## A Two Player Three Strategy Game With a Saddle Point

A payoff matrix for a two player, three strategy game is shown below.
 A\B $B_1$ $B_2$ $B_3$ $A_1$ 1 2 3 $A_2$ 0 3 -1 $A_3$ -1 -2 4
For A, none of the strategies dominate the others. For B, no strategy has a reward that cannot be increased by choosing another strategy. This is an irreducible payoff matrix. A sees that if he adopts strategy
$A_1$
, the worst that can happen is that he will receive 1 - if B plays strategy
$B_1$
. The minimum reward for A from strategies
$A_2, \: A_3$
are -1 and -2 respectively.Since A wants to maximise his payoff, he chooses the strategy that from the minimum - the row maximin, and will play strategy
$A_1$
, for which the minimum payoff is 1.
B wants to minimise the amount he must pay to A. If he plays strategy
$B_1$
, the most he has to pay A is 1, if A plays strategy
$A_1$
. The maximum payoffs for B from playing strategies
$B_2, \: B_3$
respectively are 3 and 4 respectively - payoffs for B are the negative of those for A. B chooses the strategy that will minimise the amount he has to pay A. He will choose the minimum of the column max mums and will play strategy
$B_l$
, since the column maximum is minimum for this column and will pay A 1.
The maximum and minimum payments represent the values of the game to A and B respectively. When they are equal, the value of the game is the same for both and the corresponding entry in the payoff matrix is called a saddle point.
Every two player zero sum game with full information, so that each player knows what strategy the other player has chosen before he chooses his strategy, has a saddle point. If maximin for the rows is not equal to minimax for the columns, the value of the game is not certain. By introducing the expected value of the game, even games without full information can be shown to have a unique value.