An Existence Theorem of Nash Equilibrium in Coq and Isabelle

Nash equilibrium (NE) is a central concept in game theory. Here we prove formally a published theorem on existence of an NE in two proof assistants, Coq and Isabelle: starting from a game with finitely many outcomes, one may derive a game by rewriting each of these outcomes with either of two basic outcomes, namely that Player 1 wins or that Player 2 wins. If all ways of deriving such a win/lose game lead to a game with a winning strategy for either of the players, the original game also has a Nash equilibrium.

This article makes three other contributions: first, while the original proof invoked linear extension of acyclic binary relations, here we avoid it by generalizing the relevant definition. Second, we notice that the theorem also implies the existence of a secure equilibrium, a stronger version of NE that was introduced for model checking. Third, we also notice that the constructive proof of the theorem computes secure equilibria for non-zero-sum parity games in quasi-polynomial time.

This is a joint work by Stéphane Le Roux, Erik Martin-Dorel and Jan-Georg Smaus.

Coq formalization

You can download the Winning2NE Coq library as a tar.gz archive (Winning2NE-0.2.tar.gz).

Isabelle formalization

You can download the Winning2NE Isabelle library as a theory file (Winning2NE.thy).