What is a pure-strategy Nash equilibrium in a 2×2 game?

A pure-strategy Nash equilibrium is a cell of the payoff matrix where each player's chosen strategy is a best response to the other player's strategy. In a 2×2 game, a cell is a Nash equilibrium if the row player's payoff is maximal in its column and the column player's payoff is maximal in its row.

How does the tool compute a mixed-strategy equilibrium?

In games where there is no pure-strategy Nash equilibrium or where players randomise, the solver equates each player's expected payoff from their two strategies to find mixing probabilities. It checks whether the resulting probabilities are between 0 and 1 and reports a valid interior mixed equilibrium or explains why one does not exist.

Is the 2×2 game theory solver suitable for financial or strategic decisions?

The solver is intended for teaching, intuition building, and quick checks. Real-world strategic, financial, or policy decisions typically involve more players, more strategies, and uncertainty that go beyond a 2×2 model. Always complement this tool with professional judgement and more detailed analysis.

Game Theory 2×2 Matrix Solver

Enter a two-player, two-strategy payoff matrix and this tool will find pure-strategy Nash equilibria, dominant strategies, and (when it exists) a mixed-strategy equilibrium, with clear, teaching-oriented explanations.

Built for economics, strategy, and teaching

Highlights best responses, flags dominant strategies, and solves the 2×2 mixed equilibrium by indifference conditions — ideal for classroom use and quick strategic checks.

Author: CalcDomain Game Theory Team

Reviewed by: Microeconomics instructor

Last updated: 2025

This calculator supports simple 2×2 normal-form games. Real-world strategic decisions often require richer models, more players, and uncertainty; treat outputs as intuition aids, not prescriptive advice.

Interactive 2×2 game matrix

Row player name (Player 1)

Row strategy 1

Row strategy 2

Column player name (Player 2)

Column strategy 1

Column strategy 2

Payoff matrix (first entry is Row player, second is Column player)

	Left	Right
Top	Row, Column ,	Row, Column ,
Bottom	Row, Column ,	Row, Column ,

Default example is a Prisoner’s Dilemma-style payoff structure. Feel free to override with any 2×2 normal-form game.

Decimal places for probabilities

Show detailed steps? Yes, show best-response logic and mixed-equilibrium derivation

The solver reports pure Nash equilibria, dominant strategies, and any interior mixed equilibrium.

Equilibria and strategic summary will appear here.

Step-by-step explanation of best responses and mixed strategies will appear here.

What is a 2×2 normal-form game?

In a 2×2 normal-form game there are two players. Each player chooses one of two strategies simultaneously. The outcome is represented by a 2×2 payoff matrix. Every cell contains an ordered pair of payoffs, for example:

\[ \begin{array}{c|cc} & \text{Left} & \text{Right} \\\hline \text{Top} & (a_{11}, b_{11}) & (a_{12}, b_{12}) \\ \text{Bottom} & (a_{21}, b_{21}) & (a_{22}, b_{22}) \end{array} \]

The first number in each pair is the row player’s payoff, the second number is the column player’s payoff.

Best responses and pure-strategy Nash equilibrium

A best response is a strategy that achieves the highest payoff for a player given the other player’s choice. In a 2×2 game:

For each column (Left or Right), the row player compares the two payoffs in that column and chooses the higher one.
For each row (Top or Bottom), the column player compares the two payoffs in that row and chooses the higher one.

A pure-strategy Nash equilibrium is a cell where both players are playing best responses at the same time. Neither can gain by unilaterally deviating.

Dominant strategies

A strategy is strictly dominant if it yields a strictly higher payoff than any other strategy, for all possible moves of the opponent. A strategy is weakly dominant if it does at least as well in all cases and strictly better in at least one case.

In a 2×2 game, the row player’s first strategy dominates the second if:

\[ a_{11} \ge a_{21} \quad\text{and}\quad a_{12} \ge a_{22}, \]

with at least one inequality strict for strict dominance. Analogous inequalities define dominance for the column player.

Mixed-strategy equilibrium in a 2×2 game

When no pure-strategy Nash equilibrium exists, players may randomise. A mixed-strategy equilibrium is a pair of probabilities such that each player is indifferent between their two strategies given the other player’s mixing.

Let \(p\) be the probability that the row player plays Top, and \(q\) the probability that the column player plays Left. The row player is indifferent when their expected payoff from Top equals their expected payoff from Bottom:

\[ q a_{11} + (1-q)a_{12} = q a_{21} + (1-q)a_{22}. \]

Solving this linear equation gives the equilibrium value of \(q\). Similarly, the column player’s indifference condition determines \(p\). The calculator builds and solves these equations, checks that the probabilities lie between 0 and 1, and reports a valid interior mixed equilibrium when it exists.

FAQ: 2×2 game theory solver

What if there are multiple pure Nash equilibria?

Coordination-type games often have two pure-strategy Nash equilibria. The solver lists all of them and flags that the game has multiple stable outcomes. In such cases, extra selection principles (risk dominance, focal points, learning dynamics) are typically discussed in game theory courses.

What if no pure Nash equilibrium exists?

Some games, like Matching Pennies, have no pure-strategy Nash equilibrium. The solver detects this and then looks for an interior mixed equilibrium by equating expected payoffs. If the resulting probabilities are outside the [0, 1] interval or the equations are degenerate, it will explain why a clean interior mixed equilibrium does not exist.

Can I model more than two strategies or players?

This tool is intentionally restricted to 2×2 games to remain transparent and teachable. Games with more strategies or more players require larger matrices or extensive form representations, and are better handled with specialised software or algebra systems.

Is this suitable for financial, legal, or policy decisions?

It can be a helpful sanity check or communication aid, but it cannot capture all institutional details, risk, or repeated-interaction effects in real-world settings. Treat it as a learning instrument and pair it with domain expertise and scenario analysis before making consequential decisions.