Equilibrium Points in N-Person Games.¶
Nash, J. F. (1950). Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences, 36(1), 48-49.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Competition
- Treating every interaction as zero-sum competition is a common misreading — the prime identifies where negative coupling exists, not that it exists everywhere.
This sourceFirst appearance of the existence theorem for the Nash equilibrium: every finite n-person game — including non-cooperative, non-zero-sum games — has an equilibrium in mixed strategies, establishing that strategic structure (not just zero-sum rivalry) determines stable outcomes and that non-zero-sum settings form a well-defined family.
- Treating every interaction as zero-sum competition is a common misreading — the prime identifies where negative coupling exists, not that it exists everywhere.
- Equilibrium
- Not stasis. Dynamic equilibrium (a chemical reaction, a steady-state ecosystem, a market at price-clearing) involves vigorous activity; only the net on the balanced dimension is zero. Not optimality. An equilibrium is a balance point, not necessarily a good one. Nash equilibria in non-cooperative games
This sourceThe originating existence theorem — every finite game has an equilibrium in mixed strategies — for the Nash equilibrium, a profile in which no player gains by a unilateral deviation.
- Not stasis. Dynamic equilibrium (a chemical reaction, a steady-state ecosystem, a market at price-clearing) involves vigorous activity; only the net on the balanced dimension is zero. Not optimality. An equilibrium is a balance point, not necessarily a good one. Nash equilibria in non-cooperative games
- Fixed Point
- Take a Cournot duopoly: the state space is the pair of output quantities (q1, q2); the self-map sends each firm's quantity to its profit- maximizing best response given the other's current quantity; a fixed state is a pair where neither firm wants to change — a Nash equilibrium, self-consistency under the best-response update.
This sourceProves existence of Nash equilibrium as a fixed point of the best-response correspondence, via a fixed-point theorem.
- Take a Cournot duopoly: the state space is the pair of output quantities (q1, q2); the self-map sends each firm's quantity to its profit- maximizing best response given the other's current quantity; a fixed state is a pair where neither firm wants to change — a Nash equilibrium, self-consistency under the best-response update.
- Game-Theoretic Strategy
- … the strategy type: pure (one action per information set), mixed (a probability distribution over pure strategies), behavioural (independent randomisation at each information set), or correlated (joint randomisation through a public signal); (3) the solution concept that justifies the strategy — Nash equilibrium
This sourceThe single most-cited solution concept in game theory and the foundation for nearly all subsequent equilibrium analysis.)
- … the strategy type: pure (one action per information set), mixed (a probability distribution over pure strategies), behavioural (independent randomisation at each information set), or correlated (joint randomisation through a public signal); (3) the solution concept that justifies the strategy — Nash equilibrium
- Nash Equilibrium
- The deep structural fact is that interdependent rational choice under common knowledge has fixed-point solutions — guaranteed to exist in any finite game once mixed strategies are admitted, by a fixed-point argument.
This sourceProves that every finite game has at least one equilibrium point in mixed strategies via a fixed-point argument — the existence theorem.
- The deep structural fact is that interdependent rational choice under common knowledge has fixed-point solutions — guaranteed to exist in any finite game once mixed strategies are admitted, by a fixed-point argument.
Domain-specific¶
- Mixed Strategy Equilibrium
- A mixed-strategy equilibrium (Nash, 1950) is a profile of mixed strategies — one per player — such that no player can increase their expected payoff by unilaterally changing their own distribution while holding all others fixed.
This sourceNash proves that every finite game has at least one equilibrium point by applying Kakutani's fixed-point theorem to the best-response correspondence, showing the mapping's graph is closed and its images convex.
SupportedVerified against a saved copy of the source
“the other players in the countered n-tuple. A self-countering n-tuple is called an equilibrium point.”
- A mixed-strategy equilibrium (Nash, 1950) is a profile of mixed strategies — one per player — such that no player can increase their expected payoff by unilaterally changing their own distribution while holding all others fixed.
- Nash equilibrium computation
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