Nash equilibrium

Nash equilibrium
Solution concept in game theory
Relationship
Subset ofRationalizability, Epsilon-equilibrium, Correlated equilibrium
Superset ofEvolutionarily stable strategy, Subgame perfect equilibrium, Perfect Bayesian equilibrium, Trembling hand perfect equilibrium, Stable Nash equilibrium, Strong Nash equilibrium
Significance
Proposed byJohn Forbes Nash Jr.
Used forAll non-cooperative games

In game theory, the Nash equilibrium is the most commonly-used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed).[1] The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly.[2]

If each player has chosen a strategy – an action plan based on what has happened so far in the game – and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium.

If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. In a game in which Carol and Dan are also players, (A, B, C, D) is a Nash equilibrium if A is Alice's best response to (B, C, D), B is Bob's best response to (A, C, D), and so forth.

Nash showed that there is a Nash equilibrium, possibly in mixed strategies, for every finite game.[3]

  1. ^ Osborne, Martin J.; Rubinstein, Ariel (12 Jul 1994). A Course in Game Theory. Cambridge, MA: MIT. p. 14. ISBN 9780262150415.
  2. ^ Kreps D.M. (1987) "Nash Equilibrium." In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London.
  3. ^ Nash, John F. (1950). "Equilibrium points in n-person games". PNAS. 36 (1): 48–49. Bibcode:1950PNAS...36...48N. doi:10.1073/pnas.36.1.48. PMC 1063129. PMID 16588946.