On the Randomized Metric Distortion Conjecture

TL;DR

This paper disproves a long-standing conjecture by demonstrating a specific instance of the metric single winner determination problem where any randomized social choice function has a distortion of at least 2.063164, exceeding the conjectured bound.

Contribution

It provides a concrete counterexample to the conjecture that the worst-case distortion of any randomized social choice function is at most 2.

Findings

Established a lower bound of 2.063164 for distortion in a specific instance.

Disproved the conjecture that the maximum distortion is 2 for all randomized functions.

Highlights limitations of current randomized social choice mechanisms.

Abstract

In the single winner determination problem, we have n voters and m candidates and each voter j incurs a cost c(i, j) if candidate i is chosen. Our objective is to choose a candidate that minimizes the expected total cost incurred by the voters; however as we only have access to the agents' preference rankings over the outcomes, a loss of efficiency is inevitable. This loss of efficiency is quantified by distortion. We give an instance of the metric single winner determination problem for which any randomized social choice function has distortion at least 2.063164. This disproves the long-standing conjecture that there exists a randomized social choice function that has a worst-case distortion of at most 2.