- Email: [email protected]
Maximum probabilities, information, and choice under uncertainty
Accepted Manuscript Maximum probabilities, information, and choice under uncertainty Daniel R. Burghart
PII: DOI: Reference:
S0165-1765(18)30101-0 https://doi.org/10.1016/j.econlet.2018.03.010 ECOLET 7971
To appear in:
Economics Letters
Received date : 11 February 2018 Revised date : 13 March 2018 Accepted date : 14 March 2018 Please cite this article as: Burghart D.R., Maximum probabilities, information, and choice under uncertainty. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
An Expected Utility-like model is proposed for choice under uncertainty. The model weights utilities by the maximum probability and an information term. Information depends on how much is known about probabilities. A probability triangle-like figure is introduced and used to explore the model. Applications to medical decision making and financial asset demand are explored.
*Manuscript Click here to view linked References
Maximum probabilities, information, and choice under uncertainty Daniel R. Burghart∗ March 12, 2018
Abstract This note proposes a simple, expected utility-like model for decision making under uncertainty. The model uses the maximum probability for each possible outcome and the amount of information conveyed by this upper envelope. A graphical tool is introduced and used to study the model when two outcomes are possible. The model is extended to an abstract number of outcomes in which interpersonal comparisons of preferences are considered along with applications to medical decision making and financial asset demand. JEL Classification: D81, G11. Keywords: Decision-Making, Uncertainty, Expected Utility, Portfolio Choice
∗ California State University Sacramento, Department of Economics, 6000 J Street, Sacramento, California 95819. [email protected]. The inspiration for this note came from a comment made by Mark Machina during my talk at RUD 2016 (Paris-Dauphine). This note also benefitted greatly from suggestions made by an anonymous referee.
1
For decisions involving unknown outcomes economists distinguish between risk and uncertainty. Risk is a setting where probabilities for monetary outcomes are assumed fully-known at the time of choice. Uncertainty, in contrast, relaxes this assumption. Models for decision making under risk, such as von Neumann-Morgenstern expected utility (vNMEU), can be a parsimonious way to analyze choices when outcomes are unknown (von Neumann and Morgenstern, 1947). A tradeoff for this parsimony is a lack of realism – outside of casino games and lottery tickets most decisions involve probabilities that are not fully known at the moment of choice. Such decisions are said to be made under uncertainty. One way in which models for decision making under uncertainty are distinguished from the vNMEU setting is that the events causing monetary payoffs are assumed to be important determinants of choice.1 While there are many settings in which this assumption is likely to hold, there are also situations in which these events are less relevant. For example, in the case of financial assets, the probabilities for monetary outcomes, whether known or unknown, are likely to be more relevant for determining choice. This note proposes a simple model that extends vNMEU to such a setting. The model adds a parameter that takes into account the amount of information available at the time of choice while also weighting utilities by the maximum probabilities. The objects of choice in this model thus represent the maximum probabilities for monetary outcomes and a term that captures the amount of information conveyed by this upper envelope.2 This information term can take on any value between zero and one. A value of zero implies that nothing is known about the probabilities – the maximum probability for each outcome could be unity. A value of one represents full information about the probabilities, a setting analogous to risk because it implies a unique distribution. These objects of choice are only slightly more complicated than money lotteries yet, they can accommodate unknown probabilities. The next section leverages a well-known experiment in Ellsberg (1961) to introduce and explain the model. This explication is aided by the introduction of a graphical tool. The penultimate section generalizes the setting, introduces interpersonal comparisons of preferences, and discusses some applications. The final section concludes and indicates avenues for future work. 1
Risk can also be conceptualized in such a setting (Arrow, 1964; Debreu, 1959). On the flip side, choice under uncertainty has also been considered using a “set of lotteries” approach in Olszewski (2007) and Ahn (2008). 2 The term “upper envelope” is adopted from the statistics literature used to analyze sets of probability distributions by their maximal and minimal probabilities (Dempster, 1967; Shafer, 1976).
2
1
The Ellsberg Experiment
In 1961 Daniel Ellsberg proposed several experiments designed to test subjective expected utility (Ellsberg, 1961; Savage, 1954). One of these experiments involves drawing a single ball from an urn containing 90 balls. Exactly 30 of the balls are red. Each of the remaining 60 balls could be black or yellow. So, the ratio of black to yellow balls is an unknown proportion. The decision making component of the single urn Ellsberg experiment is composed of two choice situations. The first involves choosing between a “bet on red” and a “bet on black.” The bet on red pays $100 if the ball drawn from the urn is red and $0 otherwise. The bet on black pays $100 if the ball drawn is black and $0 otherwise. The second choice situation involves choosing between a “bet on red or yellow” and a “bet on black or yellow.” The bet on red or yellow pays $100 if the ball drawn is red or yellow and $0 otherwise. The bet on black or yellow pays $100 if the ball drawn from the urn is black or yellow and $0 otherwise. A “very frequent pattern of response” has the bet on red preferred to the bet on black and the bet on black or yellow preferred to the bet on red or yellow (Ellsberg, 1961, pg. 654). This pattern of responses, commonly referred to as ambiguity aversion, violates the Sure-thing Principle of subjective expected utility. A number of models have been proposed that can accommodate ambiguity aversion. Two of the more popular types are models with multiple priors, such as Maxmin Expected Utility (Gilboa and Schmeidler, 1989) and α-Maxmin Expected Utility (Ghirardato, Maccheroni, and Marinacci, 2004), and models with non-additive probabilistic beliefs, such as Choquet/Rank-Dependent Expected Utility (Schmeidler, 1989). Section 5 in Machina and Siniscalchi (2014) provides a more thorough review of models that can accommodate ambiguity aversion.
1.1
The sets of lotteries implied by the Ellsberg experiment
For the two possible monetary outcomes, $100 and $0, two of the four bets in the Ellsberg experiment imply exact probability distributions. The bet on red implies an exact probability distribution of (1/3, 2/3) because there are exactly 30 red balls and 60 non-red balls in the urn. Similarly, the bet on black or yellow implies an exact probability distribution of (2/3, 1/3) because there are exactly 60 black or yellow balls in the urn. The other two bets in the Ellsberg experiment, however, imply a set of possible probability values for each outcome. For the bet on black the minimum and
3
maximum probabilities for the $100 outcome are 0 and 2/3 because the number of black balls is unknown and could take on any integer value 0 through 60. The minimum and maximum probabilities for the $0 outcome are 1/3 and 1. Analogously, the bet on red or yellow implies probability values that can range from 1/3 to 1 for the $100 outcome and 0 to 2/3 for the $0 outcome.
1.2
Upper envelope lotteries
Order the monetary outcomes in the Ellsberg experiment from largest to smallest: {$100, $0}. An upper envelope lottery lists the maximum probability for each of these monetary outcomes and the amount of information conveyed by these maximum probabilities.3 Denote these as (p100 , p0 , i). Similar to a traditional probability distribution, each entry in an upper envelope lottery is constrained to the unit interval: pj ∈ [0, 1] for each j, and i ∈ [0, 1]. In contrast to a traditional probability distribution, however, the entries in a two outcome upper envelope lottery must sum to two: i + p100 + p0 = 2. To see why i+p100 +p0 = 2, consider two cases: p100 +p0 = 1 and p100 +p0 = 2. The only way in which p100 +p0 = 1 is if the upper envelope lottery implies a unique probability distribution. In such a case there is full information about the probability distribution. So, i = 1 and i + p100 + p0 = 2. The second case, p100 + p0 = 2, represents the opposite extreme: only if p100 = 1 and p0 = 1 can their sum be two. That either probability could be the maximum allowable value for a probability means that nothing is known about the distribution. So, i = 0 and i + p100 + p0 = 2. The upper envelope lottery implied by the bet on red is R = (1/3, 2/3, 1) while for the bet on black it is B = (2/3, 1, 1/3). The upper envelope lottery R has full information (i = 1). The upper envelope lottery B, in contrast, has only one third of the probability information known at the time of decision. Hence, for B, i = 1/3. The upper envelope lottery implied by the bet on red or yellow is RY = (1, 2/3, 1/3) and BY = (2/3, 1/3, 1) for the bet on black or yellow. For RY there is only 1/3 of the probability information known at the time of decision. The upper envelope lottery BY , in contrast, has i = 1. 3
The working paper by Burghart, Epper, and Fehr (2015) explores lower envelope lotteries. These objects are defined as the minimum probability for monetary outcomes and the amount of probability mass missing from this lower envelope, a quantity referred to as ambiguity.
4
1.3
A graphical tool
This subsection introduces a triangular figure that allows for graphical analysis of preferences for two-outcome upper envelope lotteries. This figure draws its inspiration from the probability triangle, a tool that permits graphical analysis of preferences for three-outcome lotteries in a setting of risk (Marschak, 1950; Machina, 1982). For three-outcome risky lotteries, each of the probabilities belongs to the unit interval and their sum must be one. So, they can be plotted in an isosceles right triangle whose right angle lies at the origin (see, for example, Machina (1982), Figure 5). Each of the three entries in a two-outcome upper envelope lottery must also be one but, their sum must be two. So, upper envelope lotteries can be plotted in an isosceles right triangle such as the one in Figure 1a. Define this as the upper envelope triangle. In a Cartesian coordinate system the vertices of the upper envelope triangle correspond to the points (0, 1), (1, 1), and (1, 0). Whereas the vertices of the probability triangle represent certainty of the three possible outcomes, the vertices of the upper envelope triangle represent (1, 0, 1), certainty of $100, (0, 1, 1), certainty of $0, and (1, 1, 0), a situation in which nothing is known about the probabilities for the two outcomes (see Figure 1a). Points along the hypotenuse of the upper envelope triangle have full information (i = 1). So, they are analogous to money lotteries because they imply a unique probability distribution. Points on the vertical leg of the triangle have a maximum probability for the $0 outcome as unity (p0 = 1) while points on the horizontal leg have a maximum probability for the $100 outcome as unity (p100 = 1). Movements parallel to the hypotenuse hold i constant while trading off p100 and p0 in a one-to-one ratio. Movements parallel to the vertical leg keep p0 constant while trading off (one-to-one) p100 and i. And movements parallel to the horizontal leg keep p100 constant while trading off (one-to-one) p0 and i. Figure 1b plots the upper envelope lotteries implied by the bets in the Ellsberg experiment. Because the upper envelope lottery R has i = 1 it must lie on the hypotenuse. In contrast, B has a maximum probability for $0 of one so it lies on the vertical leg of the triangle. RY lies on the horizontal leg because the maximum probability of getting the $100 outcome is one. BY has i = 1 so it lies on the hypotenuse.
5
Figure 1: A triangular diagram can be used to plot two outcome upper envelope lotteries (a) The upper envelope triangle
Certainty of $100 (1, 0, 1)
(b) The upper envelope lotteries implied by the bets in the Ellsberg experiment
RY = (1, 2/3, 1/3)
Zero information
p100 = 1
(1, 1, 0)
BY = (2/3, 1/3, 1)
li
1 atio i = orm nf
p0 = 1
l Fu
B = (2/3, 1, 1/3)
n
R = (1/3, 2/3, 1)
)
,1 ,1 (0
Certainty of $0
1.4
A simple model
For the $100 and $0 outcomes possible in the Ellsberg experiment the upper envelope expected utility (U EEU ) of an upper envelope lottery A = (p100 , p0 , i) is
U EEU [A] = p100 u100 + p0 u0 + iui ,
(1)
where p100 and p0 are the maximum probabilities for the $100 and $0 outcomes, u100 and u0 are the utility numbers assigned to the $100 and $0 outcomes, i is the amount of information in A, and ui is a utility number assigned to information. The U EEU model is an expected utility-like expression of the entries in an upper envelope lottery. While it is tempting to think that U EEU will collapse to expected utility when i = 1, this is not the case because of the structure of upper envelope lotteries. Too see this consider R = (1/3, 2/3, 1), an upper envelope lottery with full information. Normalizing utilities so that u100 = 1 and u0 = 0, U EEU [R] = 1/3 + ui . The ui term distinguishes the U EEU [R] from an expected utility of the money lottery with probabilities identical to first two entries in R. Importantly, however, choices amongst upper envelope lotteries with i = 1 would be consistent with choices generated by an expected utility function that has the
6
same utility numbers for the monetary outcomes. Given the normalizations for u100 and u0 , ui must belong to the unit interval: ui ∈ [0, 1]. Values of ui lower than zero imply a chooser willing to trade information for increases in the maximum probability of the $0 outcome. On the flip side, values of ui larger than unity imply a chooser willing to trade maximum probability for the $100 outcome for additional information. Neither of these potential tradeoffs seem justifiable for the same reason that, in a setting of risk, a lottery that is first order stochastically dominated by another lottery would be preferred – it is not justifiable to give up probability for an outcome to get an equal amount of probability for an outcome lower on the preference scale.
1.5
Types of information preferences
From a graphical perspective the U EEU model implies indifference curves that are parallel straight lines in the upper envelope triangle. The slope of these lines is dictated solely by ui . Maintaining the normalizations u100 = 1 and u0 = 0, ui = 1/2 implies indifference curves that have a slope of 45 degrees in the triangle. To see this consider two upper envelope lotteries: (1, 1, 0), nothing is known about the probabilities, and (1/2, 1/2, 1), an upper envelope lottery analogous to a 50/50 lottery. Evaluating each of these upper envelope lotteries gives U EEU [·] = 1. Figure 2a plots these two alternatives and shows a map of indifference curves for a chooser with ui = 1/2. Such choosers are labeled information neutral. Consider again the two alternatives (1, 1, 0) and (1/2, 1/2, 1). When ui < 1/2 the ordering between these two alternatives is (1, 1, 0) (1/2, 1/2, 1) – the chooser prefers the upper envelope lottery with zero information. It seems natural, therefore, to label choosers with ui < 1/2 as information averse. The solid lines in Figure 2b illustrate a map of indifference curves for an information averse U EEU chooser. The dashed lines are at 45 degrees to serve as a reference. When ui > 1/2 the ordering between the two alternatives flips: (1/2, 1/2, 1) (1, 1, 0). Such a chooser prefers the upper envelope lottery with full information. Choosers with ui > 1/2 are labeled as information seeking. Figure 2c shows indifference curves for such a chooser. Figure 2d plots the upper envelope lotteries implied by the bets in the Ellsberg experiment (see Section 1.3). Figure 2d also shows that information seeking preferences imply R B and BY RY , a pattern of choices consistent with the Ellsberg paradox. Of course there is no 7
Figure 2: Indifference curves for various information preferences (a) Information neutrality
(b) Information aversion
(1, 1, 0)
(1, 1, 0)
(1/2, 1/2, 1)
(1/2, 1/2, 1)
g
sin ea
cr In ce en
er
ef
pr (d) Information seeking implies R B and BY RY
(c) Information seeking
RY
(1, 1, 0)
BY
B
(1/2, 1/2, 1)
R
8
guarantee that choices amongst the upper envelope lotteries implied by the bets in the Ellsberg experiment will be consistent with choices amongst bets in a traditional Ellsberg setting.
2
Generalizing to more than two outcomes
Assume J > 2 monetary outcomes ordered from largest to smallest: z1 > z2 > · · · > zJ . An upper envelope lottery is the maximum probability for each outcome and the amount of information: A = (p1 , . . . , pJ , i). The amount of information depends on the number of outcomes (J) and the maximum probability for these outcomes:
i=
PJ
J−
j=1 pj
J −1
.
(2)
Each entry in an upper envelope lottery belongs to the unit interval: pj ∈ [0, 1], and
j = 1, . . . , J,
(3)
i ∈ [0, 1].
The entries in an upper envelope lottery sum to J:
i+
J X
pj = J.
(4)
j=1
The upper envelope expected utility (U EEU ) of A = (p1 , . . . , pJ , i) is
U EEU [A] = iui +
J X
pj uj ,
(5)
j=1
where ui is the utility for information and uj is the utility for the respective outcome. Normalizing utility such that u1 = 1 and uJ = 0, information preferences are categorized as information averse, neutral, or seeking whenever ui < 1/2, ui = 1/2, or ui > 1/2, respectively.
9
2.1
Interpersonal comparisons of information preferences
Consider two choosers, u and v, with utilities normalized such that u1 = v1 = 1 and uJ = vJ = 0. We say u is more information preferring than v if ui > vi .
2.2
Applications
The U EEU model provides a simple framework for decision making under uncertainty via probability intervals. In real-world settings probability intervals are commonly invoked when probabilities are estimated or when, by their nature, precise values cannot be known. Examples of this include medical treatment decisions and choice amongst financial assets. 2.2.1
Medical treatment decisions
Following a diagnosis it is common for patients to request information regarding treatment options. For safety, legal, and billing reasons, then, clinicians provide a range of probabilities for specific outcomes. By way of an example consider a diagnosis of stage IV melanoma. The five year survival rate for such a diagnosis, under standard treatment, is 15-20%.4 Assuming binary outcomes of “survival for at least five years” and “death within five years,” the probability interval stated above implies a patient facing the upper envelope (0.20, 0.85, 0.95). Now imagine a patient considering a novel treatment regime such as a clinical trial available only to late stage cancer patients.5 In such a case nothing is known regarding the five year survival rate. So the patient faces the upper envelope (1, 1, 0) for this treatment option. The U EEU model predicts that the choice between treatments will depend on information preferences. Information seeking individuals will obviously prefer the standard treatment. People with strongly information averse preferences, however, would choose the novel treatment. 2.2.2
Asset demand
Financial asset returns are typically described with a probability distribution f (z) on a support [c, d] ⊂ R. Historical returns are used to characterize f , although the aphorism “past performance 4
See the American Cancer Society website, https://www.cancer.org. For ethical reasons clinical trails involving late stage cancer patients usually do not include a placebo control group. Instead, the efficacy of novel treatments is compared to the standard treatment regime. 5
10
is not indicative of future results” suggests this characterization is tenuous. Generalizing to a set of possible probability distributions, f1 , . . . , fN , the upper envelope is f (z) = max{f1 (z), . . . , fN (z)}. Assume an initial wealth w and the presence of a safe asset with certain return r. The chooser selects the amount to invest in the uncertain asset, a, by solving
max
i(a)ui +
a∈[0,w]
Z
u az + r(w − a) f (z)dz,
(6)
where u(·) is a twice differentiable, increasing, concave utility function for money. Letting ir and ia denote the information for the safe and uncertain assets, respectively, i(a) = ( wa )ia + (1 −
a w )ir ,
the amount of information for a particular portfolio composition. Note that i(0) = ir = 1 and i(w) = ia , so i(·) must be non-increasing. Assuming an interior solution, a∗ will satisfy the first order condition 0
∗
i (a )ui +
Z
u0 a∗ z + r(w − a∗ ) (z − r)f (z)dz = 0.
(7)
Suppose there is some probability distribution g, with g = f . This implies i(w) = 1 so that i0 (·) = 0. In this case the left hand term in (7) drops out making it equivalent to a setting of risk wherein the slope and concavity of u(·) are the preference-based determinants of portfolio composition. Assuming there is no such g, however, provides a richer characterization. The slope and concavity of u(·) still matter but information preferences (ui ) also play a role. For example, two choosers with the same u(·) could have different portfolios owing to differences in ui – a chooser who is information averse would have a higher demand for a than a chooser who is information seeking. The model can therefore accommodate heterogeneity in portfolio composition even when individuals have identical utility functions for money.
3
Conclusion
This note introduced upper envelope expected utility as a simple model for decision making under uncertainty. Future work will provide an axiomatic foundation for the model and discuss additional applications such as demand for insurance, game theory, and other situations where choices are made in the face of uncertainty.
11
References Ahn, D. S. (2008): “Ambiguity Without a State Space,” Review of Economic Studies, 75(1), 3–28. Arrow, K. J. (1964): “The Role of Securities in the Optimal Allocation of Risk-bearing,” The Review of Economic Studies, 31(2), 91–96. Burghart, D. R., T. Epper, and E. Fehr (2015): “The ambiguity triangle: uncovering fundamental patterns of behavior under uncertainty,” University of Zurich, Department of Economics Working Paper, (196). Debreu, G. (1959): Theory of Value: An Axiomatic Analysis of Economic Equilibrium, vol. 17. Cowles Foundation Monographs. Dempster, A. P. (1967): “Upper and Lower Probabilities Induced by a Multivalued Mapping,” The Annals of Mathematical Statistics, 38(2), 325–339. Ellsberg, D. (1961): “Risk, Ambiguity, and the Savage Axioms,” Quarterly Journal of Economics, 75(4), 643–669. Ghirardato, P., F. Maccheroni, and M. Marinacci (2004): “Differentiating ambiguity and ambiguity attitude,” Journal of Economic Theory, 118(2), 133–173. Gilboa, I., and D. Schmeidler (1989): “Maxmin Expected Utility With Non-Unique Prior,” Journal of Mathematical Economics, 18(2), 141–153. Machina, M. (1982): “”Expected Utility” Analysis without the Independence Axiom,” Econometrica, 50(2), 277–323. Machina, M. J., and M. Siniscalchi (2014): “Ambiguity and Ambiguity Aversion,” in Economics of Risk and Uncertainty, ed. by M. J. Machina, and W. K. Viscusi, vol. 1, chap. 13, pp. 729–808. North-Holland, 1st edn. Marschak, J. (1950): “Rational Behavior, Uncertain Prospects, and Measurable Utility,” Econometrica, 18(2), 111–141. Olszewski, W. (2007): “Preferences Over Sets of Lotteries,” Review of Economic Studies, 74(2), 567–595. Savage, L. J. (1954): The Foundations of Statistics. John Wiley & Sons, New York. Schmeidler, D. (1989): “Subjective Probability and Expected Utility without Additivity,” Econometrica, 57(3), 571–587. Shafer, G. (1976): A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ. von Neumann, J., and O. Morgenstern (1947): Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ.
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PII: DOI: Reference:
S0165-1765(18)30101-0 https://doi.org/10.1016/j.econlet.2018.03.010 ECOLET 7971
To appear in:
Economics Letters
Received date : 11 February 2018 Revised date : 13 March 2018 Accepted date : 14 March 2018 Please cite this article as: Burghart D.R., Maximum probabilities, information, and choice under uncertainty. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
An Expected Utility-like model is proposed for choice under uncertainty. The model weights utilities by the maximum probability and an information term. Information depends on how much is known about probabilities. A probability triangle-like figure is introduced and used to explore the model. Applications to medical decision making and financial asset demand are explored.
*Manuscript Click here to view linked References
Maximum probabilities, information, and choice under uncertainty Daniel R. Burghart∗ March 12, 2018
Abstract This note proposes a simple, expected utility-like model for decision making under uncertainty. The model uses the maximum probability for each possible outcome and the amount of information conveyed by this upper envelope. A graphical tool is introduced and used to study the model when two outcomes are possible. The model is extended to an abstract number of outcomes in which interpersonal comparisons of preferences are considered along with applications to medical decision making and financial asset demand. JEL Classification: D81, G11. Keywords: Decision-Making, Uncertainty, Expected Utility, Portfolio Choice
∗ California State University Sacramento, Department of Economics, 6000 J Street, Sacramento, California 95819. [email protected]. The inspiration for this note came from a comment made by Mark Machina during my talk at RUD 2016 (Paris-Dauphine). This note also benefitted greatly from suggestions made by an anonymous referee.
1
For decisions involving unknown outcomes economists distinguish between risk and uncertainty. Risk is a setting where probabilities for monetary outcomes are assumed fully-known at the time of choice. Uncertainty, in contrast, relaxes this assumption. Models for decision making under risk, such as von Neumann-Morgenstern expected utility (vNMEU), can be a parsimonious way to analyze choices when outcomes are unknown (von Neumann and Morgenstern, 1947). A tradeoff for this parsimony is a lack of realism – outside of casino games and lottery tickets most decisions involve probabilities that are not fully known at the moment of choice. Such decisions are said to be made under uncertainty. One way in which models for decision making under uncertainty are distinguished from the vNMEU setting is that the events causing monetary payoffs are assumed to be important determinants of choice.1 While there are many settings in which this assumption is likely to hold, there are also situations in which these events are less relevant. For example, in the case of financial assets, the probabilities for monetary outcomes, whether known or unknown, are likely to be more relevant for determining choice. This note proposes a simple model that extends vNMEU to such a setting. The model adds a parameter that takes into account the amount of information available at the time of choice while also weighting utilities by the maximum probabilities. The objects of choice in this model thus represent the maximum probabilities for monetary outcomes and a term that captures the amount of information conveyed by this upper envelope.2 This information term can take on any value between zero and one. A value of zero implies that nothing is known about the probabilities – the maximum probability for each outcome could be unity. A value of one represents full information about the probabilities, a setting analogous to risk because it implies a unique distribution. These objects of choice are only slightly more complicated than money lotteries yet, they can accommodate unknown probabilities. The next section leverages a well-known experiment in Ellsberg (1961) to introduce and explain the model. This explication is aided by the introduction of a graphical tool. The penultimate section generalizes the setting, introduces interpersonal comparisons of preferences, and discusses some applications. The final section concludes and indicates avenues for future work. 1
Risk can also be conceptualized in such a setting (Arrow, 1964; Debreu, 1959). On the flip side, choice under uncertainty has also been considered using a “set of lotteries” approach in Olszewski (2007) and Ahn (2008). 2 The term “upper envelope” is adopted from the statistics literature used to analyze sets of probability distributions by their maximal and minimal probabilities (Dempster, 1967; Shafer, 1976).
2
1
The Ellsberg Experiment
In 1961 Daniel Ellsberg proposed several experiments designed to test subjective expected utility (Ellsberg, 1961; Savage, 1954). One of these experiments involves drawing a single ball from an urn containing 90 balls. Exactly 30 of the balls are red. Each of the remaining 60 balls could be black or yellow. So, the ratio of black to yellow balls is an unknown proportion. The decision making component of the single urn Ellsberg experiment is composed of two choice situations. The first involves choosing between a “bet on red” and a “bet on black.” The bet on red pays $100 if the ball drawn from the urn is red and $0 otherwise. The bet on black pays $100 if the ball drawn is black and $0 otherwise. The second choice situation involves choosing between a “bet on red or yellow” and a “bet on black or yellow.” The bet on red or yellow pays $100 if the ball drawn is red or yellow and $0 otherwise. The bet on black or yellow pays $100 if the ball drawn from the urn is black or yellow and $0 otherwise. A “very frequent pattern of response” has the bet on red preferred to the bet on black and the bet on black or yellow preferred to the bet on red or yellow (Ellsberg, 1961, pg. 654). This pattern of responses, commonly referred to as ambiguity aversion, violates the Sure-thing Principle of subjective expected utility. A number of models have been proposed that can accommodate ambiguity aversion. Two of the more popular types are models with multiple priors, such as Maxmin Expected Utility (Gilboa and Schmeidler, 1989) and α-Maxmin Expected Utility (Ghirardato, Maccheroni, and Marinacci, 2004), and models with non-additive probabilistic beliefs, such as Choquet/Rank-Dependent Expected Utility (Schmeidler, 1989). Section 5 in Machina and Siniscalchi (2014) provides a more thorough review of models that can accommodate ambiguity aversion.
1.1
The sets of lotteries implied by the Ellsberg experiment
For the two possible monetary outcomes, $100 and $0, two of the four bets in the Ellsberg experiment imply exact probability distributions. The bet on red implies an exact probability distribution of (1/3, 2/3) because there are exactly 30 red balls and 60 non-red balls in the urn. Similarly, the bet on black or yellow implies an exact probability distribution of (2/3, 1/3) because there are exactly 60 black or yellow balls in the urn. The other two bets in the Ellsberg experiment, however, imply a set of possible probability values for each outcome. For the bet on black the minimum and
3
maximum probabilities for the $100 outcome are 0 and 2/3 because the number of black balls is unknown and could take on any integer value 0 through 60. The minimum and maximum probabilities for the $0 outcome are 1/3 and 1. Analogously, the bet on red or yellow implies probability values that can range from 1/3 to 1 for the $100 outcome and 0 to 2/3 for the $0 outcome.
1.2
Upper envelope lotteries
Order the monetary outcomes in the Ellsberg experiment from largest to smallest: {$100, $0}. An upper envelope lottery lists the maximum probability for each of these monetary outcomes and the amount of information conveyed by these maximum probabilities.3 Denote these as (p100 , p0 , i). Similar to a traditional probability distribution, each entry in an upper envelope lottery is constrained to the unit interval: pj ∈ [0, 1] for each j, and i ∈ [0, 1]. In contrast to a traditional probability distribution, however, the entries in a two outcome upper envelope lottery must sum to two: i + p100 + p0 = 2. To see why i+p100 +p0 = 2, consider two cases: p100 +p0 = 1 and p100 +p0 = 2. The only way in which p100 +p0 = 1 is if the upper envelope lottery implies a unique probability distribution. In such a case there is full information about the probability distribution. So, i = 1 and i + p100 + p0 = 2. The second case, p100 + p0 = 2, represents the opposite extreme: only if p100 = 1 and p0 = 1 can their sum be two. That either probability could be the maximum allowable value for a probability means that nothing is known about the distribution. So, i = 0 and i + p100 + p0 = 2. The upper envelope lottery implied by the bet on red is R = (1/3, 2/3, 1) while for the bet on black it is B = (2/3, 1, 1/3). The upper envelope lottery R has full information (i = 1). The upper envelope lottery B, in contrast, has only one third of the probability information known at the time of decision. Hence, for B, i = 1/3. The upper envelope lottery implied by the bet on red or yellow is RY = (1, 2/3, 1/3) and BY = (2/3, 1/3, 1) for the bet on black or yellow. For RY there is only 1/3 of the probability information known at the time of decision. The upper envelope lottery BY , in contrast, has i = 1. 3
The working paper by Burghart, Epper, and Fehr (2015) explores lower envelope lotteries. These objects are defined as the minimum probability for monetary outcomes and the amount of probability mass missing from this lower envelope, a quantity referred to as ambiguity.
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1.3
A graphical tool
This subsection introduces a triangular figure that allows for graphical analysis of preferences for two-outcome upper envelope lotteries. This figure draws its inspiration from the probability triangle, a tool that permits graphical analysis of preferences for three-outcome lotteries in a setting of risk (Marschak, 1950; Machina, 1982). For three-outcome risky lotteries, each of the probabilities belongs to the unit interval and their sum must be one. So, they can be plotted in an isosceles right triangle whose right angle lies at the origin (see, for example, Machina (1982), Figure 5). Each of the three entries in a two-outcome upper envelope lottery must also be one but, their sum must be two. So, upper envelope lotteries can be plotted in an isosceles right triangle such as the one in Figure 1a. Define this as the upper envelope triangle. In a Cartesian coordinate system the vertices of the upper envelope triangle correspond to the points (0, 1), (1, 1), and (1, 0). Whereas the vertices of the probability triangle represent certainty of the three possible outcomes, the vertices of the upper envelope triangle represent (1, 0, 1), certainty of $100, (0, 1, 1), certainty of $0, and (1, 1, 0), a situation in which nothing is known about the probabilities for the two outcomes (see Figure 1a). Points along the hypotenuse of the upper envelope triangle have full information (i = 1). So, they are analogous to money lotteries because they imply a unique probability distribution. Points on the vertical leg of the triangle have a maximum probability for the $0 outcome as unity (p0 = 1) while points on the horizontal leg have a maximum probability for the $100 outcome as unity (p100 = 1). Movements parallel to the hypotenuse hold i constant while trading off p100 and p0 in a one-to-one ratio. Movements parallel to the vertical leg keep p0 constant while trading off (one-to-one) p100 and i. And movements parallel to the horizontal leg keep p100 constant while trading off (one-to-one) p0 and i. Figure 1b plots the upper envelope lotteries implied by the bets in the Ellsberg experiment. Because the upper envelope lottery R has i = 1 it must lie on the hypotenuse. In contrast, B has a maximum probability for $0 of one so it lies on the vertical leg of the triangle. RY lies on the horizontal leg because the maximum probability of getting the $100 outcome is one. BY has i = 1 so it lies on the hypotenuse.
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Figure 1: A triangular diagram can be used to plot two outcome upper envelope lotteries (a) The upper envelope triangle
Certainty of $100 (1, 0, 1)
(b) The upper envelope lotteries implied by the bets in the Ellsberg experiment
RY = (1, 2/3, 1/3)
Zero information
p100 = 1
(1, 1, 0)
BY = (2/3, 1/3, 1)
li
1 atio i = orm nf
p0 = 1
l Fu
B = (2/3, 1, 1/3)
n
R = (1/3, 2/3, 1)
)
,1 ,1 (0
Certainty of $0
1.4
A simple model
For the $100 and $0 outcomes possible in the Ellsberg experiment the upper envelope expected utility (U EEU ) of an upper envelope lottery A = (p100 , p0 , i) is
U EEU [A] = p100 u100 + p0 u0 + iui ,
(1)
where p100 and p0 are the maximum probabilities for the $100 and $0 outcomes, u100 and u0 are the utility numbers assigned to the $100 and $0 outcomes, i is the amount of information in A, and ui is a utility number assigned to information. The U EEU model is an expected utility-like expression of the entries in an upper envelope lottery. While it is tempting to think that U EEU will collapse to expected utility when i = 1, this is not the case because of the structure of upper envelope lotteries. Too see this consider R = (1/3, 2/3, 1), an upper envelope lottery with full information. Normalizing utilities so that u100 = 1 and u0 = 0, U EEU [R] = 1/3 + ui . The ui term distinguishes the U EEU [R] from an expected utility of the money lottery with probabilities identical to first two entries in R. Importantly, however, choices amongst upper envelope lotteries with i = 1 would be consistent with choices generated by an expected utility function that has the
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same utility numbers for the monetary outcomes. Given the normalizations for u100 and u0 , ui must belong to the unit interval: ui ∈ [0, 1]. Values of ui lower than zero imply a chooser willing to trade information for increases in the maximum probability of the $0 outcome. On the flip side, values of ui larger than unity imply a chooser willing to trade maximum probability for the $100 outcome for additional information. Neither of these potential tradeoffs seem justifiable for the same reason that, in a setting of risk, a lottery that is first order stochastically dominated by another lottery would be preferred – it is not justifiable to give up probability for an outcome to get an equal amount of probability for an outcome lower on the preference scale.
1.5
Types of information preferences
From a graphical perspective the U EEU model implies indifference curves that are parallel straight lines in the upper envelope triangle. The slope of these lines is dictated solely by ui . Maintaining the normalizations u100 = 1 and u0 = 0, ui = 1/2 implies indifference curves that have a slope of 45 degrees in the triangle. To see this consider two upper envelope lotteries: (1, 1, 0), nothing is known about the probabilities, and (1/2, 1/2, 1), an upper envelope lottery analogous to a 50/50 lottery. Evaluating each of these upper envelope lotteries gives U EEU [·] = 1. Figure 2a plots these two alternatives and shows a map of indifference curves for a chooser with ui = 1/2. Such choosers are labeled information neutral. Consider again the two alternatives (1, 1, 0) and (1/2, 1/2, 1). When ui < 1/2 the ordering between these two alternatives is (1, 1, 0) (1/2, 1/2, 1) – the chooser prefers the upper envelope lottery with zero information. It seems natural, therefore, to label choosers with ui < 1/2 as information averse. The solid lines in Figure 2b illustrate a map of indifference curves for an information averse U EEU chooser. The dashed lines are at 45 degrees to serve as a reference. When ui > 1/2 the ordering between the two alternatives flips: (1/2, 1/2, 1) (1, 1, 0). Such a chooser prefers the upper envelope lottery with full information. Choosers with ui > 1/2 are labeled as information seeking. Figure 2c shows indifference curves for such a chooser. Figure 2d plots the upper envelope lotteries implied by the bets in the Ellsberg experiment (see Section 1.3). Figure 2d also shows that information seeking preferences imply R B and BY RY , a pattern of choices consistent with the Ellsberg paradox. Of course there is no 7
Figure 2: Indifference curves for various information preferences (a) Information neutrality
(b) Information aversion
(1, 1, 0)
(1, 1, 0)
(1/2, 1/2, 1)
(1/2, 1/2, 1)
g
sin ea
cr In ce en
er
ef
pr (d) Information seeking implies R B and BY RY
(c) Information seeking
RY
(1, 1, 0)
BY
B
(1/2, 1/2, 1)
R
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guarantee that choices amongst the upper envelope lotteries implied by the bets in the Ellsberg experiment will be consistent with choices amongst bets in a traditional Ellsberg setting.
2
Generalizing to more than two outcomes
Assume J > 2 monetary outcomes ordered from largest to smallest: z1 > z2 > · · · > zJ . An upper envelope lottery is the maximum probability for each outcome and the amount of information: A = (p1 , . . . , pJ , i). The amount of information depends on the number of outcomes (J) and the maximum probability for these outcomes:
i=
PJ
J−
j=1 pj
J −1
.
(2)
Each entry in an upper envelope lottery belongs to the unit interval: pj ∈ [0, 1], and
j = 1, . . . , J,
(3)
i ∈ [0, 1].
The entries in an upper envelope lottery sum to J:
i+
J X
pj = J.
(4)
j=1
The upper envelope expected utility (U EEU ) of A = (p1 , . . . , pJ , i) is
U EEU [A] = iui +
J X
pj uj ,
(5)
j=1
where ui is the utility for information and uj is the utility for the respective outcome. Normalizing utility such that u1 = 1 and uJ = 0, information preferences are categorized as information averse, neutral, or seeking whenever ui < 1/2, ui = 1/2, or ui > 1/2, respectively.
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2.1
Interpersonal comparisons of information preferences
Consider two choosers, u and v, with utilities normalized such that u1 = v1 = 1 and uJ = vJ = 0. We say u is more information preferring than v if ui > vi .
2.2
Applications
The U EEU model provides a simple framework for decision making under uncertainty via probability intervals. In real-world settings probability intervals are commonly invoked when probabilities are estimated or when, by their nature, precise values cannot be known. Examples of this include medical treatment decisions and choice amongst financial assets. 2.2.1
Medical treatment decisions
Following a diagnosis it is common for patients to request information regarding treatment options. For safety, legal, and billing reasons, then, clinicians provide a range of probabilities for specific outcomes. By way of an example consider a diagnosis of stage IV melanoma. The five year survival rate for such a diagnosis, under standard treatment, is 15-20%.4 Assuming binary outcomes of “survival for at least five years” and “death within five years,” the probability interval stated above implies a patient facing the upper envelope (0.20, 0.85, 0.95). Now imagine a patient considering a novel treatment regime such as a clinical trial available only to late stage cancer patients.5 In such a case nothing is known regarding the five year survival rate. So the patient faces the upper envelope (1, 1, 0) for this treatment option. The U EEU model predicts that the choice between treatments will depend on information preferences. Information seeking individuals will obviously prefer the standard treatment. People with strongly information averse preferences, however, would choose the novel treatment. 2.2.2
Asset demand
Financial asset returns are typically described with a probability distribution f (z) on a support [c, d] ⊂ R. Historical returns are used to characterize f , although the aphorism “past performance 4
See the American Cancer Society website, https://www.cancer.org. For ethical reasons clinical trails involving late stage cancer patients usually do not include a placebo control group. Instead, the efficacy of novel treatments is compared to the standard treatment regime. 5
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is not indicative of future results” suggests this characterization is tenuous. Generalizing to a set of possible probability distributions, f1 , . . . , fN , the upper envelope is f (z) = max{f1 (z), . . . , fN (z)}. Assume an initial wealth w and the presence of a safe asset with certain return r. The chooser selects the amount to invest in the uncertain asset, a, by solving
max
i(a)ui +
a∈[0,w]
Z
u az + r(w − a) f (z)dz,
(6)
where u(·) is a twice differentiable, increasing, concave utility function for money. Letting ir and ia denote the information for the safe and uncertain assets, respectively, i(a) = ( wa )ia + (1 −
a w )ir ,
the amount of information for a particular portfolio composition. Note that i(0) = ir = 1 and i(w) = ia , so i(·) must be non-increasing. Assuming an interior solution, a∗ will satisfy the first order condition 0
∗
i (a )ui +
Z
u0 a∗ z + r(w − a∗ ) (z − r)f (z)dz = 0.
(7)
Suppose there is some probability distribution g, with g = f . This implies i(w) = 1 so that i0 (·) = 0. In this case the left hand term in (7) drops out making it equivalent to a setting of risk wherein the slope and concavity of u(·) are the preference-based determinants of portfolio composition. Assuming there is no such g, however, provides a richer characterization. The slope and concavity of u(·) still matter but information preferences (ui ) also play a role. For example, two choosers with the same u(·) could have different portfolios owing to differences in ui – a chooser who is information averse would have a higher demand for a than a chooser who is information seeking. The model can therefore accommodate heterogeneity in portfolio composition even when individuals have identical utility functions for money.
3
Conclusion
This note introduced upper envelope expected utility as a simple model for decision making under uncertainty. Future work will provide an axiomatic foundation for the model and discuss additional applications such as demand for insurance, game theory, and other situations where choices are made in the face of uncertainty.
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References Ahn, D. S. (2008): “Ambiguity Without a State Space,” Review of Economic Studies, 75(1), 3–28. Arrow, K. J. (1964): “The Role of Securities in the Optimal Allocation of Risk-bearing,” The Review of Economic Studies, 31(2), 91–96. Burghart, D. R., T. Epper, and E. Fehr (2015): “The ambiguity triangle: uncovering fundamental patterns of behavior under uncertainty,” University of Zurich, Department of Economics Working Paper, (196). Debreu, G. (1959): Theory of Value: An Axiomatic Analysis of Economic Equilibrium, vol. 17. Cowles Foundation Monographs. Dempster, A. P. (1967): “Upper and Lower Probabilities Induced by a Multivalued Mapping,” The Annals of Mathematical Statistics, 38(2), 325–339. Ellsberg, D. (1961): “Risk, Ambiguity, and the Savage Axioms,” Quarterly Journal of Economics, 75(4), 643–669. Ghirardato, P., F. Maccheroni, and M. Marinacci (2004): “Differentiating ambiguity and ambiguity attitude,” Journal of Economic Theory, 118(2), 133–173. Gilboa, I., and D. Schmeidler (1989): “Maxmin Expected Utility With Non-Unique Prior,” Journal of Mathematical Economics, 18(2), 141–153. Machina, M. (1982): “”Expected Utility” Analysis without the Independence Axiom,” Econometrica, 50(2), 277–323. Machina, M. J., and M. Siniscalchi (2014): “Ambiguity and Ambiguity Aversion,” in Economics of Risk and Uncertainty, ed. by M. J. Machina, and W. K. Viscusi, vol. 1, chap. 13, pp. 729–808. North-Holland, 1st edn. Marschak, J. (1950): “Rational Behavior, Uncertain Prospects, and Measurable Utility,” Econometrica, 18(2), 111–141. Olszewski, W. (2007): “Preferences Over Sets of Lotteries,” Review of Economic Studies, 74(2), 567–595. Savage, L. J. (1954): The Foundations of Statistics. John Wiley & Sons, New York. Schmeidler, D. (1989): “Subjective Probability and Expected Utility without Additivity,” Econometrica, 57(3), 571–587. Shafer, G. (1976): A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ. von Neumann, J., and O. Morgenstern (1947): Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ.
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