# Zero-Sum Game Solver (Saddle Point / Linear Programming) Two-person zero-sum games: saddle-point test first, then mixed strategies from a single-phase simplex LP with dual shadow prices, verified by security-level checks. > Canonical page: https://elysiatools.com/en/tools/game-theory-zero-sum - **Category:** Math & Numbers - **Keywords:** game theory, zero-sum game, saddle point, mixed strategy, minimax, nash equilibrium, linear programming, shadow price, payoff matrix, operations research ## Overview Solve 2–6 × 2–6 two-person zero-sum matrix games by automatically testing for pure-strategy saddle points or computing optimal mixed strategies and game value using simplex linear programming and dual shadow prices. ## Inputs - **Payoff matrix (row player's gain, one row per line)** (textarea): Payoffs to the row player (positive = row wins). 2–6 rows × 2–6 columns; the column player minimizes. - **Decimal Places** (number) ## When to use - Evaluating competitive two-player zero-sum scenarios where one player's gain is the other player's loss. - Checking whether a payoff matrix has a pure strategy Nash equilibrium via maximin and minimax saddle point testing. - Calculating optimal randomized probability distributions and expected game value when no saddle point exists. ## How it works - Enter the payoff matrix with rows representing the maximizing row player's payoffs and columns representing the minimizing column player's choices. - The solver computes row minima (maximin) and column maxima (minimax) to detect pure-strategy saddle points. - If maximin is strictly less than minimax, the tool shifts payoffs to positive values and executes a single-phase simplex linear program to derive primal and dual mixed strategies. - The output presents optimal probability distributions for both players, the expected game value, and verified security-level bounds. ## Use cases - Operations research and game theory coursework requiring step-by-step zero-sum matrix solutions. - Strategic decision modeling for competitive zero-sum pricing or security defense scenarios. - Verifying minimax theorem conditions, dual shadow prices, and security level guarantees. ## Frequently asked questions ### What matrix dimensions are supported? The solver supports payoff matrices ranging from 2×2 up to 6×6 entries. ### How does the tool identify a saddle point? A saddle point exists when the maximum of the row minima (maximin) equals the minimum of the column maxima (minimax). ### Why does the solver shift matrix values during LP computation? Payoffs are shifted by a positive constant to make all matrix elements positive for single-phase simplex LP solving, after which the game value is shifted back. ### What does the game value represent? The game value is the expected payoff guaranteed to the row player when both players follow their optimal minimax strategies. ### Whose payoff perspective is represented in the matrix? 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