A Semi-Potential for Finite and Infinite Sequential Games (Extended Abstract)

TL;DR

This paper introduces a semi-potential approach to sequential games, demonstrating convergence to Nash equilibria in finite and certain infinite cases, even with irrational or cyclic preferences.

Contribution

It proposes a semi-potential framework for sequential games, ensuring convergence of better-response dynamics under restricted convertibility, applicable to both finite and infinite games.

Findings

Finite games converge to Nash in quadratic time.

Players with acyclic preferences stabilize despite cyclic preferences.

Infinite games with continuous payoffs also converge under the framework.

Abstract

We consider a dynamical approach to sequential games. 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 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 sequential games 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.