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(40 points) Consider the reduced normal form of the following game, in which the strategy set of Player 1 is { X, A, B}, so that the equivalent strategies XA and X B are represented by a single strategy X. X NON B 2 (a) Compute the set of rationalizable strategies. (Show your result.) (b) Compute the set correlated equilibria. (Show your result.) (c) Suppose that in addition to the type with the payoff function above, with proba- bility 0.1, Player 1 has a "crazy" type who gets 1 if he plays A and 0 otherwise. Compute the set of all sequential equilibria. 1. Ann is pregnant. According to the tests so far, there is p = 1/3000 chance that the baby has a serious disease. There is a new test that could find out for sure whether the baby has the disease but kills the baby with probability q = 1/300. Ann also has the option of aborting the baby after taking the test. Assuming that she is an expected utility maximizer, this question asks you to help her to decide whether to take the test. Here preferences are as follows. She cares only about whether she has a baby and whether that the baby is healthy. Hence, she considers the following consequences: (A) a healthy baby; (B) a baby with the disease; (C) no baby. Her utility function is given by u (A) = 1, u (B) = v and u (C) =0, where ve (-1, 1) is known. (a) As a function of v, find whether she should take the test. (b) Suppose now that Ann will learn the test result regardless of whether the baby lives and she cares about how she would feel when she learns the test results. In addition, she now considers the following two consequences: (D) baby dies during the test and she learns that the baby was healthy; (E) baby dies during the test and she learns that the baby had the disease. (Abortion still corresponds to (C).) Assume 1 > u (E) > 0 > > > u (D). Find the condition under which she takes the test. (30 points) Consider an expected profit maximizing monopolist who faces an uncertain demand. He supplies q units of goods at zero cost and sells it at price 8 - q, where e is unknown. [The price and the supply level can be negative.] (a) Assuming that 8 ~ N (y. o'), compute the monopolist's optimal supply q and his expected profit under the optimal supply. (b) Suppose that, through market research, the monopolist can learn about 0. In particular, by investing c', he can learn the value of a random variable Y before choosing his supply q, such that 0 = X + Y, X ~ N (0, 1 - c) and Y ~ N (0, c). How much should the monopolist invest? [Note that the utility function of the monopolist is (0 - q) q - c2.]4. Beatrice has initial wealth of wo and suffers from quasi-hyperbolic discounting. At any date s, her utility from a consumption stream a = (ro, 21, .. .) is U (x|s) = In(z.) + 8) 8 In(1+k). where 8, 6 E (0, 1). She gets return of r > 1 from her savings so that her wealth at t + 1 is wy+1 = " (14 - It) if her wealth at t is wy and she consumes r, at t. (b) Find a sophisticated-optimal consumption strategy for her in which the self at any given date s consumes yu's. Compute the constant y and briefly verify that this is indeed a subgame-perfect equilibrium of the multi-agent game. (c) For #