Saddle Points - Zero-Sum Games: Minimax is Duality

A saddle point appears when the row player's guarantee equals the column player's cap. In that case, pure strategies already certify the game value. The lesson contrasts that settled case with the flagship gap, where a predictable pure move can be exploited.

highlighted = computed this step

Saddle Condition

In the saddle example, maxmin and minmax both equal 2. Why: neither player can improve by switching from the saddle cell, so the adversarial guarantee from the row side and the cap from the column side meet at one pure outcome. When that happens, randomizing is unnecessary for this matrix.

\max\min=\min\max=2

Saddle Cell

The saddle is at row 1 and column 2. Why: that cell is simultaneously a row minimum and a column maximum. The row player can choose the row and guarantee the value, while the column player can choose the column and prevent anything larger.

\text{saddle value}=2

Contrast With Gap

The first game has maxmin -1 and minmax 2. Why: the gap is exactly why mixing is needed there. A fixed row can be punished by a worst-case column, and a fixed column can be punished by a best-response row, so pure strategies do not settle the value.

\text{gap game: }\max\min\ne\min\max

Diagram note

The saddle cell is a pure-strategy certificate for this matrix only. It certifies a finite zero-sum game because the two security levels match; it does not say that unrelated games or learning processes have been solved. The key interpretation is that both players can point to the same cell as their guarantee. Pixel positions are rounded for layout; every number shown is exact.

\text{matching security levels certify a saddle}