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Payoff Matrix
Click on cells to see analysis. Gold cells indicate Nash equilibria. Arrows show best responses.
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Visual representation of how players respond to opponent strategies. Arrows point toward better responses.
Mixed Strategy Analyzer
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Nash Equilibrium Finder
Automated analysis of all Nash equilibria in the current game.
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A Nash equilibrium is a concept in game theory where no player has an incentive to deviate from their chosen strategy after considering an opponent's choice.
Mathematical Definition:
Let (s₁*, s₂*, ..., sₙ*) be a strategy profile where sᵢ* is player i's strategy.
This is a Nash equilibrium if for all players i:
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*) for all sᵢ ∈ Sᵢ
Where:
- uᵢ = player i's utility function
- sᵢ* = player i's equilibrium strategy
- s₋ᵢ* = strategies of all other players
- Sᵢ = set of all possible strategies for player i
A best response is a strategy that maximizes a player's payoff given the strategies chosen by other players.
Definition:
Strategy sᵢ* is a best response to s₋ᵢ* if:
uᵢ(sᵢ*, s₋ᵢ*) = max(uᵢ(sᵢ, s₋ᵢ*)) for all sᵢ ∈ Sᵢ
Key Insight:
A Nash equilibrium is a strategy profile where every player is playing a best response to the other players' strategies.
John Nash proved that every finite game has at least one Nash equilibrium (in either pure or mixed strategies).
Theorem Statement:
Every game with a finite number of players and finite strategy sets has at least one Nash equilibrium.
Implications:
- Pure strategy Nash equilibria may not exist
- But mixed strategy equilibria always exist
- Fixed point theorem guarantees existence
Economics
Oligopoly competition, auction design, price wars, and market entry decisions.
Biology
Evolutionary stable strategies, animal behavior, and evolutionary game theory.
Politics
Voting systems, international relations, arms races, and policy choices.
Computer Science
Network routing, algorithmic game theory, and distributed systems.
While Nash equilibrium is a powerful concept, it has several limitations:
- Not Always Optimal: The Prisoner's Dilemma shows how Nash equilibrium can be Pareto inferior to other outcomes.
- Multiple Equilibria: Many games have multiple Nash equilibria, making prediction difficult.
- Rationality Assumption: Assumes all players are perfectly rational, which may not hold in reality.
- Common Knowledge: Requires that all players know the equilibrium and that others know it too.