Nash meets Łukasiewicz: computing equilibria through logic

Abstract A Nash equilibrium is a strategy profile of a game in which none of the players involved has any gain by changing alone his/her own strategy. Given a two-player game, we show a codification of all of its Nash equilibria into Łukasiewicz infinitely-valued logic, that is, we derive a propositional theory in this logic whose models codify all the Nash equilibria. Based on such propositional theory, we derive a polynomial reduction from the problem of computing a Nash equilibrium to the problem of satisfiability of (sets of) formulas of Łukasiewicz infinitely-valued logic. These applications of logic to game theory lead to new methods for computing Nash equilibria.