Soln In period T player 1 will accept any offer So player 2 offers 0 to player

Soln in period t player 1 will accept any offer so

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Soln: In period T , player 1 will accept any offer. So player 2 offers 0 to player 1. Therefore, if the game reaches period T , player 2 gets the whole pie. Knowing this, in period T ° 1 ; player 2 will accept if and only if he is offered at least ° 2 ; so player 1 will offer him exactly ° 2 . In period T ° 2 ; player 2 will accept if and only if he is offered at least ° 2 2 ; so player 1 will offer him ° 2 2 : Proceeding backwards, we see that in any period T ° k , player 2 will accept if and only if he is offered at least ° k 2 ; so player 1 will offer him ° k 2 : In the first period, player 2 will accept if and only if he is offered at least ° T ° 1 2 ; so player 1 will offer him ° T ° 1 2 : Therefore, the subgame perfect equilibrium is: ´ In any period T ° k , k D 1 ; :::; T ° 1 ; the offer strategy of player 1 is to offer ° k 2 regardless of previous play, and the acceptance strategy of player 2 is to accept if and only if he is offered at least ° k 2 I ´ In period T , the offer strategy of player 2 is to offer 0 to player 1, regardless of pre- vious play, and the acceptance strategy of player 1 is to accept any offer, regardless of previous play. The outcome of this equilibrium is agreement in the first period, with player 1 receiving 1 ° ° T ° 1 2 and player 2 getting ° T ° 1 2 : 5. (20 pts) Andy and Betty can together choose any action a 2 [0 ; 10] : Given a choice of a ; their payoffs are u A D a and u B D 100 ° a 2 : (a) (10 pts) Find all Pareto efficient values of a : Soln: Any value of a 2 [0 ; 10] is Pareto efficient: making Andy better off would require increasing a , but this would make Betty worse off, and vice versa. (b) (10 pts) Now assume transferable utility: in addition to choosing a ; a transfer of any amount t 2 R can be made from Andy to Betty, resulting in payoffs u A D a ° t and u B D 100 ° a 2 C t : Find all Pareto efficient pairs . a ; t /: Soln: Given the transferable utility assumption, we know that . a ; t / is Pareto effi- cient if and only if a maximizes the joint payoff, a C 100 ° a 2 : This is maximized by a D 1 2 : Hence, . a ; t / is Pareto efficient if and only if a D 1 2 : 3
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6. (20 pts) Find all pure strategy subgame perfect equilibria, and one other Nash equilibrium, of this game: 1 2 B S b s b s 3,1 0,0 0,0 1,3 1 C D C D c d 3,3 4,0 0,4 2,2 1 2 X Y Soln: The subgame following X is b s B 3,1 0,0 S 0,0 1,3 Its Nash equilibria are (B,b) and (S,s). Note that (B,b) yields payoff 3 to player 1, and (S,s) yields payoff 1 to player 1. The subgame following Y is c d C 3,3 0,4 D 4,0 2,2 It has a unique Nash equilibrium: (D,d). It yields payoff 2 to player 1. Player 1 prefers to play X if the equilibrium (B,b) will be played in the resulting sub- game, and otherwise prefers to play Y . Therefore, the subgame perfect equilibria are: . X BD ; bd / and . Y SD ; sd / . . Y BD ; sd / is a Nash equilibrium that is not subgame perfect. Here, a Nash equilibrium gets played in the subgame that is in fact reached after player 1 plays Y . Furthermore, deviating to X would give player 1 a payoff of at most 1 < 2, so he is playing a best response. This equilibrium is not subgame perfect because it prescribes the play of (B,s)
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