Chapter 6 A lte r n a tives to E x pect e d Utility Theory In this lecture, I describe some well-know n experimental evidence against the expected utility theory and the alternativ e theories developed in order to accommodate these experimen ts. (I hav e posted a comprehensive survey on the class web page.) 6.1 Allais Parado x and Weighted Utilit y Imagine yourself c hoosing bet ween the follow in g t wo alternativ e s: A Win 1 million dolla r for su r e. B W in 5 millio n dollar with 10% c ha nc e, 1 million dollar with 89%, nothing with 1%. Wh ich one would y ou choose? In many surveys, subjects who were offered these alternatives c h ose A. It seems that they did not want to risk the opportunity of ha vin g a million dollar for a 10% c h an ce of having five million dollar instead. Now consider the following two altern atives: C Win $1M with 11% c han ce, nothing with 89%. D Win $5M with 10% chance, nothing with 90%. 4748 CHAPTER 6. ALTERNATIVES TO EXPECTED UTILITY THEORY It seems that the probability of winning the prize is similar for the two alternatives, while the prizes are substantially different. Hence, it seems reasonable to c h oose the higher prize, choosing D rather than C. Indeed, in surveys, the subjects c hoose D. Un fortu n ately, for an expected utility maximizer, the trad e of between A and B is iden tica l to the trade of between C and D, and he prefers A to B if and only if he prefers C to D. To see this, note that for an expected utilit y maximizer with utilit y function u, A is betterthanBif andonlyif u (1) > 0.1u (5) + 0.89u (1), i.e., 0.11u (1) > 0.1u (5) , (6.1) where the unit of money is million dollar, and the utilit y from 0 is normalized to 0. Bu t for such an expected utility maxim iz er, C is better than D if and only if (6.1) holds. The above experiment against th e expected utility theory has been d esig ned by A llais. It illustrates for the subjects surv eyed that the indifference curv es are not parallel, and hence the independence axiom is violated. T h is is illustrated in Figur e 6.1. As shown in the figure, the lines connecting A to B and C to D are parallel to each other. Since A is better than B, the indifference curve through A is steeper than the line connecting A to B. Since D is better than C, the indifference curve through C is flatter than the line connecting C to D. Therefore, the indifference curve through A is steeper than the indiffer en ce curve through C. A series of other experiments also suggested that the indifference curv es are not parallel and"fanout’ as inthe figure. Consequen t ly, decision theorists have dev eloped many altern ative theories in whic h the indifference curves are not parallel. These theories often assume that the indifference curves are straight lines, called betweenness. A prominent theory among these assumes that the indifference curves are straigh t lines that fan out from a single origin. This theory is called We ig hted Utility Theory,as it assumes the follow ing general form for the utility from a lottery p: P W (p)= w (x p, g) u (x) x∈C |where g (x) p (x) w (x|p, g)= P y∈C p (y) g (y) for some function g : C R. Here, the utilities are weigh ted according to not only the → probabilities of the consequ ences but also according to the consequen ces themselv es. Of49 6.1. ALLAIS PARADOX AND WEIGHTED UTILITY Pr($0) Pr($5) 1 1 0 A ∙ ∙∙ B C D ∙ B’ ∙ Indifference curves Figure 6.1: Allais P ara dox. The prizes are in terms of millio n dollars. Probabilit y of 0 is on the horizontal axis; the probability of 5 is on the vertical axis, and the remainin g probab ility goes to the intermediate prize 1. course, if g is constant, the weighting is done only according to the probabilities, as in the expected utility theory. Exercise 7 Check that under the weig h ted utility theory, the indifference curves are str a ight lines, but the slop e of the indifference curves differ when g is not consta nt. Tak-ing C with three elem e nts, charac terize the function s g and u under which the indifference sets fan out as in the A llais paradox. In the w eighted utility theory, the decision maker distorts the probabilities using the consequences themselves and the whole probabilit y vector p. In general, probabilities need to be distorted if one wants to incorporate Allais parad ox in expected utility theory. A prominent theory that distorts the probabilities to this end is r ank-dependent expected utility theory. In this theory, one first ranks the consequences in the order of increasing utility. He then applies probability weighting function w to the cum u lative distribution function F and distorts it to a new cumulative distribution function w F .One then ◦ fina lly uses expected utility und er the distorted proba bilities in order to evalua te the lottery. Th e resulting value function is Z U (x|w)= u (x) dw (F (x)) . (6.2)50 CHAPTER 6. ALTERNATIVES TO EXPECTED UTILITY THEORY 10.750.50.250 1 0.75 0.5 0.25 0 p w pwFigure 6.2: Probability Weigh ting Function; w (p)= e−(− ln p)α for some α ∈ (0, 1). The surv ey results in the Allais paradox suggest that the subjects overestimate the small probab ilit y events with extreme value, such as getting nothing with a small proba-bilit y. In order to capture suc h a behav ior, one often uses an in verted S shaped probabil-it y weighting function as in Figure 6.2. Here, w is an increasing function with w (0) = 0 and w (1) = 1, and it crosses the diagon al once at some p∗. The general functional form w (p)= e−(− ln p)α for some α ∈ (0, 1) has man y desirable properties. Example 2 Consider the lotteries in the Allais paradox. Set u (0) = 0 and u (1) = 1. The value of lottery B is com puted as follows: U (B|w)= w (0.01) u (0) + [w (0.9) − w (0.01)] u (1) + (1 − w (0.9)) u (5) = w (0.9) − w (0.01) + (1 − w (0.9)) u (5) . Similarly, the values of the other lotteries are U (A|w)=1 U (C|w)=1 − w (0.89) U (D|w)=(1 − w (0.9)) u (5) . Now take u (5) ∈ (1,e/ (e − 1)) and w (p)= e−(− ln p)α . N ote that αlim 0 U (A|w)=1 > (1 − 1/e) u (5) = αlim 0 U (B|w) = αlim 0 U (D|w) > 1−1/e = αlim 0 U (C|w) . → → → →Thus, for small α, the preferences …