Optimal bounds for dissatisfaction in perpetual voting

TL;DR

This paper investigates perpetual approval voting, establishing conditions under which voters' dissatisfaction grows sublinearly over time, and introduces a voting method with guarantees based on expert advice techniques.

Contribution

It identifies a bounded conflicts condition ensuring sublinear dissatisfaction growth and develops a voting method with such guarantees using Kolmogorov complexity and prediction techniques.

Findings

Bounded conflicts condition guarantees sublinear dissatisfaction growth.

A tight upper bound on dissatisfaction growth is established.

A voting method with sublinear dissatisfaction guarantees is proposed.

Abstract

In perpetual voting, multiple decisions are made at different moments in time. Taking the history of previous decisions into account allows us to satisfy properties such as proportionality over periods of time. In this paper, we consider the following question: is there a perpetual approval voting method that guarantees that no voter is dissatisfied too many times? We identify a sufficient condition on voter behavior -- which we call 'bounded conflicts' condition -- under which a sublinear growth of dissatisfaction is possible. We provide a tight upper bound on the growth of dissatisfaction under bounded conflicts, using techniques from Kolmogorov complexity. We also observe that the approval voting with binary choices mimics the machine learning setting of prediction with expert advice. This allows us to present a voting method with sublinear guarantees on dissatisfaction under bounded conflicts, based on the standard techniques from prediction with expert advice.