Calculator
Enter a 2×2 zero-sum payoff matrix for the row player. Results appear above this form after submission.
Plotly Graph
The chart tracks the row player's expected payoff for each pure row as the column player changes the probability of choosing the first column.
Example Data Table
This example uses an interior mixed equilibrium.
| Row \ Column | Column Strategy 1 | Column Strategy 2 |
|---|---|---|
| Row Strategy 1 | 4 | 1 |
| Row Strategy 2 | 0 | 3 |
For this matrix, the row player mixes the first row with probability 0.5, the column player mixes the first column with probability 0.3333, and the game value is 2.
Formula Used
[ [a, b], [c, d] ]
Let p be the probability of the first row. Then the second row has probability 1 − p.
Column payoff equalization for the row player gives:
p = (d − c) / (a − b − c + d)
Let q be the probability of the first column. Then the second column has probability 1 − q.
Row payoff equalization gives:
q = (d − b) / (a − b − c + d)
V = (ad − bc) / (a − b − c + d)
Also check pure equilibrium conditions using:
maximin = max(min(a,b), min(c,d))
minimax = min(max(a,c), max(b,d))
How to Use This Calculator
- Enter names for the two row strategies and two column strategies.
- Fill in the four payoff values for the row player’s 2×2 matrix.
- Select the decimal precision you want in the output.
- Press Calculate Mixed Strategy.
- Read the result summary above the form for equilibrium type, value, and probabilities.
- Review the graph to see how the row payoffs move as the column mix changes.
- Export the calculation with the CSV or PDF buttons.
FAQs
1. What does this calculator solve?
It solves a 2×2 zero-sum payoff matrix for the row player. It checks for a saddle point, finds mixed-strategy probabilities when needed, and reports the game value.
2. What are mixed strategies?
Mixed strategies assign probabilities to pure actions. Instead of always choosing one row or column, a player randomizes so the opponent cannot exploit predictable behavior.
3. What do p and q represent?
The symbol p is the row player’s probability of choosing the first row. The symbol q is the column player’s probability of choosing the first column.
4. Why might the result show a pure equilibrium?
If maximin equals minimax, the matrix already has a saddle point. In that case, neither player needs an interior mixed strategy because one pure outcome is stable.
5. Can I use negative or decimal payoffs?
Yes. The calculator accepts integers, decimals, and negative values. That makes it useful for gains, losses, costs, penalties, and other row-player payoff setups.
6. What is the game value?
The game value is the expected payoff to the row player when both players use optimal strategies. In a zero-sum game, the column player’s expected value is the negative counterpart.
7. What does the graph show?
It shows the row player’s expected payoff for each pure row as the column player changes the probability of using the first column. Their intersection indicates an indifference point.
8. What do the CSV and PDF buttons export?
They export the entered matrix, equilibrium type, value, and strategy summary from the current result. This is helpful for saving calculations or sharing them in reports.