Nash Equilibria in Mixed Strategies
Here is a general method for finding Nash equilibria in mixed strategies. The key is to
think of a game as a list of functions which specify the payoffs for each of the players
as functions of the action taken by the given player and the actions taken by the other
player(s). A bestreply for a player then is a simply a choice that
maximizes
the payoff
of that player given the actions taken by the other players. A bestreply function is just
that function which describes the bestreply for the player as a function of the actions
taken by the other players. Nash equilibria are then combinations of strategies which
are mutually best replies to one another.
This "maximization" approach works for mixedstrategies as well as purestrategies  all
that's required is to think of what's being chosen in the right way. Let's consider the
game of chicken:
Dean
James
Swerve
Straight
qMix
Swerve
0
0
1
1
1q
q1
Straight
1
1
2
2
q2
3q2
pMix
p1
1p
3p2
p2
2+3p+q2pq
2+p+3q2pq
The "qMix" column gives the payoffs if Dean plays "swerve" with probability q and
"straight" with probability 1q. Similarly, the "pMix" row gives the payoffs if James
plays "swerve" with probability p and "straight" with probability 1p. The payoffs
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 Spring '09
 Physics, Game Theory, Dean

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