74 non-expected utility theory Bibliography Schafer, W. and Sonnenschein, H. 1975. Equilibrium in Baye, M., Tian, G. and Zhou, J. 1993. Characterizations of Abstract economies without ordered preferences. Journal the existence of equilibria in games with discontinuous of Mathematical Economics 2, 345–8. and non-quasiconcave payoffs. Review of Economic Simon, L. 1987. Games with discontinuous payoffs. Review Studies 60, 935–48. of Economic Studies 54, 569–97. Bertrand, J. 1883. The´orie mathe´matique de la richesse Simon, L. and Zame, W. 1990. Discontinuous games and sociale. Journal des Savants 67, 499–508. endogenous sharing rules. Econometrica 58, 861–72. Billingsley, P. 1968. Convergence of Probability Measures.New Sion, M. 1958. On general minimax theorems. Pacific York: John Wiley and Sons. Journal of Mathematics 8, 171–6. Cournot, A. 1838. Researches into the Mathematical Vives, X. 1990. Nash equilibrium with strategic Principles of the Theory of Wealth, ed. N. Bacon. 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A further generalization of the Kakutani standard theory of individual choice under objective and fixed point theorem. Proceedings of the American subjective uncertainty, experimental work by both psy- Mathematical Society 3, 170–4. chologists and economists has uncovered systematic Hotelling, H. 1929. The stability of competition. Economic departures from the expected utility hypothesis, which Journal 39, 41–57. has led to the development of alternative models of Jackson, M. and Swinkels, J. 2005. Existence of equilibrium preferences over uncertain prospects. in single and double private value auctions. Econometrica 73, 93–139. The expected utility model Jackson, M., Simon, L., Swinkels, J. and Zame, W. 2002. In one of the simplest settings of choice under economic Communication and equilibrium in discontinuous uncertainty, the objects of choice consist of finite- games of incomplete information. Econometrica 70, outcome objective lotteries of the form P ¼ðx1; p1; ...; 1711–40. x ; p Þ, yielding a monetary payoff of x with probability Milgrom, P. and Roberts, J. 1990. Rationalizability, n n i pi, where p1 þ ...þ pn ¼ 1. In such a case, the expected learning, and equilibrium in games with utility model of risk preferences assumes (or posits axi- strategic complementarities. Econometrica 58, oms sufficient to imply) that the individual ranks these 1255–77. prospects on the basis of an expected utility preference Milgrom, P. and Weber, R. 1982. A theory of auctions function of the form and competitive bidding. Econometrica 50, 1089–122. V EU ðPÞV EU ðx1; p1;...;xn; pnÞ Milgrom, P. and Weber, R. 1985. Distributional strategies for Uðx Þp þ ...þ Uðx Þp games with incomplete information. Mathematics of 1 1 n n Operations Research 10, 619–32. in the standard economic sense that the individual Nash, J. 1950. Equilibrium points in n-person games. n n n n n prefers lottery P ¼ðx1 ; p1 ;...;xnn ; pnn Þ over lottery P ¼ Proceedings of the National Academy of Sciences 36, n ðx1; p1;...;xn; pnÞ if and only if V EU ðP Þ4V EU ðPÞ, and 48–9. n is indifferent between them if and only if VEU ðP Þ¼ Nash, J. 1951. Non-cooperative games. Annals of V EU ðPÞ. U( Á ) is termed the individual’s von Neumann– Mathematics 54, 286–95. Morgenstern utility function (von Neumann and Osborne, M. and Rubinstein, A. 1994. A Course in Game Morgenstern, 1944; 1947; 1953), and its various math- Theory. Cambridge, MA: MIT Press. ematical properties serve to characterize various features Reny, P. 1999. On the existence of pure and mixed strategy of the individual’s attitudes toward risk, for example: Nash equilibria in discontinuous games. Econometrica 67, 1029–56. VEU( Á ) exhibits first-order stochastic dominance pref- Robson, A. 1994. An ‘informationally robust’ equilibrium in erence (a preference for shifting probability from lower two-person nonzero-sum games. Games and Economic to higher outcome values) if and only if U(x)isan Behavior 2, 233–45. increasing function of x. non-expected utility theory 75 VEU( Á ) exhibits risk aversion (an aversion to all mean- 1 preserving increases in risk) if and only if U(x)isa concave function of x. n V EU ðÞ is at least as risk averse as VEU( Á ) (in several equivalent senses) if and only if its utility function UÃ( Á ) is a concave transformation of U( Á ) (that is, if Increasing preference and only if U nðxÞrðUðxÞÞ for some increasing concave function r( Á )). As shown by Bernoulli (1738), Arrow (1965), Pratt (1964), Friedman and Savage (1948), Markowitz (1952) and others, this model admits of a tremendous p3 flexibility in representing attitudes towards risk, and can be applied to many types of economic decisions and markets. But in spite of its flexibility, the expected utility model has testable implications which hold regardless of the shape of the utility function U( Á ), since they follow from the linearity in the probabilities property of the preference function V ( Á ). These implications can be best EU p expressed by the concept of an a : ð1 À aÞ probability 011 mixture of two lotteries P ¼ðx1; p1;...;xn; pnÞ and n n n n n Figure 1 Expected utility indifference curves in the probability P ¼ðx ; p ;...;x n ; p n Þ, which is defined as the single- 1 1 n n n triangle stage lottery a Á P þð1 À aÞP ¼ðx1; a Á p1;...; xn; a Á n n n n pn; x1 ; ð1À aÞp1 ;...;xnn ; ð1 À aÞpnn Þ. The mixture a Á P þð1 À aÞPn can be thought of as a coin flip yielding à lotteries P and P with probabilities a : ð1 À aÞ, where fixed p1) represent shifting probability from outcome x¯2 the uncertainty in the coin and in the subsequent lottery up to x¯3, and leftward movements represent shifting is resolved simultaneously. Linearity in the probabilities probability from x¯1 up to x¯2, such movements constitute is equivalent to the following property, which serves as first-order stochastically dominating shifts and will thus the key foundational axiom of the expected utility model always be preferred. Expected utility indifference (Marschak, 1950): curves (loci of constant expected utility) are given by the formula Independence Axiom If lottery Pà is preferred (indiffer- n ent) to lottery P, then the probability mixture a Á P þ Uðx¯1Þp1 þ Uðx¯2Þ½1 À p1 À p3 ð1 À aÞPÃà is preferred (indifferent) to a Á P þð1 À aÞ þ Uðx¯3Þp ¼ constant PÃà for every lottery PÃà and every mixture probability 3 a 2ð0; 1. and are thus seen to be parallel straight lines of slope ½Uðx¯2ÞUðx¯1Þ=½Uðx¯3ÞUðx¯2Þ, as indicated by the This axiom can be interpreted as saying ‘given an a : solid lines in the figure. The dotted lines in Figure 1 are à ð1 À aÞ coin, the individual’s preferences for receiving P loci of constant expected value, given by the formula versus P in the event of a head should not depend upon x¯ Á p þx¯ ½1Àp À p þx¯ Á p ¼ constant, with slope Ãà 1 1 2 1 3 3 3 the prize P that would be received in the event of a tail, ½x¯2 À x¯1=½x¯3 À x¯2. Since north-east movements along nor upon the probability a of landing heads (so long as the constant expected value lines shift probability from x¯2 this probability is positive)’. The strong normative appeal down to x¯1 and up to x¯3 in a manner that preserves the of this axiom has contributed to the widespread adoption mean of the distribution, they represent simple increases of the expected utility model. in risk (Rothschild and Stiglitz, 1970; 1971). When U( Á ) The property of linearity in the probabilities, as well as is concave (that is, risk averse), its indifference curves the senses in which it has been found to be empirically will have a steeper slope than these constant expected violated, can be illustrated in the special case of prefer- value lines, and such increases in risk move the individual ences over all lotteries P ¼ðx¯1; p1; x¯2; p2; x¯3; p3Þ over a from more to less preferred indifference curves, as illus- fixed set of outcome values x¯1ox¯2ox¯3. Since we must trated in the figure. It is straightforward to show that the have p2 ¼ 1 À p1 À p3, each such lottery can be com- indifference curves of any expected utility maximizer pletely summarized by its pair of probabilities (p1, p3), as with a more risk-averse (that is, more concave) utility plotted in the ‘probability triangle’ of Figure 1. Since function UÃ( Á ) will be steeper than those generated upward movements in the diagram (increasing p3 for by U( Á ). 76 non-expected utility theory Systematic violations of the expected utility 1 hypothesis In spite of its normative appeal, researchers have uncov- ered several types of widespread systematic violations of the expected utility model and its underlying assumptions.