Game Theory Calculator | Nash Equilibria, Minimax & Dominant Strategies
Analyze 2×2 and 3×3 two-player strategic games. Finds all pure Nash equilibria by best-response check, computes mixed Nash equilibria for 2×2 games, identifies strictly and weakly dominant strategies, determines minimax and maximin values for zero-sum games, and classifies the game type.
| Col 1 | Col 2 | |
|---|---|---|
| Row 1 | , | , |
| Row 2 | , | , |
What Is the Game Theory Calculator | Nash Equilibria, Minimax & Dominant Strategies?
A Nash equilibrium is a strategy profile where no player can improve their payoff by changing their strategy unilaterally. Pure Nash equilibria are strategy pairs that are mutual best responses. Mixed Nash equilibria (2×2) are found by solving for probabilities that make the opponent indifferent. Dominant strategies are always rational to play; minimax values bound zero-sum game outcomes.
Formula
Pure NE: (i*,j*) where payoff_row(i*,j*) ≥ payoff_row(i,j*) ∀i and payoff_col(i*,j*) ≥ payoff_col(i*,j) ∀j
How to Use
- 1
Choose matrix size: 2×2 for standard two-strategy games, 3×3 for richer strategy sets.
- 2
Choose game type: General for non-zero-sum (enter both payoffs per cell), Zero-Sum for competitive (enter row payoff only).
- 3
Use a preset button to load a classic game, or type your own payoffs directly in the grid.
- 4
For general mode, enter Row payoff and Col payoff (comma-separated) in each cell.
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Click "Analyze Game" to compute Nash equilibria, dominant strategies, and minimax.
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Nash equilibrium cells are highlighted with a star (★ NE) in the matrix.
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For 2×2 games, a mixed Nash equilibrium is also computed if it exists in (0,1)×(0,1).
Select matrix size (2×2 or 3×3) and game type (general or zero-sum). Enter payoffs in each cell — general mode takes (row payoff, col payoff) per cell; zero-sum mode takes one value (col gets the negative). Click Analyze Game to find all Nash equilibria, dominant strategies, and minimax values.
Example Calculation
Prisoner's Dilemma: payoffs (Cooperate,Cooperate)=(−1,−1), (Cooperate,Defect)=(−3,0), (Defect,Cooperate)=(0,−3), (Defect,Defect)=(−2,−2). Pure NE: (Defect, Defect) since Defect strictly dominates Cooperate for both players. Pareto optimum (−1,−1) is not a Nash equilibrium — the classic social dilemma.
Understanding Game Theory | Nash Equilibria, Minimax & Dominant Strategies
Classic games reference
| Game | Type | Nash Equilibrium | Key insight |
|---|---|---|---|
| Prisoner's Dilemma | Non-zero-sum | (Defect, Defect) | Individual rationality leads to collective suboptimality |
| Matching Pennies | Zero-sum | Mixed: (1/2, 1/2) | No pure NE; players must randomize |
| Stag Hunt | Cooperation | (Stag,Stag) or (Hare,Hare) | Two pure NE; coordination required for Pareto-optimal |
| Battle of Sexes | Coordination | (Opera,Opera) or (Football,Football) | Two pure NE with different preferred outcomes |
| Chicken / Hawk-Dove | Anti-coord. | (Straight,Swerve) and (Swerve,Straight) | Asymmetric NE; also mixed NE exists |
| Rock-Paper-Scissors | Zero-sum 3×3 | Mixed: (1/3, 1/3, 1/3) | Unique mixed NE; no pure NE |
Strategy and equilibrium concepts
| Concept | Definition | Implication |
|---|---|---|
| Pure Nash Equilibrium | Each player's strategy is the best response to the other's strategy | Neither player wants to deviate unilaterally |
| Mixed Nash Equilibrium | Players randomize so the opponent is indifferent between strategies | Always exists in finite games (Nash 1950) |
| Strictly dominant | Strategy always gives strictly higher payoff regardless of opponent | Rational players always play dominant strategies |
| Weakly dominant | Strategy gives ≥ payoff in all cases and > in at least one | May survive iterated dominance elimination |
| Minimax (zero-sum) | Each player minimizes the maximum loss (or maximizes minimum gain) | Minimax = Nash for zero-sum games (von Neumann) |
| Pareto optimum | No player can be made better off without making another worse off | NE is often Pareto-inefficient (Prisoner's Dilemma) |
Real-world applications of game theory
- ›Economics and auctions: Spectrum auctions, bid strategies, oligopoly pricing (Cournot, Bertrand), and market entry games use Nash equilibrium analysis.
- ›Evolutionary biology: Evolutionary stable strategies (ESS) generalize Nash equilibria to population dynamics: hawk-dove, cooperation/defection in repeated games.
- ›Political science: Voting models, arms races, nuclear deterrence, and coalition formation all rely on strategic equilibrium concepts.
- ›Network routing: Selfish routing in networks (Braess paradox) shows that adding capacity can worsen equilibrium outcomes; price of anarchy quantifies this.
- ›Algorithmic mechanism design: Designing auctions and platforms so that rational agent behaviour aligns with social welfare (second-price auction, VCG mechanism).
- ›Machine learning: Generative adversarial networks (GANs) formalize image generation as a two-player zero-sum game between a generator and discriminator.
Frequently Asked Questions
What is a Nash equilibrium and how is it found?
A Nash equilibrium is a combination of strategies where each player is playing the best response to the other's strategy — no one benefits from switching. To find pure Nash equilibria, check every cell (i,j): is it the maximum in its column for the row player AND the maximum in its row for the col player? If both conditions hold, it is a pure NE.
How is mixed strategy Nash equilibrium computed for 2×2 games?
In a mixed NE, each player randomizes to make the opponent indifferent between their strategies. For a 2×2 game with Row payoffs a,b,c,d and Col payoffs e,f,g,h: the row player mixes with p = (d−c)/(a−b−c+d) (making col indifferent), and col mixes with q = (h−f)/(e−f−g+h) (making row indifferent). A valid mixed NE requires both p and q in (0,1).
What is the difference between zero-sum and non-zero-sum games?
In a zero-sum game, one player's gain is the other's loss: payoffs in every cell sum to zero. Examples: chess, poker (heads-up), matching pennies. In non-zero-sum games, payoffs can vary independently — both players can win or both can lose. Most real-world situations (trade, contracts, negotiations) are non-zero-sum, making Nash equilibrium richer than minimax.
What are dominant strategies and why do they matter?
A strategy is strictly dominant if it gives a strictly higher payoff than any other strategy regardless of what the opponent does. Rational players always play strictly dominant strategies — so they can be identified and eliminated from consideration (iterated dominance). In the Prisoner's Dilemma, Defect strictly dominates Cooperate for both players, leading to the unique Nash equilibrium.
What does minimax mean and when does it equal the Nash equilibrium?
Minimax is a decision rule for zero-sum games: the row player maximizes the minimum payoff they can guarantee (maximin); the col player minimizes the maximum the row player can achieve (minimax). By the minimax theorem (von Neumann, 1928), for finite zero-sum games minimax = maximin in mixed strategies, and this value equals the Nash equilibrium value. A saddle point exists when these values match in pure strategies.
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