2x2 Nash Equilibrium Calculator
Enter payoffs for a normal-form game with Row strategies Up/Down and Column strategies Left/Right. This tool finds pure equilibria and checks for a mixed-strategy equilibrium candidate.
What this Nash equilibrium calculator does
This calculator solves a classic 2x2 strategic game where each player has two actions. It helps you quickly answer three practical questions:
- What are each player’s best responses?
- Does the game have one or more pure-strategy Nash equilibria?
- Is there a valid mixed-strategy equilibrium (with probabilities between 0 and 1)?
Whether you’re studying game theory, building business strategy models, or preparing for economics exams, this gives you immediate structure and interpretation.
Quick refresher: what is a Nash equilibrium?
A Nash equilibrium is a strategy profile where no player can improve their payoff by unilaterally changing strategy, holding the other player’s strategy fixed.
Pure-strategy Nash equilibrium
A pure equilibrium is a single action pair such as (Up, Right). It works as an equilibrium only if:
- Row’s action is a best response to Column’s choice, and
- Column’s action is a best response to Row’s choice.
Mixed-strategy Nash equilibrium
In some games, no pure equilibrium exists. Then players randomize. Row chooses Up with probability p, and Column chooses Left with probability q. In equilibrium, each player makes the other player indifferent between their available actions.
How to use the calculator
- Enter the eight payoffs: four for Row and four for Column.
- Keep strategy labels fixed as Up/Down (Row) and Left/Right (Column).
- Click Calculate Equilibrium.
- Read best responses first, then pure equilibria, then the mixed candidate.
If you want a quick test, use the sample buttons. Prisoner’s Dilemma usually gives a single pure equilibrium; Matching Pennies gives a mixed equilibrium at 50/50.
How the mixed equilibrium is computed
For a 2x2 game, the mixed candidate is computed by indifference conditions:
q* = (rDR - rUR) / (rUL - rUR - rDL + rDR)
If both probabilities are in the interval [0,1], the candidate is feasible. If either is outside that range, then no interior mixed equilibrium exists for both players simultaneously.
Interpreting outcomes correctly
- Multiple pure equilibria: coordination problems often have more than one stable outcome.
- No pure equilibrium: look for mixed behavior, common in competitive settings.
- Ties in best responses: can indicate weak equilibria or continuum-like behavior in degenerate cases.
- Context matters: equilibrium predicts strategic stability, not fairness or social optimality.
Typical applications
People use Nash equilibrium analysis in:
- Pricing decisions between competitors
- Negotiation strategy design
- Political campaign positioning
- Cybersecurity attacker-defender models
- Behavioral economics classroom experiments
Limitations
This page focuses on two-player, two-strategy normal-form games. It does not solve larger games, dynamic extensive-form games, or repeated-game equilibria. Still, mastering 2x2 analysis is the best foundation before moving into broader equilibrium concepts.