Decisions under risk: A descriptive model and a technique for decision making

European Journal of Political Economy 7 (1991) 38145.

North-Holland

Decisions under risk: A descriptive model and a technique for decision making* Ole Hagen Norwegian School of Management (Handelsh0yskolen BI), P.O. Box 580, N-1301 Sandvika, Norway

Accepted for publication March 1991

Empirical evidence against EU maximising as descriptive model is conclusive. Alternatives based on transformed probabilities has absurd implications. A more realistic alternative to EU is introduced: A function of expectation, dispersion and skewness of the probability distribution over utilities, which takes account of emotions, tension and hope fear. Applied (1969) to outcomes defined as after-game situations it solves the well known paradoxes. It is now redefined to concern utility of the change. EU maximizing as prescriptive rule is not operative. An alternative decision technique is.

1. The descriptive expected utility model I .I. A selected reference:

The von Neumann-Morgenstern

version

There are many expected utility theorems, if we include in the concept of a theorem the set of axioms on which the proof of the conclusion rests. Under this concept they may all be true, regardless of empirical facts. But if we understand by theorem the conclusion, proved in n different ways from n different sets of axioms it may be wrong. If the common conclusion is contradicted by facts it is wrong. The validity of the internal logic in the n proofs only shows that in each case at least one of the axioms is false. *The basic contents of this paper were presented at the Fifth International Conference on the Foundations and Applications of Utility and Risk Theory, FUR V, at Duke University 1989 [Hagen (1991)]. During the long pre-history of this paper the author has benefited from encouragement and advice from a long row of persons beginning with Maurice Allais and Werner Leinfellner through the discussions at the FUR conferences and seminars at The Norwegian School of Management in Oslo/Bekkestua/Sandvika, and in a later stage during a visit at the University of Bonn, particularly Professors Reinhard Selten and Wilhelm Krelle. This visit was made possible by a Ruhrgas Stipend. At a still later stage, during a visiting professorship at Fern UniversitHt Hagen, BDR. I benefited from discussions in a staff seminar, in particular from remarks by Professor Michael Bitz. This final version has benefited extensively from editorial comments, corrections and inspiring suggestions. All remaining errors and omissions are my sole responsibility. This period has seen an explosive expansion in the literature presenting ‘alternative’ decision theory, and I should probably hand out apologies for neglecting interesting contributions. 01762680/91/$03.50 0 1991-Elsevier

Science Publishers B.V. (North-Holland)

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Under this concept each ‘axiom’ can be seen as an independent scientific hypothesis if its characteristics are identifiable with objective characteristics. What is more important: Is the conclusion that is common to all the proofs resting on different sets of axioms falsitiable, and is so, is it falsified? I select the original von Neumann-Morgenstern (NM) version as a point of reference. Let me focus on two partly forgotten facts: (1) Von Neumann and Morgenstern (1944) do not define the outcomes as anything but utilities. (2) They admit that they avoid the issue of a specific utility of risk and so to speak ‘define utility’ as that concept which lends itself to their mathematical operations. The values of this utility are again defined by the choices under risk. I leave the question open whether this implies that the conclusion of the theorem is ‘true’ in some metaphysical sense which is above empirical testing. 1.2. The current interpretation

Generations of followers have forgotten their masters’ reservation about the specific utility of risk. That is a loss. These followers tend to describe outcomes of games by objective characteristics, which is a gain. A person’s wealth is an attractive example since it may perhaps be described by a scalar, such as an amount of money. At least in games about money the change in a decision maker’s wealth is measurable. What could we, in common sense language and in the context of games about money, mean with a risk averter, a risk neutral person, and a risk seeker? Let me suggest the following: Risk auerter: Refuses a game unless expected (positive) gains are large enough. Risk neutral: Accepts a game with positive expectation. Risk seeker: Accepts a game if expected loss is small enough. Assuming EU maximisation this leaves out many, in fact most, people. For example, the insurance-lottery paradox and its apparent solution by Friedman and Savage (1948) shows that one must describe persons to be risk aversive, etc., over specific intervals of wealth. However, what is sufficiently small/large? A difference in expectation is mentally weighed against a risk. But there are many different risks, and there must be some kind of a scalar measurement of risk. Could it be the standard deviation? If not, there is no necessity to go into the reasons given for this, is there any alternative? If for example a person would rather have SO.9M than a 50-50 chance of winning $2 M he is risk averse. Here a difference in expectation of SO.1M is outweighed by a risk whose standard deviation (SD) is $1 M. How can the expected utility theory (EU) predict that the same person will prefer a 50% chance of winning $0.9 M to a 25% chance of winning $2 M? Should now a

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difference in expectation of 0.05 M necessarily be outweighed by a difference in SD of about minus O.O2M, i.e. in favour of what would generally be considered the most risky alternative? Could there be another single scalar expression of risk in money, more relevant than SD? If so I am not aware of it. Perhaps Lola Lopes hit the nail on the head when she said that risk is ‘The dog that did not bark’ in the EU theory (Lopes, 1988). So far we have humoured the preference of the contemporary NM followers for dealing with games about money. There is a trap here that they have fallen into. When outcomes are described as amounts of a numeraire, it apparently makes sense to speak of risk aversion etc. in terms of that. The ‘political marriage’ between expected utility and ordinalism [Hagen (1988)] made it urgent to explain the NM utility index not as an index of utility but an index of risk attitude. Now if we forget about amounts of money and think of the baskets of goods that will be bought for them, the concept of risk aversion in reference to a numeraire becomes problematic. (Unless of course you admit cardinal utility, which is taboo.) Let us exemplify this: We start with a choice situation between alternatives expressed in money: A: $50,000 with certainty, B: A fifty-fifty game between $100,000 and nothing. If Ann prefers A, she is, in the current usage of the NM followers, a risk averter. If Ben prefers B, he is a risk seeker. Both characterisations are absolute, and also mutually independent. Suppose now we asked them: What would you buy for the two amounts of money if it were given to you. Suppose we got the same answer from both: $50,000: A cabin cruiser Alpha, $1OO,ooO: A cabin cruiser Beta, which has all the qualities of Alpha plus excessive safety devices. The risk seeker seems to be the one who is more afraid of drowning. Suppose now that we knew the answers to the second question in advance, and offered alternatives as above but with Alpha in lieu of $50,000 and Beta in lieu of $100,000. The choices would now have the same end results. But, assuming the expected utility theory and ordinal utility who can now tell who is risk averse, Ann and Ben or Ann only or none of them? All we can say is that Ben is not the only risk seeker, if at all. We can say this on the given theoretical basis because we implicitely assumed that Beta is preferred to Alpha which is preferred to being alive (ashore) with no boat which is preferred to being dead on the bottom of the sea.

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Let us now take another theoretical set of assumptions as a starting point: there exists a cardinal utility u, and we assume that u(Beta) > u(Alpha) > u(alive with no boat) > u(dead). Further we assume that the utility of a game is not (necessarily) equal to the expectation of the probability distribution over these utilities, but allows for a specific utility of risk in terms of the cardinal utility (our alternative numeraire). It is then plausible to assume that for Ben, u(Beta)/u(Alpha)>2, so much so that even if he is more risk averse in terms of utility than Ann, he prefers B. And for Ann: even if she is less risk averse in terms of utility than Ben, her u(Beta)/u(Alpha) is so low that she prefers A. It is fully compatible with this theoretical layout that Ann may be a daredevil and Ben a prude in any respect. 1.3. WeN known paradoxes and some new ones Allais (1952, 1979 and all other references) was the first to explain, and also show empirically, that in general the expected utility model is not descriptive of the actual preferences of presumably rational persons. His paradoxes, particularly the one bearing the proper name ‘The Allais Paradox’, are well known. They will probably stand in the history of risk theory on line with the St. Petersburg Paradox. Here I assume that the paradoxes are known and present a few comments and introduce some other paradoxes. 1.3.1. The insurance-lottery paradox This is an old one and so is the alleged solution [Friedman and Savage (1948)], but they are still kicking, as we shall see. In Schmeidler (1989, p. 586), the following sentence is found: ‘One of the puzzling phenomena of decisions under uncertainty is people buying life insurance and gambling at the same time.’ To this sentence is attached the following note: ‘It is not puzzling, as a referee has pointed out (my italics - O.H.), if one accepts the Friedman-Savage (1948) explanation of this phenomenon.’ The explanation referred to is the assumption that the curve describing a person’s NM utility of wealth is (roughly) concave over an interval covering the loss against which the insurance protects and convex over an interval covering the possible prizes in the lottery. This - it is understood - saves the expected utility hypothesis from empirical rejection through this observation. But accepting this assumption implies accepting another direct implication of it, see below. 1.3.2.

The lottery-bet paradox

A general trend of change in the structures

of money

lotteries

is the

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increase in the largest prizes, if not in real at least in nominal value. This means that the size of the first prize is continuously considered, so we must assume that in general it is not larger than most of the ticket buyers want. That means that most of the ticket buyers would rather have the lotteries as they are than having the first prize replaced by two prizes of half the size. If an expected utility maximizer wins a prize which is not higher than half the highest prize, he would, giaen the circumstances, under the expected utility principle be happy to risk it in a fifty-fifty double-or-nothing bet.

We know that in nearly all cases a corresponding offer will be rejected. (The insurance-lottery paradox is substituted by a lottery-bet paradox.) Any solution must imply that the persons who want the lottery to maintain its highest prize and at the same time will not exchange an amount of money of half that prize for half a chance of double that amount (= the highest prize) are not expected utility maximizers. 1.3.3.

The Allais paradox

This paradox is well known, but for the purpose of reference I give it in the terms I find practical. A game is described as (probability, prize; probability, prize): Choice situation 1, A or B: (1, lM), B: (0.01,OM;0.89,1M;0.10, 5M).

A:

Choice situation 2, X or Y: x: (0.89,OM;0.11, lM), Y: (0.90, OM; O.l0,5M). M is an amount of money considered very large by most people. According to EUT (> means ‘preferred to’, and u(O) =0): u(A)-u(B)=u(lM)-CO.89

u(lM)+O.lO u(5M),

=O.ll u(lM)-0.10 u(X)-u(Y)=O.ll So u(A)-u(B)=u(X)-u(Y) A>B

o

u(5M)<

u(5M)]

u(lM)-0.10

u(5M).

and >O if u(5M)cl.l 1.1 u(lM) 0

Choice combinations obeying EUT: The Allais Paradox: AY

u(lM). Hence

X>Y

AX (i.e. A>B

and X>Y)

and BY

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This is a set of choices stated by many respondents, (1954); see comment below (1.3.6.)

including Savage

1.3.4. The paradox of the inverted Allais paradox Consider the following sequence of games: First step, a game between the choice situations indicated above: (0.50, AB; 0.50, XY). As a result of this chance fork we arrive at one out of the two hypothetical decision forks described above. We shall either have a choice between A and B or between X and I: I do not intend to contribute to the discussion on dynamic consistence, which is implied in EUT but not depending on it. I take it for granted that if the respondents are asked to make their decision for each possible choice before the chance move is made in the first step, the decision will be the same as if it were made when and if that choice situation was a fact. We ask them to decide before the chance move is made in the first step. Their choice is then in fact between AX, AI: BX and BY (meaning: if I get the choice between A and B my choice is A, and if I get the choice between X and Y my choice is X, and so on). They have again the opportunity to show the Allais Paradox, as well as the EU-consistent sets of choices and the inverted Allais paradox. An experiment like this is not likely to bring any new information. But to focus more sharply on the issue: tell the respondents that their choice is restricted to AY and BX. Now the percentage choosing BX can for two obvious reasons not be smaller than in the experiments already carried out and reported. But there is a possibility that it will be larger. I shall in the future probably not have the facilities for carrying out this experiment, but challenge those who have. I do not think I risk much if I bet on an overwhelming majority for AI: Now what is interesting about that? According to von Neumann and Morgenstern’s axiom 3:C:b (1944) the utility of composite games depend solely on the probability distribution over final results. Let us find out what the probabilities are in this case: A Y: Probability of 0 =0.5 * 0.90 Probability of 1M = 0.5 * 1 Probability of 5M = 0.5 * 0.10 BX:

Probability of 0 = 0.5 * 0.01 + 0.5 * 0.89 Probability of 1M = 0.5 * 0.89 + 0.5 * 0.11 Probability of 5M = 0.5 * 0.10

= 0.45, = 0.50, = 0.05, = 0.45, = 0.50, = 0.05.

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According to the EUT the two lotteries should have the same value as their expected utility, i.e., the expected utility of two identical probability distributions. Why then is it possible to expect that AY is preferred? The key concept might be regret. Assume that all games are mutually stochastically independent. Game A gives no possibility of regret since one does not know how B would have turned out. Nor do games X and I: But game B does. If this game comes out with zero, one knows that the choice of A would have given lM, and there is cause for regret. So the choice BX, but not AY, can give cause for regret, and the fear of regret favours AI: It should be noted, though, that Allais type paradoxes also grow on trees where there is no possibility of regret. The described Allais Paradox is just one of several ‘Allais’ and many ‘Allais type’ paradoxes. It belongs to a class named by MacCrimmon and Larsson (1979) ‘common consequence paradoxes’ as opposed to ‘common ratio paradoxes’.

I .3.5. The Bergen paradox Imagine the following choice situation: Given a set of games by Probability choose n.

[winning $ 2”] =OS” (n= 1,2,3,. . . , N or ad infinitum),

We would expect that, more or less, all buyers of lottery tickets would choose an n much higher than 3. We define k as the chosen n and ask the respondent to choose A or B: A: The certainty of $ 2&-r, B: Probability [winning $ 2k] =OS. In conversational informal experiments respondents with k > 2 mostly choose A, which is contrary to the prediction of the EU model. This preference structure may have deep biological roots, because it has also been observed in the behaviour of rats [Battalio, Kagel and McDonald (1985)]. I have difficulties in understanding those who find this irrelevant on the ground that the EUT does not pretend to concern animals. As far as I am concerned, I am an animal. I have more understanding for the view that it is no wonder that people violate EUM, because they descend from apes. If we ‘Allaisians’ are allowed to describe the behaviour of animals, including the

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descendants of the apes, let the EU theory describe the behaviour of whoever that might be.

I .3.6. The juxtaposition paradox The concept of state dependent utility is now well known and so is the effect of juxtaposition. Many highly intelligent persons, professing their belief in EUT have ‘committed’ the above described Allais Paradox. They stick to the theory and explain their spontaneous choice as a mistake. When Savage (1954) explained his own (‘paradoxical’) choices as a mistake he did it in the following way: All games are arranged so that all outcomes were obtained at given numbers (‘states of the world’) out of the same 100, and so that in each choice situation the outcome will be the same for 89 numbers, regardless of which game is chosen. Game A and game B would both give 1M for the same 89 numbers, 12-100. Game X and game Y would both give 0 for the same 89 numbers. Table 1 in Savage (1954) with changed designations is: No.

1

2-11

12-100

z

1M 0

1M 5M

1M 1M

x y

1M 0

IM 5M

0 0

The numbers 12-100 could then be ignored and the choice could be made as between games with 11 numbers which are arranged in the following way, game A represented by game a’ and so on: No.

1

2-11

a’ 6’

1M 0

IM SM

x’ y’

1M 0

1M 5M

Savage now found it possible to correct his ‘mistake’ of preferring game A to game B and game Y to game X by preferring x to y and in consequence X to Y Still, he admitted, his original choices had some intuitive attraction for him. In Hagen (1972) it was pointed out that one could also arrange the games so that the outcomes were choice independent over 79 numbers. The remaining numbers can then be arranged as follows: No.

l-10(10)

11(l)

12-21 (10)

;’

1M 1M

1M 0

1M 5M

j

01M

01M

0 5M

’

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[or: x’

l-11 (11)

12-21(10)

1M

0 51M

Y’ 0

1

Here, by Savage’s own logic game A can be represented by game a’ and so on. The question was asked if this could not reswitch the preference. MacCrimmon (1979) and Pope (1985) have both supported this viewpoint. Since Savage was uncertain, even if in principle he was an EU maximiser, the ratio between his utility of winning the two prizes could not be far from 11: 10. If the monetary prizes are replaced by utility values, u(SiM)= 11 and u(lM) = 10, the minimax regret principle could explain the results. Both arrangements can be understood under the concept of framing, but then Savage’s frame (89 states) is broader than mine (79 states). I was not aware that I had stumbled onto a concept that was going to be important in the discussion of decisions under risk [see, for example, Stahl (1980) and Pope (1985)], but I had the sense to put into the ‘restrictions on the domain of the theory’ a condition for its validity that the utility of a prize was independent of the number on which it was won; which amounted to the restriction that utilities must not be state dependent.

2. An alternative descriptive model In this section there is just a short mention of Allais’ foundation, then my own model, a comparison between the models and a short comment on the most celebrated of other alternative models.

2.1. Allais' foundation Allais (1952, 1979) explained why, in general, the EU model falls short of describing actual preferences of presumably rational persons. This was exemplified through mental experiments like the one of ‘the’ Allais paradox. Allais also presented the axioms that should form the base of a more realistic theory. The maximand should be a functional of the whole shape of the probability distribution over utilities. His present views are still essentially the same [see his contributions to Allais and Hagen (1979)]: There exists a cardinal utility and a specific utility of risk. There is an absolute preference which is now generally known as stochastic preference of the first order. The degree of dispersion in the probability distribution of a game has a negative effect on its utility. There is

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a complementarity of certainty.

risk

effect in increasing the probability

in the neighbourhood

2.2. Hagen’s ‘three moments model’ of 1969 with modifications 2.2.1.

The 1969 version

The essence of the first version of the model in Hagen (1969) was the same as that of Allais. The definition of the absolute preference was formulated in terms of utility instead of money. It postulated the existence of a cardinal utility u, and in relation to money, x, the function u(x) had the derivatives: u’>O,

u”
u”‘>o,

..,

An important factor in the evaluation of a game is the expectation of ex post cardinal utility, also the tension caused by dispersion as such. Hagen (1969) puts particular weight on the strong psychological motivation of hope and fear. Therefore the skewness of the distribution was important. Even a small probability for something very good gives a person hope that can help alleviate the suffering of the dreariness of a grey life. On the other hand, even a small probability for a disaster (compared to the expectation) would cause fear, which means suffering in itself even if the possibility of disaster never materializes. In Hagen (1969) I did not fall into the trap of distorting the probability distribution itself. Skipping details, the essence of the mathematical formulation was: U( ii, s, z) = max

where U =utility of a game, =cardinal utility of outcome, ir =mean of 2.4, s = standard deviation, Z = a special measure of skewness = m3/s2 (except, of course, when s = 0 and m3 = 0; then z = 0), m3 = third central moment U

and

au/air> 0, au/as < 0,

a2u/as2 c au/az > 0.

0,

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The special measure of skewness is chosen because it has an important characteristic: in a two-outcome game it increases in absolute value as the smallest of the probabilities decrease [Hagen (1979)]. When u = u(x), and x =monetary gain/loss then odd number derivatives of u are positive and even number derivatives are negative. It has been shown that this model absorbs the well-known paradoxes of that time. In fact, it solves all the paradoxes mentioned above with the exception of the juxtaposition paradox.

2.2.2. The 1972 additive version Werner Leinfellner suggested that my model in Hagen (1969) might be lacking an axiom which captures the effect if outcomes undergo the same change. It seemed reasonable to me to assume that, if they all had the same change in utility then the value of the game would change with the same amount. I added an axiom stating that, if all outcomes were improved by the same amount of utility, the utility of the game would increase by the same amount. This made it possible to separate the expectation of the ex post utility and the specific utility of risk in an additive function [Hagen (1972)] as follows: Value of game is U = E(u) + f(s, 2) where f(s,z) represents the specific utility of risk. Obviously, if all u values increase by the same amount, then E(u) increases with that same amount, and f(s,z) remains constant, as a result, U increases with the same amount. If we make the mental experiment that both cardinal utility exists and that the von Neumann-Morgenstern utility index exist (no specific utility of risk), then f(s,z) evaporates, and the two utility concepts must be identical. Allais has (1979b) given this axiom the name: The axiom of cardinal isouariation, and applied it in a similar way. It is perhaps also worth noting that any linear operation on u implies the same linear operation on U. The 1972 version [Hagen (1972)] contains some theorems that concern games about money which can be tested against reality, in experiments or in observation of market behavior. Most of them explain well known paradoxes. Some explain more, like the following.

T(AAA) 6a. If games with more than two outcomes, say n+ 1, are represented by points in an n-dimensional space where coordinates represent probabilities of

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winning each ‘prize’ (excepting the least desirable outcome) indifference hypersurfaces may contain convexities towards the origin.

This again implies that, if there is a linear restriction in the n-dimensional space, the restricted optimum solution may not be a corner solution. The practical importance of this is that the current strategy of the large lotteries may be rational. The fact that they are multiprize lotteries is contrary to the assumptions of the EU descriptive model applied to the gamblers. Under the EU assumptions the utility of a game over any number of prizes is a linear function of the probabilities with the utilities as coefficients. A budget restriction on prizes (as percentage of the ticket sales) is a linear function of the probabilities with the utilities as coefficients. For the individual gambler the optimal solution would be a corner solution with one prize only. The gamblers’ tastes differ, but that is no reason for offering them ail the same full course menu. A better solution, if the EU theory were true, would be to offer them an a la carte menu, where each gambler could go for one prize. It has been shown [Hagen (1979)] how this could be done in a simple way. For reasons best known to themselves those who believe in EU maximization have not campaigned for a reform in this direction. The money lotteries as they actually are, present the multi prize lottery paradox, which is also solved as shown in Hagen (1979).

2.2.3. The development during the last two decades After Allais (1952, 1953) there was a period of thundering silence concerning the doubts about EU theory until the publications of Krelle (1968) and Hagen (1969). Krelle pointed out very clearly that there are two factors at work in the motivation of choices under risk, utility and risk attitude. He expressed this as the choice function under risk being a function of the utility function. This line of thought has been developed further by Bitz (1976). It is a different approach from the one I am partial to, but the presentations are certainly elucidating and touch on the core of the matter. In the seventies and eighties there has been springtime and ‘a hundred flowers’ have blossomed in the garden of alternative risk theory. Let me quote Sugden (1989): ‘For many years almost all economic analysis of choice under uncertainty was based on expected utility theory. The validity of this theory, both as a prescription as to how people ought to choose and as a description of how they do choose, was hardly questioned. The few sceptics, led by Maurice Allais, were generally perceived as eccentrics, outside the main stream of economic thought. Now it seems that their time has come. There is a rapidly-growing literature criticizing expected utility theory and suggesting alternative approaches. Let me

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say at the outset that this paper does not purport to be a general survey of this literature. [Two excellent surveys are available, written by Schoemaker (1982) [up to 1967 as far as Europe is concerned? - O.H.] and Machina (1983).] Rather it describes one class of alternative theories that has been developed in the last

few years by, among others, Bell, Chew, Fishburn, Hagen, Loomes, MacCrimmon and myself. I believe that this work provides one promising avenue for progress

but it is certainly

not the only one.’

His list of excellent surveys could be updated with, e.g. Fishburn (1988), Hey and Lambert (1987), of course Sugden (1989) and finally Bohren (1990). For the further development below two points made by members of the class dealt with by Sugden could be of special importance to the following development. Catchwords are: (1) disappointment, (2) regret. Disappointment is the effect of what might have been if, given his choice and the outcome, the gambler had had more luck. Regret is the effect of what might have been if, given the state of the world after the game, the gambler had made a choice that would have turned out to be better. It depends on how the probability distributions over states determine the probability distributions over the outcomes. It is a case of state dependent utility and of the juxtaposition problem, which has been commented on above (1.3.6). [See also for example Stahl (1980) and all references to Loomes and Sugden.] To take this into account it would be necessary to deal with the utility of change caused by the outcome of the game (and of what might have been). Outside this class the ‘Prospect Theory’ of Kahneman and Tversky has received more attention than any o&her alternative model. Their main points are: (1) They deal with utility of change. (2) The utility of a game is a product sum of weights w and utilities u, cwu, where the weight of the utility of each outcome is the value of one general function of probability, w(p), applied to the probability of that particular outcome. This general function is such that outcomes with small probabilities are given much higher weights, relative to the probability, than outcomes with greater probability. The transformation of probabilities to weights are independent of the utility of the outcome. The weights are not adjusted to add up to unity. That means that a group of several outcomes with small utility differences between them and small probabilities can add more to the value of a game than one outcome with the sum of their probabilities and a utility like the lowest in the group. The transformation described under point (2) above is the weak spot of the model. Let us take an example where the mentioned effect in the transformation function is oery moderate compared to the radical transformation illustrated by the graphical presentation given by the authors: w(p):

J.Pol.E-F

w(O.01) =

0.02, w(O.40)= 0.34, w(O.50)= 0.45.

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Prospect A in terms of utility: p(0) = OSO,p(I) = 0.50. Prospect B: = 0.50,

P(0)

p(O.91)=0.01, ~(0.92) =O.Ol,

w = 0.45, w=o.o2, w = 0.02,

and so on till p(O.99)=O.Ol, p( 1JO) = 0.40, p(1.01)=0.01,

w = 0.02, w = 0.34, w=o.o2.

Now, the utility of Prospect A is u(A) = 0.45 * 0 + 0.45 * 1= 0.45.

In Prospect B the sub mean of the utilities 0.91 to 0.99 is 0.95, and the sum of their weights is 0.18, so u(B)=0.95*0.18+1*0.34+1.01*0.02=0.5502. In fact there split in this know if this probabilities

is no limit to the utility value a game can get if probabilities are way. Tversky is going to revise prospect theory, and I do not objection will still apply. It is in any case my conviction that should be left well alone.

2.2.4. A reformulation of the Hagen model (1972) Let me first emphasise that when stochastically independent games are concerned I see little reason for any substantial change in this model. Its formulation and interpretation [Hagen (1979)] can stand. I feel that disappointment or the fear of it (or the opposites), which can occur in this case, is taken care of in my model by the skewness effect. However, regret requires a reformulation. The utility of outcome is redefined: Ui= that change in the utility of the decision maker’s ex post situation if outcome No. i occurs (named Vi)+ the possible state dependent utility effect (regret) (named Si). Using the names u and S, and dropping the subscripts we now have

u=u+s.

0. Hagen, Decisions under risk

When u = u(x), where x = monetary gain/loss, the mathematical in A.3. and A.4. [Hagen (1972)] become A.31 v’>O

and

395

formulations

A.4 v”-cO.

In the lemmas derived from these hypotheses, u is replaced by u. L(A) 1. All odd number derivatives of v are positive. L(A) 2. All even number derivatives of v are negative. My version of ‘Preference Absolue’ or ‘Stochastic Preference of the First Order’, concerning u as defined here, needs no reformulation. But other formulations, concerning x or v, can be violated by the subset of the set of state dependent utility effects which is identical to the set of juxtaposition effects. In the next step the symbol S should be given something like a concrete content. The most serious and extensive contributions in this field known to me are those given by Loomes and/or Sugden. I think it still stands that the concept can only be dealt with in a ‘coarse’ way [Sugden (1985)] if it is to be included in a mathematical and testable model. The usual definition of regret seems to be the difference between the utility of the outcome of the ex post best choice and the utility of the outcome of the actual choice. (Incidentally: minimising the expected regret in this sense gives the same choices as maximising expected utility.) I feel more for a related, but somewhat different concept which I have chosen to name ‘blame’ (meaning potential blame). By this I roughly mean the utility difference between the worst outcome of any other admissible choice, given the state, and the actual outcome (if the difference is positive, otherwise zero). This concept does not take care of all the qualitative considerations mentioned by Sugden (1985), absorbs some of them. In the following our decision maker might not be an ideal ‘rational man’ but we assume that he is not a muddlehead either. Let us take an example from gambling. A person puts a chip on red at a roulette with one zero and where, if zero will result all will be lost. For simplicity we assume that gain or loss in utility is identical with gain or loss of a chip. Table 1 will show the outcomes, and regret and blame resulting from betting on red, on black, and not at all. I feel that the table for blame might apply to a reasonably sensible person, but the table for regret could only apply to a muddlehead. If we include the possibility of placing the chip

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Table 1 Outcome The roulette shows: Red Black No game (alt. red) No game (no ah.)

R

B

Regret 0

RBO

+1 -1 -1 -1 +1 -1 0 0 0 0 0 0

0 2 2 0 100 000

Blame RBO

1

0 1 0

1

1 0 0 0 0 0 0 0 0

on a number, we shall get a ridiculous regret table, but still a reasonable blame table. The restriction to ‘admissible’ alternatives is meant to exclude dominated alternatives. But I will go a step further: If the chosen action could have been elected on the maximin principle, there is by definition no blame. If we removed the possibility ‘No game’, blame would be the same as regret, and ridiculous, if it were not for this last modification: Since the maximin principle indicates Red or Black, there can be no blame. An example from business: An industrial company had made a deal, subject to shareholders’ approval, with a foreign government which would bring it a petrol drilling concession in return for an industrial investment. The manager of an investment fund successfully opposed the deal and won. To his misfortune the block that had been assigned for the company was specified and later turned out a thundering success. One might think that had it not been specified the manager might still hold his chair. The blame effect on the utility of the outcome can ‘coarsely’ be defined as b * B (B for ‘Blame’ as above). My preliminary conclusion is: S = bB. In the ranking of stochastically independent games my 1969172 model [Hugen (2969, 1972)] is still undefeated by facts. It may be necessary to

introduce a regret/blame effect when one alternative is a certainty, like game A in the Allais paradox above.

3. The normative EUT

3.1. A curmudgeon’s view Since the von Neumann-Morgenstern utility index is defined as a measure which is derived from actual choices of a decision maker whose expectation is supposed to be maximised, to advise a decision maker to maximize its expectation is void of meaning. A more liberal curmudgeon could allow this interpretation: Make your choices in such a way that it could be consistent with maximising the expectations of some utility index. But any single choice could comply with this demand. Now suppose that Castor is faced with the choice between A and B in the

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Allais paradox games, and thinks he will choose A. The EUT believing consultant asks him what he would have chosen if the alternatives had been X and K We know that if Castor is EUM consistent he will prefer X, but this implication cannot be inverted: characterised by identical historical experience there may be a third choice situation where his choice would be ‘inconsistent’. If Castor answers x we know that he is ‘inconsistent’. But an EUM consultant cannot tell him which choice is ‘irrational’; he can only advise him to reconsider. Suppose further that Pollux, Castor’s identical twin who is with the same experience, is faced with the choice between X and I: Suppose he chooses I! He is asked to choose hypothetically between A and B. He ‘should’ choose B. If he prefers A, he is inconsistent. Again, all the EUM consultant can do is to advise him to reconsider. What will be the effect of the reconsideration? These are the possibilities: 1. 2. 3. 4.

No change. Both will reverse their choice. Castor will reverse and Pollux not. Pollux will reverse and Castor not.

1 and 2 means that just as without ‘consultation’ one decision is ‘right’, the other ‘wrong’ in the EUM sense. 3 or 4 may mean that ‘wrong’ decision is changed to a ‘right’, or the opposite. If anything good comes out of this, a more general and equivalent technique would be simply to ask any decision maker to reconsider any decision! 3.2. A nicer person’s view A nicer person might suggest the following interpretation: A decision maker’s utility index is defined through choices between simple games and the choices between complex games should be calculated from this index. But, alas, this will not hold water. Assume for example that we have defined the present level of wealth as having utility 0, and some gain M as having utility 1. The certainty equivalent (c.e.) of any game (1 -p,O; p, M) has now by definition the utility p. For example the certainty equivalent of (0.875, 0; 0.125, 1) would have the utility 0.125. We name this c.e. Md. Then u(M,)=0.125. Now suppose we approach the amount of money that has the utility 0.125 stepwise in the following way. First we find the c.e. of (0.50, 0; 0.50, M). We name it M,. Now u(M,)=0.50. We then proceed correspondingly to find the c.e. for (0.50,0;0.50, M2), which we name M,. Now according to EUT, u(M,) =0.50 * 0.50 = 0.25. We repeat this procedure with Ma in the place of M2 to get M,. Then according to EUT, by definition u(MJ = 0.25 * 0.50 = 0.125. It follows that EUT * M,, = M,. If this equation

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does not hold, not only is EUT as descriptive theory falsified [pointed out in Hagen (1969)], in Krzystofowicz (forthcoming) this has given occasion to empirical testing. In order to make EUM an operational normative device, a convention is required to tell us how to select one of the equivalent methods of determining the utility index. Not a generalised but a more specified EUT would be required. Let us try. It seems natural to proceed as strong believers in expected utility maximizing have done [see Raiffa (1968)]: Given a set of games for the decision maker to choose between. The most preferred outcome is named xi. The least preferred outcome is named x0. We define, without loss of generality: u(xJ

=o,

u(x,) = 1.

Now for any x (x,,
P($25) = 1,

B:

P($50) = 0.5.

In Raiffa (1968, fig. 4.13, p. 67), we find the utility function of Mr D.M. which displays a function containing values of equilibrating probabilities for -$25: 0, for 0: 0.50, for +$50: 0.81, for $100: 1.00. For our purpose we perform a positive linear transformation: EP(0) =0,

EP(25) =0.62,

EP(50) = 1.

This gives, since by definition u(x) = EP(x), the expected utilities EU(A) = 0.62, so A>B.

EU(B) = 0.50,

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Assume now that alternative C adds to the menu: c: P(S25,OOO)= 0.0004. Since most people buy lottery tickets that are ‘unfair’ it seems not unreasonable to assume that we might have EP(S25) = 0.5 * 25/25,000 = 0.0005. Since this ‘lottery effect’ must necessarily weaken with increasing price of the lottery ticket, it is not unreasonable to assume that EP(S50) > 0.5 * 50/25,000,

for example = 0.0011,

which would mean EU( A) = 0.0005,

EU( B) = 0.00055.

EU( C) = 0.0004,

and the order of preference would be B> A> C. This means that the preference between A and B is reversed through the introduction of an ‘irrelevant’ alternative C. One could maybe find new restrictions on the application of the EU principle to avoid absurdities. But how could it be avoided that it would be arbitrary? 3.3. Survival criterion, not for the ELT Time and again EUT successfully passes some rationality test which upon closer examination turns out to be no test at all. The examiner often forgets that the ability to produce on some occasion decisions that are rational is no criterion of consistent and exclusive rationality. Karni and Schmeidler (1987) referring to Borch (1966) argue (1) that maximising the probability of financial survival implies maximising expected utility, and (2) ‘... that violations of the independence axiom, e.g. the Allais paradox, imply that decision makers choose a probability of survival smaller than the maximum possible given the set of acts.’ It is stated that Borch’s analysis uses discounted future dividends, whereas the authors use probability of survival as utility. In Borch (1968, 2.6) it is suggested that the probability of survival can serve as a utility function. The authors stress that the two analysis have in common that they are dynamic. Now, in the context

of EUT, to maximize the expectation

of a utility

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which is defined as the probability of survival is the same as maximizing the expectation of a utility function with only two levels: Survival:

1,

Death:

0.

With only two levels of utility EUM is nothing but absolute preference, analogue to Allais ‘preference absolute’ and to stochastic preference of first orer. Moreover it is identical with the version in Hagen (1969). This may be an implication of RUM but is not its exclusive property. But there is still a remaining part of the allegation in Karni and Schmeidler (1987) which should be investigated. The Allais paradox, at least in the original form, is about wealth, and violates the EUT in so far as the utility in question is a unique function of wealth, which seems to be the current interpretation. For illustration, we use the following scenario for a decision maker: Initial wealth: IM, Expenditure each period: IM. The income of each period the decision maker and its decision maker depend on Allais paradox (as described Situation

derives from the outcome of the game chosen by outcome. The games which are available to the the two alternative situations 1 and 2 from the in 1.3.3):

1, A or B:

A: (l,lM), B: (0.01,0;0.89, lM;O.l0,5M). Situation 2, X or Y: X:

(0.89,O; 0.11 1M),

y:

(0.90,O; O.l0,5M).

Financial death occurs when wealth=O. Objective function: probability of surviving 2 periods (‘POS 2’). Situation 1.

Here the choice is obvious: A in both periods:

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POS 2= 1.00. Any other procedure: POS 2< 1.00. Situation 2. X gives maximum probability of surviving the first period (0.11). In case of survival the situation before the second period is identical with the first; maximum probability of surviving second period is again X. This gives POS 2=(0.11)*=0.0121. If Y is chosen before the first period, the probability of surviving period 1 is only 0.10, but in case of survival the decision maker’s wealth is 5M and he is certain to survive period 2 regardless of which choice he makes before that period. Thus POS 2 = 0.10. A decision maker maximising his probability of surviving two periods will make exactly the decisions of the Allais paradox: He will prefer A to B in both periods in situation 1, and in situation 2 he will prefer Y to X, at least in the first period. The above conclusions can of course be extended to more than two periods, but the additional information would be rather trivial. While bearing in mind that it was not this author who brought in the Allais paradox as a test example it may be of interest to investigate whether this is a unique and perverse example. This paradox is an example of what has been named common consequence paradoxes as opposed to common ratio paradoxes [MacCrimmon and Larson (1979)]. We will now test an example of the latter type. The same scenario as above is used but we introduce two new choice situations. Situation 23, Q or R:

Game

Prize

Probability

Q

1M

1.0

R

0

0.50 0.50

2M

Situation 4, S or T:

Game

Prize

Probability

S

0

0.98 0.02

1M T

0

2M

0.99 0.01

Here the expected utility model predicts that preferences should be either for Q and S respectively or for R and ‘I: whereas some people prefer Q and T [Hagen (1979), MacCrimmon and Larson (1979)]. We will see what preferences maximize POS 2.

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Situation 3.

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By the same reasoning as in Situation 1 the choice must be Q.

Situation 4. If S is chosen before the first period, the probability of surviving that period is 0.02. Given survival of the first period, choosing S again gives the highest probability of surviving period 2, (0.02). Thus POS 2 = o.022 = 0.0004. If T is chosen before the first period, the probability of surviving that period is 0.01, but given survival of period 1, survival of period 2 is certain for both S and YKThus POS 2=0.01. Thus the choice must be T before period 1. The next choice is indifferent to the object function. Conclusion: There are situations where maximizing the probability of survival implies decisions as described by the Allais paradox which violate the expected utility theory as usually understood.

Here is a friendly test of rationality where the EUT fails.

4. The proxy method

In Hagen (1983) it is pointed out that one assumption concerning risk behavior held by some adherents of EUT is not unlike some assumptions underlying the three-moment model in Hagen (1969, 1972). Those EUT adherents who believe in diminishing absolute risk aversion implicitly believe that the third-order derivative of the von Neumann-Morgenstern utility index with respect to monetary wealth is positive. This again implies that the expected utility of games with the same expectation and variance is higher when the third central moment is higher. In the three-moment model one assumption is that the third derivative of the cardinal utility of wealth is positive. It is also assumed that the utility of a game increases with increasing skewness of the probability distribution over utility. This suggests that a coarse approximation to a ranking function for games about money could be some function of expectation, variance and third central moment in the probability distribution over monetary values. A two-outcome game has three degrees of freedom and is determined by, e.g., the two values and the probability of one of them. Excluding the degenerate case (two equal outcomes), any one element in the set of two-outcome games can be identified through its mean, its variance and its third central moment. We must assume that any reasonably intelligent decision maker can rank all two-outcome games (which are the simples possible games). If he is sincere (and does not apply a rule which he thinks will help him to look intelligent) his ranking will give some information about his tastes, or preference function. The same decision maker may, however, be unable to

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rank very complex games in the same direct way. He will need some kind of technique, but this technique should enable him to rank the gams in accordance with his own personal tastes. If we assume that a coarse approximation to a subject ranking function for games about money could be some function of expectation, variance and third central moment in the probability distribution over monetary values, the following technique should be of use to help him find a ranking that would be in approximate accordance with his personal subjective preferences. 1. For each alternative, compute expectation, variance and the third central moment, and find the ‘proxy’: a two-outcome game with the same values of the three statistics. 2. Ask the decision maker which of these proxies he prefers. 3. Recommend the corresponding original alternative. Please note that the decision maker does not have to know the three statistics, neither their definitions nor their values. In general, decision models based on subjective preferences cannot be scientifically tested. One cannot compare the direct ranking within a set, there this is possible to the direct ranking in a set where this is impossible. But both sets are fuzzy. There is an interface, or let us put it this way: most people can rank two-outcome games directly. Some of these can also rank three-outcome games directly. This opens for empirical testing. In Blasche and Dorfner (1987) a test is reported. ‘Conclusion

(1) The null-hypothesis, that corresponding pairs of games are chosen no more often than by coincidence, is rejected in favour of a weaker formulation of our initial hypothesis to the extent that not all, but many decision makers chose corresponding pairs of games. (2) The percentage of corresponding pairs chosen was higher in the second test, when the games were presented in an easy-to-see manner and the use of decision rules was discouraged. (3) This test is only valid when the original games are as simple as the ones we used in our test. But it is an indication that the same decision technique could also be recommendable when the original games are so complicated that

an empirical test is not possible.’ The null hypothesis was in both cases tested on a 99% criterion and was in both cases rejected with very broad margins. The games had approximately the same expectations, and the two moments were chosen so that there could be no simple generally known rule that could suggest the ranking. To the extent that experimental evaluation is possible, the method has, I feel, stood up well. It has however one foreseable weakness for practical use: even if dominated alternatives are weeded out in the original games,

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domination can pop up among the proxies. For example a little increase in the highest prize in the Allais Paradox would lead to the proxy of B dominating A. Domination created by simplification would be less likely to occur if the proxies had three outcomes. Another device might be to keep the extreme values and their probabilities from the originals in the proxies, and apply the method above to determine two more outcomes with probabilities. References Allais, M., 1952, 1979a. Fondements dune theotie positive des choix comportrant un risque et critique des postulats et axiomes de I’ecole americaine, Colloques Internatiooaux du Centre National de la Recherche, Vol. XL, Paris 1953. Also in: Allais and Hagen (1979): English translation. Allais, M., 1979b. The so-called Allais paradox and rational decisions under uncertainty, in: Allais and Hagen (1979). Allais, M., 1983, The foundation of the theory of utility and risk, in: Hagen and Wenstoep (1984). Allais, M., 1985a, Three theorems on the theory of cardinal utility and random choice, in: Eberlein and Berghel, eds. (Reidel, Dordrecht, 1987). Allais, M., 1985b, The Allais paradox, in: The new Palgrave. A dictionary of economics, vol. 1 (Macmillan, London, 1987). Allais, M., 1986, The general theory of random choices in relation to the invariant cardinal utility function and the specilic probability function, The (V, 0) model, in: B.R. Munier, ed., Risk, decision and rationality (Reidel, Dordrecht, 1988). Allais, M., 1988, Scientific papers on risk utility theory-Theory, experience, and applications (Reidel, Dordrecht), forthcoming. Ch. XVIII: Cardinal utility-History, empirical findings and applications. Allais, M. and 0. Hagen, eds., 1979, Expected utility hypotheses and the Allais paradox (Reidel, Dordrecht). Allais, M. and 0. Hagen, eds., 1992, Cardinal Utility (Kluwer, Dordrecht), forthcoming. Battalio, R.C., J.H. Kagel and D.N., McDonald, 1985, Animals choices over uncertain outcomes: Some initial experimental results, American Economic Review 75. Bell, D., 1982, Regret in decision making under uncertainty, Operations Research 33, l-27. Bernard, J., 1974, On utility functions, Theory and Decision 5, 205-242. .- Bitz, M. and M. Rogusch, 1976, Risiko-Nutzen, Geld-Nutzen und Risikoeinstellung. Zur Diskussion urn das Bernoulli-Prinzip, Zeitschrift fiir Betriebswissenschaft. Vol. 46.12, 853-868. Blasche, H. and E. Dorfner, 1987, Selection by proxy: A model for the simplification of decision under risk and under uncertainty, Theory and Decision 23, no. 3, 283-300. Bohren, O., 1990, Theory development processes in the social sciences: The case of stochastic choice theory. Journal of Economic Psychology 11, l-34. Borch, K., 1966, A utility function derived from a survival game, Management Science 12, 287-295. Borch, K., 1968, Decision rules depending on the probability of ruin, Oxford Economic Papers 20, no. 1, l-10. Chew, S. and K. MacCrimmon, 1979, Alpha-nu choice theory: A generalization of expected utility theory, Working paper no. 669, University of British Columbia. Chikan, A., J. Kindler and I. Kiss, 1991, Progress in decision, utility and risk theory, Proceedings of the 4th FUR Conference (Kluwer, Dorcrecht). Daboni et al. eds.. 1985, Recent developments in the foundations of utility and risk theory (Reidel, Dordrecht). Fishbum, P.C., 1987, Reconsiderations in the foundations of decision under uncertainty, The Economic Journal 97,825-841. Fishbum, P.C., 1988, Nonlinear preference and utility (Wheatsheaf Books, Brighton).

0. Hngen,

Decisions

under risk

405

Friedman, M. and L. Savage, 1948, The utility analysis of choices involving risk, Journal of Political Economy 56, 279-304. Geweke. J.. ed.. 1992. Decision making under risk and uncertaintv: New models and emoirical findings. (Kluwer, Dordrecht). Hagen, O., 1969, Separation on cardinal utility and specific utility of risk in theory of choices under uncertainty, Statsakonomisk Tidsskrift 3. Hagen, 0.. 1972, A new axiomatisation of utility under risk, Teorie a metoda, Also Reprint _ i987/4 Norwegian School of Management. Hagen, 0.. 1979, Towards a positive theory of decisions under risk, in: Allais and Hagen (1979). Hagen, O., 1983, Paradoxes and their solutions, in: Stigum and Wenstoep (1983). Hagen, 0.. 1984, Relativity in decision theory, in: Hahen and Wenstop (i984), 237-249. Hagen, 0.. 1985, Rules of behavior and expected utility theory. Comparability versus dependence, Theory and Decision 18, no. 1, January, 31-46. Hagen, O., 1991, Expected utility theory, The ‘confirmation’ that backfires, in: Chikan, Kindlooer and Kiss (1991). Hagen, O., 1992, A true descriptive theory and a useful decision technique, in: Geweke (forthcoming). Hagen, 0. and F. Wenstop, eds., 1984, Progress in utility and risk theory (Reidel, Dordrecht). Hey, J.D. and P.J. Lambert, eds., 1989, Surveys in the economics of uncertainty (Basil Blackwell, Oxford). Kahneman, D. and A. Tversky, 1979, Prospect theory: An analysis of decisions under risk, Econometrica 47, no. 2, 261-291. Karni, E. and D. Schmeidler, 1986, Self-preservation as a foundation of rational behavior under risk, Journal of Economic Behavior and Organisation 7, 71-81. Krelle, K., 1968, Preferenz- und Entscheidungstheorie (J.C.B. Mohr, Tiibingen). Krelle, K., 1984, Remarks to Professor Allais’ contributions to the theory of expected utility and related subjects, in: Hagen and Wenstoep (1984). Krzysztofowicz, R., forthcoming, Filtering risk effect in standard-gamble utility measurement, in: Allais and Hagen (forthcoming). Loomes, G. and R. Sugden, 1985, Regret theory: An alternative theory of rational choice under uncertainty, Economic Journal 92, 805-824. Loomes, G. and R. Sugden, Some implications of a more general form of regret theory, Available from authors (not read). Lopes, L., 1988, Economics as a psychology: A cognitive assay of the French and American schools of risk theory, in: B.R. Munier, eds., Risk, decision and rationality (Reidel, Dordrecht). MacCrimmon, K.R. and S. Larsson, 1979, Utility theory: Axioms versus ‘paradoxes’, in: Allais and Hagen (1979). Machina, M.J., 1983, Generalized expected utility analysis and the nature of observed violations of the independence axiom, in: Stigum and Wenstop (1983). Pope, R., 1985, Timing contradictions in von Neumann and Morgenstern’s axioms and in Savage’s ‘sure-thing’ proof, Theory and Decision 18, no. 3, 229-261. Raiffa, H., 1968, Decision analysis (Addison-Wesley, New York). Savage, L.J., 1954, The foundations of statistics (John Wiley and Sons, New York). Schoemaker, P., 1982, The expected utility model: Its variants, purposes, evidence and limitations, Journal of Economic Literature 20, 529-563. Snow, P., 1987, Maximizing expected utilty is a survival criterion, Theory and Decision 22, no. 2, 143-154. Sugden, R., 1985, Regret, recrimination and rationality, Theory and Decision 19, no. 1, July, 77-100. Sugden, R., 1989, New developments in the theory of choice under uncertainty, in: Hey and Lambert (1987). Stigum, B.P. and F. Wenstop, eds., 1983, Foundations of utility and risk theory with applications (Reidel, Dordrecht). Stihl, I., 1980, Review of Allais and Hagen (1979), Scandinavian Journal of Economics, 413-417.

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