Infinite sequential Nash equilibrium

TL;DR

This paper generalizes existing results on the existence of Nash equilibria in infinite sequential games, providing a transfer theorem under ZF+DC that applies to multi-agent, multi-outcome settings with well-behaved set classes.

Contribution

It introduces a unifying transfer theorem that extends finite and infinite game Nash equilibrium results to more complex multi-agent, multi-outcome games under ZF+DC.

Findings

Proves a transfer theorem for Nash equilibria in infinite sequential games.

Extends finite game results to multi-agent, multi-outcome games.

Establishes conditions under which Nash equilibria exist in complex game settings.

Abstract

In game theory, the concept of Nash equilibrium reflects the collective stability of some individual strategies chosen by selfish agents. The concept pertains to different classes of games, e.g. the sequential games, where the agents play in turn. Two existing results are relevant here: first, all finite such games have a Nash equilibrium (w.r.t. some given preferences) iff all the given preferences are acyclic; second, all infinite such games have a Nash equilibrium, if they involve two agents who compete for victory and if the actual plays making a given agent win (and the opponent lose) form a quasi-Borel set. This article generalises these two results via a single result. More generally, under the axiomatic of Zermelo-Fraenkel plus the axiom of dependent choice (ZF+DC), it proves a transfer theorem for infinite sequential games: if all two-agent win-lose games that are built using a well-behaved class of sets have a Nash equilibrium, then all multi-agent multi-outcome games that are built using the same well-behaved class of sets have a Nash equilibrium, provided that the inverse relations of the agents' preferences are strictly well-founded.