Saddle Point

Maximin Criterion

The row player picks the strategy maximizing their guaranteed minimum payoff: $\max_i \min_j a_{ij}$.

Minimax Criterion

The column player picks the strategy minimizing the row player’s guaranteed maximum payoff: $\min_j \max_i a_{ij}$.

Saddle Point

Exists when $\max_i \min_j a_{ij} = \min_j \max_i a_{ij}$. The corresponding entry is the game’s value, and both players have an optimal pure strategy: neither benefits from deviating alone.

If the maximin and minimax values differ, no saddle point exists, and pure strategies alone don’t give an optimal solution.