## Graphical Representation of Method to Find Optimal Strategy for Two Player Game

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

If player B chooses strategy U with probability and strategy V with probability and if A chooses strategy Y, the expected gain for B is If A chooses strategy Z, the expected gain for B is We can plot these two functions. The point of intersection will give the proportion of the time that player B choose strategy U. The optimal value for occurs 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 probability and 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 We can plot these. The intersection give give the the proportion of games in which player A should play strategy Y. The optimal value for which minimises the losses for play A, occurs when these expected losses are equal, so  