Payoff Matrix Calculator

Change the incentives. See who wants to move. Explore pure and mixed Nash equilibria in a two-player, two-strategy game — free, without signing in.

Two firms decide simultaneously whether to launch. These illustrative profits make each prefer being the only new entrant.

Game title

Payoff unit

Row player

Column player

Two players choose simultaneously. Each cell gives the row player’s payoff first, then the column player’s. Higher is better for each player. Use whole numbers from −1,000,000 to 1,000,000.

Edit both strategies and all eight payoffs.
	Column player’s first strategy	Column player’s second strategy
Row player’s first strategy	Row payoff, row 1, column 1 Column payoff, row 1, column 1	Row payoff, row 1, column 2 Column payoff, row 1, column 2
Row player’s second strategy	Row payoff, row 2, column 1 Column payoff, row 2, column 1	Row payoff, row 2, column 2 Column payoff, row 2, column 2

Submit edits to update the analysis and lesson. The address bar then holds a shareable link to this exact game; do not put private information in its labels.

Analyzed game

Competing launches

A boxed cell is a pure Nash equilibrium. Underlining marks a best response, including ties. Payoffs are in profit units.

Established firm chooses a row; New entrant chooses a column. Each pair reads (Established firm, New entrant).
	Enter	Stay out
Launch	-2,-1	6,0 Best response for Established firm. Best response for New entrant. Nash equilibrium
Wait	0,4 Best response for Established firm. Best response for New entrant. Nash equilibrium	0,0

2 pure Nash equilibria

In each boxed cell, neither player gains by switching alone. An underline may tie another payoff: a best response need not be unique.

All equilibrium probabilities

p = probability Established firm chooses Launch. q = probability New entrant chooses Enter. The other strategy gets 1 − p or 1 − q.

p = 1; q = 0
p = 0; q = 1
p = 4/5; q = 3/4

An interval includes both endpoints. Every combination within a listed pair of intervals is an equilibrium; the full list includes pure and mixed strategies.

Why randomize? Try a mixture.

Move one player’s probability and watch the other player’s incentives change. These are trial choices, not necessarily an equilibrium. They do not alter your game or lesson. Interactive controls require JavaScript.

Established firm: choose Launch 50% of the time

New entrant’s expected payoff for each pure strategy

Enter: 1.50 profit units
Stay out: 0.00 profit units

New entrant: choose Enter 50% of the time

Established firm’s expected payoff for each pure strategy

Launch: 2.00 profit units
Wait: 0.00 profit units

What this calculator does — and does not tell you

A Nash equilibrium is a pair of strategies where neither player can improve their expected payoff by changing alone. It need not be fair, best for the pair, or a prediction of what students will actually choose.

This solver covers two players with two strategies each, including ties, weak best responses and infinitely many equilibria. It assumes each player maximizes their stated numerical payoff. It does not solve larger games, sequential games or repeated-game incentives.

The mixed-strategy calculation makes each player indifferent between the strategies they randomize over. Equilibrium probabilities are exact reduced fractions; the optional trial-mixture display is rounded.

For a worked mathematical introduction, see MIT’s mixed-strategy lecture. Ready to teach? Explore game theory classroom activities, then use the exact analyzed game below.

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