第六讲 Extensive games with perfect information: illustrations,一、Stackelberg’s 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 firm’s preferences are represented by its profit.,基本假定,Constant unit cost： Linear inverse demand function： Assume：,Backward induction,First compute firm 2’s reaction to an arbitrary quantity by firm 1: Next firm 1’s problem in the first stage of the game amounts to: Thus the outcome of the game is:,Conclusion,The outcome of the equilibrium output is: Firm 1’s profit is , and firm 2’s profit is . By contrast, in the unique Nash equilibrium of Cournot’s (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 subgame 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 payoff of the union is and the payoff of the firm is .,Backward induction,First, we can characterize the firm’s best response in stage 2, to an arbitrary wage demanded by the union in stage 1, . The union’s problem at the first stage amounts to:,,,,,,,R,L,,L*(w),R(L),Slope=w,,,,,,,w,,L,L*(w),Firm’s best response function,Firm’s isoprofit curves,,,,,,L,w,,Union’s 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.,Firm’s isoprofit curve,三、Bargaining,The ultimatum game (最后通谍博弈) A finite horizon game with alternating offers and impatient players An 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 receives x1; if she rejects it, neither player receives any pie. Preferences：Each person’s 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 2’s strategy either accepts all offers (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 1’s 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 equilibrium of the game is the strategy pair in which player 1 offers 0 and player 2 accepts all offers. In this equilibrium, player 1’s payoff is all the pie and player 2’s payoff is zero.,Extensions of the ultimatum game,Conclusion,In this game, player 1 is powerless; her proposal at the start of the game is irrelevant. Every subgame following player 2’s 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 backward induction, player 2’s 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 perfect 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 1’s initial proposal is Player 2 accepts all 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 1 Player 1 accepts all proposals of player 2 at the end of the game (after a history in which player 2 rejects player 1’s opening proposal). The outcome of this equilibrium is that player 1 proposes ,which player 2 accepts; player 1’s payoff is and player 2’s 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。