Acyclic Games and Iterative Voting

TL;DR

This paper analyzes convergence properties of iterative voting within acyclic game frameworks, providing conditions under which voting processes converge or not, especially in Plurality voting with different tie-breaking rules.

Contribution

It offers a complete characterization of acyclicity in Plurality voting variations, resolving open questions about restricted and weak acyclicity and refuting a conjecture on strongly-acyclic rules.

Findings

Convergence under lexicographic tie-breaking with weak voter restrictions.

Non-guaranteed convergence with randomized tie-breaking, but existence of paths to Nash equilibrium.

First separation between restricted-acyclicity and weak-acyclicity in game forms.

Abstract

We consider iterative voting models and position them within the general framework of acyclic games and game forms. More specifically, we classify convergence results based on the underlying assumptions on the agent scheduler (the order of players) and the action scheduler (which better-reply is played). Our main technical result is providing a complete picture of conditions for acyclicity in several variations of Plurality voting. In particular, we show that (a) under the traditional lexicographic tie-breaking, the game converges for any order of players under a weak restriction on voters' actions; and (b) Plurality with randomized tie-breaking is not guaranteed to converge under arbitrary agent schedulers, but from any initial state there is \emph{some} path of better-replies to a Nash equilibrium. We thus show a first separation between restricted-acyclicity and weak-acyclicity of game forms, thereby settling an open question from [Kukushkin, IJGT 2011]. In addition, we refute another conjecture regarding strongly-acyclic voting rules.