Two-Player Game Analyzer
Solve a two-player 2×2 zero-sum payoff matrix for a saddle point or an interior mixed-strategy solution. Processing takes place locally in your browser.
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Two-Player Game Analyzer
Solve a two-player 2×2 zero-sum payoff matrix for a saddle point or an interior mixed-strategy solution.
Two-Player Game Analyzer
Solves a two-player 2×2 zero-sum payoff matrix for the row player's payoff. Input must contain exactly two rows and two finite numeric values per row. It first compares row minima and column maxima to detect a pure-strategy saddle point.
Verified pure example [3,2;1,0] has maximin=minimax=2 and reports row 1, column 2 with value 2.00000. If there is no saddle point, the tool computes the unique interior completely mixed solution when its denominator a-b-c+d is nonzero and both player probabilities lie within (0,1).
For [3,0;1,2], row player first-row probability is(d-c)/den=0.25, column player first-column probability is(d-b)/den=0.5 and game value(ad-bc)/den=1.5. Results display five decimal places.
A zero denominator reports no unique completely mixed solution. Calculated probabilities outside (0,1) report no interior completely mixed solution rather than displaying invalid probabilities. The tool does not enumerate every equilibrium, eliminate dominated strategies or solve general-sum, cooperative or matrices larger than 2×2.
Calculation is browser-local and assumes the entered payoffs form one zero-sum game. Rounding can hide small differences; verify strategic assumptions and original payoffs before using the result in economic, legal or operational decisions. It is an educational calculator, not professional advice.
The mixed calculation scales payoffs by their largest absolute value to reduce overflow. Floating-point arithmetic and display rounding remain: a displayed zero is not proof of an exact zero. Large values may use scientific notation. If several saddle points exist, only the first is reported.
Enter exactly two rows of two complete finite payoffs. Scientific notation and negative values are accepted; fractions, expressions, brackets and extra text are rejected. Use the local decimal example below and spaces between columns. Nonzero inputs too small to remain nonzero are rejected. Clear empties the input and result. Failure to compute a mixed solution is not proof that a zero-sum game has no equilibrium; numerical limits still apply.
3.5 0 1 2.5
Internal mixed probabilities must be strictly between 0 and 1. A displayed 0 or 1 can still result from rounding a valid internal probability to five decimal places.
Processing takes place locally in your browser.