# Nash Equilibrium

**Type:** Concept  
**Domain:** Applied Mathematics  
**Proved:** 1950, John Nash  
**Codex URL:** /mathematics/applied/nash-equilibrium/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A stable outcome where no player benefits from unilaterally changing strategy. Nash proved existence in every finite game (1950), aged 22. 1994 Nobel Memorial Prize in Economic Sciences.

## Definition
No player can improve their payoff by changing only their own strategy, given others' strategies fixed.

## Mixed strategies
Matching pennies has no pure-strategy equilibrium; both players randomizing 50-50 is the unique Nash equilibrium.

## Existence proof
Uses Kakutani's fixed-point theorem (1941) — a Nash equilibrium is a fixed point of the "best response" function.

## Refinements
Selten's subgame perfect equilibrium (1965) rules out non-credible threats. Harsanyi (1967-68) extended to incomplete information. All three shared 1994 Nobel Prize.

## The 27-page thesis
Nash's complete 1950 doctoral thesis was just 27 pages, containing the full existence proof.

## Correlated equilibrium
Aumann (1974) — allows correlated strategies via shared signals, potentially better for all players. 2005 Nobel Prize.

## Sources
### Tier 1
- Nash, J.F. (1950). Equilibrium points in n-person games. *PNAS*.
- Nash, J.F. (1951). Non-cooperative games. *Annals of Mathematics*.
### Tier 2
- Osborne, M.J. and Rubinstein, A. (1994). *A Course in Game Theory*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*