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 subgameperfect: 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 subgameperfect.
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
5period 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 25;
–
if (B,R) was played in period 1, play (A,L) in periods 25.
To check if this strategy proFle is subgameperfect, 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 singleperiod deviation. Therefore, the
strategy proFle described above is subgameperfect.
In the same manner, we can construct a subgameperfect 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 subgameperfect:
1
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•
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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 Spring '05
 MuhammadYildiz
 Game Theory, Period, Goliath, Goliath Software

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