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Soln:In periodT, player 1 will accept any offer.So player 2 offers 0 to player 1.Therefore, if the game reaches periodT, player 2 gets the whole pie. Knowing this, inperiodT°1;player 2 will accept if and only if he is offered at least°2; so player 1 willoffer him exactly°2. In periodT°2;player 2 will accept if and only if he is offered atleast°22;so player 1 will offer him°22:Proceeding backwards, we see that in any periodT°k, player 2 will accept if and only if he is offered at least°k2;so player 1 will offerhim°k2:In the first period, player 2 will accept if and only if he is offered at least°T°12;so player 1 will offer him°T°12:Therefore, the subgame perfect equilibrium is:´In any periodT°k,kD1; :::;T°1;the offer strategy of player 1 is to offer°k2regardless of previous play, and the acceptance strategy of player 2 is to accept ifand only if he is offered at least°k2I´In periodT, 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, regardlessof previous play.The outcome of this equilibrium is agreement in the first period, with player 1 receiving1°°T°12and player 2 getting°T°12:5. (20 pts) Andy and Betty can together choose any actiona2[0;10]:Given a choice ofa;their payoffs areuADaanduBD100°a2:(a) (10 pts) Find all Pareto efficient values ofa:Soln:Any value ofa2[0;10] is Pareto efficient: making Andy better off wouldrequire increasinga, but this would make Betty worse off, and vice versa.(b) (10 pts) Now assume transferable utility: in addition to choosinga;a transfer ofany amountt2Rcan be made from Andy to Betty, resulting in payoffsuADa°tanduBD100°a2Ct: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 ifamaximizes the joint payoff,aC100°a2:This is maximized byaD12:Hence,.a;t/is Pareto efficient if and only ifaD12:3
6. (20 pts) Find all pure strategy subgame perfect equilibria, and one other Nash equilibrium,of this game:12BSbsbs3,10,00,01,31CDCDcd3,34,00,42,212XYSoln:The subgame followingXisbsB3,10,0S0,01,3Its 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 followingYiscdC3,30,4D4,02,2It has a unique Nash equilibrium: (D,d). It yields payoff 2 to player 1.Player 1 prefers to playXif the equilibrium (B,b) will be played in the resulting sub-game, and otherwise prefers to playY. 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 equilibriumgets played in the subgame that is in fact reached after player 1 playsY. Furthermore,deviating toXwould give player 1 a payoff of at most 1<2, so he is playing a bestresponse. This equilibrium is not subgame perfect because it prescribes the play of (B,s)