ps 9 solution

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Problem Set 9 Solution 17.881/882 December 1, 2004 1 Gibbons 2.11 (p.135) In the stage-game, there are three Nash Equilibria ((T,L), (M,C ), (1/2T + 1/2M, 1/2L +1/2C)). Note that (B,R) is not an equilibrium of the stage game, since then player 1 has an incentive to deviate to T . In order to enforce compliance by player 1, let us think of ’punishments/rewards’ in the second stage. Of the three Nash Equilibria, (M,C ) ((T,L)) is that which gives player 1 his/her lowest (highest) payo. So let us devise the following strategies: -In the rst stage: play (B,R) -In the second stage: if (B,R) obtained in the rst stage, play (T,L); oth- erwise, play (M,C ) Then, we obtain the following payomatrix as a function of rst-period strategy: L C R T (4, 3) (1, 2) (6, 2) M (3, 3) (2, 4) (4, 3) B (2, 4) (1, 3) (7, 5) There, we see that neither player 1 nor player 2 has an incentive to deviate from their prescribed strategy. We have thus constructed a subgame perfect Nash Equilibrium with (B,R) as the outcome of the rst stage. 1

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  • Problem Set 9 Solution

    17.881/882December 1, 2004

    1 Gibbons 2.11 (p.135)

    In the stage-game, there are three Nash Equilibria ((T,L), (M,C), (1/2T +1/2M, 1/2L+ 1/2C)).

    Note that (B,R) is not an equilibrium of the stage game, since then player1 has an incentive to deviate to T . In order to enforce compliance by player 1,let us think of punishments/rewards in the second stage. Of the three NashEquilibria, (M,C) ((T,L)) is that which gives player 1 his/her lowest (highest)payo. So let us devise the following strategies:-In the first stage: play (B,R)-In the second stage: if (B,R) obtained in the first stage, play (T,L); oth-

    erwise, play (M,C)

    Then, we obtain the following payo matrix as a function of first-periodstrategy:

    L C RT (4, 3) (1, 2) (6, 2)M (3, 3) (2, 4) (4, 3)B (2, 4) (1, 3) (7, 5)

    There, we see that neither player 1 nor player 2 has an incentive to deviatefrom their prescribed strategy.We have thus constructed a subgame perfect Nash Equilibrium with (B,R)

    as the outcome of the first stage.

    1