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TL;DR

In two-player zero-sum games, the maximum of the row player's minimum guaranteed payoff equals the minimum of the column player's maximum loss: \( \max_x \min_y x^T A y = \min_y \max_x x^T A y \).

By Valenke Exam Prep Team·Last updated 2026-06-18

Minimax Theorem

In two-player zero-sum games, the maximum of the row player's minimum guaranteed payoff equals the minimum of the column player's maximum loss: \( \max_x \min_y x^T A y = \min_y \max_x x^T A y \).

Why it matters for interviews

The minimax theorem is foundational for optimal decision-making under adversarial conditions. It connects to robust optimization, worst-case portfolio design, and competitive trading strategies.

Definition and Mathematical Foundation

In two-player zero-sum games, the maximum of the row player's minimum guaranteed payoff equals the minimum of the column player's maximum loss: \( \max_x \min_y x^T A y = \min_y \max_x x^T A y \).

Application in Quantitative Finance

The minimax theorem is foundational for optimal decision-making under adversarial conditions. It connects to robust optimization, worst-case portfolio design, and competitive trading strategies.

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Frequently Asked Questions

What is a saddle point in a game matrix?▼
An entry that is simultaneously the minimum of its row and the maximum of its column. If a saddle point exists, it is the game value and no mixing is needed.
How does the minimax theorem relate to linear programming?▼
Solving a zero-sum game is equivalent to a linear program: the row player maximizes their minimum payoff subject to strategy constraints. LP duality proves the minimax theorem.
Does the minimax theorem extend to continuous games?▼
Yes, under compactness and continuity conditions (Sion's minimax theorem). This is relevant to continuous auction design and robust portfolio optimization over continuous parameter spaces.