Enter Payoff Matrix
Use row payoffs for a two-player zero-sum game. Enter values by row. Separate numbers with spaces or commas.
Example Data Table
This sample payoff matrix represents the row player’s gains in a zero-sum game.
| Strategy | C1 | C2 | C3 |
|---|---|---|---|
| R1 | 3 | -1 | 2 |
| R2 | 0 | 4 | -2 |
| R3 | 1 | 2 | 1 |
Click Use Example Matrix to load these values into the calculator instantly.
Formula Used
For a zero-sum payoff matrix A, the row player solves:
Maximize v
Subject to pTA ≥ v1T
pi ≥ 0 for all i
Σpi = 1
The column player solves the dual problem:
Minimize u
Subject to Aq ≤ u1
qj ≥ 0 for all j
Σqj = 1
At equilibrium, u = v. The solver searches binding constraints exactly through support enumeration, then falls back to an iterative approximation only when needed.
How to Use This Calculator
- Choose the number of row and column strategies.
- Optionally enter custom labels like Attack, Defend, Hold.
- Paste the payoff matrix using one row per line.
- Use positive or negative payoffs as needed.
- Set your preferred decimal precision.
- Click Solve Game to calculate equilibrium results.
- Review the game value, mixed strategies, dominance findings, and strategy chart.
- Download CSV or PDF reports for documentation or sharing.
Frequently Asked Questions
1) What does this calculator solve?
It solves two-player zero-sum matrix games. It estimates optimal mixed strategies for both players, the equilibrium game value, pure dominance findings, and stability indicators using linear programming logic.
2) What do the matrix values represent?
Each cell is the row player’s payoff when a specific row strategy meets a specific column strategy. Because the game is zero-sum, the column player’s payoff is the negative of that value.
3) Can I use negative payoffs?
Yes. Negative values are valid and common. They usually represent losses for the row player and gains for the column player in the same position.
4) What is the game value?
The game value is the expected payoff when both players use optimal strategies. It represents the long-run average result for the row player if rational play continues repeatedly.
5) Why are mixed strategies important?
Mixed strategies matter when no pure saddle point exists. Randomizing across strategies prevents exploitation and produces a balanced equilibrium between the players.
6) What does pure dominance mean?
A pure strategy is dominated if another pure strategy always performs better for the same player. Dominated strategies can often be removed before deeper analysis.
7) Is the solution always exact?
The calculator first attempts an exact equilibrium using support-based equalities from the linear program. If that fails because of degeneracy, it uses a strong iterative fallback approximation.
8) What matrix size is supported?
This version supports payoff matrices from 2×2 up to 5×5. That range balances practical flexibility, responsive layout, and reliable server-side solving in one file.