Study payoff matrices with confidence and clarity. Check maximin, minimax, and equilibrium value in seconds. Use practical exports for assignments, exams, revision, and teaching.
| Column 1 | Column 2 | |
|---|---|---|
| Row 1 | 4 | 1 |
| Row 2 | 2 | 3 |
Example maximin: 2
Example minimax: 3
Example optimal p: 0.25
Example optimal q: 0.50
Example game value: 2.50
Row minima: min(a, b) and min(c, d)
Maximin: max[min(a, b), min(c, d)]
Column maxima: max(a, c) and max(b, d)
Minimax: min[max(a, c), max(b, d)]
Saddle point rule: If maximin equals minimax, a pure strategy solution exists.
Mixed strategy denominator: a - b - c + d
Optimal p: (d - c) / (a - b - c + d)
Optimal q: (d - b) / (a - b - c + d)
Game value: (ad - bc) / (a - b - c + d)
Custom expected payoff: aqp + bp(1-q) + c(1-p)q + d(1-p)(1-q)
Enter the four payoffs for the 2x2 zero sum matrix.
Use the row player payoffs only.
Click the calculate button.
Read the maximin and minimax comparison first.
If both match, the game has a saddle point.
If they differ, check the mixed strategy result.
Optional custom probabilities let you test any strategy pair.
Use the export buttons to save the result summary.
Zero sum games describe direct conflict. One player gains exactly what the other loses. A zero sum game value calculator helps you study this conflict through a payoff matrix. This page focuses on a 2x2 matrix. It checks pure strategy stability first. It then solves the mixed strategy case. That makes the tool useful for homework, revision, classroom examples, and quick decision analysis.
The game value is the expected payoff when both players act optimally. It is the fair result of the contest. A positive value favors the row player. A negative value favors the column player. A zero value suggests balance. The calculator compares maximin and minimax values first. That step reveals whether a saddle point already exists. When those values match, the best answer comes from pure strategies.
Many zero sum games do not have a saddle point. In that case, players randomize. The row player chooses Row 1 with probability p. The column player chooses Column 1 with probability q. These probabilities make the opponent indifferent. That is the key equilibrium idea in game theory. The calculator uses standard 2x2 formulas to find p, q, and the game value. It also checks custom probabilities, so you can test your own trial strategies.
This zero sum game value calculator supports matrix game practice in maths, economics, and operations research. It helps learners see maximin, minimax, dominance hints, expected payoff, and equilibrium value together. The example table shows how the numbers fit. The export options help you save results for notes or assignments. Use the tool to verify textbook answers, compare strategy choices, and understand why optimal play often requires randomization.
A zero sum game is a contest where one player’s gain equals the other player’s loss. The total payoff always balances to zero.
Game value is the expected payoff under optimal play. It shows the fair long run result of the game for the row player.
A saddle point is a payoff that is the minimum of its row and the maximum of its column. It creates a pure strategy solution.
Mixed strategies are needed when no saddle point exists. Randomization prevents the opponent from exploiting a predictable pure move.
Yes. A negative value means the row player expects a loss under optimal play. It means the column player has the advantage.
No. This version is built for 2x2 zero sum payoff matrices. It is ideal for fast learning, practice, and standard examples.
Maximin shows the row player’s safest guaranteed result. Minimax shows the column player’s best damage limit. Their equality signals a saddle point.
Exports help you save results, submit examples, and keep revision notes. They also make classroom sharing and record keeping easier.
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.