Dominance and Replay - Zero-Sum Games: Minimax is Duality

Dominance is a quick reduction before solving a game. A dominated column is never useful for the minimizer, so removing it can expose a smaller game with the same value. The lesson uses dominance as a certificate that an option can be removed before replaying the exact minimax calculation.

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Dominated Column

Column 3 is dominated by column 1. Why: the minimizer never prefers a column that is entrywise no smaller. A dominated column gives the row player at least as much payoff in every row, so it cannot improve the column player's worst-case protection.

\text{dominated column}=3

Entrywise Check

The top-row comparison is 3 at most 4, and the bottom-row comparison is -2 at most 3. Why: both rows favor removing the dominated column for the minimizer. The check is entrywise, so it does not depend on guessing the opponent's mixture or solving the game first.

3\le 4\quad -2\le 3

Replay the Reduced Game

After removing that column, the reduced game has value 1/2. Why: the remaining matrix is the same exact game solved earlier. Dominance is therefore a preprocessing certificate: it removes an option that cannot be part of an optimal minimizing strategy, then minimax replays on the smaller matrix.

v_{\text{reduced}}=1/2

Diagram note

These are exact values for pinned finite two-player zero-sum games; general-sum equilibria and learning dynamics are outside this model. Dominance here is an exact entrywise reduction for this finite zero-sum payoff matrix, and replaying minimax after the reduction keeps the same value. It is not a claim about general-sum Nash equilibria or learning dynamics. Pixel positions are rounded for layout; every number shown is exact.

\text{dominance removes an option before replaying minimax}