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1、第六讲 Extensive games with perfect information:illustrations,一、Stackelbergs model of duopoly:constant unit cost and linear inverse demand,Players：the two firms(a leader and a follower)；Timing：(1)firm 1 chooses a quantity；(2)firm 2 observes and then chooses a quantity;Preferences：Each firms preferences
2、 are represented by its profit.,基本假定,Constant unit cost：Linear inverse demand function：Assume：,Backward induction,First compute firm 2s reaction to an arbitrary quantity by firm 1:Next firm 1s problem in the first stage of the game amounts to:Thus the outcome of the game is:,Conclusion,The outcome o
3、f the equilibrium output is:Firm 1s profit is,and firm 2s profit is.By contrast,in the unique Nash equilibrium of Cournots(simultaneous-move)game under the same assumptions,each firm produces units of output and obtains the profit.Thus firm 1 produces more output and obtains more profit in the subga
4、me perfect equilibrium of the sequential game,and firm 2 produces less output and obtains less profit.,二、Wages and employment in a unionized firm,Players:a firm and a monopoly union;Timing:the union makes a wage demand w;the firm observes(and accepts)w and then chooses employment L.Preferences:the p
5、ayoff of the union is and the payoff of the firm is.,Backward induction,First,we can characterize the firms best response in stage 2,to an arbitrary wage demanded by the union in stage 1,.The unions problem at the first stage amounts to:,R,L,L*(w),R(L),Slope=w,w,L,L*(w),Firms best response function,
6、Firms isoprofit curves,L,w,Unions indifference curves,w,L,L*(w*),w*,L*(w),Conclusion,Thus(w*,L*(w*)is the backwards-induction outcome of this wage-and-employment game.It is straightforward to see that(w*,L*(w*)is inefficient.,Firms isoprofit curve,三、Bargaining,The ultimatum game(最后通谍博弈)A finite hori
7、zon game with alternating offers and impatient playersAn infinite horizon game with alternating offers and impatient players,三（1）The ultimatum game(最后通谍博弈),Players：the two players；Timing：player 1 proposes a division(x1,x2)of a pie,where x1+x2=1.If 2 accepts this division,she receives x2 and player 1
8、 receives x1;if she rejects it,neither player receives any pie.Preferences：Each persons preferences are represented by payoffs equal to the division of pie she receives.,Figure,Backward induction,First consider the subgame,in a subgame perfect equilibrium player 2s strategy either accepts all offers
9、(including 0),or accepts all offers and rejects the offer.Now consider the optimal strategy of player 1:If player 2 accepts all offers(including 0),then player 1s optimal offer is 0;If player 2 accepts all offers except zero,then no offer of player 1 is optimal.,Conclusion,The only subgame perfect e
10、quilibrium of the game is the strategy pair in which player 1 offers 0 and player 2 accepts all offers.In this equilibrium,player 1s payoff is all the pie and player 2s payoff is zero.,Extensions of the ultimatum game,Conclusion,In this game,player 1 is powerless;her proposal at the start of the gam
11、e is irrelevant.Every subgame following player 2s rejection of a proposal of player 1 is a variant of the ultimatum game in which player 2 moves first.Thus every such subgame has a unique subgame perfect equilibrium,in which player 2 offers nothing to player 1,and player 1 accepts all proposal.Using
12、 backward induction,player 2s optimal action after any offer(x1,x2)of player 1 with x21 is rejection.Hence in every subgame perfect equilibrium player 2 obtains all the pie.In the extension of this game in which the players alternate offers over many periods,a similar result holds:in every subgame p
13、erfect equilibrium,the player who makes the offer in the last period obtains all the pie.,三（2）A finite horizon game with alternating offers and impatient players,Two-period deadline,Conclusion,The game has a unique subgame perfect equilibrium in which:Player 1s initial proposal is Player 2 accepts a
14、ll proposals in which she receives at least and rejects all proposals in which she receives less than Player 2 proposes(0,1)after any history in which she rejects a proposal of player 1Player 1 accepts all proposals of player 2 at the end of the game(after a history in which player 2 rejects player
15、1s opening proposal).The outcome of this equilibrium is that player 1 proposes,which player 2 accepts;player 1s payoff is and player 2s is.,1,1,2,Y,N,N,Y,2,Third-period deadline,1,2,N,Y,This game has a unique perfect equilibrium:.,三（3）An infinite horizon game with alternating offers and impatient players,Subgame perfect equilibrium,The structure of the game is stationary.A stationary strategy pair:,1,2,Y,N,N,Y,2,1,2,1,Y,N,N,Y,1,2,Subgame perfect equilibrium,Proposition,Properties of subgame perfect equilibrium,
