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. New complementarities. Journal of Mathematical Economics York: Macmillan, 1897. 19, 305–21. Dasgupta, P. and Maskin, E. 1986. The existence of von Neumann, J. 1928. Zur Theorie der Gesellshaftspiele. equilibrium in discontinuous economic games, I: theory. Mathematische Annalen 100, 295–320. Trans. S. Bargmann Review of Economic Studies 53, 1–26. [On the theory of games of strategy] in Contributions to Debreu, G. 1952. A social equilibrium existence theorem. the Theory of Games, vol. 4, ed. R. Luce and A. Tucker. Proceedings of the National Academy of Sciences 38, Princeton: Princeton University Press, 1959. 386–93. Fudenberg, D., Gilbert, R., Stiglitz, J. and Tirole, J. 1983. Preemption, leapfrogging, and competition in patent races. European Economic Review 22, non-expected utility theory 3–31. Although the expected utility model has long been the Glicksberg, I. 1952. 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 ðPÞ V EU ðx1; p1;...;xn; pnÞ Milgrom, P. and Weber, R. 1985. Distributional strategies for Uðx Þ p þ ...þ Uðx Þ p 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( ) 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( ) 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( ) 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 ð Þ is at least as risk averse as VEU( ) (in several equivalent senses) if and only if its utility function UÃ( ) is a concave transformation of U( ) (that is, if Increasing preference and only if U nðxÞ rðUðxÞÞ for some increasing concave function r( )).
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( ), since they follow from the linearity in the probabilities property of the preference function V ( ). These implications can be best EU p expressed by the concept of an a : ð1 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 P þð1 aÞ P ¼ðx1; a p1;...; xn; a n n n n pn; x1 ; ð1 aÞ p1 ;...;xnn ; ð1 aÞ pnn Þ. The mixture a P þð1 aÞ Pn can be thought of as a coin flip yielding à lotteries P and P with probabilities a : ð1 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 P þ Uðx¯1Þ p1 þ Uðx¯2Þ ½1 p1 p3 ð1 aÞ P is preferred (indifferent) to a P þð1 aÞ þ Uðx¯3Þ p ¼ constant P for every lottery PÃà and every mixture probability 3 a 2ð0; 1 . and are thus seen to be parallel straight lines of slope ½Uðx¯2Þ Uðx¯1Þ =½Uðx¯3Þ Uðx¯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 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¯ p þx¯ ½1 p p þx¯ 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 x¯1 =½x¯3 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( ) 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 p1 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( ) will be steeper than those generated upward movements in the diagram (increasing p3 for by U( ). 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. Increasing preference These can be categorized into (a) violations of the Inde- pendence Axiom (such as the common consequence and common ratio effects), (b) violations of the hypothesis of probabilistic beliefs (such as the Ellsberg Paradox) and (c) violations of the model’s underlying assumptions of descriptive and procedural invariance (such as reference- point and response-mode effects). p3
Violations of the Independence Axiom The best-known violation of the Independence Axiom is the so-called Allais Paradox, in which individuals are a2 a3 asked to rank the lotteries in each of the following pairs, where $1 M ¼ $1; 000; 000: 0 a a : f1:00 chance of $1M versus 1 p1 a4 1 1 8 > : $ Figure 2 Expected utility indifference curves and the Allais <> 10 chance of 5M Paradox choices a2 : > :89 chance of $1M :> :01 chance of $0 ( 1 :10 chance of $5M a3 : versus :90 chance of $0 ( :11 chance of $1M Increasing preference a4 : :89 chance of $0
Researchers such as Allais (1953), Morrison (1967), Raiffa (1968), Slovic and Tversky (1974) and others have found that the modal if not majority preference of subjects is for a1 over a2 in the first pair of choices and for a3 over a4 in p the second pair. However, such preferences violate 3 expected utility, since the first ranking implies the inequality Uð$1 MÞ4:10 Uð$5 MÞþ:89 Uð$1 MÞþ :01 Uð$0Þ whereas the second implies the inconsistent inequality :10 Uð$5 MÞþ:90 Uð$0Þ4:11 Uð$1 MÞ a2 a3 þ:89 Uð$0Þ. By setting x¯1 ¼ $0, x¯2 ¼ $1 M and x¯3 ¼ $5 M, the lotteries a1, a2, a3 and a4 are seen to form a parallelogram when plotted in the probability 0 triangle (Figure 2), which explains why the parallel a1 p1 a4 1 straight line indifference curves of an expected utility maximizer must either prefer a1 and a4 (as illustrated for Figure 3 Allais Paradox choices and indifference curves which the relatively steep indifference curves of the figure) or ‘fan out’ else prefer a2 and a3 (for relatively flat indifference curves). Figure 3 illustrates non-expected utility indiffer- ence curves which fan out, and are seen to exhibit psychologists, economists and others have uncovered a the typical Allais Paradox rankings of a1 over a2 and a3 similar pattern of violations over a range of probability over a4. and payoff values, and the Allais Paradox is now seen to Although the Allais Paradox was originally dismissed be a special case of a widely observed phenomenon as an isolated example, subsequent experimental work by known as the common consequence effect. This effect non-expected utility theory 77 involves pairs of prospects (probability mixtures) of the the form: form: ( ( p chance of $X a chance of x c1 : versus 1 p b1 : versus chance of $0 1 a chance of P ( ( q chance of $Y a chance of P c2 : b2 : 1 q chance of $0 1 a chance of P ( ( a p chance of $X a chance of x c3 : versus b3 : versus 1 a p chance of $0 1 a chance of Pn ( ( a q chance of $Y a chance of P c4 : b4 : 1 a q chance of $0 1 a chance of Pn where p4q,0oXoY and a 2ð0; 1Þ. (The term where the lottery P involves outcomes both greater and Ãà ‘common ratio effect’ comes from the common value less than the amount x, and P first order stochastically of prob($X)/prob($Y) in the upper and lower pairs.) dominates Pà (in Allais’s example, x ¼ $1M, P ¼ n Setting fx¯1; x¯2; x¯3g¼f$0; $X; $Yg and plotting these ð$5M; 10=11; $0; 1=11Þ, P ¼ð$0; 1Þ, P ¼ð$1M; 1Þ prospects in the probability triangle as in Figure 4, the and a ¼ :11). Although the Independence Axiom clearly line segments c1c2 and c3c4 are seen to be parallel, so that implies choices of either b1 and b3 (if x is preferred to P) the expected utility model again predicts choices of c1 or else b2 and b4 (if P is preferred to x), researchers have and c3 (if the indifference curves are relatively steep) or found a tendency for subjects to choose b1 in the first else c and c (if they are flat). However, experimental pair and b in the second. When the distributions P, Pà 2 4 Ãà 4 studies by MacCrimmon (1968), Tversky (1975), and P are each over a common outcome set MacCrimmon and Larsson (1979), Kahneman and fx¯1; x¯2; x¯3g with x¯2 ¼ x, the prospects {b1, b2, b3, b4} Tversky (1979), Hagen (1979), Chew and Waller (1986) again form a parallelogram in the (p1, p3) triangle, and and others have found a systematic tendency for choices a choice of b1 and b4 again implies indifference curves to depart from these predictions in the direction of which fan out. preferring c1 over c2 and c4 over c3, which again suggests The intuition behind this phenomenon can be that indifference curves fan out, as in the figure. For described in terms of the above ‘coin-flip’ scenario. According to the Independence Axiom, one’s preferences over what would occur in the event of a head ought not 1 depend upon what would occur in the event of a tail. However, they may well depend upon what would otherwise happen (as Bell, 1985, p. 1, notes, ‘winning the top prize of $10,000 in a lottery may leave one much Increasing preference happier than receiving $10,000 as the lowest prize in a lottery’). The common consequence effect states that the better off individuals would be in the event of a tail (in the sense of stochastic dominance), the more risk averse their preferences over what they would receive in the Ãà event of a head. That is, if the distribution P in the c2 p3 pair {b1, b2} involves very high outcomes, one may prefer not to bear further risk in the unlucky event that one doesn’t receive it, and hence opt for the sure outcome x over the risky distribution P (that is, choose b over b ). à 1 2 But, if P in {b3, b4} involves very low outcomes, one might be more willing to bear risk in the lucky c4 event that one doesn’t receive it, and prefer going for the lottery P rather than the sure outcome x (choose b4 over b3). 0 c p c 1 A second type of systematic violation of linearity in 1 1 3 the probabilities, also noted by Allais and subsequently Figure 4 Common ratio effect and fanning out indifference termed the common ratio effect, involves prospects of curves 78 non-expected utility theory example, Kahneman and Tversky (1979) found that, persistency rate of such choices (1974, p. 234, Tables 4, 6). while 86 per cent of their subjects preferred a .90 chance In experiments where subjects who responded to Allais- of winning $3,000 to a .45 chance of $6,000, 73 per cent type problems were then presented with written argu- preferred a .001 chance of $6,000 to a .002 chance of ments both for and against the expected utility position, $3,000. Kahneman and Tversky (1979) observed that, neither MacCrimmon (1968), Moskowitz (1974) nor when the positive outcomes $3000 and $6000 in the Slovic and Tversky (1974) found predominant net swings above gambles are replaced by losses of these magnitudes, toward the expected utility choices. 0 0 0 0 to obtain the lotteries c1; c2; c3 and c4, preferences Further descriptions of these and other violations of 0 0 0 0 typically ‘reflect,’ to prefer c2 over c1 and c3 over c4. the Independence Axiom can be found in Camerer Setting x¯1 ¼ $6000, x¯2 ¼ $3000 and x¯3 ¼ $0 (to (1989), Machina (1983; 1987), Starmer (2000), Sugden preserve the ordering x¯1ox¯2ox¯3) and plotting as in (1986) and Weber and Camerer (1987). Figure 5, such preferences again suggest that indifference curves in the probability triangle fan out. Battalio, Kagel Non-existence of probabilistic beliefs and MacDonald (1985) found that laboratory rats Although the expected utility model was first formulated choosing among gambles involving substantial variations in terms of preferences over objective lotteries P ¼ in their daily food intake also exhibited this pattern of ðx1; p1;...;xn; pnÞ with pre-specified probabilities, it choices. has also been applied to preferences over subjective acts One criticism of this evidence has been that individuals f ð Þ ¼ ½x1 on E1;...;xn on En , where the uncertainty whose initial choices violated the Independence Axiom in is represented by a set {E1,y, En} of mutually exclusive the above manners would typically ‘correct’ themselves and exhaustive events (such as the alternative outcomes once the nature of their violations was revealed by an of a horse race) (Savage, 1954). As long as an individual application of the above type of coin-flip argument. Thus, possesses well-defined subjective probabilities m(E1),y, while even Leonard Savage chose a1 and a3 when first m(En) over these events, their subjective expected utility presented with such choices by Allais, he concluded upon preference function takes the form reflection that these preferences were in error (Savage, 1954, pp. 101–3). Although Moskowitz found that W SEU ðf ð ÞÞ WSEU ðx1 on E1;...;xn on EnÞ allowing subjects to discuss opposing written arguments Uðx1Þ mðE1Þþ...þ UðxnÞ mðEnÞ: led to a decrease in the proportion of violations, 73 per cent of the initial fanning-out type choices remained However, researchers have found that individuals may unchanged after the discussions (1974, pp. 232–7, not possess such well-defined subjective probabilities, in Table 6). When written arguments were presented but even the simplest of cases. The best-known example of no discussion was allowed, there was a 93 per cent this is the Ellsberg Paradox (Ellsberg, 1961), in which the individual must draw a ball from an urn that contains 30 red balls, and 60 black or yellow balls in an unknown 1 proportion, and is offered the following bets based on the c′ 4 colour of the drawn ball: ′ c3 30 balls 60 balls Increasing preference red black yellow
′ ƒ1( ) $100 $0 $0 c2
ƒ2( ) $0 $100 $0 p 3 ƒ3( ) $100 $0 $100
ƒ4( ) $0 $100 $100 c′ 1 Most individuals exhibit a preference for f1( )over f2( ) and f4( )overf3( ). When asked, they explain that the chance of winning under f2( ) could be anywhere from 0 to 2/3 whereas under f1( ) it is known to be 0 exactly 1/3, and they prefer the bet that offers the known p1 1 probability. Similarly, the chance of winning under f3( ) Figure 5 Common ratio effect for losses and fanning out could be anywhere from 1/3 to 1 whereas under f4( )itis indifference curves known to be exactly 2/3, so the latter is preferred. non-expected utility theory 79
However, such preferences are inconsistent with any changes from the reference point of current wealth. In his assignment of subjective probabilities m(red), m(black), discussion of this phenomenon, Markowitz (1952, p. m(yellow) to the three events. If the individual were to be 155) suggested that certain circumstances may cause the choosing on the basis of such probabilistic beliefs, the individual’s reference point to temporarily deviate from choice of f1( )overf2( ) would ‘reveal’ that m(red)4 current wealth. If these circumstances include the man- m(black), but the choice of f4( )overf3( ) would reveal ner in which a problem is verbally described, then differ- that m(red)om(black). A preference for gambles based on ing risk attitudes towards gains and losses from the probabilistic partitions such as {red, black [ yellow} over reference point can lead to different choices, depending gambles based on subjective partitions such as {black, red upon the exact description of an otherwise identical [ yellow} is termed ambiguity aversion. problem. A simple example of this, from Kahneman and In an even more basic example, Ellsberg presented Tversky (1979, p. 273), involves the following two subjects with a pair of urns, the first containing 50 red choices: balls and 50 black balls, and the second with 100 red and In addition to whatever you own, you have been given black balls in an unknown proportion. When asked, a 1,000 (Israeli pounds). You are now asked to choose majority of subjects strictly preferred to stake a prize on between a 1/2:1/2 chance of a gain of 1,000 or 0 or a drawing red from the first urn over drawing red from the sure chance of a gain of 500. second urn, and strictly preferred staking the prize on drawing black from the first urn over drawing black from and the second. It is clear that there can exist no subjective In addition to whatever you own, you have been given probabilities p:(1 p) of red:black in the second urn, 2,000. You are now asked to choose between a 1/2:1/2 including 1/2:1/2, which can simultaneously generate chance of a loss of 1,000 or 0 or a sure loss of 500. both of these strict preferences. Similar behaviour in this and related problems has been observed by Raiffa (1961), These two problems involve identical distributions Becker and Brownson (1964), MacCrimmon (1965), over final wealth. But, when put to two different groups Slovic and Tversky (1974) and MacCrimmon and of subjects, 84 per cent chose the sure gain in the first Larsson (1979). problem but 69 per cent chose the 1/2:1/2 gamble in the second. Violations of descriptive and procedural invariance In another class of examples, not based on reference Researchers have also uncovered several systematic point effects, Moskowitz (1974), Keller (1985) and Carlin violations of the standard economic assumptions of (1990) found that the proportion of subjects choosing in stability of preferences and invariance with respect to conformity with the Independence Axiom in examples problem description in choices over risky prospects. In like the Allais Paradox was significantly affected by particular, psychologists have found that alternative whether the problems were described in the standard means of representing or framing probabilistically matrix form, decision tree form, roulette wheels, or as equivalent choice problems lead to systematic differ- minimally structured written statements. Interestingly, ences in choice. Early examples of this were reported by the form judged the ‘clearest representation’ by the Slovic (1969), who found that offering a gain or loss majority of Moskowitz’s subjects (the tree form) led to contingent on the joint occurrence of four independent the lowest degree of consistency with the Independence events with probability p elicited different responses Axiom, the highest proportion of Allais-type (fanning than offering it on the occurrence of a single event with out) choices, and the highest persistency rate of these probability p4 (all probabilities were stated explicitly). In choices Moskowitz (1974, pp. 234, 237–8). comparison with the single-event case, making a gain In other studies, Schoemaker and Kunreuther (1979), contingent on the joint occurrence of events was found Hershey and Schoemaker (1980), Kahneman and Tversky to make it more attractive, and making a loss contin- (1982; 1984), and Slovic, Fischhoff and Lichtenstein gent on the joint occurrence of events made it more (1977) found that subjects’ choices in otherwise identical unattractive. problems depended upon whether they were phrased as One class of framing effects exploits the phenome- decisions whether or not to gamble as opposed to non of a reference point. According to economic whether or not to insure, whether statistical information theory, the variable which enters an individual’s von for different therapies was presented in terms of cumu- Neumann–Morgenstern utility function should be total lative survival probabilities or cumulative mortality (that is, final) wealth, and gambles phrased in terms of probabilities, and so on (see the references in Tversky gains and losses should be combined with current wealth and Kahneman, 1981). and re-expressed as distributions over final wealth levels Whereas framing effects involve alternative descriptions before being evaluated. However, risk attitudes towards of an otherwise identical choice problem, alternative gains and losses tend to be more stable than can be response formats have also been found to lead to different explained by a fixed utility function over final wealth, and choices, leading to what have been termed response-mode utility functions might be best defined in terms of effects. For example, under expected utility, an individual’s 80 non-expected utility theory von Neumann–Morgenstern utility function can be a variety of settings involving real-money gambles, assessed or elicited in a number of different manners, patrons in a Las Vegas casino, group decisions and which typically involve a sequence of pre-specified lotter- experimental market trading. By expressing the implied ies P1, P2, P3, y, and ask for (a) the individual’s certainty preferences as ‘$-bet CE($-bet) CE(p-bet) p-bet equivalent CE(Pi) of each lottery Pi,(b) the gain equiv- $-bet’, some economists have categorized this phenom- alent Gi that would make the gamble (Gi,1/2; $0,1/2) enon as a violation of transitivity and tried to model it as indifferent to Pi,or(c) the probability equivalent Yi that such (see the ‘regret theory’ model below). However, would make the gamble ($1000,Yi; $0,1 Yi) indifferent most psychologists and economists now view it as a to Pi. Although such procedures should generate equiv- response-mode effect: specifically, that the psychological alent assessed utility functions, they have been found to processes of valuation (which generates certainty equiv- yield systematically different ones (for example, Hershey, alents) and direct choice are differentially influenced by Kunreuther and Schoemaker, 1982; Hershey and the probabilities and payoffs involved in a lottery, and Schoemaker, 1985). that under certain conditions this can lead to choices and In a separate finding now known as the preference valuations which ‘reveal’ opposite preference rankings reversal phenomenon, subjects were first presented with a over a pair of gambles. number of pairs of lotteries and asked to make one choice out of each pair. Each pair of lotteries took the following form: Non-expected utility models of risk preferences ( Non-expected utility functional forms p chance of $X p-bet : versus Researchers have responded to departures from linearity 1 p chance of $0 in the probabilities in two manners. The first consists of ( replacing the expected utility form V EU ðPÞ¼Uðx1Þ q chance of $Y p1 þ ...þ UðxnÞ pn by some more general form for the $-bet : preference function VðPÞ¼Vðx ; p ;...;x ; p Þ. Several 1 q 1 1 n n chance of $0 such forms have been proposed (for the Rank Depend- ent, Dual and Ordinal Independence forms, the payoffs where 0oXoY and p4q. The terms ‘p-bet’ and ‘$-bet’ must be labelled so that x1 ... xn, and G( ) must derive from the greater probability of winning in the first satisfy Gð0Þ¼0 and Gð1Þ¼1):
P Prospect theory n u x p p Edwards (1955; 1962), Kahneman–Tversky (1979) P i¼1 ð iÞ Pð iÞ n n Subjectively weighted utility uðxiÞ pðp Þ= pðp Þ Karmarkar (1978; 1979) P i¼1 hi P i i¼1 Pi Rank-dependent expected utility n i i 1 Quiggin (1982) i¼1uðxiÞ G j¼1pj G j¼1pj P hi P P Dual expected utility n i i 1 Yaari (1987) i¼1xi G j¼1pj G j¼1pj P P hi P P Ordinal independence n i i i 1 Segal (1984), Green–Jullien (1988) i¼1hxi; j¼1pj G j¼1pj G j¼1pj P P Moments of utility n n 2 Mu´nera–de Neufville (1983), Hagen (1979) M uðxiÞ pi; uðxiÞ pi; ... Pi¼1 Pi¼1 Weighted utility n u x p = n t x p Chew (1983) Pi¼1 ð iÞ i i¼1 ð iÞ i Optimism–pessimism n u x g p ; x ; ...; x Hey (1984) Pi¼1 Pð iÞ ð i 1 nÞ n n Quadratic in the probabilities i¼1 j¼1Kðxi; xjÞ pi pj Chew, Epstein and Segal (1991) P P n Regret theory n n n n Loomes–Sugden (1982) i¼1 j¼1Rðxi; xj Þ pi pj bet, and greater possible gain in the second bet. Subjects Most of these forms have been formally axiomatized, were next asked for their certainty equivalents of each of and, under the appropriate monotonicity and/or curva- these bets, via a number of standard elicitation tech- ture assumptions on their constituent functions u( ), niques. Standard theory predicts that, for each such pair, G( ), and so on, most are capable of exhibiting first- the prospect selected in the direct choice problem would order stochastic dominance preference, risk aversion, and also be assigned the higher certainty equivalent. However, the above types of systematic violations of the Inde- subjects exhibit a systematic departure from this predic- pendence Axiom. Researchers such as Konrad and tion in the direction of choosing the p-bet in a direct Skaperdas (1993), Schlesinger (1997) and Gollier (2000) choice, but assigning a higher certainty equivalent to the have used these forms to revisit many of the applications $-bet (Lichtenstein and Slovic, 1971). Although this previously modelled by expected utility theory, such as finding initially generated widespread scepticism, it has asset and insurance demand, in order to determine which been replicated by both psychologists and economists in expected-utility-based results are, and which are not, non-expected utility theory 81
robust to departures from linearity in the probabilities, UðxnÞ pn,thevalueU(xi)canbeinterpretedasthecoeffi- and which additional properties of risk-taking behaviour cient of pi, and that many theorems involving a linear can be modelled. P function’s coefficients continue to hold when generalized to n Although the form i¼1uðxiÞ pðpiÞ was the earliest a nonlinear function’s derivatives. By adopting the notation non-expected utility model to be proposed, it was largely Uðx; PÞ @VðPÞ=@probðxÞ and the term ‘local utility abandoned when it was realized that, whenever the function’ for the function U( ;P), standard expected utility weighting function p( ) was nonlinear, the generic ine- characterizations such as those listed at the beginning of a qualities pðpiÞþpðpjÞ pðpi þ pjÞ and pðp1Þþ...þ this article can be generalized to any smooth non-expected a pðpnÞ 1 implied discontinuities in the payoffs and utility preference function V(P) in the following manners inconsistency with first-order stochastic dominance pref- (Machina, 1982): erence. BothP problemsP were corrected by adopting weights ½Gð i p Þ Gð i 1p Þ based on the cumula- V( ) exhibits global first order stochastic dominance j¼1 j j¼1 j preference if and only if, at each lottery P, its local tive probability values p1; p1 þ p2; p1 þ p2 þ p3; ...,to obtain the Rank-Dependent form. Under the above- utility function U(x; P) is an increasing function of x. mentioned restrictions on this form, these weights nec- V( ) exhibits global risk aversion (aversion to small or essarily sum to unity, and the Rank-Dependent form has large mean-preserving increases in risk) if and only if, emerged as the most widely adopted model in both the- at each lottery P, its local utility function U(x; P)isa concave function of x. oretical and applied analyses. The Dual Expected Utility à and Ordinal Independence forms are based on similar V ( ) is globally at least as risk averse as V( ) if and only if, at each lottery P, VÃ( )’s local utility function weighting formulas. à Unlike the other models, the regret theory form U (x; P) is a concave transformation of V( )’s local dispenses with the assumption of a preference function utility function U(x; P). over lotteries, and instead derives choice from the psy- Similar generalizations of expected utility results and chological notions of rejoice and regret – that is, the characterizations can be obtained for general compara- reaction to receiving outcome x when an alternative tive statics analysis, the theory of asset demand, and the decision would have led to outcome xÃ. The primitive of demand for insurance. With regard to the Allais Paradox this model is a regret:rejoice function R(x, xÃ) which is and other observed violations of the Independence positive if x is preferred to xÃ, negative if xà is preferred Axiom, it can be shown that the indifference curves of to x, zero if they are indifferent, and satisfies the skew- a smooth preference function V( ) will fan out in the symmetry condition Rðx; xnÞ Rðxn; xÞ. In a choice probability triangle if and only if U(x; PÃ) is a concave n à between lotteries P ¼ðx1; p1;...;xn; pnÞ and P ¼ transformation of U(x; P) whenever P first-order stoc- n n n n ðx ; p ;...; x n ; p n Þ which are realized independently, hastically dominates P. This analytical approach has been 1 1 n n à the individual’sP expectedP rejoice from choosing P over P extended to larger classes of preference functionals and n nn n n is given by i¼1 j¼1Rðxi; xj Þ pi pj , and the individual distributions by Chew, Karni and Safra (1987), Karni is predicted to choose P if this value is positive, Pà if it is (1987; 1989) and Wang (1993), formally axiomatized by negative, and be indifferent if it is zero (various proposals Allen (1987), and applied to the analysis of choices under for extending this approach beyond pairwise choice uncertainty by Chew, Epstein and Zilcha (1988), Chew have been offered). Since this model specifies choice in and Nishimura (1992), Dekel (1989), Green and Jullien pairwise comparisons rather than preference levels of (1988), Machina (1984; 1989; 1995) and others. individual lotteries, it allows choice to be intransitive, so the individual might select P over PÃ, Pà over PÃÃ, and PÃà over P. Though some have argued that such cycles Non-expected utility preferences under subjective allow for the phenomenon of ‘money pumps’, it has uncertainty allowed the model to serve as a proposed solution to the Recent years have seen a growing interest in models of Preference Reversal Phenomenon. choice under subjective uncertainty, with efforts to rep- resent and analyse departures from both expected utility Generalized expected utility analysis risk preferences and probabilistic beliefs. A non-expected An alternative approach to non-expected utility preferences utility preference function Wðf ð ÞÞ Wðx1 on E1;...; does not rely upon any specific functional form, but links xn on EnÞ over subjective acts f ð Þ ¼ ½x1 on E1;...; properties of attitudes toward risk directly to the proba- xn on En is said to be probabilistically sophisticated if it bility derivatives of a general ‘smooth’ preference function takes the form Wðf ð ÞÞ Vðx1; mðE1Þ;...;xn; mðEnÞÞ for VðPÞ¼Vðx1; p1;...;xn; pnÞ. Such analysis reveals that some subjective probability measure m( ) over the space the basic analytics of the expected utility model are in of events and some non-expected utility preference func- fact quite robust to general smooth departures from lin- tion VðPÞ¼Vðx1; p1;...;xn; pnÞ. Such preferences earity in the probabilities. This approach is based on have been axiomatized in a manner similar to Savage’s the observations that for the expected utility function (1954) axiomatization of the subjective expected utility V EU ðx1; p1;...;xn; pnÞþ Uðx1Þ p1 þ ...þ form WSEU ðf ð ÞÞ Uðx1Þ mðE1Þþ...þ UðxnÞ mðEnÞ 82 non-expected utility theory
(Machina and Schmeidler, 1992). Although such See also Allais paradox; expected utility hypothesis; prefer- preferences can be consistent with Allais-type depar- ence reversals; prospect theory; risk; risk aversion; Savage’s tures from linearity in (subjective) probabilities, they subjective expected utility model; uncertainty. are not consistent with Ellsberg-type departures from probabilistic beliefs. Efforts to accommodate the Ellsberg Paradox and the general phenomenon of ambiguity aversion have led to Bibliography the development of several non-probabilistically sophis- Allais, M. 1953. Le comportement de l’homme rationnel ticated models of preferences over subjective acts (see the devant le risque: critique des postulats et axiomes de analysis of Epstein, 1999, as well as the surveys of Camerer l’Ecole Ame´ricaine. Econometrica 21, 503–46. and Weber, 1992, and Kelsey and Quiggin, 1992). One Allen, B. 1987. Smooth preferences and the local such model, the maximin expected utility form, replaces expected utility hypothesis. Journal of Economic Theory the unique probability measure m( ) of the subjective 41, 340–55. expected utility model by a finite or infinite family M of Arrow, K. 1965. Aspects of the Theory of Risk Bearing. such measures, to obtain the preference function Helsinki: Yrjo¨ Jahnsson Sa¨a¨tio¨. Battalio, R., Kagel, J. and Macdonald, D. 1985. Animals’ Wmaximinðx1 on E1;...;xn on EnÞ choices over uncertain outcomes. American Economic
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