The Column LP Is the Dual - Zero-Sum Games: Minimax is Duality

The column player solves the matching minimax envelope. Its optimum agrees with the row player's guarantee because the two LPs are a primal-dual pair. This lesson interprets the column LP as the same fair price seen from the opponent's cap rather than from the row player's floor.

highlighted = computed this step

Column Envelope

The column player chooses q and minimizes w above both shifted row-payoff lines. Why: w is the cap on the row player's payoff, so it must be at least as large as either pure row's expected payoff. The column player's problem is the same conflict seen from the minimizing side.

\min w\quad w\ge 4q + 2\quad w\ge - 4q + 5

Column Optimum

The optimal point is (3/8, 7/2). Why: the minimizer's best cap is also at the intersection of the two active payoff bounds. At that point the row player is indifferent between the active rows, so no pure row can break the cap.

(q^*,w^*)=(3/8, 7/2)

Duality

The duality certificate gives shifted objective 7/2 on both sides. Why: the column player's cap matches the row player's guarantee. This is the minimax theorem in LP language: the maximizing lower bound and minimizing upper bound close to the same exact number.

z_{\text{row}}=z_{\text{column}}=7/2

Original Value

Subtracting shift 3 gives original value 1/2. Why: the LP was solved on the shifted matrix, so the last step must undo the bookkeeping. The shift made the envelope convenient; it did not change the original strategic value.

7/2 - 3 = 1/2

Diagram note

The column LP diagram and duality table are both recomputed from the shifted envelope formulation. Read the equal shifted objectives as the same optimum from both players' sides, then read the subtraction as the return to the original payoff scale. Pixel positions are rounded for layout; every number shown is exact.

\text{minimax is now an LP equality}