations with a lack of (2) were emphasised by Knight (1933) according to Langlois and Cosgel’s (1993) re-interpretation of Knightian uncertainty. Such situations must be ignored to use standard gambles to elicit utilities.
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involving a change in knowledge from risk endured followed by risk resolved. This is because the health care recipient may experience a radically different level of utility during the two phases of this sequence. She may for instance while facing risk about her health outcome, have her utility depressed by fear below what it would otherwise have been. Then after that risk is resolved—if it is resolved in the form of the risky health intervention act having turned out to be successful—she may have her utility elevated by relief above what it would otherwise have been. 1.5. Utilities independent of knowledge ahead Let utility be measured on the vertical axis with ‘0’ for the worst quality of life derivable from a health outcome during each year. Let this utility of 0 be derived from a health outcome of death. Let the utility scale end with a maximum value of ‘1’ for the best quality of life health outcome during each year. Let this utility of 1 be reaped from perfect health. Let a set of health outcomes from the worst to the best be denoted along the horizontal axis (a big enough finite set that visually the dots denoting each distinct outcome appear as a continuous line). For those with already damaged hips, let us in Fig. 1 denote four of possible health intervention acts with their associated health outcomes: utility best 1
0.9
0.05
worst 0
Y0
death (possible) (iv) operation
. .
Y1
low mobility much pain (i) nothing
.
A
B
.
.
Y2 low mobility limited pain
Y3 moderate mobility limited pain
(ii) pain killers
(iii) Al'r Technique
health
YF outcomes full mobility (possible or certain) no pain (possible) (iv) operation
interventions
Fig. 1. Discovering utilities of outcomes—is A true or B true? Or does this knowledge-ahead-independent x-axis preclude true utilities? Note: (1) There is only a one-to-one correspondence between outcomes and sure acts. Risky act (iv) of an operation has two possible outcomes Y0 and YF . (2) If B is true, the utility increment from a successful (iv) (operation) over (i) (do nothing) is large 1 − 0.05 = 0.95—maybe large enough to prefer the operation despite its cost and risk of death. But if A is true, that increment is a modest 1 − 0.9 = 0.1—maybe too small to prefer the operation, particularly if the risk of death is sizable. Also (iii) (Alexander Technique education with no risk of death) is nearly as good as a successful operation if B not A is true. (3) The x-axis of Fig. 1 limits probabilities to being aggregation weights—weights for aggregating the utilities of alternative possible outcomes into an overall valuation of a risky act. The x-axis excludes utility function C of Table 1 wherein knowledge ahead (about which outcome will occur) affects the utilities of outcomes.
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(i) nothing with the health outcome Y1 of low mobility and much pain; (ii) pain killers with the health outcome Y2 of low mobility but limited pain; (iii) Alexander Technique education to enhance mobility with the health outcome Y3 of moderate mobility and limited pain; (iv) risky hip replacement operation with the outcome YF of full mobility and no pain (if successful, or outcome Y0 of death (if unsuccessful). The recommendation of (iv) (an expensive risky operation) depends partly on the increment to utility of a successful health outcome of operating over (i) (do nothing). The importance of discovering the true utilities of the individual health outcomes is illustrated in Fig. 1. The utility increment from the expensive operation over doing nothing could be a modest 0.1 units. This would be the case if utility is already at 0.9 in the case of doing nothing because the true curve is A. But the utility increment from the expensive operation could be a hefty 0.95 units. This would be the case if utility is a miserable 0.05 in the case of doing nothing because the true curve is B. 1.6. Utilities based on knowledge ahead It could be that the utility increment from an operation cannot be depicted by any curve on Fig. 1 because situation C of Table 1 holds. In Fig. 1, utilities of outcomes are independent of knowledge ahead. This means that each outcome causes a unique utility, allowing a curve to denote how the set of outcomes map into utility. In situation C by contrast utilities of outcomes are knowledge ahead based. In situation C, the utility of an outcome differs depending on how much the health recipient knows ahead about her health outcome—about which acts are available to her, which act will be chosen to replace the status-quo; her likelihood of a successful outcome from this intervention, and what is the case if the intervention is unsuccessful. That is, in situation C a recipient’s utility from an outcome is not based exclusively on the health outcome after it becomes known. Table 1 Utility increment from outcome YF over outcome Y1
A B C
Utility function
Increment in utility
The concave–convex curve A in Fig. 1 is the true utility shape The convex–concave curve B in Fig. 1 is the true utility shape Factors like fear mean that the utility of each individual outcome Yi is not caused by that outcome alone. It is caused partly by what the health care recipient knows ahead about: (i) available acts; (ii) which act will be chosen to replace the status-quo; and (iii) the other possible outcomes of the chosen and rejected acts
This is a mighty 0.95 This is a modest 0.10 Factors like fear mean that the utility increment is not depictable in Fig. 1. It is based also on what the health care recipient knows ahead about the set of available acts and associated other possible outcomes replacing the status quo
Note: The information in Table 1 concerns the utilities of the individual outcomes. The value of an act is constructed by aggregating (e.g. by probability weights) the utilities of individual outcomes.
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An example of situation C is Betty’s utility from the best health outcome YF . For Betty there is no unique utility caused by YF . Betty’s utility from YF is the maximum utility of 1 as depicted in Fig. 1 if YF is the outcome of a guaranteed-to-be successful hip replacement operation. But she gets a much lower utility from YF if it is the outcome of a risky operation under which she has a chance of dying. Her lower utility from YF (should it eventuate under a risky operation) is partly because of her fear of dying during the time in between choosing the operation and learning that the outcome was YF . There is an almost universal belief that utilities for constructing QALYs can be depicted with an outcomes axis as in Fig. 1 wherein outcomes determine utilities independently of what the health care recipient knows ahead definitely and probabilistically about the outcome and the entire decision situation. Situation C is rarely contemplated. Prominent minority dissenting views from the health economics perspective are in Richardson (1994, 2001) and Sen (2002). 1.7. Axiomatised generalised expected utility and Fig. 1 Axiomatised expected utility theory is not any formula that values acts as a probability weighted sum of their component utilities of outcomes. It is the formula that obeys all the restrictions that an axiom set implying this formula imposes. These are restrictions on how acts and outcomes are evaluated and given values (utilities). These restrictions include the requirement that utilities from the five health outcomes Y0 to YF can be depicted on Fig. 1—a variant of the figure in Friedman and Savage (1948). This 1948 article expounds how under axiomatised expected utility theory, outcomes are evaluated independently of the chooser’s degree of knowledge ahead of the outcome. It explains how an outcome is evaluated as having exactly the same utility if the chooser knows in advance it is guaranteed to occur, as if the chooser has in advance merely probabilistic knowledge of learning later that it has occurred. This precludes a chooser’s risk attitude affecting the utility of an outcome. From full knowledge that she has the guaranteed outcome YF of full mobility and no pain, Betty derives no more utility than if she only learns later that outcome YF has occurred. Learning later occurs whenever that outcome YF was not guaranteed, but merely probabilistic knowledge ahead at the time of choosing the operation. At that time of choosing the operation it might be that Betty only learns later whether the outcome is full mobility and no pain, YF , or death, Y0 . According to axiomatised expected utility, living through a post-choice period in which she might die in no way reduces her total utility! Expected utility theory’s constraint that the utilities of outcomes be identical under certainty and under risk recurs with most generalisations of expected utility theory. It recurs for instance in the Quiggin and cumulative prospect versions of rank dependent theories that are applied to health economics issues (Bleichrodt and Quiggin, 1999; Bleichrodt et al., 2001). This identity treatment of the utilities of risky and riskless outcomes enters such rank dependent theories via their imposition of a preference for acts with stochastically dominant (de)cumulative distributions of outcomes. Imposition of such a preference precludes choosers from considering utility derived from anything other than the outcome alone. It precludes the utility of an outcome being affected by whether a chooser’s knowledge ahead of that outcome is merely probabilistic (risky) or certain.
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1.8. The use of standard gambles to elicit utilities Utilities of outcomes that are “knowledge-ahead independent”, in conjunction with the other features of axiomatised expected utility theory, enable standard gambles to be used to deduce utilities. The numbers so elicited are the chooser’s true utilities only if the participant’s choices obey all the restrictions of axiomatised expected utility theory—including its restriction that the utilities of outcomes are “knowledge-ahead independent”. Use of standard gambles to deduce the utilities used in QALYs is widespread, Torrance (1986). The typical way health economists use expected utility theory’s standard gamble procedure is in the probability equivalent version, as follows. Describe to a participant the two possible health outcomes Y0 of death and YF of full health. Next ask that participant to select a probability number p (between 0 and 1) that would make choosing this risky operation as attractive as choosing a guaranteed outcome Y1 of low mobility + much pain. This number p1 is the utility for Y1 . Repeat the exercise for guaranteed outcomes Y2 and Y3 . The resulting probability numbers p2 and p3 are the utilities of the health outcomes Y2 and Y3 , respectively. 1.9. Systematic bias in standard gamble utilities If the standard gamble procedure for eliciting utilities is valid, the three utility numbers p1 , p2 and p3 lie on an identical utility curve under the reverse “certainty equivalent” version of the standard gamble procedure. Under this version participants select from the continuum of health outcomes intermediate safe (i.e. certain) health outcomes that they find as attractive as having the probability numbers p1 , p2 and p3 of full health and thus a probability 1 − pi , i = 1, 2, 3 of death. If choosers obey all the restrictions of expected utility theory so that the utilities elicited are their true utilities, their certainty equivalent choices are respectively Y1 , Y2 and Y3 . But this check on the appropriateness of utilities so elicited has not been made by health economists. It would have been a check to alert them to the procedure’s inappropriateness. Hershey et al. (1982) showed that the utility curve of money nearly always twists from concave to convex like A in Fig. 1 when scientists use the “probability equivalence” version, from convex to concave like B when scientists use the reverse “certainty equivalence” version. Thus, the two versions of the standard gamble: 1, by probability equivalences, and 2, by certainty equivalences, systematically and predictably yield opposite shapes of utilities in the lower outcome range, and then both inflect to yield opposite shapes for the higher outcome range. To check, put on the board a figure scaled 0 to 1 on the vertical utility axis, US$ 0–1 billion on the horizontal money axis. Then ask one colleague for three probability equivalences and insert them in this figure. Draw another figure scaled 0–50 on the vertical utility axis, £0–50 on the horizontal money axis. Ask another colleague for three certainty equivalences and insert them in the second figure.2 This normally suffices to identify a marked twist from concave to convex under probability equivalence, and from convex to concave under 2
The new scale reduces the likelihood that the second asked seeks to proffer answers consistent with the first.
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certainty equivalence, with the twists in each case occurring at the point of the respondent’s starting intermediate outcome (under probability equivalence) and respondent’s starting probability (under certainty equivalence). Subsequent work reinforces the seriousness of the discrepancies generated by these reverse twists in the utility curve under the two versions, Hershey and Schoemaker (1985). In health applications, this twisting would mean the utility of some good health outcomes are arbitrarily stretched or compressed depending on which way the standard gamble is used. In Fig. 1, curve B illustrates such stretching (curve A such compressing) for the utilities of health outcomes Y2 and Y3 relative to those of health outcomes Y0 and Y1 . Health economists typically use the probability equivalence version of the standard gamble procedure and thus attribute huge utility improvements to some expensive interventions that improve health outcomes toward full health. Were they instead to use the certainty equivalence version, they would, as in curve A of Fig. 1, tend to attribute minor utility improvements to some expensive interventions that improve health outcomes toward full health.3 Thus, any user of the standard gamble procedure in health economics can predictably twist the policy conclusions to advocate dearer interventions with better health outcomes, or the converse. Standard gamble utilities are manipulable. Major predictable changes in estimates are a hall-mark of major omitted variables bias. Progress has already been made in identifying the omitted variables in standard gamble utilities (e.g. Krzysztofowicz, 1994), and in other expected utility theory techniques for eliciting utilities (e.g. Inder and O’Brien, 2003). A decision theory is usable as an approximation if it omits minor causal chains that rarely have a discernible impact on choice so that gigantic data bases are required to detect their minute effects out of background noise. To omit causal factors that generate such major errors that their effects are transparent above random noise—even in small samples—is to virtually guarantee large scale misdirection of health budgets. The omitted variables bias in standard gamble utilities stems from expected utility theory’s omission of the knowledge-ahead-based cause–effect chains yielding pre and post-choice risk effects. 1.10. Paper format Section 2 introduces the stages of knowledge periodisation necessary to delineate the sources of the risk effects excluded under expected utility theory and many of its generali3 Hershey and Schoemaker (1985) propose that the certainty equivalent curve A is the true shape. Their erroneous belief that the standard gamble procedure includes risk attitude (Schoemaker, 1982), causes them to begin their search for an explanation with the premiss that the discrepancies between the two versions must be due to irrationality on the part of the participants (not an error in the elicitation procedure itself in irrationally excluding risk attitude). The only form of irrationality they find that fits the facts is if participants reframe the probability equivalence questions in a way that, in conjunction with some irrational loss aversion, results in their true concave convex curve being twisted into a convex–concave curve. Such an account fails to explain the twist in their perceived true curve at the point of the first answer elicited. The example they give is that if participants are asked to choose between US$ 100 for sure and a 50–50 chance of either US$ 200 or zero, participants might reframe it as a choice between $0 for sure and a 50–50 chance of either gaining or losing US$ 100, p1224. Had Hershey and Schoemaker understood that the standard gamble procedure excludes risk attitude, they would have appreciated that neither shape can be true in the sense of representing utilities of a reasonable rational person who, as they themselves state, should take risk attitude into account. Their hunt for the explanation would not have confined them to seeking some irrationality account.
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sations. Section 3 illustrates with a non-emotional risk effect, inability to commit. Section 4 examines proposals to include stages of knowledge risk effects within axiomatised expected utility theory and shows how each proposal fails. Section 5 explains how the omitted risk effects cause the systematic biases inherent in expected utility theory’s standard gamble procedure. Section 6 concludes. 2. Choice allowing for changing knowledge and risk effects 2.1. Von Neumann and Morgenstern omit risk attitude because of a contradiction For expected utility theory, Ramsey’s 1926 lecture presented a partial axiomatisation (1950), and von Neumann and Morgenstern’s 1944 book, Theory of Games and Economic Behaviour presented a full axiomatisation. Von Neumann and Morgenstern (1944, 1947, 1953, 1972, p. 28) state their regret that they have “practically defined numerical utility” . . . as excluding knowledge ahead risk effects. Morgenstern reported in 1974 shortly before his death on the deep conviction of himself and von Neumann that such knowledge ahead risk effects exist and should be included (Morgenstern, 1979). Von Neumann and Morgenstern (1947, 1953, 1972, pp. 628–632) had added an appendix to the 1947 edition of their book to explain to their critics why they had had to leave to future researchers the task of including these knowledge ahead risk effects. They reported that these “cannot be formulated free of contradiction on this level”. The contradiction encountered and described in this appendix is their puzzlement that mutually exclusive outcomes (e.g. the mutually exclusive outcomes of death and full health) interact with each other (as they do in the case of fear and inability to commit). The contradiction is solved by recognising that risky acts have at least a two period future (Pope, 1985). There is a pre-outcome period during which the mutually exclusive outcomes interact with each other since both are possibilities in the mind of the chooser. There is a sequel post-outcome period, beginning when the outcome is learned. During the pre-outcome period, people can reap a utility of gambling from juggling these mutually exclusive possible outcomes in their minds. During the post-outcome period, by memory and physical heritage that there used to be other possible outcomes, there remain effects of the prior lack of certainty. For risk to be contrasted with the certainty of sure acts at the time of choice, the risky acts require this epistemic two-period future. This two period future is one demarcated by the change in knowledge between the two periods, from a risky future, to full knowledge (in simple cases). Periodising by when in the future risk is resolved, reveals a timing contradiction in von Neumann and Morgenstern’s (1944, 1947, 1953, 1972, p. 19) assumption that all outcomes of all acts are known at the same time. In a choice set comprising a sure and risky act, the timing contradiction is as follows. Let Betty choose between these two acts at t = 0. If Betty chooses the sure act, she knows the outcome at time t = 0. But if instead she chooses the risky act, she knows the outcome at a different later time, t > 0. At time t = 0, she merely probabilistically knows the outcome of the risky act. It is only at the later time t > 0 that the risk is resolved, and she will know that one of the previously possible outcomes is the actual outcome, and that the other previously possible outcomes are now impossible.
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In order to (1) avoid the timing contradiction of assuming that choosers learn the outcomes of sure and risky acts simultaneously, and (2) reach the higher level von Neumann and Morgenstern sought (wherein mutually exclusive outcomes can interact), we need to move to a new framework. We need to move from expected utility theory’s atemporal framework (backwards decomposable into an atemporal set of axioms) to a dynamic framework in which changes in knowledge demarcate the future from the point of choice. We do this in the next sections. 2.2. Anticipated changes in knowledge and risk effects At the time of deciding under risk, choosers are ignorant of which of various possible outcomes will ensue. Choosers can however anticipate changes in what they know, even though they cannot be confident whether the new knowledge will be good or bad. When Betty encounters the problem of a mushy hip, she can anticipate that after not knowing what are her available choices, she will later ascertain these, and after not knowing what to choose, that she will later have chosen. If she chooses a risky hip replacement operation, she can anticipate knowing later what she does not know now, that it is successful, and also can anticipate knowing nothing later, because her risky hip operation is unsuccessful and she dies. See Table 2. Table 2 Three anticipated changes in knowledge Ki , i = 1, 2, 3 after encountering a problem Time
Knowledge with respect to a future or past event
K1 From
Pre-choice set period
To
Choice set identified
Only probabilistic knowledge of what the sub-acts of search/negotiation in ascertaining and creating available acts will identify as the choice set, e.g. unclear which interventions are feasible for Betty’s damaged hip Knowing with a probability of 1 the choice set, e.g. which intervention acts are feasible
K2 From
Pre-act choice period
To
Act chosen
K3 From
Pre-outcome period
To
Post-outcome period
Only probabilistic knowledge about what the sub-acts of evaluating the available acts will identify as the act to choose, e.g. uncertain which intervention act will turn out to be the best Knowing with a probability of 1 the act chosen, e.g. chose a hip replacement operation Only probabilistic knowledge of which will be learned to be the chosen act’s actual outcome, e.g. the outcome could be full mobility and no pain, or it could be death Knowing with a probability of 1 the chosen act’s actual outcome, e.g. knowing that the risky operation was successful, or dead
Risk effects (such as those arising out of Betty’s fear) arise out of anticipated possible changes in probabilistic knowledge from being uncertain whether the operation will be successful to knowing whether the operation was successful. If there could never be this change in knowledge, Betty has no fear she will die. Without any anticipated future changes in knowledge, Betty and all of us have chosen a future perceived as certain with regard to
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its effects on us—chosen a sure act. Without any anticipated future changes in knowledge, we experience no risk effects from unknown outcomes. In the simplest risky choices illustrated in Table 2, anticipated changes in knowledge causing risk effects are changes from merely probabilistic knowledge to full certainty. In this table, each future change in knowledge creates at that future date a disparity between: 1, what was previously a set of possible mutually exclusive events and associated utilities (of which the chooser had a non-degenerate probabilistic knowledge); and 2, the chooser’s new knowledge at that future date, namely, (a) full certainty about the actual occurrence of one previously possible event, and (b) full certainty of the non-occurrence of all the other previously possible mutually exclusive events. Anticipated future changes in knowledge generate anticipated risk effects such as fear. Risk effects influence choices if and only if anticipated effects of changes in the probabilistic knowledge impact on the chooser’s evaluation of acts and outcomes. As shown in Section 1, in axiomatised expected utility theory (and in its generalisations to the rank dependent models of Quiggin and cumulative prospect theory) acts and outcomes are evaluated independent of Ki , i = 1, 2, 3—i.e. independent of knowing ahead the choice set, the act, and which outcome will occur—and thus excluding all pre- and post-decision risk effects. Theories that exclude anticipated pre- and post-decision risk effects are simpler. But as in the well-known comment of Einstein, a theory should be simple, but not too simple. By today, we know that expected utility theory’s standard gamble procedure to elicit utilities has a hallmark of being too simple, a hallmark of major omitted risk effects. 2.3. Old misleading and new neutral terminology Misleading terminology has hindered scientists discerning the seriousness of omitting the anticipated effects of pre- and post-decision risk effects. The name most often used for the post-decision risk effects of K3 excluded under expected utility theory is the “(dis)utility of gambling”. This term has the disadvantage that Friedman and Savage (1948) created confusion by altering it to mean a (concave) convex “as if certain” utility of monetary outcomes function. This name has the further disadvantage of frivolous (even derogatory, unethical, irrational) connotations. These misleadingly create the impression that the (positive and negative) satisfactions omitted under expected utility theory are minor—or even desirable ones to omit. There are disadvantages in using terms “risk attitude” and “risk aversion” (preference) for the downgrading (upgrading) of the value of assets that have higher riskiness as measured by the dispersion of the assets returns. The disadvantages are that Marschak (1950) confusingly changed risk aversion (preference) to mean a concave (convex) “as if certain” utility function of asset outcomes. Marschak’s terminological changes became ensconced in the construction and naming of the Arrow–Pratt measures of risk aversion (e.g. Arrow, 1971; Pratt, 1964). Pope (1996/1997, 2001) lists other names used for the anticipated post-decision risk effects of K3 and confusing changes in terminology or misleading connotations associated with each, then introduces a new terminology to avoid the confusions/misleading connotations. Under this new terminology, satisfactions derived from the outcomes alone
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are termed primary. Primary satisfactions are knowledge ahead independent in the sense of the utilities depicted in Fig. 1. Such utilities are satisfactions from learning that a particular outcome has occurred (or will occur) which are independent of whether that outcome was at the point of choice: certain having a probability of 1; very unlikely having only a probability of 0.05; and better or worse than other possible outcomes that instead might have occurred (yet occur). Secondary satisfactions by contrast are based on such degrees and aspects of knowledge (Table 3). Table 3 Secondary satisfactions anticipated at the time a problem requires action Event
1 Pre-act search 2 Choice set identified 3 Pre-act evaluation
4 Act chosen 5 Background conditions
6 Learn outcome 7 Background conditions
When satisfaction reaped
Source of secondary satisfaction, probabilistic knowledge of an event or its wisdom
Pre-choice set period: before stop Uncertainty about which acts are search for available acts available and what will ensue Time point following event 1 Pre-choice period: before end Uncertainty about which act to valuation of available acts choose and certainty or uncertainty on wisdom of 1–2 and what will ensue Time point following event 3 Pre-outcome period: before Uncertainty or certainty about outcome becomes known outcomes and wisdom of 1–4 and what will ensue Time point following event 5 Post-outcome period: after Prior uncertainty or certainty about outcome has become known outcomes and wisdom of 1–4
Primary satisfactions from the events 1–7 themselves, i.e. independent of the knowledge ahead based sources listed in the last column. Secondary satisfactions from the events are those dependent on chooser’s prior or current probabilistic knowledge of events 1–7 or their wisdom, i.e. dependent on the knowledge ahead based sources listed in the last column. Note: (1) Satisfactions include both positive ones and negative ones (i.e. dissatisfactions). (2) Satisfactions denote levels of attainment of the chooser’s goals. These need not be solely or at all related to personal degrees of pleasure or pain, but could be entirely satisfactions with respect to attainments or features of another individual or of society or the planet, etc. (3) Making a choice involves a series of prior choices, here termed pre-acts. (4) It is only in the simplest risky choice situation that there is a single pre- and post-outcome period. In more complex cases, there are multiple pre-outcome periods. Multiple pre-outcome periods occur for instance whenever aspects of the outcome get learned at different times in the future. (5) For a sure act, the pre-outcome uncertain period is degenerate, of zero duration. (6) Axiomatic derivations of expected utility theory contain no entities denoting background conditions. Hence, a pre-requisite for its axiomatic justification is that all relevant background conditions must be subsumable into an elaborated specification of individual outcomes of an act. Background conditions which are sources of primary satisfactions can be so subsumed. By contrast background conditions which are sources of secondary satisfactions cannot be so subsumed: elaborating outcomes to include secondary satisfactions destroys that theory’s axiomatic basis, as also its applicability to multi-stage acts (Pope, 1983, 1984, 1989, 2000).
In Table 3, the positive and negative satisfactions from the risk effects that “dominance preserving” theories exclude are no longer derogatorily termed utilities of gambling. Rather they are given the more neutral name of (positive and negative) secondary satisfactions.
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The pair of names is not intended to give the impression that secondary satisfactions are unimportant relative to primary satisfactions, ones that as a first approximation can be ignored. To the contrary in many decision contexts, secondary satisfactions are quantitatively more important. Table 4 Changing knowledge demarcating six sources of emotional secondary satisfactions Problem at time t = −1 s ↑ Pre-decision period (act unknown) ↓
Anticipated future 1
Bad stress, e.g. time lost choosing and Good stress, e.g. brain growth from challenge of choosing
Act choice at time t = 0 ↑
2
Pre-outcome period (act’s outcome unknown)
3
Fear that one of the act’s possible bad outcomes may come and Hope that one of the act’s possible good outcomes may come Depression at guaranteed bad aspects of the future and Joys in guaranteed good aspects of the future
↓ Learn outcome at time t = 1 ↑
4
Disappointment that an act’s good outcome did not occur or Thankfulness that an act’s bad outcome did not occur
5
Regret and blame since rejected acts had better outcomes or Rejoicing and praise since rejected acts had worse outcomes
Post-outcome period (act’s outcome known) 6
Bad shock at act’s completely unforeseen bad outcome or Good shock at acts’ completely unforeseen good outcome
↓ New problem at time t = 2 1–6 help in this next pre-decision period, in making the next choice at t = 3. We reason, choose partly through the emotions. Brain damage removing fear removes good decision making (Demasio, 1984) See on emotional sources of secondary satisfactions in the pre-decision period: Janis and Mann (1977), Lazarus (2000), Somerfield and McCrae (2000) and Hardie (2002); pre-outcome period: Plato (1967) and Smith (1778), Darwin (1874), Canaan (1926), Bernard (1963), Hart (1930), Pope (1983, 1984), Wu (1991), Rothenberg (1992, 1994), Schade and Kunreuther (2000) and Inder and O’Brien (2003); post-outcome period: von Neumann and Morgenstern (1944), Savage (1954), Markowitz (1959), Loomes and Sugden (1982), Bell (1983), Hagen (1985) and Berns et al. (2001).
Table 4 instances emotional sources of secondary satisfactions and locates these in the evolving stages of knowledge anticipated on encountering a problem. In Table 4, and henceforth, we have collapsed the two pre-choice periods (that of selecting a choice set and choosing within it) into one period. This collapse is for brevity, not to downplay the significance of pre-decision risk effects. Consider for instance the evidence that some health insurance is taken out to avoid the stress of making financial decisions after getting ill and needing health help.
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There is extensive medical evidence of the impact of emotional states on health, including those pertaining to the emotional sources of secondary satisfactions in Table 4. But since some believe that emotional effects should not enter public health decisions, it is important that health economists appreciate that secondary satisfactions stem also from non-emotional sources. One such source is an (in)ability to commit. To delineate its role, let us look at an amalgam of the decisions that recently faced the author’s mother Betty (who had a mushy hip) and a brother-in-law helping administer AIDS projects in Africa.
3. A non-emotional source of secondary satisfaction in a health choice Betty’s problem of a “mushy” hip joint yields her QALY (quality adjusted life years) with the two features depictable on the knowledge ahead independent health outcomes axis of Fig. 1, namely: (a) pain and (b) low mobility, plus (c), a knowledge ahead based commitment effect of her QALY not depictable on the x-axis of Fig. 1. A person committing at time t = 0—as distinct from a person lying and merely saying that they will commit—must know at t = 0 with probability 1 that nothing ahead will preclude that commitment. (We shall deal later in Section 4.1 with the secondary satisfactions associated with partial, conditional commitment effects.) In Betty’s case, let the commitment concern the project described in (c) of Table 5. It encapsulates a vivid commitment effect problem in communities where the AIDS inroads are severe: all the qualified care givers are infected. Their considerable risk of being too ill to serve for long precludes them from being able to elicit the level of co-operation from others in the community to introduce some sorts of AIDS amelioration projects. Correspondingly, inability to commit creates problems for organisations fostering community health, such as the World Bank, the World Health Organisation, local governments and non-government organisations. Table 5 Betty’s decision situation (a) (b) (c)
(d)
Her hip hurts, reducing her QALY below that gained from no pain She has restricted mobility, reducing her QALY below that gained from full mobility Her other potential source of QALY is a community job. To do it, she must tomorrow commit unconditionally to work on it for the next 5 years (apart from 1 month per annum of holiday/sick leave), something she has already decided to do if her hip decision does not preclude her committing She must decide tomorrow on her health intervention act
In reading (c) bear in mind commitment’s knowledge ahead prerequisites. In particular, note that a necessary condition for Betty tomorrow to have her desired ability to commit to the project (i.e. guarantee that she will do it), is that tomorrow Betty knows with probability 1 that for these next 5 years, nothing will prevent her working full time on it. Let us exclude four common features of actual decisions that expected utility theory excludes: (i) inherently multi-dimensional goals, (ii) erroneous anticipations about an act’s possible outcomes and the satisfactions to be reaped from these outcomes and associated acts and related costs of decision making,
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(iii) satisfactions indices that cannot be mapped into real numbers, (iv) choosers ignorant of how to rank acts in terms of the satisfaction yielded them. Let us for simplicity also exclude choosers who have considerable merit with respect to being practicable and rational, namely choosers who satisfice—i.e. take any act that reaches a satisfactory level of utility (see, e.g. Simon, 1983). Also let us exclude another common feature of many actual decisions, namely choosers whose satisfactions we deem irrational. With these exclusions, we can offer an intuitive decisive gold standard, a sufficient condition for identifying when a choice theory selects an irrational act (Table 6). Table 6 Intuitive decisive gold standard: sufficient condition for an irrational choice theory If Then
For any choice set, we know with certainty at time of choice that another available act than that chosen by our theory would yield Betty a better QALY Our theory is irrational
What makes Table 6 uncontroversial is that it avoids the contentious issue of: 1. whether the only rational summation rule for forming an overall valuation of a risky act is to aggregate its mutually exclusive utilities by their probabilities (as happens with a number of decision procedures including expected utility theory) or 2. whether alternative aggregation rules may be as reasonable or even more reasonable in some circumstances as Keynes (1921), Allais (1952, 1979a,b, 1986) and others have argued. Other aggregation rules are used in many generalisations of expected utility theory and in some non-nested alternatives to expected utility theory. We shall show that axiomatised expected utility theory fails this intuitive gold standard under both versions that are internally consistent with its own implication that utilities be a function of outcomes alone as in Fig. 1. These are the versions of Ramsey (1950), that outcomes be evaluated independently of knowledge ahead, and of Friedman and Savage (1948), that outcomes be evaluated “as if certain”. We shall show that the failure to meet this intuitive gold standard stems from the omission of actual secondary satisfactions. We do this in stages. First, we specify Betty’s decision procedure for choice sets comprising only sure acts and make simplifying assumptions to allow an easy partition of Betty’s satisfactions into primary and secondary ones. We present a scenario 1, a choice set under certainty, and show that, whereas the Ramsey version of expected utility theory fails it, both Betty’s decision procedure and the Friedman–Savage version of expected utility theory survive it. But under certainty there is no need for expected utility theory. The Friedman–Savage version survives the gold standard since it does not impute illusory secondary satisfactions under its “as if certain” rule. All outcomes really are certain. We next introduce a scenario 2, a new choice set which includes a risky act, something realistic in health choices, and also essential for the application of the standard gamble procedure. We elaborate on Betty’s decision procedure to include the evaluation of risky acts. We show that for the particular scenario 2 choice set examined, both the Friedman–Savage version and the Ramsey version fail the intuitive gold standard whereas Betty’s decision procedure survives it. The intuitive gold standard has then been used for two particular choice sets, with a particular constellation of satisfactions. We then explain the general implications of this finding of the irrationality of axiomatised expected utility theory. The
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sequel section (Section 4) examines proposals to overcome the irrationality by means of opportunity cost, elaborated outcomes, and different valuing of outcomes. It shows that each proposal for including fear or commitment destroys expected utility theory’s axiomatic base—destroys its unique mapping from outcomes into utilities that is required in order to use standard gamble procedures to elicit utilities. First, let us introduce Betty’s way of choosing among sure acts, Table 7. Table 7 Betty’s choice procedure under certainty 1
Betty decides on her objective, to maximise U = U1 + U2 + U3 + U4 + U5
2 3
4
(1)
where U is an index of the QALY caused by her act choice, Ut is that act’s satisfaction in year t, t = 1, . . . , 5, and Ut is normed with 0 her worst conceivable satisfaction, and 1 her highest Betty discovers which acts are available to her Betty, who has no constraints, costs or benefits from her pre-decision analysis, anticipates today precisely her U caused by each available act. The fact that she anticipates satisfactions from all the actual cause–effect chains initiated if she were to choose each act and traces these through to ascertain each act’s U is indicated with the script typeface. (An ordinary typeface U will denote what satisfactions Fig. 1 and expected utility theory permits/requires Betty to include.) At her time of choice, t = 0, Betty chooses an act with highest U
For simplicity in depicting the partitioning of Betty’s total QALY index U between its component primary and secondary satisfactions, let these be additively separable, i.e. let: Ut = Pt + St (2) where Pt is a cardinal index of anticipated primary satisfactions in year t, St is a cardinal index of anticipated secondary satisfactions in year t. We can now decompose U—the sum of Ut over 5 years and index of Betty’s QALY caused by her chosen act. We can decompose it into her primary satisfaction over the 5 years plus her secondary satisfaction over the 5 years. That is, ifs P = P1 + P2 + P3 + P4 + P5 (3) and S = S1 + S2 + S3 + S4 + S5
(4)
U=P +S
(5)
then
Let Betty discover that her available acts are those of Table 8. Table 8 Betty’s choice set: scenario 1 under certainty HR+ Hip replacement operation plus intensive physiotherapy: This yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation followed by two consecutive months of sick leave doing supervised exercises, then full health HR Hip replacement operation with minor physiotherapy: This yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation requiring only 2 weeks of sick leave, then nearly full health AT Alexander Technique education to reduce her asymmetric posture and movement: This yields her an outcomes flow of small steady increments in mobility and decreases in pain from her after work education in this technique
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Table 9 Betty’s anticipations of her primary and secondary satisfactions when choosing tomorrow Available sure acts
Outcomes, knowledge effects, satisfactions flows
QALY
Hip replacement operation plus intensive physiotherapy Outcome flow, YHR+ Year 1, queue Year 2, new hip, long leave Years 3–5, full health P1 = 0.1 P2 = 0.3 P3 = 0.5 P4 = 0.5 Primary satisfactions, Pt Effect flow of YHR+ known Years 1–5, no job since knows she will take long sick leave in year 2 S1 = 0 Secondary satisfactions, St S2 = 0 S3 = 0 S4 = 0 = P t + St U1 = 0.1 U2 = 0.3 U3 = 0.5 U4 = 0.5 Satisfaction in year t, Ut QALY, U = Ut = P + S
P5 = 0.5
P = 1.9
S5 = 0 U5 = 0.5
S=0 P + S = Ut U = 1.9
P = 1.48
HR Hip replacement operation with minor physiotherapy Outcome flow, YHR Year 1, queue Primary satisfactions, Pt P1 = 0.1 Effect flow of YHR known Secondary satisfactions, St S1 = 0.5 = P + S U1 = 0.6 Satisfaction in year t, U t t t QALY, U = Ut = P + S
Year 2, new hip, short leave Years 3–5, nearly full health P2 = 0.33 P3 = 0.35 P4 = 0.35 Years 1–4, progress on full time 5 year job S2 = 0.5 S3 = 0.5 S4 = 0.5 U2 = 0.83 U3 = 0.85 U4 = 0.85
P5 = 0.35 Year 5, job done S5 = 0.5 U5 = 0.85
AT Alexander Technique Outcome flow, YAT Primary satisfactions, Pt Effect flow of YAT known Secondary satisfactions, St Satisfaction in year t, Ut = Pt + St QALY, U = Ut = P + S
Years 1–5, annual, ↑ in mobility↓ in pain as better gait, no leave P2 = 0.15 P3 = 0.2 P4 = 0.25 Years 1–4, progress on full time 5-year job S2 = 0.5 S3 = 0.5 S4 = 0.5 U2 = 0.65 U3 = 0.7 U4 = 0.75
P5 = 0.3 Year 5, job done S5 = 0.5 U5 = 0.8
P1 = 0.1 S1 = 0.5 U1 = 0.6
S = 2.5 P + S = Ut U = 3.98
P=1 S = 2.5 P + S = Ut U = 3.5
HR > AT HR+ : HR better than AT which is much better than HR+ . Betty’s rational choice is HR. In calculating her QALY (quality of life U), the Ramsey version of axiomatised expected utility theory limits Betty to P, primary satisfactions, i.e. has her irrationally choose her worst act HR+ .
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Let the cause–effect chains initiated by each available act, the contributions to Betty’s satisfaction of each of these effects and her QALY index U from each act be as in Table 9. Betty’s utility from primary satisfactions is what she derives each year from her amount of pain, mobility and sick leave independently of knowing beforehand what these amounts would be. The amount of such utility is recorded in the second row of Table 9 for each of her three available acts, HR+ , HR and AT. Any utility from positive secondary satisfactions is what she derives from already knowing at her point of choice that her future hip outcome flow involves sufficiently little or no sick leave and so is able to commit and take on her project job. The amount of such utility is recorded in the fourth row for each of the three interventions, HR+ , HR and AT. Her total utility from the combined effects of her primary plus secondary satisfactions is recorded in the last row for each act. Comparing rows 2 and 4 for each act, it can be seen that Betty’s secondary (knowledge ahead based) satisfactions are quantitatively bigger than her primary satisfactions. Readers may realise from introspection that there are many occasions and choosers for whom this will be the case. Step 3 of Betty’s procedure of Table 7 enjoins her to include in her evaluation of each act all cause–effect chains initiated by the act that enter her QALY. Thereby she includes in her calculation of her QALY both primary and secondary satisfactions. Betty makes her best choice of HR and obtains her highest possible QALY of 3.98. Consider now Betty’s choice if in step 3 of Table 7 she excludes her secondary satisfactions. This is Ramsey’s version of expected utility theory. His version allows her to have the requisite feature of Fig. 1, namely that (for a given origin and scale), her mapping from outcomes into utilities is unique. She would choose HR+ since it is the act with the highest amount of primary satisfactions. But due to its associated extended sick leave, she would be unable to commit and undertake the project job. She would be choosing the act with the guaranteed lowest total QALY. The Ramsey version thus fails the intuitive gold standard. Consider next the Friedman–Savage version of expected utility theory. This is that outcomes be evaluated “as if certain”. It meets the intuitive gold standard and chooses exactly as does Betty. This is because the choice set comprises three sure acts, rendering irrelevant the “as if certain” means of constraining utilities of outcomes to be identical under risk and certainty. But once we alter the choice set to include a risky act, “as if certain” evaluations can impute satisfactions that are illusory under risk. If the illusorily imputed positive satisfactions are large enough, these constrain Betty to make a choice guaranteed to yield her a lower number of utils than had she made a different choice. To illustrate, let Betty be faced with a choice between the risky act RHR and the sure act AT of scenario 2, Table 10. Table 10 Betty’s choice set: scenario 2 under risk RHR
AT
Risky hip replacement operation with minor physiotherapy: This yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation in which she has a 5% chance of dying, and a 95% chance of success in which case she will require only 2 weeks of sick leave, then nearly full health Alexander Technique education to reduce her asymmetric posture and movement: This yields her an outcomes flow of small steady increments in mobility and decreases in pain from her after work education in this technique
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Betty now needs a more elaborate decision procedure than that of Table 7 wherein, step 4, she chooses the act with the highest U, her index of QALY (sum of Ut over 5 years). This is because for risky act RHR, her U lacks a single guaranteed value: there are two mutually exclusive Us that may occur. In order to compare her overall valuation of risky RHR with her guaranteed level of QALY from sure act AT, Betty modifies step 4 of Table 7 to incorporate a rule for aggregating RHR’s two mutually exclusive Us, Table 11. Table 11 Betty’s choice procedure under risk 1
Betty decides on her objective, to maximise U = U1 + U2 + U3 + U4 + U5
2 3
4
(1)
where U is an index of the QALY caused by her act choice, Ut is that act’s satisfaction in year t, t = 1, . . . , 5, and Ut is normed with 0 her worst conceivable satisfaction, and 1 her highest Betty discovers which acts are available to her Betty, who has no constraints, costs or benefits from her pre-decision analysis, anticipates today precisely her U caused by each available act. The fact that she anticipates satisfactions from all the actual cause–effect chains initiated if she were to choose each act and traces these through to ascertain each act’s U, is indicated with the script typeface. (An ordinary typeface U will denote what satisfactions Fig. 1 and expected utility theory permits/requires Betty to include.) At her time of choice, t = 0, Betty chooses an act with highest expected U
In step 4 of Table 11, it can be seen that, like axiomatised expected utility theory, Betty uses probability weights to aggregate mutually exclusive utilities (QALYs) of a risky act to form her overall valuation of a risky act. But do not be deceived into thinking that she has adopted that theory. In understanding how axiomatised expected utility theory places a value on risky act RHR, it is critical that health economists bear in mind the difference between the value of a risky act and the utilities of its component utilities of outcomes. Step 3 in Table 11 is what distinguishes Betty’s evaluation of a risky act from that of axiomatised expected utility theory. In step 3, she rationally anticipates and includes (in her evaluation of their impact on her QALY index) all the actual cause effect chains of each possible mutually exclusive outcome. By contrast axiomatised expected utility theory’s constraint that utilities of outcomes to be the same under risk and certainty, requires Betty to omit some actual cause–effect chains of outcomes (the Ramsey version) and in addition to impute some illusory cause–effect chains of outcomes (the Friedman–Savage version). Because the physiotherapy is minor, not needing a long absence from work, under sure act HR, Betty could commit. Under risky act RHR by contrast, these positive secondary satisfactions of being able to commit are non-existent. Instead Betty has negative secondary satisfactions in the pre-outcome year from fear of dying. Let Betty’s primary satisfactions outcomes flow under RHR be identical to that under sure act HR in year 1 (the pre-outcome period), and thereafter: zero if she is dead; those of HR for years 2 to 5 if the operation is successful. Then from Table 8, we can see that even if Betty learns afterwards that the outcome of the hip replacement operation is success, RHR will net her 1.48 utils in primary satisfactions, and that her QALY will be less than this by the amount of her negative secondary satisfactions from fear. Betty’s decision procedure, Table 11, avoids adding in certainty effects when these are absent, and ensures that risk effects are added in when present. Under risky RHR, her Table 11 decision procedure recognises that her utils are reduced (compared to sure HR)
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each year by her inability to commit, and in addition further reduced by her fears in her pre-operation year (her pre-outcome period). Her Table 11 decision procedure computes her utility as it actually is, namely less than 0.1 in her pre-operation year (due to her fear), and an expected 0.95 × (0.33 + 3[0.35]) ≈ 1.31 utils out of her three post-operation years. This amount of less than 0.1 + 1.31 ≈ 1.41 utils, is below her guaranteed 3.4 utils under the safe Alexander Technique, sure act AT. Betty’s decision procedure for valuing outcomes allows her to reasonably choose AT. But the Friedman–Savage “as if certain” evaluation of outcomes would force Betty to ignore risk in evaluating the outcome of a successful operation under RHR. From Table 9, it would force Betty to impute to RHR the “as if certain” utility of 3.98 that she would reap under sure act HR. The Friedman–Savage “as if certain” evaluation of RHR has two sources of error. One is that its imputed utility to risky act RHR excludes actual negative satisfactions, Betty’s fear. It ignores how fear reduces Betty’s pre-outcome utils. The other source of error is that it includes non-existent positive secondary satisfactions from Betty’s ability to commit to her full time 5-year job. This commitment does not exist. The combination of excluding actual negative secondary satisfactions from a risk effect (fear) plus adding in illusory positive secondary satisfactions from a non-existent certainty effect (ability to commit) causes Betty to inflate the value of RHR to 3.81 and choose it, even though RHR is in reality guaranteed to yield her a lower sum of utils than AT, to in reality yield less than 1.48 utils even if successful, as against AT’s guaranteed 3.4 utils. The Friedman and Savage version of expected utility theory thus fails the intuitive gold standard. On analogous illusory effects of certainty in Savage’s sure-thing principle, see Pope (1991a). Under a choice between RHR and AT, the Ramsey version of expected utility theory, similarly ranks RHR higher and thus fails the intuitive gold standard. It would consider only primary satisfactions. From Table 9, these would be 1.48 utils under RHR, only 1 util under AT. It might be thought that there could be other ways of imposing expected utility theory’s constraint that utilities of outcomes are identical when outcomes are risky and certain. It might be thought that at the time of evaluation and choice of act, there are intermediate probability values above zero—which excludes valuing any outcomes—and less than 1—the probability number used in Friedman and Savage (1948)—that meet this constraint. For instance it might be thought that the constraint can be met by having each outcome, no matter what its actual probability of occurrence, evaluated as if its probability of occurring were 0.5. To see that this cannot be the case, note that to be afraid of dying, death and better outcomes must be within the act’s set of possible outcomes. To be unable to commit to a job, extensive time off work must be within the set of outcomes. Thus, an outcome’s associated secondary satisfactions depend not only on itself, but partly also on the other possible outcomes—on whether the alternative possible outcomes (whose probabilities sum 0.5) lie above or below, or on both sides of this outcome, and on their range. There are no other alternative unspecified outcomes to consider in the Friedman–Savage version (each outcome is evaluated as if it is certain). But evaluating each outcome as if it has a probability of 0.5 means that there are alternative unspecified outcomes. Each different set of alternative outcomes will yield a different set of associated secondary satisfactions. The mappings from outcomes into utilities will be non-unique (for a given origin and scale). Their being
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non-unique precludes such utility mappings being versions of axiomatised expected utility theory. Expected utility theory thus fails the intuitive gold standard on the only two versions that preserve its required unique mapping from outcomes into utilities (for a given origin and scale). It failed in the particular decision situations considered. These were engineered to demonstrate that theory’s irrationality in general. The examples focussed on positive secondary satisfactions from certainty effects since these will be central to the biases generated in the particular choice sets used in standard gamble elicitations. This does not mean that secondary satisfactions are primarily negative in the case of risk effects (i.e. positive in the case of certainty effects). To the contrary, as expounded and empirically demonstrated elsewhere (Scitovsky, 1976; Albers et al., 2000; Pope, 2001, 2002), in some decision situations the predominance of secondary satisfactions are positive for risk effects. In the spectrum of positive and negative satisfactions generated by risk effects and illustrated in Table 4 above, it is an empirical question whether the positive or the negative dominate in particular a decision situation. Human physiology and associated financial and other resource issues aid in predicting in which decision situations the positive secondary satisfactions from risk effects dominate. The fact that both versions of expected utility theory here failed the intuitive gold standard for rationality does not mean that both invariably fail it. Whether they fail it depends on the relative magnitude of primary and secondary satisfactions under different acts in the choice set. There are situations in which, relative to primary satisfactions, the secondary ones (actual and illusory) are sufficiently minor that both versions pass the intuitive gold standard. What the above failures of expected utility theory prove is that it is an accident of the choice set and the chooser’s level of secondary satisfactions whenever that theory passes the intuitive gold standard, not a consequence of expected utility theory being rational. In a similar way, the accident of a dice throw can sometimes select an act that survives this intuitive gold standard. But we do not suggest that reasonable choosers make their decisions by throws of a dice.
4. Proposals for expected utility theory to meet rationality standards 4.1. Opportunity cost included by elaborating acts, outcomes in the given situation Some may feel that sources of secondary satisfactions like commitment effects can be incorporated into expected utility theory by means of treating them as opportunity costs. An opportunity cost of an act is the best alternative. To determine the best alternative to each act in the choice set, it is first necessary to rank the acts, and thus evaluate (determine the utility of) each act separately. The value of an act minus its opportunity cost is its utility minus that of its best alternative. This value is positive in the case of a uniquely first ranked act, zero for equally first ranked acts and negative for acts ranked below the first.4 Under 4 The concept of opportunity cost is thus redundant if, as in this paper, the alternative acts are specified explicitly. The concept of opportunity cost is widely used in economics texts and cost benefit analyses when instead the
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certainty (scenario 1 and Table 8), the Ramsey version of expected utility theory failed the intuitive gold standard, and thus failed to correctly evaluate the best alternative act. Under risk (scenario 2 and Table 10), both versions of expected utility theory failed to identify the best alternative act. Now to specify an act involves specifying its outcome flow(s). Are there other ways of specifying Betty’s outcome flow(s)—and hence acts—than as in Tables 8 and 10? If we specify some acts as involving a commitment outcome flow, can expected utility correctly incorporate the opportunity cost and reach a rational choice? An answer of yes is one interpretation of Luce and Raiffa’s claim that under axiomatised expected utility theory “the given situation” is taken into account in the utilities, 1957, p. 15. This claim is correct if interpreted as pertaining to primary satisfactions. These can be included in how we define (or re-define or elaborate) acts and hence outcomes (sources of primary satisfaction) without destroying the axiomatic base of expected utility theory, e.g. the amount of sick leave. In Fig. 1, there is no mention of the amount of time unavailable for work. Table 12 gives elaborated outcomes, elaborated to include this amount. Table 12 Betty’s conceivable sure acts and associated outcome flows over her 5-year time horizon specified independently of her knowing whether they are to occur when she chooses tomorrow Sure act
Outcome flow
Suicide Operation dirty hospital Do nothing Pain killers Alexander Technique Hip replacement operation with minor physiotherapy Hip replacement operation + intensive physiotherapy Hip replacement operation now
Y0 Y0 + YDN YPK YAT YHR YHR+ YF
Death the day after tomorrow Year’s wait, elective surgery, death within year of surgery Annual mobility decrements, pain increments, ever able to work Annual mobility decrements, limited pain, ever able to work Annual mobility increments, pain decrements, ever able to work Year’s wait for elective surgery as public patient, short period unable to work, thereafter nearly full health Year’s wait for elective surgery as public patient, 2 months unable to work, thereafter full health Surgery today, 1 day unable to work, thereafter full health
Time unavailable for work is a source of primary satisfaction as is a hip mobility and pain level. Any configuration of a degree of hip mobility, pain and time unavailable for work can be combined with any set of probabilities. None creates contradictions between the probabilities denoting knowledge of outcomes and the outcomes themselves. Table 12’s specification of sure acts and associated outcome flows does list some aspects of Betty’s decision situation that affect her satisfaction. But the list defining these sure acts in terms of their outcome flows makes no mention of those sources of satisfaction for Betty that depend on her knowing which act or outcome she will choose tomorrow. Thereby the choice set remains unspecified. These are situations in which it is assumed implicitly that the alternative act to a project under consideration can be decomposed into independent decisions about multiple sub-acts, e.g. three sub-acts concerning the use of labour, capital, land if these three resources were not employed on the project under investigation. It may be deemed that each would otherwise be used in competitive markets in which case their respective opportunity costs can be separately assessed as their market values. This saves the analyst the trouble of identifying the particular alternative usage to which each resource would be put and constructing explicitly the other acts (and associated sub-acts) in the choice set.
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list excludes aspects of the situation which depend on Betty’s probabilistic knowledge at her time of choice tomorrow of whether: (a) she will either die over the next 5 years, (b) be unable to work for any extended time interval. But a necessary condition for Betty having her desired ability tomorrow to commit to a project full time over the next 5 years (i.e. guarantee that she will do it) over this entire interval, is that she knows tomorrow with probability 1 that she has avoided all acts which imply a positive probability of (a) or (b). Table 12’s specifications of sure acts and associated outcome flows includes whether (a) and (b) will ensue, but does not include the necessary condition for Betty to commit, namely her knowledge ahead with probability 1 that (a) and (b) are excluded. By excluding information about this knowledge ahead, Table 12 excludes reference to Betty’s commitment source of secondary satisfaction. Axiomatised expected utility theory requires the specifications of acts (and hence outcomes) to be knowledge ahead independent in precisely the manner of Table 12. See e.g. Savage (1954) and Tables 14 and 16 below. Knowledge ahead independence precludes including commitment as an opportunity cost by inserting it in the specification of the act (i.e. specifying it as part of the act’s outcomes flow). To see that expected utility theory’s standard gamble procedure requires this independence, suppose we augment outcomes with knowledge ahead based sources of secondary satisfaction such as commitment considerations. Commitment in Betty’s case stems from knowing that she has a health outcome sufficiently good to guarantee her starting work within a week and taking no more than a month off each year for 5 years. Suppose we include such knowledge in the definition of her acts and associated outcomes, as in Table 13. Table 13 Sure acts and health outcome flows conflated with ability/inability to commit Y0C
.. . YFC
(i) Death immediately after tomorrow + (ii) + inability to commit, no job Implication of (ii): Betty tomorrow at time t = 0 will lack knowledge with probability 1 that under her chosen an act she will be alive for 5 years without an extended period out of work .. . (i) Immediate elective surgery, 1 day unable to work, thereafter full health + (ii) Ability to commit, from tomorrow at time t = 0 committed to and doing a 5-year full time job Implication of (ii): Betty at time t = 0 knows with probability 1 that under her chosen act she will be alive for 5 years without an extended period out of work
Note: For brevity, we list here only the conflations from Table 12 for the extreme outcome flows which yield the best and worst utilities.
Precede the best outcome in Table 13, YFC , with a probability number less than one with the complementary probability for the worst outcome in Table 13, Y0C . We must do this under the standard gamble procedure. But in so doing we have the internal contradiction of simultaneously stating that tomorrow at her time of choice, time t = 0, she knows with complete certainty she can commit (Table 13’s definition of YFC ) and knows with complete certainty she cannot commit (the death outcome has a positive probability).
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It might be thought that some forms of partial, conditional commitment effects (less than Betty’s case where the commitment has to be 100%) might avoid the contradiction. This however is never the case. To see this, take another decision situation in which Betty could execute the community project job if she can 90% commit tomorrow at time t = 0. Then Betty would undertake the job if she had a 90% commit tomorrow at time t = 0. In this decision situation: (a) the extreme elaborated worst outcome has an implication (ii)
that Betty knows with probability more than 10% that she will not be alive for 5 years; and (b) the extreme elaborated best outcome has an implication (ii)
that Betty knows with probability 90% or more that she will live be alive for 5 years. We now seek to apply a common probability number to these two extreme outcomes of Table 13, as required under the standard gamble procedure. Our problem is that no common probability number can be applied without encountering a contradiction. Any probability of being alive for 5 years big enough to enable the partial commitment and its secondary satisfactions of being committed and having a job, is too big a probability for the other extreme elaborated outcome of an inability to commit, thus being uncommitted and having no job. We would also be lacking in general a cardinal utility scale unique apart from scale and origin as is required for axiomatised expected utility theory to uniquely rank acts (Pope, 1983, 1984, 1989, 2000, 2002a). 4.2. Fear included by elaborating acts, outcomes in the “gestalt” It is widely believed that outcomes can be elaborated to include emotional sources of secondary satisfactions. For instance Samuelson (1952a) claimed that acts and outcomes could be elaborated within axiomatised expected utility theory to include emotional sources of secondary satisfactions. His example was a source of positive secondary satisfaction, suspense (1952a, p. 676). This belief is evident in Luce and Raiffa’s claim that the outcomes of axiomatised expected utility theory can be elaborated to include the “gestalt” of the situation, where gestalt refers to secondary satisfactions (1957, p. 26). Their example of such a “gestalt” was a positive secondary satisfaction from thrills. The claim recurs in Markowitz who asserted that elaborated outcomes to include two positive sources of secondary satisfaction, wonder and surprise, were within axiomatised expected utility theory (1959, pp. 225–226). But the claim is incorrect. To insert emotional sources of secondary satisfactions into the specifications of outcomes is to destroy axiomatised expected utility theory and its standard gamble procedure just as completely as does the insertion of non-emotional sources of secondary satisfactions. To see this, take the case of Betty’s secondary (dis)satisfaction from an emotional source, fear. Elaborate the outcomes of Fig. 1 to include this fear. Add to the worst and best outcomes of death and full health the elaboration that for Betty this is preceded by an amount of fear, setting this amount of fear at what Betty says she would experience if the chance is 50/50 of these two outcomes. So elaborating, we have set the probabilities and thus have to adopt the certainty equivalent version to ask Betty a question. We ask her which intermediate guaranteed outcome is equivalent. We cannot however keep our elaborated outcomes axis and ask her any more questions. Why? Because if we alter
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the probability of death, this will alter her amount of her fear. If for instance, we reduce the probability of death, we now have an even better outcome at the top, full health with a smaller amount of fear (now that the risk of death is lower), and at the bottom, the worst outcome is now better than before, for the same reason. The standard gamble procedure of eliciting multiple certainty equivalents is thus unavailable. It depends on the worst and best outcomes remaining the same as we ask successive questions. That is, it depends on being able to identify the worst and the best outcomes independently of knowledge ahead of whether they will occur. The probability equivalent method faces the same problems. Suppose we start with a sure health outcome, say limited mobility, elaborated to include its certainty effects. For elaborated outcomes, we need to ask Betty to state two things: (1) the risky act’s probability of death (and conversely full health) that would make it as good for her as this sure outcome, and (2) the amount of fear associated with this probability. Her answers are then our elaborated worst and best outcomes. But to proceed, we must ask Betty about a second probability number for death and full health that would leave her indifferent between another sure health outcome and another risky act. This second risky act will have a different amount of fear, and so our worst and best outcomes have shifted. This renders the standard gamble procedure of eliciting multiple probability equivalents unavailable. It depends on the worst and best outcomes remaining the same as we ask successive questions. That is, it depends on being able to identify the worst and the best outcomes independently of knowing ahead whether they will occur. Elaborated outcomes to include emotional sources of secondary satisfaction like fear, it can be seen, render the standard gamble procedure unavailable. Some readers might nevertheless think that under expected utility theory the effects of commitment and fear and other sources of secondary satisfactions do not need to enter these lists of how acts and outcomes are specified (defined) to get included. They might think that these can enter a decision theory through how Betty values the acts and their associated outcomes. They might think that, by giving their outcome flows higher total utilities (QALYs), Betty can take into account the fact that some sure acts (e.g. YDN , YPK , YAT , YHR and YF of Table 12) furnish her scope for commitment. They might think that, by giving their outcome flows lower total utilities (QALYs) Betty can take into account the fact that some risky acts cause her fear. This is correct—and precisely what Betty’s decision procedure of Table 11 does—but axiomatised expected utility theory irrationally precludes this. Below are five ways (set forth in Sections 4.3–4.7) of seeing that expected utility theory excludes Betty’s rational inclusion of secondary satisfactions in how she values her knowledge ahead independently specified acts and outcomes flows. 4.3. The lack of distinction between a sure act and its corresponding outcome flow One way is to observe that axiomatised expected utility theory requires identical utilities for: (a) a sure act (which embeds knowing at the point of choice that the outcome is certain) and (b) the corresponding outcome flow (which has to be specified in a knowledge ahead independent way to avoid contradictions when preceded by a non-degenerate probability).
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The identity of (a) and (b) means that many axiomatisations lack distinct symbols for acts and outcomes (e.g. that of von Neumann and Morgenstern). It means that the same probability weighted sum of utilities duals for valuing acts and outcomes, as discussed further in Section 4.6 and Table 16. It means that Luce and Raiffa (1957) can and do shift almost page to page in whether an expression refers to a sure act or to an individual outcome. The context has to be read (often several times) to ascertain which. The difference between sure acts and individual outcomes is deemed irrelevant because the chooser’s probabilistic knowledge of whether an act will happen in the future is deemed irrelevant to how the chooser should value that act or outcome. Knowledge of this probability is the essence of commitment. 4.4. Inconsistency in the standard gamble QALYs from secondary satisfactions A second way of discerning that a secondary satisfaction from a certainty effect like ability to commit cannot be taken into account under axiomatised expected utility theory by how outcomes are valued is to look carefully at the standard gamble elicitation procedure. We have shown in Sections 4.1 and 4.2, that standard gamble procedures fail and are inconsistent if we attempt to include secondary satisfactions by elaborating the outcomes and hence acts along the x-axis of Fig. 1. We now show that analogous inconsistencies arise in standard gamble QALYs if we do not elaborate the x-axis, but instead attempt to include secondary satisfactions in how Fig. 1’s knowledge ahead independent acts and outcomes are valued. In Fig. 1 and Table 12, Betty has already ranked the set of outcomes independently of knowing ahead whether they will occur. Such a ranking is by definition with respect to primary sources of satisfactions alone. In this knowledge-ahead independent ranking of outcomes, Y0 , death 2 days’ hence, is Betty’s worst outcome and her best, YF , full health over her entire 5 year time horizon from 2 days’ hence. In Section 1, we have normalised these worst and best utilities from primary satisfactions as 0 and 1 respectively. Now we attempt to change the vertical axis to include secondary satisfactions in the utilities. Let us term this Fig. 2, even though we shall not depict it in the paper, merely visualise it in our minds. Let us normalise Fig. 2’s utilities origin and scale to have 0 the worst and 1 the best from primary plus secondary satisfactions, and use the information in Tables 7, 9 and 12 to compute these QALYs (total utilities). From Table 7, we note that Betty’s minimum primary plus secondary satisfaction is 0 utils per year, and her maximum primary plus secondary satisfactions for Betty is 1 utils per year, and thus over her 5 year horizon, 5. From Table 9, she would get half of the maximum of 5 utils from her primary satisfactions if in full health, and the other half from her secondary satisfaction if able to commit. From the last act listed in Table 12, we can see that she would reap this maximum of 5 utils under the conceivable but unavailable sure act of a safe hip replacement operation now. So as to have the maximum 1 not 5 utils in our new Fig. 2, all the QALYs in Table 9 must be rescaled, divided by 5. We then choose the probability equivalence version of eliciting utilities of outcomes. We ask Betty to nominate a probability between 0 and 1 that will give her as much QALY from a chance of instant death or full health as to make her indifferent between that lottery and getting the guaranteed outcomes flow YAT afforded by AT, the sure act of Alexander Technique of education. But immediately we encounter a problem. Once the full health
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outcome is risky, it no longer yields Betty her maximum conceivable utility. This outcome now yields her primary satisfactions (which as rescaled are 1/2 a util) and negative secondary satisfactions from fear, with no positive secondary satisfactions from an ability to commit. Our extreme best outcome has slipped from being at the top to being somewhere in the middle. We cannot apply the standard gamble procedure. It needs a stable best outcome. We try again. We construct a Fig. 3 that we shall not depict in the paper, merely visualise it in our minds. This time we set the maximum on Fig. 3’s vertical utility axis as Betty’s maximum satisfactions from primary satisfactions alone. These are from full health each year. From Table 9 this would sum to 2.5 over her 5 year horizon. When we rescale to have a maximum of 1 util on our vertical axis, this means dividing all the other numbers in Table 9 by 2.5. We again choose the probability equivalence version of eliciting outcomes. We ask Betty to nominate a probability number between 0 and 1 that will give her as much QALY from a chance of instant death or full health that would make her indifferent between that lottery and getting the guaranteed outcomes flow YAT afforded by AT, the sure act of Alexander Technique of education. We look in Table 9 to form the answer. It should be the QALY for AT divided by 2.5, namely 3.5/2.5 = 1.75. But 1.75 exceeds her maximum QALY from primary satisfactions of 1. An answer of 1.75 would exceed the upper bound of our vertical utility axis due to Betty valuing her positive secondary satisfactions (from ability to commit enabled under this act) sufficiently highly relative to her primary satisfactions from its intermediate health outcome. This also violates another requirement for using the standard gamble procedure, namely that the extreme outcome has the maximum utility. Her answer of 1.75 is in addition an improper fraction when we asked Betty to scale her probabilities to lie between 0 and 1. No probability number between 0 and 1 exists to render Betty indifferent to a guaranteed YAT with its beneficial commitment effects, and the lottery of either almost instant death or almost instant full health. Under the standard gamble probability equivalence method of eliciting utilities, the utility of outcome YAT is elicited by choosing at t = 0 a positive probability of death (and complementary probability of full health) that leaves the chooser indifferent between this lottery and a guaranteed outcome YAT . The lottery’s positive probability of death robs her of the probability of 1 of staying alive at her time of making this standard gamble choice, t = 0. This probability of 1 at t = 0 for not dying is a pre-requisite of her committing at t = 0, and reaping her satisfactions from commitment effects. Remember, to commit to her job, Betty must know ahead with probability 1 that nothing will preclude fulfilling her job commitment at t = 0. Since commitment is the dominating source of her total satisfaction and hence QALY, she lacks any answer consistent with expected utility theory. The analogous conclusion follows from attempting to implement the certainty equivalent version retaining unelaborated outcomes on the horizontal axis and including secondary satisfactions on the vertical utility axis. Betty’s answers would be thrown out as “inconsistent”, those of a recalcitrant irrational chooser. After devoting our time to filling out this or a related questionnaire for a colleague, some of us turn in answers that the researcher tells us disappointedly they had to discard since our answers were “inconsistent”. The researcher sees the alternatives as either (1) an expensive, time-consuming and not always successful procedure of “educating” the participant to answer “consistently”, or (2) to correct for the participant’s perceived “biases”, e.g. with use of prospect theory, Bleichrodt, Pinto and Wakker (2001).
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Not many participants are as recalcitrantly “inconsistent” as Betty. This is because the researchers employing the standard gamble technique (and all others that focus on primary satisfactions) fail to alert their participants that secondary satisfactions exist and should be taken into account. Such researchers fail to lay out all risks and future changes in knowledge involved in risky acts, and how these generate risk and certainty effects. Unaware of the existence of these effects themselves, such researchers misinterpret the imposition of a preference for stochastically dominating acts and sets of outcomes as a mere “consistency” requirement. But Betty (and any other participant perceptive enough to discern the secondary satisfactions that they reap out of risky decision situations), takes into account “the given situation” in valuing outcomes with respect to secondary satisfactions. Betty disobeys the knowledge ahead independent evaluation rules of expected utility theory. 4.5. No room for commitment in expected utility theory’s act-value equation A third way of seeing that knowledge ahead based sources cannot be included in how an outcome is defined and still preserve expected utility theory is to look at its equation for valuing an act, Eq. (6) in Table 14 for the case where there are seven conceivable outcomes, Yi , i = 1, 2, 3, AT, HR, HR+ , F. Table 14 Axiomatised expected utility theory’s rule for valuing individual outcomes EU implies that the value of any act regardless of aspects of its given situation is V, V = E[U(Y)] = p0 U(Y0 ) + p1 U(Y1 ) + p2 U(Y2 ) + pAT U(YAT ) + pHR U(YHR ) + pHR+ U(YHR+ ) + pF U(YF )
(6)
where pi is the probability of outcome Yi , i = 1, 2, 3, AT, HR, HR+ and F, V must hold for all sets of pi that sum to 1. That is, the utility attached to an outcome has to be the same regardless of whether its probability number is 1 (enabling commitment in the case of the intermediate sure acts/outcomes) or less than 1 (precluding commitment). This precludes modifying the valuation of outcomes to incorporate any aspect of the actual “given situation” to reflect commitment and any other post-decision risk or certainty effect in the valuation of acts and outcomes which are themselves defined in a knowledge-independent way
In this table, it can be seen that under axiomatised expected utility theory the outcome YF of almost instant full health + only 1 day unable to work, has the same utility number regardless of whether it is: 1, guaranteed to occur, as in the case of the sure act of the unavailable instant operation; or 2, merely possible since the outcome of almost instant death, Y0 , is also possible. But if death is a possibility, this precludes Betty from giving her 5 year job commitment, and undertaking her project job. This identity of the utilities into which outcomes map regardless of whether the outcomes are ultra risky, fairly safe or absolutely certain, precludes commitment, fear, or any other knowledge ahead based aspect of the actual decision situation being included in the valuation of the outcome. 4.6. Commitment effects and the utility of multi-stage acts Any (non-degenerate) multi-stage act is risky. Let Betty’s choice be between multi-stage act MHR and the sure act AT of Table 15.
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Table 15 Betty with a choice set including a multi-stage act Betty conceives choosing tomorrow between the two acts, a multi-stage act MHR in which the risk will only be resolved just over a year later, and the sure act AT MHR A 20% chance of learning next year that she must do HR+ Hip replacement operation plus intensive follow-up physiotherapy which yields her an outcomes flow of a year’s wait in the surgery queue then the operation followed by two consecutive months doing supervised exercises most of her waking hours, then full health An 80% chance of learning next year that she must do HR Hip replacement operation with minor physiotherapy which yields her an outcomes flow of a year’s wait in the elective surgery queue then the operation requiring only 2 weeks of sick leave, then nearly full health AT Alexander Technique education to reduce her asymmetric posture and movements that caused the “mushy” hip joint, which yields her an outcomes flow of small steady increments in mobility and decreases in pain from her post-work once weekly solo session with a teacher trained in this technique
In MHR, we conceive the situation in which Betty has no choice about whether to follow a hip replacement operation with intensive physiotherapy. The health providers will dictate to her after her hip replacement operation a year hence whether her life takes the HR+ branch of two consecutive months doing supervised exercises most of her waking hours, or her life takes the HR branch, leaving her free to resume work with less than a month’s sick leave. Risky MHR destroys Betty’s opportunity to combine HR with her ardently desired full-time 5-year job. As Table 16 shows axiomatised expected utility theory requires that the value of a multi-stage act be a probability weighted aggregation of its constituent utilities of sub-acts, outcomes, and that it can be decomposed back into these constituents. Table 16 Axiomatised expected utility theory’s knowledge-independence feature which excludes secondary satisfactions Axiomatised EU implies that V, the rule for aggregating mutually exclusive possibilities by their associated probability weights, p1 , p2 , . . . into an overall value, (7) V = E[U(H)] = p0 U(h0 ) + p1 U(h1 ) + · · · holds when H refers to Either (a) An outcome space, in which case its possible values hi are the possible outcomes of an act, i.e. the Yi ’s of Eq. (5) and Table 9, the possible outcomes of an act Or (b) An act space, in which case its possible values h0 , h1 , . . . , are themselves possible acts and hence in Table 15, (8) V(MHR) = 0.2 U(HR+ ) + 0.8 U(HR) This identity of formulae in the valuation of outcomes and acts over all probability distributions of acts and outcomes means that a multi-stage (compound) act decomposes into a weighted sum of the utilities of its component acts. In turn such decomposition requires knowledge ahead independence of which act the chooser will select and of the chooser’s array of different conceivable “given situations” for each probability distribution over acts and outcomes in the (i) specification of acts and outcomes, and (ii) U(·), the utility transform for evaluating satisfaction from acts and outcomes
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This requirement affords a bird’s eye view of why axiomatised expected utility theory precludes commitment or any other pre- or post-risk or certainty effect entering either how acts and outcomes are valued and defined (Table 16). The knowledge ahead independence features (a) and (b) of Table 16 preclude Betty taking her job commitment ability into account in evaluating her risky multi-stage act MHR and comparing it with simple act AT. From Eq. (7), Betty evaluates and defines MHR as the probability weighted sum of the utilities of two component “acts” HR+ and HR. She gives each of these sub-acts the same value (utility number) as they would have if at the point of choice they were sure acts, definitely coming up in year 2 (the case in Table 8), or merely probabilistically going to occur (the case when these are embedded in the multi-stage act MHR of Table 15). There are only two means of obtaining axiomatised expected utility theory’s required identity in an act’s value when it is guaranteed, and when it will merely be probabilistically undertaken, because a sub-act in another multi-stage act. One is to apply the Friedman–Savage “as if certain” evaluation rule to acts (as distinct from outcomes). Under it Betty would value the risky component acts HR+ and HR “as if sure” acts. From Table 9, she evaluates them respectively as 1.9 and 3.98. Under this “as if certain” approach, Betty would then proceed to compute MHR’s value as the probability weighted sum is 0.2 × 1.9 + 0.8 × 3.98. This gives MHR a value of about 3.57. This value exceeds that of her available sure act AT whose value is 3.4, and so causes Betty to choose MHR. Choosing MHR is irrational. It is not merely irrational in the weak sense that its actual expected value (0.2 × 1.9 + 0.8 × 1.48 ∼ 1.57) is below that of AT. It is irrational in the strong sense that it fails the intuitive gold standard. The best possible outcome under MHR is only 1.9 utils, i.e. MHR is guaranteed to yield less utils than sure act AT. This has happened because expected utility theory’s “as if certain” imputation for MHR irrationally includes positive secondary satisfactions from HR’s ability to commit that she lacks. Let us now consider the alternative way of achieving axiomatised expected utility theory’s requirement that acts have identical values when risky and certain at the time of choice—i.e. when it is known and when merely it is merely probabilistically known at the time of choice that the act will be undertaken. This is the Ramsey-style version of obeying axiomatised expected utility theory’s Eq. (7) in Table 16. This version excludes all secondary satisfactions, all knowledge ahead based effects such as ability to commit, fear.5 Excluding all secondary satisfactions overstates the value of MHR. It overstates the value of MHR because it ignores Betty’s negative secondary satisfaction from fear of dying (something absent when the operations were guaranteed to be successful and thus absent from Table 9). Conversely, it understates the value of AT by ignoring the 2.5 utils she obtains from the secondary satisfaction of being able to commit to her job. This understatement alone, even apart from its overstatement of the value of MHR, is enough to make MHR seem better when in fact it is guaranteed to yield her a lower expected QALY. Thus, the Ramsey-style version also fails the intuitive gold standard. It would not lead Betty to the rational choice of AT any more than would Friedman and Savage (1948) “as if certain” evaluations. 5 The qualifier style is added since Ramsey’s pre-axiomatisation does not consider multi-stage acts, and thus this is an extension of his interpretation that choosers must value outcomes independently of knowledge ahead, independently of the probabilities of the outcomes occurring.
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In summary then, neither by how acts and outcomes are valued, nor by elaborating, redefining acts and outcomes, can axiomatised expected utility theory include the risk effects of Betty’s inability to commit, and her fear when these effects exist. It cannot do this in a choice set including only a simple risky act (as is required to apply the standard gamble procedure) or in a choice set including a multi-stage act. 4.7. Job and commitment inseparable Some might feel that Betty’s need for total commitment to a full-time 5-year job is esoteric, that expected utility theory as used in health economics should not cater for unusual sources of satisfaction. They may grant that jobs and feeling useful give choosers finance and improve their health, but feel that the commitment aspect can be side-stepped, that what Betty wants is the job (which it might seem that expected utility theory can readily include), not commitment to it. This is to fail to unpack the meaning of a job. A job is a causal word. The person doing a job deliberately causes something to change—the job to get done. Causation takes time and the job only gets done by that doer if that doer stays committed for the requisite duration. Commitment is thus an integral part of specifying and defining any job, and thus in determining QALYs.6 The suggestion that we might replace the commitment factor with a job factor in elaborating outcomes, in no way eliminates the knowledge ahead based factor of commitment and the destruction of expected utility theory’s standard gamble process for eliciting QALYs. 5. Post-decision risk effects yield the standard gamble utilities bias 5.1. Pre-choice planning problems and risky choice investment compromises Our next task is to see how one particular class of secondary satisfactions yields predictable concave-convex twists when expected utility theory’s standard gamble procedure is presumed to map out utility curves. The answer relates to why individuals and organisations typically check their own ability to commit (and also seek commitment from relevant parties before deciding on a job (whether a community project or an operation on a patient or . . . ) a partnership (whether personal or business), having babies, hiring workers, and numerous other activities that directly and indirectly impinge on QALYs. Our answer to these questions should recognise that two major factors are: 1, the difficulty of foreseeing the multiple possible future choices and outcome flows entailed in the presence of risk, and determining what satisfaction might be gained from future unanticipatable choices in the chaos of no commitment, no guaranteed continuity in anything; and 2, the investment compromises entailed in the face of multiple alternative outcome flows. 6 By contrast some outputs (outcomes) of jobs can be specified and defined without reference to commitment. We can define the outcome that Betty receives an artificial hip in a year’s time as a result of a job done by a surgery team, independently of Betty knowing now at her time of choice that this will happen. The surgery team’s job, though, of putting in the artificial hip requires its commitment for that period of time.
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5.2. Pre-choice planning problems Central to both versions of the standard gamble procedure is a risky act in which the extreme outcomes of death and full health, each have a positive probability. Every alternative includes a comparison with such an extreme risky act. The situation of RHR, Betty’s risky hip operation with death a possible outcome after a lengthy pre-outcome period (a year’s wait for elective surgery) is appropriately close to the extreme risk situation used in the standard gamble procedure. Indeed it was selected to allow us to imaginatively understand how the standard gamble procedure functions. Under the standard gamble procedure Betty must, for each utility obtained, ascertain a sure act that is equivalent to an extreme risky act. To choose among acts that are far apart might require little planning and analysis.7 But the task is to select equivalent ones. In making such close comparisons, Betty would, as a standard gamble participant, have to assess fairly accurately her satisfactions over her pre-outcome year of uncertainty before knowing whether she has avoided dying on the operating table or soon after. Determining these involves determining the appropriate way to spend the pre-outcome year of uncertainty under the extreme risky act. Deciding this involves her in numerous planning problems. How much time should she spend doing those activities one would do if one knew the outcome will be death? How much should she spend doing those completely different activities she would do if she knew the outcome will be virtually full health? Ascertaining which such sub-acts are available, and choosing among them the best compromise course of action for the uncertain pre-outcome year may involve assessing long futures of branched possibilities, and extensive negotiations with family, doctors, hospitals. Betty needs to repeat this exercise for every different probability mix (or certainty equivalent) of the extreme outcomes. These pre-choice planning problems of arriving at QALYs, the overall values of the different extreme risky acts, are ignored under axiomatised expected utility theory and its standard gamble procedure. We did above also. We assumed Betty could ascertain these with perfect accuracy within her decision deadline of a day hence. Even that day is rather longer than we normally offer participants in a standard gamble procedure before they have to proffer their QALY answers. Savage (1954) recognised the long chains of branched possibilities inherent in using axiomatised expected utility theory to give a utility to any risky act (let alone one so risky as to have extreme outcomes). Savage defended expected utility theory requiring this with the argument that to do otherwise than “look before you leap” is to be irrationally myopic. But as Simon responded, it cannot be rational to have a decision theory that irrationally ignores the extensive time successful business men devote to negotiating and creating alternatives, 7 This was the case for the real life Betty who unlike the one in this paper, had such pain and immobility that she did not need detailed planning and analysis to choose an operation, even though her risk of death from that operation was 30%. Her problem was not that of assessing whether its possible primary and secondary satisfactions would render it her best alternative. Rather for her the difficulty lay in the pre-choice set problem of negotiating with family, doctors and hospitals that this act be made available to her, something she only succeeded in achieving after years of lobbying, and the added negotiation power of having been awarded a Queens Medal for her community service. In the case of eliciting QALYs under standard gamble procedures, participants wearing the hat of the health provider, have the complication of branched futures on the alternative uses of funds, especially when the recipient has a good chance of dying on the operating table.
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and irrationally ignores the complexity of an integrated analysis of all future implications of acts (e.g. Simon, 1991a,b). Commitment reduces the number of branched possibilities facing a chooser. A commitment-free world involves ultra risky futures with enormous numbers of alternatives of what others will later decide to do. It is a veritable impossibility at the physiological, emotional and financial levels of doing the Savage (1954) pre-choice “look before you leap” analysis of such futures. As Savage (1954) owned, not even choice over a family picnic could really be analysed in this way. Mere mortal Betty has very limited anticipatory powers for seeing the future out of even the small number of potential alternative acts we identify. To solve many of the dynamic programming problems her whole life would not be long enough to ascertain all the ramifications of a single sure act. She doubles what she has to evaluate to obtain a precise answer if the risky act has two possible outcomes, trebles it if there are three possible outcomes, and so forth.
5.3. Risky choice and investment compromise Keynes’ General Theory (1936) generated our modern macro-economics investmentconsumption distinction. Consumption yields its satisfaction now and thus is independent of risk, of what may or may not happen in the future. Investment by contrast yields satisfaction in the future, and thus is subject to risk, to what may or may not happen in the future. Keynes objected to expected utility theory which excludes pre and post-decision risk and certainty effects. Wold (1952a,b) provides an investment example showing that in the presence of investment expected utility theory yields arbitrary implausible choices as admitted in the Econometrica responses (Samuelson, 1952b; Savage, 1952a,b). By the later 1950s (e.g. Markowitz, 1959) it had been recognised that expected utility can address only a single choice in a life, and cannot deal with investment. It was thought that expected utility theory can deal with consumption flows such as QALYs. This however was to fail to realise the role of investment in all consumption activity, whether physiological or emotional. Betty’s autonomous nervous system invests (prepares) when it anticipates she will eat, drink, go into the cold or heat, have sex, sleep and so forth. Beyond a certain range, the more unpredictable the future, the more inefficient her bodily preparatory investment, and the lower her physiological consumption of available items. If the alternative outcome flows are extreme with death too likely—she can perform none of these functions. Her body freezes. Conversely, there is less compromise and more efficiency in Betty’s emotional investments (and thereby subsequent emotional consumption) if the risk in her future stays within a given range. Her desired emotional and financial investments, and the split between investment and consumption is very different if she is guaranteed to die within a year as against she is guaranteed full health for 5 years. This is likewise the case for health providers, governments, business, spouses, parents and so forth. The investment compromises for catering for huge risk (vastly different possible futures) greatly detract from the satisfaction flows that can be reaped. This is so even when such problems are tackled with the best dynamic programming algorithms run on the fastest super computers in the world. On how to im-
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prove such compromise investments by building in flexibility to reduce the risk of ruin or extremely low returns (e.g. De Neufville and Munéra, 1983; De Neufville and McCord, 1983; De Neufville and Odoni, 2002). 5.4. Commitment, sure acts to alleviate planning and investment costs For simplicity in Table 5, Betty’s project was impossible without full commitment tomorrow—without this the project was guaranteed to bear no fruit. In other cases, with a worker (and other resources) less than 100% committed, there may be a possibility of some useful project output. It is costly and difficult however for a Betty to do the planning to determine whether it is worthwhile proceeding with such partially committed workers, costly too for the other side of the coin, the funding agency.8 Take the World Bank as the potential funder of the Bettys doing community aid amelioration projects, and considering whether to allocate all its funds to Bettys who cannot fully commit, or to allocate some funds to different sorts of projects that have some chance of bearing some fruits when run by Bettys who cannot fully commit. The World Bank faces financial and time constraints. It is not necessarily desirable that it allocate resources to assess what else it will do if it later discovers it has lost two fifths of the money that would have gone on this project’s results. Money and time are required to compute what else to do to help the community if Betty has a high chance of dying during the project. This requires assessing whether hiring someone likely to die and implementing the back-up plan is better than allocating the funds only if Betty can fully commit having a negligible chance of dying. There is nothing necessarily irrational about the World Bank making commitment a pre-requisite for funding some community projects. It cannot use its entire budget up on feasibility studies for dealing with inabilities to commit. The World Bank must use its judgement on how much of its budget to spend on feasibility studies to deal with hires who cannot commit. It might deem it worthwhile to do a feasibility study on whether it is worthwhile to hire those with only a third ability to commit, as is the practice of a major car manufacturer in South Africa, which, recognising that AIDS precludes about one in three workers continuing in production for long, trains three for each assembly line position with two paid to be standbys for those on the line who have to drop out due to AIDS. But the World Bank might also deem it better to save the funds on such a feasibility study, and instead devote those funds to other health providers who can offer the full commitment. 8 If in the paper’s example Betty had not cared intrinsically about the project, but only getting paid, and if the World Bank required full commitment of her, Betty’s ability to commit would be likewise a major source of her secondary satisfactions from her salary obtained via her ability to commit. The analysis would be identical to that above if the World Bank shared the same probabilistic knowledge as Betty on her ability to commit (risk of dying or being out of work for an extended time), or if Betty did not lie. If Betty were ready to lie and had a capacity to deceive the World Bank, then the general conclusion still holds, but the analysis requires inserting for the job secondary satisfactions, a transform from the actual probability distribution over outcomes from different health interventions to the distribution that Betty through lies has the capacity to convince the World Bank is the case. Betty’s own perception of her probability distribution over outcomes from her alternative acts is then the pertinent one only for her ascertaining her fear of dying (and if pertinent, fear of being discovered later to have lied to the World Bank).
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Nor is there anything irrational about the commitments the government imposes in allocating a garbage contract. Nor is there anything irrational in a host seeking a commitment that the guest will arrive tonight to have the dinner being prepared. The optimal investments in organising the community, the garbage, the dinner are generally radically different under alternative outcomes. Discovering and evaluating compromise investments for coping with risky acts is a costly time consuming process. It would be an endless regress were a chooser so irrational as to attempt cognitively to optimise this complex process. It is an endless regress that expected utility theory simply ignores and assumes implicitly that chooser’s have the answers to all these calculations. On the disasters introduced by attempting to optimise in complex decision situations (e.g. MacKinnon and Wearing, 1983). These planning and compromise investment costs inherent in evaluating and choosing risky acts make their inroads into financial, time and physiological stress resources. They result in most people making many choices that seem likely to avoid large scale post-choice risk effects. Genes, as well as environmental factors make for individual differences in the reluctance for novelty inherent in risky acts. Two thirds of the population are averse to novelty and post-decision risk effects, one third extremely so. On the work in discovering the ten sequences on DNA pertaining to the genetic component in this dislike of post-decision risk effects, see Hamer and Copeland (1998). On adverse decisions made by those lacking enough aversion to post-decision risk effects through damage to pertinent segments of the brain, see Demasio (1984). 5.5. Post-decision risk effects yield the standard gamble utilities bias Axiomatised expected utility theory excludes the chooser’s pre-decision planning costs and those of relevant other parties who contribute to the chooser’s environment of penalties for failing to commit and imposing risk and thereby planning costs on others. On Ramsey’s version, axiomatised expected utility theory similarly excludes all other pre- and post-decision risk effects, and their mirror image, certainty effects. It limits satisfactions to post-decision primary satisfactions. Under the alternative “as if certain” version, it again excludes all risk effects, though not the converse certainty effects. Expected utility theory’s standard gamble procedures elicit what we erroneously infer are utilities. They elicit them with pre- and post-decision risk effects at their maximum. They always involve the extreme outcomes, the best and the worst. They furthermore typically begin with risk effects at their maximum for most people, i.e. a 50% chance of the extreme values (in a QALY context) either death or full health. (The risk effects would be by comparison more modest for many choosers, were the probabilities 99.99 and 0.01%, since then one outcome is approaching one of the extreme outcomes.) For these omitted risk effects to yield unbiased utilities, the ignored pre- and post-decision risk effects would need to be orthogonal to the included primary satisfactions derived from the outcomes alone. Then the effect of omitting these risk effects would be to generate non-systematic variation in the utility curve elicited under the certainty equivalence and the probability equivalence methods. Then omitting these risk effects would yield merely random differences making it impossible to retrodict from the shape of the utility curve whether the certainty equivalence or the probability equivalence method of elicitation had been used.
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But we know from Hershey et al. (1982) that choosing the probability equivalence version of the standard gamble systematically tends to a concave curve for low monetary outcomes, twisting into a convex shape for high monetary outcomes, and vice-versa for the certainty equivalence version. To describe how these reverse twists relate to reverse bunchings in the answers elicited, we can use Fig. 1 as we do not need outcomes that lie some objectively determined distance from each other, merely that their positions on the x-axis are pre-specified. Let us draw a line between 0,0 (denoting the zero utility of the worst outcome) and 1,1 (denoting the maximum utility from the best outcome), and call its slope S. Let us split the outcome space into high and low sub-sets at a middle health outcome Ym (between Y2 and Y3 ). The certainty equivalent’s convexity below, concavity above, means that relative to S, the elicited utility curve’s slope is steeper for intermediate outcomes, flatter near the extreme outcomes—i.e. relative to certainty equivalents elicited for low probabilities of either extreme outcome, there is a bunching together of the certainty equivalents elicited for intermediate probabilities (for when the probabilities of both extreme outcomes are sizable). Conversely, the probability equivalent procedure’s concavity below, convexity above, means that relative to S, the elicited utility curve’s slope is flatter for intermediate outcomes, steeper near the extreme outcomes—i.e. relative to those for very low or very high outcomes, there is a bunching together of the probability equivalents elicited for intermediate outcomes. Thus under the two procedures there are: A systematic (albeit opposite) reversals in what is bunched at exceedingly low outcomes relative to intermediate ones, and B systematic (albeit opposite) reversals in what is bunched at intermediate relative to exceedingly high outcomes. First for the certainty equivalent, and then for the probability equivalent, procedure we explain below how the pre- and post-decision risk effects (that are assumed to be non-existent under these elicitation procedures) in fact contaminate the answers and cause A and B. Under the certainty equivalent procedure, let us begin by putting participants in the situation of a 50/50 chance of death or full health, and ask for their certainty equivalent. Risk’s planning problems and some other pre- and post-decision risk effects give the typical participant a desire to switch out of their current situation of a 50/50 chance of death or full health, and say that a low health outcome is worth accepting to avoid the risk of death. But the causal chain underlying this answer is not representable on Fig. 1 or any other mapping from primary sources of satisfaction into utilities. Fig. 1 says nowhere there is a 50/50 chance of death or full health. Mis-representing the answer by plotting it on Fig. 1 is spurious, misleading, since the answer represents an escape from the risk of death as well as that very low health outcome. That is, the utility plot suffers from omitted variables bias. The utility plot is a conflation of a primary satisfaction (from the low health outcome) and omitted causes, the pre- and post-decision risk effects. The set of three utilities 0, this fraction and 1, do not, as implied by Fig. 1 and axiomatised expected utility theory, represent utilities that participants would place on death independent of knowing they are about to die, of low health independent of knowing this will be their fate, and full health independent of knowing whether this will really be their fate (with their alternative fate not some slightly low health outcome, but the other extreme, death).
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Let us next put participants in the situation of 75/25 chance of death or full health and identify its certainty equivalent. Since the adverse pre- and post-decision risk effects are fairly similar to the initial situation since both extreme outcomes still have substantial probabilities, the typical answer is fairly close to the earlier one. Bunching is only absent when we ask for certainty equivalents to gambles in which the probabilities of one of the extreme outcomes is sufficiently tiny that the typical chooser might neglect pre- and post-decision risk effects. The converse conflation and omitted variable bias occurs under the probability equivalence method. Let it begin by putting participants in the situation of being guaranteed the middle health outcome Ym . We ask them to nominate a probability of death or full health that would be on the borderline of warranting switching out of security and undertaking this sure act. Risk’s planning problems and some other pre- and post-decision risk effects give many a desire to avoid switching out of security into a chance of death or full health. The typical answer p is accordingly a high chance of full health. They require this high chance to warrant loss of security and taking on board all the pre- and post-decision risk effects. But these causal chains are not representable on Fig. 1 or any other mapping from primary sources of satisfaction into utilities. Fig. 1 says nowhere there is a 1-p chance of death, and a p chance of full health. Mis-representing p by plotting it on Fig. 1 is spurious and misleading since p represents the cost of incurring the risk as well as satisfactions from that middle health outcome Ym . That is, the utility plot suffers from omitted variables bias. The utility plot is a conflation of a primary satisfaction (from the middle health outcome Ym ) and omitted causes in the form of pre- and post-decision risk effects. The set of three utilities 0, p and 1, do not—as implied by Fig. 1 and axiomatised expected utility theory—represent the three utilities that participants would place on: 1 death independent of knowing they are about to die; 2 low health independent of knowing this will be their fate, and 3 full health independent of knowing whether this or death will be their fate. Let us next put participants in the situation of being guaranteed another intermediate health outcome, Y2 . We ask them to nominate a probability of death or full health that would be on the borderline of warranting switching out of security and undertaking this sure act. Concern about pre- and post-decision risk effects cause most participants to demand a high price of giving up even this level of security, and thus to nominate a probability fairly close to p, their original answer. This bunching is only absent when they are asked to nominate probability equivalences sufficiently close to the extreme outcomes, that one of the probabilities is sufficiently tiny for the typical chooser to neglect pre- and post-decision risk effects. 5.6. Bias from other omitted risk effects including fear For simplicity the analysis of the biases in standard gamble utilities in Sections 5.1–5.5 has focussed on a single source of omitted secondary satisfaction, that of the certainty effect of ability to commit. In their omission of secondary satisfactions, these elicitations omit all other secondary satisfactions as well. These include a range of financial risk cost effects (i.e. certainty benefit effects) explored elsewhere (Pope, 1983, 1991b). They also encompass a range of emotional risk effects, some of which are negative (e.g. Betty’s fear)
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and others positive (Table 4) explored in Pope (1983, 1989, 1995, 1998). If the post-choice negative emotional secondary satisfactions under risky acts outweigh the corresponding positive emotional secondary satisfactions, these exacerbate the bias in standard gamble utility elicitations.
6. Conclusions There are pronounced biases in standard gamble utilities for use in QALYs and other decision making purposes. These arise out of omitted risk and certainty effects. Such risk and certainty effects reflect planning difficulties at the neurophysiological level. The autonomous nervous system requires a degree of predictability for its planning and investment processes. This is manifest in and paralleled by emotional, financial and other physical planning difficulties and investment compromises necessitated when there is risk. The standard gamble procedure for eliciting utilities involves contrasting extreme risk (both extreme outcomes possible) with the minimum risk (certainty). The procedure is valid only under the implausible constraint that outcomes yield identical utilities under extreme risk and certainty. This implausible constraint is imposed by axiomatised expected utility theory. This paper has offered steps toward restoring the chooser’s anticipated satisfaction (including that from health outcomes) to its necessarily central place in any descriptive or rational choice procedure. It has taken the next steps of demonstrating what expected utility theory includes, what it omits, and of appraising what it omits and includes, showing that it excludes factors critical to health. To indicate their criticality, it has introduced a new more neutral terminology for what expected utility theory includes and excludes. The new term for what this theory includes is primary satisfactions. The new term for what it entirely excludes under the Ramsey version of axiomatised expected utility theory, is secondary satisfactions, namely pre- and post-decision risk and certainty effects. Both primary and secondary satisfactions are critical to health. The empirical magnitude of the two will vary across decision situations and choosers. The paper has further shown that the Friedman and Savage “as if certain” version of the utilities of outcomes under axiomatised expected utility theory differs from that of Ramsey. It differs by including a subset of secondary satisfactions, namely certainty effects, even when these effects are non-existent! Like the Ramsey version, the Friedman and Savage version excludes secondary satisfactions arising from risk effects. Central to health economists including risk effects in QALYs is to set out the changing stages of knowledge inherent in any situation of risk and uncertainty. Changing knowledge is the hall mark and defining characteristic of risk. But axiomatised expected utility theory and most of its rank dependent generalisations ignore this and model risky choice as an atemporal procedure. This has resulted in many health economists treating probabilities exclusively as if they are atemporal shares of simultaneously existing objects or events, like slices of a cake. They play this atemporal role in the aggregation of outcomes to form an overall valuation of an act. The other role of probabilities, namely as denoters of our knowledge ahead, has a fundamental temporal dimension In this regard, a probability of for instance 0.5 for two outcomes
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denotes that the chooser now has merely probabilistic knowledge ahead of the future outcome. The risk of which of the two outcomes is the case may later be resolved. When this risk is fully resolved, the probability of one of the two outcomes becomes 1 and the other 0. This paper has shown how to divide up the future by stages of knowledge to identify the changing risk and certainty effects caused by our anticipated changing stages of knowledge about the future. Such risk and certainty effects include fear and ability to commit. Meaningful QALY estimates under risky futures entail laying out these changing stages of knowledge facing choosers. It further entails alerting choosers to these effects in the same manner that utility elictations alert choosers to knowledge ahead independent features of health states. This paper has shown that to include anticipated consequences of future changes in knowledge, we must move beyond the knowledge-wise atemporal standard decision framework adopted by expected utility theory and most of its generalisations (e.g. the Quiggin and cumulative prospect rank dependent theories). In moving to the new stages of knowledge framework we can consistently (without contradictions) include: 1, QALY consequences independent of knowing ahead and thus independent of risk effects (primary satisfactions) and that are included in the standard decision framework; plus 2, QALY consequences based on knowing ahead about risks, certainties (secondary satisfactions) and that are excluded in the standard framework. Thus, this new framework, unlike that of axiomatised expected utility theory, includes all consequences—all cause–effect chains affecting satisfactions—and thereby provides the solid comprehensive consequentialism required for QALY applications.
Acknowledgements I thank for valued comments Jeff Richardson and Lillian Bulfone of CHPE, Monash University; Reinhard Selten of the Experimental Economics Laboratory, Bonn University, John Quiggin, Department of Economics, University of Queensland, Tony Culyer, Department of Economics, York University, Duncan Foley, New School University, Vivian Walsh, Muhlenberg College, two anonymous referees of this journal; and for earlier background discussions hugely assisting my understanding, Maurice Allais, Ken Arrow, Will Baumol, the late Ole Hagen, Hans Jensen, Duncan Luce, Roy Radner, Howard Raiffa, Paul Schoemaker, the late Herb Simon and Paul Samuelson. Remaining errors are my own.
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