Zero-Sum Game Solver (Saddle Point / Linear Programming) | Online Free Tool | Elysia Tools
Elysia Tools
Navigation
Math & Numbers
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.
Execution
Run this tool
Fill in the form, run the tool, and review the result in one place.
Samples
Examples that match this tool
Related
Continue with connected tools and hubs
Result
Ready for a run
Run the tool to preview files, text, structured data, or streamed output here.
Tool usage guide
Learn when to use this tool, what it supports, and how real users apply it.
Key facts
Category
Math & Numbers
Input types
textarea, number
Output type
text
Sample coverage
4
API ready
Yes
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.
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
1Enter the payoff matrix with rows representing the maximizing row player's payoffs and columns representing the minimizing column player's choices.
2The solver computes row minima (maximin) and column maxima (minimax) to detect pure-strategy saddle points.
3If 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.
4The 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.
A student needs to solve the standard 2x2 matching pennies game to identify the equilibrium and randomized strategies.
Problem
Verify that no pure-strategy saddle point exists and compute the exact mixed strategy probabilities for both players.
How to use
Enter the 2x2 payoff matrix into the text area and set decimal precision to 4.
matrix: "1, -1\n-1, 1"
decimalPlaces: 4
Outcome
Confirms maximin (-1) < minimax (1), and outputs optimal 50/50 mixed strategy probabilities with an expected game value of 0.
2. Skew-Symmetric 3x3 Matrix Game
Operations Research Analyst
Background
An analyst is evaluating a 3-strategy competitive interaction represented by a skew-symmetric payoff matrix.
Problem
Determine optimal strategy vectors and confirm that both players share identical guarantees in a fair game.
How to use
Paste the 3x3 matrix into the payoff matrix input and run the linear programming solver.
FAQ
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?
All matrix values represent payoffs to the row player, which the row player maximizes and the column player minimizes.