Calculator Inputs
This tool solves a 2×2 zero-sum game using the row player payoff matrix.
Example Data Table
Use the following sample matrix to test a valid interior mixed-strategy equilibrium.
| Scenario | Top-Left (a) | Top-Right (b) | Bottom-Left (c) | Bottom-Right (d) | Row mix on Top | Column mix on Left | Game value |
|---|---|---|---|---|---|---|---|
| Sample 2×2 zero-sum game | 4 | 1 | 0 | 3 | 50% | 33.33% | 2 |
| Matching pennies style test | 1 | -1 | -1 | 1 | 50% | 50% | 0 |
Formula Used
Let the row player's payoff matrix be:
[ c d ]
For a 2×2 zero-sum game, the standard interior mixed-strategy formulas are:
Here, p is the probability that the row player chooses the first row, q is the probability that the column player chooses the first column, and v is the expected value to the row player.
The calculator also checks row maximin, column minimax, saddle points, and simple dominance patterns.
How to Use This Calculator
- Enter names for the two row strategies and two column strategies.
- Fill the four payoff cells using the row player’s payoff values.
- Press Calculate Mixed Strategy to solve the game.
- Read the result cards shown above the form.
- Check whether the tool found an interior mixed equilibrium or a pure saddle point.
- Use the CSV or PDF buttons to export your matrix and result summary.
Frequently Asked Questions
1) What kind of game does this calculator solve?
It solves a 2×2 zero-sum game using the row player’s payoff matrix. The column player is assumed to gain the exact opposite value.
2) What does mixed strategy mean?
A mixed strategy assigns probabilities to pure actions. Instead of always choosing one move, a player randomizes between available options.
3) Why can probabilities become negative or exceed one?
That usually means no interior mixed equilibrium exists for the entered matrix. The actual solution may sit on a boundary or appear as a pure strategy equilibrium.
4) What is the game value?
The game value is the expected payoff to the row player when both players use equilibrium strategies. In zero-sum games, the column player gets the opposite value.
5) What is a saddle point?
A saddle point is a pure strategy outcome where maximin equals minimax. When it exists, players need not randomize to achieve equilibrium.
6) Why does the calculator ask only for one payoff matrix?
Because zero-sum games need only one matrix. The other player’s payoff is the negative of the row player’s payoff at every cell.
7) Can I use decimals or negative numbers?
Yes. The tool accepts integers, decimals, and negative values, which is useful for penalty models, losses, and normalized payoff systems.
8) Can this handle larger games?
No. This version is designed specifically for 2×2 games. Larger matrices require linear programming or more advanced equilibrium methods.