Minimax Is Strong Duality - Zero-Sum Games: Minimax is Duality

The minimax equality is the strong-duality equality of the two envelope LPs. Complementary slackness then explains why the optimal supports are active. The payoff is conceptual: the row player's guarantee and the column player's cap are not two different answers, but the same LP optimum seen from opposite sides.

highlighted = computed this step

Strong Duality

The row guarantee and column cap both equal 7/2 in the shifted game. Why: the two envelope LPs are primal and dual. The row side raises a floor, the column side lowers a ceiling, and strong duality says those two exact bounds meet for this finite zero-sum game.

v^*=w^*=7/2

Original Value

Subtract shift 3 to get the original game value 1/2. Why: the shift changed every payoff by the same constant. The shifted equality is useful for the LP proof, but the actual fair price of the original game is the value after this subtraction.

7/2 - 3 = 1/2

Optimal Mixes

The row mix is 1/2, 1/2 and the column mix is 3/8, 5/8. Why: complementary slackness pins the active support. The positive probabilities sit on payoff constraints that are tight, so the strategies and equalized payoffs tell the same story.

x^*=1/2,1/2\quad y^*=3/8,5/8

Complementary Slackness

There are 4 zero complementary-slackness products. Why: all active payoff constraints are tight in this fully mixed solution. A zero product is the certificate form of the idea that unused slack and positive strategic weight cannot both appear on an active support.

\text{CS products}= 4\text{ zeros}

Diagram note

Minimax here is LP strong duality for this finite zero-sum game; the diagram does not claim anything about general-sum games. The shifted equality proves the shifted value, and subtracting the shift gives the original value. No learning dynamics or behavioral convergence claim is being made. Pixel positions are rounded for layout; every number shown is exact.

\text{minimax equality is duality equality}