Game Theory & the Nash Equilibrium
Set the payoffs in a 2×2 game and watch each player’s best responses highlight — then the cell where neither wants to move locks in as the Nash equilibrium.
Most economic decisions depend on what someone else will do. Game theory is the mathematics of those interdependent choices, and its central idea — the Nash equilibrium — is the point where no player wants to move. Enter payoffs into the grid below and watch it light up.
Choices that depend on others
A firm setting a price, two countries choosing whether to arm, flatmates deciding whether to wash up — each player’s best move depends on the others’ moves. We capture that in a payoff matrix: the rows are one player’s strategies, the columns the other’s, and each cell holds the pair of payoffs (Player A, Player B).
You can’t just pick your favourite outcome — you can only pick a strategy. The outcome is decided by both players at once.
Find the equilibrium
Edit any payoff and watch the analysis update live. Each player’s best responses are highlighted; a cell that is a best response for both players is a Nash equilibrium. Load the classic games and try to break the equilibrium by changing a single number.
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An editable 2×2 payoff matrix. Player A chooses a row, Player B chooses a column, and each cell contains two editable payoffs (A’s, then B’s); higher numbers are better. For each column the tool highlights the row giving Player A the larger payoff (A’s best response), and for each row it highlights the column giving Player B the larger payoff (B’s best response). A cell highlighted for both players is a Nash equilibrium, labelled as such. Preset games include the prisoner’s dilemma (single equilibrium where both defect: 1, 1), the stag hunt (two equilibria), and the battle of the sexes (two equilibria).
Best responses and dominant strategies
A best response is the strategy that gives a player their highest payoff, taking the other player’s choice as fixed. To find Player A’s best response, cover Player B’s payoffs and, column by column, pick A’s larger number.
The Nash equilibrium
A Nash equilibrium is a combination of strategies where every player is simultaneously playing a best response — so no one can gain by unilaterally switching. On the grid, it is the cell where bothnumbers are highlighted. It is a resting point: given what everyone else is doing, you wouldn’t want to change.
A game can have one Nash equilibrium (prisoner’s dilemma), several (the stag hunt and battle of the sexes both have two in pure strategies), or none in pure strategies — in which case the equilibrium involves randomising, a mixed strategy.
The prisoner’s dilemma
Two suspects are questioned separately. Each can stay silent (cooperate) or confess (defect). Whatever the other does, confessing gives a better individual payoff — so both confess. Yet if both had stayed silent they’d both be better off.
Using the preset payoffs (higher = better): if your opponent cooperates, defecting gives you 5 instead of 3. If your opponent defects, defecting gives you 1 instead of 0. Defect wins in bothcolumns, so it’s dominant. Both reason identically, both defect, and they land on (1, 1) — worse for each than the (3, 3) they could have shared. Individually rational, collectively poor: the dilemma at the heart of cartels, arms races and climate negotiations.
Common mistakes
In a 2×2 game, both players have a dominant strategy. How many pure-strategy Nash equilibria must the game have, and where is it?
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Exactly one, at the cell where both dominant strategies meet. Since each dominant strategy is a best response to everything, their intersection is automatically a best response for both — a Nash equilibrium.
Frequently asked questions
What is a Nash equilibrium?+
What is a dominant strategy?+
Why is the prisoner’s dilemma important?+
Can a game have more than one Nash equilibrium?+
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Written and reviewed by ScholarsGate tutors who teach A-Level and undergraduate economics. Every explainer is checked against the AQA, Edexcel, OCR and Eduqas specifications.
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