An Existence Theorem of Nash Equilibrium in Coq and Isabelle

TL;DR

This paper formally proves a theorem on the existence of Nash equilibria in finite outcome games using Coq and Isabelle, generalizes the proof, and explores implications for secure equilibria and computational complexity.

Contribution

It formalizes a published NE existence theorem in proof assistants, generalizes the proof to avoid linear extension, and links the theorem to secure equilibria and quasi-polynomial time algorithms.

Findings

Formal proof of NE existence in Coq and Isabelle

Generalization of the proof avoiding linear extension

Identification of secure equilibria and efficient computation

Abstract

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.