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Nash Equilibrium
A core concept in game theory proposed by John Nash, describing stable states in multi-player games. Awarded the Nobel Prize in Economics.
📖 Standard Introduction
Nash Equilibrium refers to a state in a game where each player's strategy is the optimal response to other players' strategies. At the Nash equilibrium point, no player can benefit by unilaterally changing their strategy.
Mathematical Definition: A strategy profile (s₁*, s₂*, ..., sₙ*) is a Nash equilibrium if and only if for every player i:
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*) 对所有 sᵢ 成立
💬 Plain Language Explanation
Nash equilibrium is a state where "no one wants to change their mind". Although it may not be the best outcome, if you change your strategy alone, you'll only make things worse for yourself.
🎯 Classic Examples:
- Prisoner's Dilemma: Both betraying is the Nash equilibrium (though both cooperating would be better)
- Traffic Choice: Everyone takes the fastest route, resulting in traffic jams
- Price Competition: All businesses lower prices, reducing profits for everyone
🎮 Interactive Nash Equilibrium Finder
Click on cells to check if it's a Nash equilibrium
Player 2: Cooperate
Player 2: Defect
Player 1: Cooperate
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Player 1: Defect
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⭐ Nash Equilibrium
🏆 Nobel Prize
John Nash won the 1994 Nobel Prize in Economics for the Nash equilibrium theory. The movie "A Beautiful Mind" tells his story.
💼 Practical Applications
Widely applied in market competition, auction design, international relations, network routing, resource allocation and other fields.
🎲 Mixed Strategies
Nash proved that every finite game has at least one Nash equilibrium (which may be a mixed strategy).