# Optimal Strategies in Games That Are Not Stable When Players Have Two Possible Strategies Each

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If a player uses the same strategy in every game, he is using a ‘pure strategy’. Pure strategies provide the best results for both players in a game which has a stable solution. If no stable solution exists, each player will maximise his winnings by adopting a mixed strategy in which each available strategy is used a certain proportion of the time.

Suppose we have the payoff matrix below for player B in a two player game between players A and B.

 Payoff Matrix for B A Y Z B U 7 -4 V -5 3

Suppose that player B chooses strategy U with probabilityand strategy V with probability

Then if A chooses strategy Y, the expected gain for B is

If A chooses strategy Z, the expected gain for B is

The optimal value foroccurs when these expected gains are equal, so

We can find the probability that player A should choose strategies Y and Z similarly.

Suppose that player A chooses strategy Y with probabilityand strategy Z with probability

Then if B chooses strategy U, the expected loss for A is

If B chooses strategy V, the expected loss for A is

The optimal value foroccurs when these expected losses are equal, so

Hence player B should play strategy U with probabilityand player A should play strategy Y with probability