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PS4_Sol - 14.12 Problem Set 4 Solution Ruitian Lang 1 The...

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14.12: Problem Set 4 Solution Ruitian Lang November 14, 2010 1 The stage game has three Nash equilibria: ( S, S ) , ( M, M ), and ( 2 3 S + 1 3 M, 1 3 S + 2 3 M ) , and the payoffs associated with these equilibria are (2 , 1) , (1 , 2) and ( 2 3 , 2 3 ) . Therefore, the players can coordinate on different Nash equilibria to punish anyone who deviates. Consider the following strategy profile for example. Player 1 plays M and Player 2 plays S in the first period. If the outcome of the first period is ( M, S ), then on odd dates they play ( M, M ) and on even dates they play ( S, S ). If the outcome of the first period is not ( M, S ), then they play ( 2 3 S + 1 3 M, 1 3 S + 2 3 M ) thereafter. To check whether it is a subgame perfect equilibrium, it suffices to check that neither player has incentive to deviate in the first period, since the outcome in the other periods does not affect continuation strategies and they are playing Nash equilibria of the stage game from period 2 on. In period 1, playing S gives Player 1 2 + 2 k 2 3 , and playing M gives her 0 + k + 2 k . Therefore, she has no incentive to deviate when 2 + 2 k 2 3 0 + k + 2 k, or k 6 5 . Similarly, Player 2 has no incentive to deviate when k 6 5 too. Therefore, the above strategy profile is a subgame perfect equilibrium when k 2. 2 We conjecture that the strategy profile is the following: at each date t = 3 n + k , the proposer proposes wage w k , and the responder accepts a wage off w if and only if her payoff from wage w is greater than or equal to her payoff from wage w k . (i.e., the union accepts w if and only if w w k for k = 2, and the employer accepts w if and only if w w k for k = 0 , 1.) At t = 3 n , when the union offers wage w , the employer knows that if agreement is not reached this period they will agree on w 1 in the following period, so he accepts w when 1 - w > δ (1 - w 1 ) and reject w when 1 - w < δ (1 - w 1 ). He has no incentive to deviate from the plan ”accepting if and only if w w 0 ” if and only if 1 - w 0 = δ (1 - w 1 ). The similar analysis for t = 3 n + 1 and 1
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t = 3 n + 2 yields two similar conditions. Therefore, no responder has incentive to deviate if and only if
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