Chapter 29
NAME
Game Applications
Introduction.
As we have seen, some games do not have a “Nash equi
librium in pure strategies.” But if we allow for the possibility of “Nash
equilibrium in mixed strategies,” virtually every game of the sort we are
interested in will have a Nash equilibrium.
The key to solving for such equilibria is to observe that if a player is
indiFerent between two strategies, then he is also willing to choose ran
domly between them. This observation will generally give us an equation
that determines the equilibrium.
Example:
In the game of baseball, a pitcher throws a ball towards a batter
who tries to hit it. In our simpli±ed version of the game, the pitcher can
pitch high or pitch low, and the batter can swing high or swing low. The
ball moves so fast that the batter has to commit to swinging high or
swinging low before the ball is released.
Let us suppose that if the pitcher throws high and batter swings low,
or the pitcher throws low and the batter swings high, the batter misses
the ball, so the pitcher wins.
If the pitcher throws high and the batter swings high, the batter
always connects. If the pitcher throws low and the batter swings low, the
batter will connect only half the time.
This story leads us to the following payoF matrix, where if the batter
hits the ball he gets a payoF of 1 and the pitcher gets 0, and if the batter
misses, the pitcher gets a payoF of 1 and the batter gets 0.
Simplifed Baseball
Pitcher
Batter
Swing Low
Swing High
Pitch High
1
,
0
0
,
1
Pitch Low
.
5
,.
5
1
,
0
This game has no Nash equilibrium in pure strategies. There is no
combination of actions taken with certainty such that each is making the
best response to the other’s action. The batter always wants to swing
the same place the pitcher throws, and the pitcher always wants to throw
to the opposite place. What we can ±nd is a pair of equilibrium mixed
strategies.
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GAME APPLICATIONS
(Ch. 29)
In a mixed strategy equilibrium each player’s strategy is chosen at
random. The batter will be willing to choose a random strategy only if
the expected payoF to swinging high is the same as the expected payoF
to swinging low.
The payoFs from swinging high or swinging low depend on what the
pitcher does. Let
π
P
be the probability that the pitcher throws high and
1
−
π
P
be the probability that he throws low. The batter realizes that if
he swings high, he will get a payoF of 0 if the pitcher throws low and 1
if the pitcher throws high. The expected payoF to the batter is therefore
π
P
.
If the pitcher throws low, then the only way the batter can score is
if pitcher pitches low, which happens with probability 1
−
π
P
.Ev
enthen
the batter only connects half the time. So the expected payoF to the
batter from swinging low is
.
5(1
−
π
P
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 Summer '09
 JOHNG.SESSIONS
 Microeconomics, Equilibrium, Game Theory

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