Study zero sum choices with dependable row guards. Find saddle points and secure values quickly. Plot outcomes for clearer planning across repeated matrix decisions.
For a payoff matrix \(A = [a_{ij}]\), the row player uses the maximin rule and the column player uses the minimax rule. The row security value for each row is the minimum number in that row. The column pressure value for each column is the maximum number in that column.
Maximin: maxi minj aij
Minimax: minj maxi aij
If maximin equals minimax, a saddle point exists and a pure strategy equilibrium is present. If they differ, no pure saddle point exists. For a 2 by 2 game without a saddle point, this page also computes the mixed strategy probabilities:
Row 1 probability: (d - c) / (a - b - c + d)
Column 1 probability: (d - b) / (a - b - c + d)
Mixed strategy value: (ad - bc) / (a - b - c + d)
| Strategy | Col 1 | Col 2 | Col 3 | Row Minimum |
|---|---|---|---|---|
| Row 1 | 4 | 1 | 3 | 1 |
| Row 2 | 2 | 5 | 0 | 0 |
| Row 3 | 6 | 2 | 4 | 2 |
| Column Maximum | 6 | 5 | 4 | 4 |
In this example, the row minima are 1, 0, and 2, so the maximin is 2. The column maxima are 6, 5, and 4, so the minimax is 4. Because they differ, there is no pure saddle point.
Minimax analysis helps you study competitive decisions when one side wants to maximize payoff and the other side wants to reduce it. It is common in game theory, economic choice, operations research, matrix games, and strategic planning. A payoff matrix makes each possible outcome visible before any move is chosen.
The maximizing player first studies the worst outcome in every row. Those worst outcomes are row minima. The best of these worst outcomes is the maximin value. The minimizing player studies the worst threat in every column. Those threats are column maxima. The smallest of them is the minimax value.
When maximin equals minimax, both players can settle on pure strategies without regret. That shared value is the saddle point value of the game. It shows a stable decision. Neither side can improve by changing alone once the saddle point is reached. This is the clearest result in a matrix game.
If the two values do not match, the game has no pure equilibrium. That means a single fixed row or column is not fully stable. The solver still shows a guaranteed lower bound and upper bound. For 2 by 2 matrices, it also estimates mixed strategy probabilities so each player can randomize efficiently.
A dominated row is never better than another row. A dominated column is never safer than another column. Removing dominated choices can simplify the decision structure and make later analysis easier. Even when the full mixed solution is not shown for larger matrices, dominance still helps you identify weak strategies quickly.
The graph compares row minima against column maxima. Large gaps usually suggest uncertainty or the need for mixed play. A direct meeting point between the two sets supports a pure strategy saddle point. This visual layer makes classroom examples, strategic audits, and decision reviews easier to explain.
It calculates row minima, column maxima, maximin, minimax, saddle points, and dominance checks. For suitable 2 by 2 games, it also computes mixed strategy probabilities and expected game value.
A saddle point is a matrix entry that is both the minimum of its row and the maximum of its column. It appears when maximin equals minimax, which creates a stable pure strategy equilibrium.
Row minima measure the worst outcome the maximizing player may face in each row. Comparing them helps identify the safest row under cautious play.
Column maxima show the largest payoff the minimizing player could allow in each column. The minimizing player prefers the column with the smallest of these values.
This version gives exact mixed strategy probabilities for 2 by 2 games when no saddle point exists. Larger matrices still receive minimax bounds and dominance diagnostics.
A strategy is dominated when another strategy is always at least as good and sometimes better. Dominated strategies are usually poor choices and can often be removed from consideration.
Yes. The matrix is treated as a zero sum payoff table for the maximizing player. The minimizing player is assumed to lose the same amount the maximizer gains.
Use CSV when you want spreadsheet friendly data. Use PDF when you want a printable result sheet for class notes, reports, or strategy reviews.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.