Game Expectation

Let the elements in a Payoff Matrix be denoted , where the s are player A's Strategies and the s are player B's Strategies. Player A can get at least

 (1)

for Strategy . Player B can force player A to get no more than for a Strategy . The best Strategy for player A is therefore
 (2)

and the best Strategy for player B is
 (3)

In general,
 (4)

Equality holds only if a Saddle Point is present, in which case the quantity is called the Value of the game.