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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 2026-06-03

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.