A semi-potential for finite and infinite sequential games

TL;DR

This paper introduces a semi-potential framework for extensive form games, demonstrating convergence of better-response dynamics to Nash equilibria in finite and certain infinite games, even with cyclic preferences.

Contribution

It develops a semi-potential approach for sequential games, showing convergence properties and rationality notions applicable to both finite and infinite extensive form games.

Findings

Finite games converge to Nash equilibria in quadratic time.

Players with acyclic preferences stabilize despite cyclic preferences of others.

Infinite games with continuous payoffs also exhibit convergence under certain conditions.

Abstract

We consider a dynamical approach to game in extensive forms. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic (finite) time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies. For infinite games in extensive form we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are $\Delta^0_2$-sets.