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EC 992 ADVANCED MICROECONOMICS
1. [50 marks] Answer all parts (a)-(c) of this question.
(a) Let X = {x1 , x2 , x3 , x4 } denote a set of monetary outcomes. A lottery L = (p1 , p2 , p3 , p4 ) over X specifies probability pn ∈ [0, 1] of outcome xn, n = 1, 2, 3, 4. Suppose decision- maker J has the following preferences over lotteries. (0, 0, 1, 0) ∼ (0.2, 0, 0, 0.8) and (0, 1, 0, 0) ∼ (0.4, 0, 0.6, 0), where ∼ denotes indifference. If J’s preferences have an expected utility representation, what would be J’s preferences between the lotteries (3/1, 0, 0, 3/2) and (0, 0.5, 0.5, 0)? Explain your answer.
(b) Consider the following two-player simultaneous move game. Each player’s pure strategy set is T = {1, 2,..., T} where T > 0 is an finite integer. The payoff to player i = 1, 2 under pure strategy profile (t1 , t2 ) where ti denotes player i’s pure strategy is as follows. If t1 
t2 then u1 (t1 , t2 ) = u2 (t1 , t2 ) = 0. If t1 = t2 then u1 (t1 , t2 ) = 1 and u2 (t1 , t2 ) = −1. Does there exist a mixed strategy equilibrium of this game where player 1 plays the mixed strategy which puts probability T/1 over each element of T ?
Explain your answer.
(c) Suppose the following two-player simultaneous-move stage game is infinitely repeated. Let δ ∈ (0.5, 1) denote a common discount factor and t = 1, 2, 3, . . . , denote time periods for the infinitely repeated game.
Table 1: The simultaneous move stage game
(i) Does the strategy profile in which player i chooses A at t = 1 and in every period thereafter if the action of player j ≠ i in each past period is A, and chooses B oth- erwise, where i,j ∈ {1, 2}, constitute a Nash equilibrium of the infinitely repeated game? Explain your answer.
(ii) Consider the subgame following the action profile (A, B) in period t = 1 and sup- pose player 1 uses the strategy described in part (c)(i). Does player 2 have a strictly profitable deviation in this subgame from the strategy described in part (c)(i)? Ex- plain your answer.
2. [50 marks] Answer all parts (a)-(c) of this question.
(a) Let X = {x1 , x2 , x3 } denote a set of monetary outcomes with x1 < x2 , x3 . A lottery L = (p1 , p2 , p3 ) over X specifies probability pn ∈ [0, 1] of outcome xn, n = 1, 2, 3. Let
0 <ˆ(p) <¯(p) < 1 and λ ∈ [0, 1]. Suppose decision maker J has the following preferences
over lotteries: (1 − ¯(p), ¯(p), 0) ≻ (1 − ˆ(p), 0, ˆ(p)) and (1−λˆ(p), 0,λˆ(p)) ≻ (1−λ¯(p),λ¯(p), 0), where ≻
denotes strict preference. Can J’s preferences over lotteries be represented by an expected utility function? Explain your answer.
(b) Consider the following two player simultaneous move game. Each player’s pure strategy set is [0, X] where X > 0. Let H(·) denote a continuous and strictly increasing function over [0, X] with H(0) = 0, H(X) = 1, and H(x* ) = 0.5 for some x* ∈ [0, 1]. The payoff to player i = 1, 2 under strategy profile (x1 , x2 ) where xi denotes player i’s strategy is as follows.
where j = 1, 2;j ≠ i. Does there exist a unique pure strategy Nash equilibrium of this game where x1 = x* ? Explain your answer.
(c) Two players bargain over the division of 1 GBP. t = 0, 1, 2, 3, . . . , ∞ denotes time periods. In even periods t = 0, 2, 4, . . . player 1 (P1) makes an offer which player 2 can accept or reject. In odd periodst = 1, 3, 5, . . . player 2 (P2) makes an offer which P1 can accept or reject. Any offer (s, 1 − s) specifies shares ∈ [0, 1] for P1 and share 1 − s for P2. If an offer (s, 1 − s) is accepted in period t, then the game ends and P1’s discounted utility is δts and P2’s discounted utility is δt(1 − s), where δ ∈ (0, 1), otherwise the game moves to the next period.
Does the strategy profile (σ1 ,σ2 ) specified below constitute a Nash equilibrium of the game? Does (σ1 ,σ2 ) constitute a subgame perfect Nash equilibrium of the game? Ex- plain your answers.
