# ps4sol - 14.12 Game Theory Muhamet Yildiz Fall 2005...

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14.12 Game Theory Muhamet Yildiz Fall 2005 Solution to Homework 4 1. Note that (A,L) is a Nash equilibrium of the stage game. Thus, for the game where the stage game is repeated Fve times, to play (A,L) in each period is a strategy proFle that is subgame-perfect: neither player has an incentive to deviate in any period. Similarly, since (B,R) is a Nash equilibrium of the stage game, to play (B,R) in each period is a strategy proFle that is subgame-perfect. Note that (B,L) is not a Nash equilibrium of the stage game. Both players have an incentive to deviate from (B,L). Therefore, to ensure that they play (B,L) in the Frst period, it is necessary to punish them in future periods if they deviate. Consider the following strategy proFle for the 5-period game. play (B,L) in period 1; if (B,L) or (A,R) was played in period 1, play (A,L) in periods 2 and 3 and (B,R) in periods 4 and 5; if (A,L) was played in period 1, play (B,R) in periods 2-5; if (B,R) was played in period 1, play (A,L) in periods 2-5. To check if this strategy proFle is subgame-perfect, we can apply the single- deviation principle: we check, for each information set, if a player can gain by deviating in that period but following the prescribed strategy in future periods. At all information sets in periods 2 through 5, the players are required to play a Nash equilibrium of the stage game. Therefore, the players do not have an incentive to deviate in these periods. (and such deviation does not lead to future gains). As for period 1, if player 1 deviates he gains 2 in the current period but loses 2 in the future as the strategy proFle requires them to play (BR, BR, BR, BR) instead of (AL, AL, BR, BR). Similarly, if player 2 deviates in period 1, she gains 2 in the current period but loses 2 in the future as the strategy proFle requires them to play (AL, AL, AL, AL) instead of (AL, AL, BR, BR). Therefore, neither player can proFt from a single-period deviation. Therefore, the strategy proFle described above is subgame-perfect. In the same manner, we can construct a subgame-perfect equilibrum where the players play (A,R) in the Frst period. 2. In each of the following cases, we apply the single deviation principle to check if the given strategy proFle is subgame-perfect: 1

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(a) Suppose Goliath Software has updated X every time a startup produced a browser in the past. If the current startup produces a browser instead of a search engine, and Goliath updates ac- cording to its strategy, the startup would receive 0 instead of 1. Therefore, the startup loses by deviating. Suppose Goliath Software has updated X every time a startup produced a browser in the past. If Goliath chooses not to update after the startup has developed a browser, but all parties play according to the given strategy proFle in future periods, then the payo± to Goliath equals 2+2(0 . 9) + 2 (0 . 9) 2 + ...
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ps4sol - 14.12 Game Theory Muhamet Yildiz Fall 2005...

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