Game Input Form
Use the responsive input grid below. Large screens show three columns, smaller screens show two, and mobile shows one.
Example Data Table
This sample shows a 3×3 payoff matrix for the row player.
| Strategy | C1 | C2 | C3 |
|---|---|---|---|
| R1 | 4 | -1 | 3 |
| R2 | 2 | 0 | 5 |
| R3 | 1 | -2 | 2 |
Formula Used
For pure strategy testing, compute the minimum payoff in each row and the maximum payoff in each column. The row player selects the maximum of row minima, while the column player selects the minimum of column maxima.
If maximin equals minimax, the game has a saddle point and the common value is the game value.
For a reduced 2×2 payoff matrix [[a, b], [c, d]], the mixed strategy formulas are:
- Row player probability of first row: p = (d - c) / (a - b - c + d)
- Column player probability of first column: q = (d - b) / (a - b - c + d)
- Game value: v = (ad - bc) / (a - b - c + d)
For larger games without a simple closed form, this calculator also uses fictitious play. The repeated best-response process estimates mixed strategies and narrows upper and lower value bounds.
How to Use This Calculator
- Select the number of row and column strategies.
- Enter the payoff matrix values for the row player.
- Choose an iteration count for the approximate mixed strategy solver.
- Click Solve Game to display results above the form.
- Review maximin, minimax, saddle point status, dominance notes, exact 2×2 results, and iterative estimates.
- Use the CSV or PDF buttons in the results area to save your outputs.
Frequently Asked Questions
1. What is a zero sum game?
A zero sum game is a competitive situation where one player’s gain equals the other player’s loss. The total payoff across players always balances to zero.
2. What does the payoff matrix represent?
The matrix lists payoffs to the row player for every strategy combination. Positive numbers favor the row player, while negative numbers favor the column player.
3. What is a saddle point?
A saddle point exists when the row player’s maximin equals the column player’s minimax. In that case, both players can use pure strategies and the game value is exact.
4. Why does the calculator remove dominated strategies?
Dominated strategies are never rational choices because another strategy always performs better. Removing them simplifies the matrix and can reveal a smaller equivalent game.
5. When is the exact mixed strategy shown?
The exact closed-form mixed strategy appears when the reduced game becomes a valid 2×2 matrix. Larger games generally need iterative or linear programming methods.
6. What is fictitious play?
Fictitious play is an iterative best-response method. Each player reacts to the opponent’s historical choices, and the observed frequencies can approximate equilibrium strategies.
7. How many iterations should I use?
More iterations usually improve the stability of the mixed strategy estimate. Start with 500 and increase the count when upper and lower bounds remain far apart.
8. Can this replace a full linear programming solver?
It is very useful for learning, diagnostics, and many practical cases. However, exact solutions for larger games are better handled by formal linear programming techniques.