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464 Chapter 15. Repeated Games: General Results A Masin (1986,1991),who establish also a result for a class of multiplayer games. reslt of Wen (1994) covers llmultiplayer games. The folk heorems for itely repeated games,Proposiions 460.1 and 461.1,are due to Benoit and Krisn. 1985,1987).Games n wich he players altenate moves,like he one in Exercise 459.3,are sudied by Lanoff and Matsui (1997);Rubnsten and Wolnsky (1995) study a closely related class of games. he idea in Section 15.4 is due to Green and Porter (1984),who study a variant of Comot's oigopoly game. he formulaion I use is taken rom Tuole (1988, Section Bargaining 16.-1 Bargaining as ale:e�sive'gime 465 16.2 Illustration: ral� In a-:na rkt 477 16;3 Nash's axlomatiCiodel,"81 14 ,Relation bew�en stit�g k "and axiomatic models 489 Prerequisite; Chapter5an� Sections 4.1.3,6.1.1, and 7.6 'N MANY siuaions,parties divide a "pie".A capitalist and he workers she hires .diide he total revenue generated by he output produced; legislators divide tax revenue among spendng programs valued by th�ir constituents; a buyer of an object and a seller ivide the amont by wich he buyer's valuaion of he object exceeds he seler's. n s chapter I discuss two very dfferent models hat are intended to capture "bargang" between he paries in such situations. One model is an extensive game (see Chapter 5).The oher model takes an approach not previously used in s book: t considers he outcomes compaible wih a ist of apparenly sensible properties. hough the models are Very diferent, he outcomes hey isolate are closely related. I 16.1 16: 1. 7 Bargaining as an extensive game Extensions of the ultimatum game One pont of departure for a theory of bargang is the "ultimatum game" studied n Section 6.1.1. n tis game,two players split a pie' of sze c that hey both value. hroughout s section I take c 1. Player 1 proposes a division (XVX2) of he pie, where Xl +X2 1 and 0 5 Xi 5 1 for i I, 2. Player 2 eiher acceps =s division, n wich case she receives X2 and player 1 receives Xl, or rejects it, n wich case neiher player receives any pie. is game has a ique subgame perfect equilibrium, in wich player 1 proposes the division (1,0),and player 2 accepts all ofers. The outcome of he equiibrium is hat player 1 receives all he pie. What accounts for tis one-sided outcome? Player 2 is powerless because her only altenative to he acceptance of player l's proposal is rejection, wich yields her no pie. Suppose,nstead, hat we give player 2 he option of makng a con­ terproposal after rejecting player l's proposal,wich player 1 may accept or reject. hen we have the game illusrated n Figure 466.1,where Y means "accept" and N means "reject".465 Chapter 16. Bargaining 466 16.1 1 Figue Figur� 467.1 A tw-peiod bargang game of altenang oers n whih each player j uses the actor 01 to dScont fuue payofs. x, y),x,N,y, Y),nd (x, N,y,N))n s game, player 1 is powerless; her proposal at he start of he game is irrelevant. Every subgame following player 2's rejecion of a proposal of player 1 is a variant of he ulimatum game n which player 2 moves irst .hus evey suh subgame has a uique subgame perfect equilibrium, n wich player 2 offers noting to player 1,and player 1 accepts all proposals. Using backward induction, player 2's opimal acion ater any offer (xv X2) of player 1 with X2
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