Calculator
Enter payoffs as ordered pairs. The first value belongs to the row player. The second belongs to the column player.
Example Data Table
This example uses a classic prisoner’s dilemma style game. It contains one pure Nash equilibrium at Defect, Defect.
| Row \ Column | Cooperate | Defect |
|---|---|---|
| Cooperate | (3, 3) | (0, 5) |
| Defect | (5, 0) | (1, 1) |
Formula Used
1) Best Response Rule
A strategy is a best response when it gives the highest payoff against the opponent’s chosen strategy. The calculator checks maxima column-by-column for the row player and row-by-row for the column player.
2) Pure Nash Equilibrium Condition
A payoff cell is a pure Nash equilibrium when both players are simultaneously playing best responses. In symbols, a profile (s₁, s₂) is stable if neither player can improve by deviating alone.
3) Mixed Equilibrium for 2×2 Games
For a 2×2 payoff matrix with row payoffs a₁₁, a₁₂, a₂₁, a₂₂ and column payoffs b₁₁, b₁₂, b₂₁, b₂₂:
Probability that the row player chooses the first row:
p = (b₂₂ - b₂₁) / (b₁₁ - b₁₂ - b₂₁ + b₂₂)
Probability that the column player chooses the first column:
q = (a₂₂ - a₁₂) / (a₁₁ - a₁₂ - a₂₁ + a₂₂)
4) Pareto Efficiency Check
An outcome is Pareto efficient if no other outcome improves one player without hurting the other. The tool scans every cell against every alternative cell.
How to Use This Calculator
- Choose the number of row and column strategies.
- Enter optional strategy labels, separated by commas.
- Fill each cell with two payoffs: row player first, column player second.
- Click Update Grid after changing matrix size.
- Click Analyze Game to calculate pure equilibria, best responses, dominated strategies, Pareto efficient outcomes, and a mixed solution for 2×2 games.
- Use the CSV or PDF buttons after analysis to save your result summary.
FAQs
1) What does this calculator find?
It identifies pure Nash equilibria, best responses, strictly dominated pure strategies, Pareto efficient outcomes, and mixed equilibrium probabilities for 2×2 games.
2) Can I use negative or decimal payoffs?
Yes. The inputs accept integers, decimals, and negative values, which is useful for costs, losses, penalties, and utility differences.
3) Does it support more than two players?
No. This version is built for two-player normal-form games. Each cell stores one payoff for the row player and one for the column player.
4) What happens when there are ties?
Tied maximum payoffs are treated as multiple best responses. Because of that, the calculator can report more than one equilibrium or more than one best response.
5) Why is no pure equilibrium shown?
Some games only have mixed equilibria, or your matrix may create cycles in best responses. In those cases, no cell satisfies both players simultaneously.
6) What does strictly dominated mean here?
A strategy is strictly dominated when another pure strategy gives a higher payoff in every comparable case. This version checks pure-strategy domination only.
7) When is the mixed result available?
The mixed calculation appears only for 2×2 games. It also requires a nondegenerate indifference equation and probabilities that stay between zero and one.
8) What do CSV and PDF exports include?
They include the current analysis summary, best-response information, payoff details, equilibrium findings, and mixed-strategy notes when applicable.