Chapter 2 Decision M aking under Risk In the previous lecture I considered abstract c h oice problems. In this section, I will focus on a special class of cho ice problems and impose more structure on the decision maker’s preferences. I will consider situations in which the decision maker cares only about the consequences, suc h as the amount of money in his bank accoun t, but he may not be able to c hoose directly from the set of consequences. Instead, he chooses from alternativ es that determ ine the consequences probabilistically, suc h as a lottery tic ket. In this lecture, I assume that, for any alterna tive x, the probability distribution on the set of consequences induced by x is giv en. That is, although decision maker does not know the consequence of choosing a giv en alternativ e, he is giv en the probability of each consequence from choosing that action. This is called decision making under risk. Such assumptio ns can be plausible in relatively few situations, such as ch an ce games in a casino, in which there are objective probabilities. In most cases of economic interest, the alternatives do not come with proba bilities. The decision maker forms his subjective beliefs about the consequences of his cho ices. T his is called decision makin g under uncertainty. I will analyze the decision making under risk as an in termediary step to ward analyzing decision making under uncertain ty. 2.1 Consequences and Lotteries Consider a finite set C of c onsequences.A lotte ry is a probability distribution p : P C [0, 1] on C,where p(c)=1. The set of all lotteries is denoted b y P .The → c∈C 910 CHAPTER 2. DECISION MAKING UNDER RISK consequences are embedded in P as point masses on single lotteries. For any c ∈ C,I will write c for both the consequence c and the prob ab ility distribution that puts probab ility 1on c. The decision mak er cares about the consequence that will be realized, but he needs to choose a lottery. In the language of the the previous lecture, the set X of alternatives is P . A lottery can be dep icted b y a tree. For example, in Figure 1, Lottery 1 dep icts a situation in whic h the decision maker gets $10 with probability 1/2 (e.g. if a coin toss results in H ead ) and $0 w ith probability 1 /2 (e.g. if the c oin toss re sults in Tail). A lottery can be simple as in the figure, assigning a probability to each consequ ence, or compound as in Figure 2.3, con taining successive branches. Th e represen tation of all lotteries as prob ab ility distributio ns incorporates the assumptions that the decision maker is consequentialist, meanin g that he cares only about the consequences, and that he can compu te the probabilit y of each consequen ce under compounding lotteries. Lottery 1 Figure 2.1: Representing the lotteries p as vectors, note that P is a |C| − 1 dimensional simplex. Hence, I will regard P as a subset of R|C| (one can envision it as a subset of R|C|−1 as w ell). Endo wing R|C| with the standard Euclidean metric, note that P is a convex and compact subset. 2.2 Ex pected Utility M ax im izatio n – Rep re senta-tion We would like to ha ve a theory that constructs the decision maker’s preferences on the lotteries from his pr eferences on the lotteries. There are many of them . Th e most 1/2 1/2 10 011 2.3. EXPECTED UTILITY MAXIMIZATION – CHARA CTERIZATION well-known– and the most canonical and the most useful– on e is the theory of expected utility max imization by Von Neu m an n and Mor g enste rn. In this lecture, I will focus on this theory. Definition 5 A relation º on P is said to be represen ted b y a von N eumann-Morgenstern utility function u : C R if and only if → X X p º q ⇐⇒ U(p) ≡ u(c)p(c) ≥ u(c)q(c) ≡ U(q) (VNM) c∈C c∈C for each p, q ∈ P . This statement has t wo crucial parts: 1. U : P → R represents º in the ordinal sense. That is, if U (p) ≥ U (q), then the decision maker finds lottery p as good as lottery q. And conversely, if the decision maker finds p at least as good as q,then U (p) must be at least as high as U (q). 2. The function U takes a particular form: for each lottery p, U (p) is the expected P value of u under p.That is, U(p) ≡ c∈C u(c)p(c). In other w ord s, th e decision maker acts as if he wants to maximize th e ex pected value of u. For instance, the expected utilit y of Lottery 1 for the decision maker is E(u(Lottery 1)) = 1 2u(10) + 1 2u(0).1 2.3 Expected Utilit y M aximization – C haracteri-zation The ma in objective of this lecture is to explore the conditions on preferences un der w hich the v on -Neumann Morgenstern representation in (VNM) is possible. In this w ay, one may ha ve a better insights in to what is in volv ed in expected utility maxim ization . First, as explained above, representation in (V N M ) implies that U represents º in the ordinal sense as well. Bu t, as w e have seen in the previous lecture, ordinal representation imp lies that º is a preference relation. Th is gives the first necessary condition. Axiom 2 (Preference) º is comp le te and transitive. R1If C were a contin uum, like R, we would compute the expected utility of p by u(c)p(c)dc.12 CHAPTER 2. DECISION MAKING UNDER RISK Second, in (VNM), U is a linear fun ction of p, and hence it is con tin uous. That is, (VNM ) in volves contin u ous ord inal representation. Hence, by Theorem 3 of the previous section, it is also necessary that º is continuo us. This g ives the second necessary condition. Ax iom 3 (Continuit y ) º is contin u o u s. Recall from the previous lecture that contin uit y means that the upper- and lo wer-contour sets {q|q º p} and {q|p º q} are closed for every p ∈ P . In this special setu p, a slightly weak er v ersion of the continuity assu m ption suffices: for any p, q, r ∈ P ,the sets {α ∈ [0, 1] |αp +(1 − p) q º r} and {α ∈ [0, 1] |r º αp +(1 − p) q} are closed. Yet another v ersion of this assumption is that for any p, q, r ∈ P ,if p  r,then there exist a, b ∈ (0, 1) suc h that ap +(1 − a)r  q  bp +(1 − r)r. By Theorem 3, Axiom s 2 an d 3 are necessary and sufficient for a rep resentation b y acontinuous function U. The …