What is a Nash Equilibrium in game theory?

A Nash equilibrium is a state in a game where each player has chosen a strategy and neither player can increase their own payoff by unilaterally changing their strategy while other players keep theirs unchanged.

What is the difference between a pure strategy and a mixed strategy?

In a pure strategy, a player chooses one specific action with 100% certainty. In a mixed strategy, a player randomizes over multiple actions according to a specific probability distribution (e.g., playing Heads 50% and Tails 50% in Matching Pennies) to make the opponent indifferent.

Why is the Prisoner's Dilemma equilibrium Pareto-suboptimal?

In the Prisoner's Dilemma, both players have a dominant strategy to 'Defect' (yielding payoff 1, 1). However, mutual cooperation yields (3, 3), which is strictly superior for both players. The Nash equilibrium traps rational actors in a suboptimal outcome.

What is a strictly dominated strategy?

A strategy is strictly dominated if another strategy yields a strictly higher payoff for that player regardless of what action the opposing player selects. Rational players will never choose strictly dominated strategies.

How does the tool calculate mixed-strategy probabilities in 2x2 games?

It sets Player 1's expected payoff from Strategy A equal to their expected payoff from Strategy B under Player 2's probability distribution q, solving the linear equation for q* that makes Player 1 indifferent.

Mobile & Tech

2x2 & 3x3 Game Theory Payoff Matrix & Nash Equilibrium Solver

Name: 2x2 & 3x3 Game Theory Payoff Matrix & Nash Equilibrium Solver
Author: AnshuTechy

Solve normal-form non-cooperative games. Identify pure and mixed-strategy Nash equilibria, strictly dominated strategies, and Pareto-optimal outcomes across Prisoner's Dilemma, Battle of the Sexes, Stag Hunt, and custom 2x2/3x3 payoff matrices.Documentation & FAQs ↓

2x2 & 3x3 Game Theory Payoff Matrix & Nash Equilibrium Solver — Interactive Console

Runs locally in your browser • Instant output

Classic Game Theory Games

2x2 Strategic Payoff Matrix (Player 1, Player 2)

P1 Action 1 (Row 1)

P1 Action 2 (Row 2)

P2 Action 1 (Col 1)

P2 Action 2 (Col 2)

P1 \ P2

Cooperate

Defect

Cooperate

Defect

★ Nash Equilibrium

Pure Strategy Nash

★ (Defect, Defect)

Both players play mutual best responses. No player has incentive to unilaterally deviate.

Dominant Strategies

Player 1: Always play Defect

Player 2: Always play Defect

A strategy strictly superior to all alternatives regardless of opponent choice.

Mixed Strategy Equilibrium

No strictly interior mixed strategy exists on (0, 1).

Probabilities that render the opposing player completely indifferent.

Step-by-Step Instructions

4 Steps to Completion

1Phase 1

Select your game dimensions (2x2 or 3x3) or pick a canonical preset (e.g., Prisoner's Dilemma or Stag Hunt).

2Phase 2

Enter player strategies and assign payoff coordinates (Player 1 payoff, Player 2 payoff) in each matrix cell.

3Phase 3

Click 'Solve Game' to compute best-response intersections, dominated strategy eliminations, and mixed-strategy equilibrium probabilities.

4Phase 4

Inspect highlighted equilibrium cells, Pareto-efficient outcomes, and read the step-by-step game theoretic explanation.

Practical Use Cases & Real-World Scenarios

Strategic Business Pricing & Market Entry

Model competitive pricing battles between rival firms to predict whether destructive price wars or tacit price collusion emerge.

Economics & Game Theory Academic Study

Understand mathematical foundations of simultaneous non-cooperative games and verify problem set solutions.

Cybersecurity & Defense Modeling

Model attacker vs defender incentives in network vulnerability patching, zero-day disclosure, and intrusion detection.

2x2 & 3x3 Game Theory Payoff Matrix & Nash Equilibrium Solver

Step-by-Step Instructions

Practical Use Cases & Real-World Scenarios

Pure & Mixed Strategy Equilibrium Solvers

Iterated Elimination of Dominated Strategies (IEDS)

Pareto Optimality & Social Optimum Mapping

Classic Game Theory Presets Library

Frequently Asked Questions

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