Game Theory Payoff Calculator
Analyze a 2x2 payoff matrix to find dominant strategies, Nash equilibria (pure and mixed), minimax values, and Pareto optimality.
About this calculator
This calculator analyzes a 2x2 payoff matrix for dominant strategies (lines 18-24), pure-strategy Nash equilibria (lines 37-45), a mixed-strategy equilibrium (lines 48-59), minimax values (lines 61-70), and Pareto-optimal outcomes (lines 72-86) — all from the same eight payoff numbers. A genuinely useful nuance the mixed-strategy formulas don't check for themselves: Mixed Strategy P1's probability is computed entirely from Player 2's four payoffs (line 51's `denom1`), and Mixed Strategy P2's probability is computed entirely from Player 1's payoffs (line 56's `denom2`) — each player's mixing probability is set by what makes the *other* player indifferent, not by their own payoffs. At this calculator's defaults, though, neither mixed-strategy result is a usable probability at all: both formulas return exactly −1, because Player 1 has a dominant strategy (Down, since 5≥3 and 1≥0 with Down strictly better in one cell) and so does Player 2 (Right), and the mixed-equilibrium algebra assumes a genuine mixing incentive that a dominant-strategy game doesn't have.
The calculator does not detect or flag that case — it returns whatever the formula computes, including a value outside the valid [0,1] probability range, so a negative or above-1 Mixed Strategy result should be read as "no meaningful mixed equilibrium here" rather than a real probability. What this does not account for: games with more than two strategies per player, and it never checks its own mixed-strategy output for validity before displaying it.
Inputs
Results
P1 Dominant (0=None,1=Up,2=Down)
2
P2 Dominant (0=None,1=Left,2=Right)
2
How to Use This Calculator
- Enter the payoff matrix values for Player 1 and Player 2 for each of the four strategy combinations (Up-Left, Up-Right, Down-Left, Down-Right).
- Review the dominant strategies for each player if they exist.
- Check for pure-strategy Nash equilibria, where neither player benefits from unilaterally changing strategy, along with the payoffs at that equilibrium.
- Review the mixed-strategy Nash equilibrium probabilities for each player.
- Use the minimax values and Pareto optimal outcome count to guide strategy selection in adversarial scenarios.
What each input means
- P1 Payoff: Up-Left
- Player 1's payoff when P1 plays Up and P2 plays Left
- P1 Payoff: Up-Right
- Player 1's payoff when P1 plays Up and P2 plays Right
- P1 Payoff: Down-Left
- Player 1's payoff when P1 plays Down and P2 plays Left
- P1 Payoff: Down-Right
- Player 1's payoff when P1 plays Down and P2 plays Right
- P2 Payoff: Up-Left
- Player 2's payoff when P1 plays Up and P2 plays Left
- P2 Payoff: Up-Right
- Player 2's payoff when P1 plays Up and P2 plays Right
- P2 Payoff: Down-Left
- Player 2's payoff when P1 plays Down and P2 plays Left
- P2 Payoff: Down-Right
- Player 2's payoff when P1 plays Down and P2 plays Right
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersP1 Payoff: Up-Left = 3, P1 Payoff: Up-Right = 0, P1 Payoff: Down-Left = 5, P1 Payoff: Down-Right = 1 = 8 input(s) provided
- Calculate P1 DominantP1 Dominant2 = 2
- Calculate P2 DominantP2 Dominant2 = 2
- Calculate Pure Nash EquilibriaPure Nash Equilibria1 = 1
- Calculate Nash P1 PayoffNash P1 Payoff1 = 1
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why do Mixed Strategy P1 and Mixed Strategy P2 both show -1 at the calculator's defaults?
Because both players have a dominant strategy at these defaults — Player 1's Down row and Player 2's Right column each dominate outright — and a mixed-strategy equilibrium only makes sense when neither player has one. The formula (lines 51-59) doesn't check for that case; it just returns whatever the algebra produces, which here is a value outside the valid 0-to-1 probability range rather than a warning.
Does Mixed Strategy P1's probability depend on Player 1's own payoffs?
No — it depends entirely on Player 2's four payoffs. The formula sets the probability that makes Player 2 indifferent between Left and Right (line 51's `denom1`, built purely from p2UL, p2UR, p2DL, p2DR), which is standard game-theory logic: a player's mixing probability is chosen to neutralize the *other* player's incentive to deviate, not their own.
Is there a pure-strategy Nash equilibrium at the default payoff matrix?
Yes, exactly one: (Down, Right), where Player 1 earns 1 and Player 2 earns 1. The calculator checks all four cells for mutual best-response (lines 37-45), and only the Down-Right cell satisfies both players' conditions simultaneously at these defaults, which is also why Nash P1 Payoff and Nash P2 Payoff both echo the Down-Right cell.
How many of the four outcomes are Pareto optimal at the defaults?
Three of the four. The calculator marks an outcome as Pareto optimal when no other cell makes both players at least as well off with one strictly better (lines 72-86); only Down-Right, where both players get just 1, is dominated in that sense — by Up-Left, which gives both players 3, strictly more on both dimensions at once — leaving Up-Left, Up-Right, and Down-Left as the three Pareto-optimal cells.
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