Game Expectation

Let the elements in a Payoff Matrix be denoted $a_{ij}$, where the $i$s are player A's Strategies and the $j$s are player B's Strategies. Player A can get at least

$$\min_{j\leq n} a_{ij}$$ (1)

for Strategy $i$. Player B can force player A to get no more than $\max_{j\leq m} a_{ij}$ for a Strategy $j$. The best Strategy for player A is therefore

$$\max_{i\leq m}\min_{j\leq n} a_{ij},$$ (2)

and the best Strategy for player B is

$$\min_{j\leq n}\max_{i\leq m} a_{ij}.$$ (3)

In general,

$$\max_{i\leq m} \min_{j\leq n} a_{ij} \leq \min_{j\leq n}\max_{i\leq m} a_{ij}.$$ (4)

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

See also Game, Payoff Matrix, Saddle Point (Game), Strategy, Value