Beyond the worst case: Distortion in impartial culture electorates

TL;DR

This paper investigates how voting rules perform in average-case scenarios with randomly distributed agent preferences, showing that simple mechanisms with minimal cardinal queries can significantly reduce distortion.

Contribution

It introduces a stochastic model for electorate preferences, refines distortion analysis for this setting, and proposes mechanisms that achieve near-optimal average distortion with minimal queries.

Findings

All voting rules have high average distortion in the stochastic model.

A mechanism with a single binary query per agent nearly optimizes average distortion.

Tradeoffs between distortion and number of queries are characterized in worst-case settings.

Abstract

{\em Distortion} is a well-established notion for quantifying the loss of social welfare that may occur in voting. As voting rules take as input only ordinal information, they are essentially forced to neglect the exact values the agents have for the alternatives. Thus, in worst-case electorates, voting rules may return low social welfare alternatives and have high distortion. Accompanying voting rules with a small number of cardinal queries per agent may reduce distortion considerably. To explore distortion beyond worst-case conditions, we use a simple stochastic model according to which the values the agents have for the alternatives are drawn independently from a common probability distribution. This gives rise to so-called {\em impartial culture electorates}. We refine the definition of distortion so that it is suitable for this stochastic setting and show that, rather surprisingly, all voting rules have high distortion {\em on average}. On the positive side, for the fundamental case where the agents have random {\em binary} values for the alternatives, we present a mechanism that achieves approximately optimal average distortion by making a {\em single} cardinal query per agent. This enables us to obtain slightly suboptimal average distortion bounds for general distributions using a simple randomized mechanism that makes one query per agent. We complement these results by presenting new tradeoffs between the distortion and the number of queries per agent in the traditional worst-case setting.