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