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1. Person 1 cares about both her income and person 2's income. Precisely, the value she attaches to each unit of her own income is the same as the value she attaches to any two units of person 2's income. For example, she is indifferent between a situation in which her income is 1 and person 2's is 0, and one in which her income is 0 and person 2's is 2. (a) (5 points) How does she order the outcomes (1,4), (2,1), and (3,0), where the rst component in each case is her income and the second component is person 2's income? (b) (10 points) Give a payoff function consistent with these prefer- ences. 2. A DM's preferences over the set A = {(1, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, u(c) = 4. (a) (5 points) Are the DM's preferences also represented by the func- tion 1) for which v(a) = 1, 'u(b) = 0, 'u(c) = 2'? Why or why not? (b) (5 points) Are the DM's preferences also represented by the func- tion 11) for which w(a) = w(b) = 0, w(c) = 8? Why or why not? 3. Consider the following game. n={1,2} S,- = {H,T} for 2' 6 {1,2} u1(H,H) =3 u1(T,T) =1, u1(T,H) = -9 u1(1',)=9 U1(H,T) = 2, u2(T, H) = u2(H,T) = 1, and \"2(T1T)= \"2(H1H)= 1 (a) (5 points) Draw the game's matrix representation. (b) (10 points) Draw the best response correspondence for the game. (c) (10 points) What is (are) the Nash equilibrium (equilibria)? . Elroy and Judy play a game that Elroy calls the \"race to 100.\" Elroy goes rst, and the players take turns choosing numbers between one and nine. On each turn, they add the new number to a running total. The player who brings the total exactly to 100 wins the game. (a) ( 5 points) If both players play optimally, who will win the game? (b) (10 points) Does this game have a rst move advantage? Explain your reasoning. (c) (10 points) What are SPNE strategies for both players? . Assume that there are three voters with Euclidean preference in two dimensions with ideal points at (1,0), (0,1), and (1, 0) respectively. (a) (5 points) Identify the median voter on the :L' and y axes. (b) (10 points) Construct an agenda to get from (0,0) to (1,0). (c) (10 points) Construct an agenda to get from (1,0) to (-1,0)