Page 204 - Probability Demystified

CHAPTER 11 Game Theory 193

3
The optimal strategy for Player A is to play the black card 10 of the time
and the white card 7 of the time. The optimal strategy for Player B is the
10
same in this case.
The optimal strategy for Player A is defined as a strategy that can
guarantee him an average payoff of V(the value of the game) no matter what
strategy Player B uses. The optimal strategy for Player B is defined as a
strategy that prevents Player A from obtaining an average payoff greater
than V(the value of the game) no matter what strategy Player A uses.

Note: When a player selects one strategy some of the time and another
strategy at other times, it is called a mixed strategy, as opposed to using the
same strategy all of the time. When the same strategy is used all of the time,
it is called a pure strategy.

EXAMPLE: Two generals, A and B, decide to play a game. General A can
attack General B’s city either by land or by sea. General B can defend either
by land or sea. They agree on the following payoff.

General B (defend)

Land Sea

General A (attack) Land $25 $75

Sea $90 $50

Find the optimal strategy for each player and the value of the game.

SOLUTION:

Let p ¼ probability of attacking by land and 1 p ¼ the probability of attack-
ing by sea.