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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]
^Osborne, Martin J.; Rubinstein, Ariel (12 Jul 1994). A Course in Game Theory. Cambridge, MA: MIT. p. 14. ISBN 9780262150415.
^ Kreps D.M. (1987) "Nash Equilibrium." In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London.
^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.
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