Zero-Sum Game Value Calculator

It first checks whether the matrix contains a saddle point, where both players can use pure strategies without either gaining from a unilateral change. When no saddle point exists, it calculates the equilibrium mixing probabilities for both players and the expected game value. The result identifies the method used, reports the value, and gives each strategy as probabilities that correspond to the first and second row or column.

Enter the payoff matrix from one player's perspective

Represent the game as two rows and two columns, with every entry giving the payoff to the row player for that combination of choices. Because the game is zero-sum, the column player's payoff is the negative of the same entry; you should not submit a second matrix. Keep the order meaningful: the first pair of values belongs to row strategy one, the second pair belongs to row strategy two, and positions within each pair correspond to column strategies one and two. Positive values favor the row player, negative values favor the column player, and zero is neutral. Decimal values are allowed as long as every entry is a finite number. The calculator deliberately accepts exactly four payoffs because its formulas and strategy labels are specific to a two-by-two game. If your game has more actions, eliminating strictly dominated strategies may reduce it to this form, but that reduction must be justified separately rather than assumed by the calculator. Careful ordering matters because the returned probabilities follow the same row and column order you supplied.

Understand the saddle-point test and pure solution

The row player wants to maximize the payoff that can still be guaranteed after the column player responds. For each row, the calculator therefore finds its smallest payoff and then takes the larger of those row minima, producing the maximin. The column player works in the opposite direction: for each column, the calculator finds its largest payoff and then chooses the smaller of those column maxima, producing the minimax. When maximin and minimax are equal, their shared entry is a saddle point and is the value of the game. Neither player can improve by changing strategy alone, so a pure-strategy equilibrium is sufficient. The output names the saddle point using one-based row and column positions and returns probability one for the selected equilibrium action and zero for the other. In a degenerate matrix there can be more than one valid saddle point; the calculator returns the first consistent row and column in input order. That selected pair remains optimal even though other optimal pure or mixed combinations may also exist.

Read mixed strategies and the game value

If the maximin is below the minimax, no pure action pair is stable, so each player must randomize. The calculator solves the indifference equations: the row player chooses probabilities that make the column player indifferent between its two columns, while the column player chooses probabilities that make the row player indifferent between its two rows. Each returned strategy is a two-number array whose entries sum to one. The first number is the probability of choosing the first row or column, and the second belongs to the second row or column. These probabilities are strategic frequencies, not predictions that one specific round must follow. The reported value is the row player's expected payoff when both players use their optimal strategies; the column player's value is its negative. A positive value favors the row player over repeated play, a negative value favors the column player, and zero describes a fair game in expected-value terms. Results are rounded deterministically for stable API output, but the calculation uses the submitted numeric payoffs before formatting. The automated API request price is $0.002.