From winning strategy to Nash equilibrium

TL;DR

This paper demonstrates that determinacy results in game theory can be transferred to the existence of pure Nash equilibria in multi-outcome games, providing theoretical insights and algorithms applicable to various game classes.

Contribution

It introduces an equilibrium-transfer theorem linking determinacy to Nash equilibria, with conditions and algorithms applicable to complex game structures.

Findings

Transfer of determinacy results to Nash equilibria

Algorithm for computing Nash equilibria in finite-outcome games

Generalization of Borel and parity game determinacy

Abstract

Game theory is usually considered applied mathematics, but a few game-theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e. the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi-outcome setting, the notion of winning strategy is irrelevant yet usually replaced faithfully with the notion of (pure) Nash equilibrium. This article shows that every determinacy result over an arbitrary game structure, e.g. a tree, is transferable into existence of multi-outcome (pure) Nash equilibrium over the same game structure. The equilibrium-transfer theorem requires cardinal or order-theoretic conditions on the strategy sets and the preferences, respectively, whereas counter-examples show that every requirement is relevant, albeit possibly improvable. When the outcomes are finitely many, the proof provides an algorithm computing a Nash equilibrium without significant complexity loss compared to the two-outcome case. As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite-memory determinacy of Muller games.