An Existence Theorem of Nash Equilibrium in Coq and Isabelle

Stéphane Le Roux
(Université Libre de Bruxelles)
Érik Martin-Dorel
(IRIT, Université de Toulouse)
Jan-Georg Smaus
(IRIT, Université de Toulouse)

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 where one player has a winning strategy, the original game also has a Nash equilibrium.

This article makes three other contributions: first, while the original proof invoked linear extension of strict partial orders, 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 priority games (generalizing parity games) in quasi-polynomial time.

In Patricia Bouyer, Andrea Orlandini and Pierluigi San Pietro: Proceedings Eighth International Symposium on Games, Automata, Logics and Formal Verification (GandALF 2017), Roma, Italy, 20-22 September 2017, Electronic Proceedings in Theoretical Computer Science 256, pp. 46–60.
Published: 6th September 2017.

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