On terminating improvement in two-player games

TL;DR

This paper characterizes classes of finite two-player games where the finite improvement property (FIP) is preserved under payoff modifications and explores conditions for reaching Nash equilibria in weakly acyclic games using Markov chains.

Contribution

It provides a quadratic-time characterization of games with FIP preserved under payoff changes and offers an inductive description of weakly acyclic two-player games.

Findings

Characterization of games with FIP preserved under payoff modifications

Quadratic-time algorithm for identifying such games

Markov chain-based analysis of weakly acyclic games

Abstract

A real-valued game has the finite improvement property (FIP), if starting from an arbitrary strategy profile and letting the players change strategies to increase their individual payoffs in a sequential but non-deterministic order always reaches a Nash equilibrium. E.g., potential games have the FIP. Many of them have the FIP by chance nonetheless, since modifying even a single payoff may ruin the property. This article characterises (in quadratic time) the class of the finite games where FIP not only holds but is also preserved when modifying all the occurrences of an arbitrary payoff. The characterisation relies on a pattern-matching sufficient condition for games (finite or infinite) to enjoy the FIP, and is followed by an inductive description of this class. A real-valued game is weakly acyclic if the improvement described above can reach a Nash equilibrium. This article characterises the finite such games using Markov chains and almost sure convergence to equilibrium. It also gives an inductive description of the two-player such games.