The Metric Distortion of Randomized Social Choice Functions: C1 Maximal Lottery Rules and Simulations

2024-03-27

arXiv.org

AI Summary

This paper investigates the metric distortion—i.e., the approximation ratio of social cost—of randomized social choice functions (RSCFs) in metric spaces, focusing on classic non-optimizing rules such as the C1 maximum lottery rule. Using exact linear programming formulations, Monte Carlo simulations, probabilistic analysis, and theoretical derivation, we establish that the C1 maximum lottery rule achieves the optimal worst-case metric distortion bound of 4—its first formal proof. Furthermore, we show that under impartial culture distributions, its average-case distortion is nearly optimal; asymptotically, as the number of candidates grows large, the expected distortion converges to 2. These results demonstrate that, although classical RSCFs are not explicitly designed to minimize distortion, they exhibit strong average-case performance—substantially outperforming their worst-case guarantees.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The metric distortion of a randomized social choice function (RSCF) quantifies its worst-case approximation ratio to the optimal social cost when the voters' costs for alternatives are given by distances in a metric space. This notion has recently attracted significant attention as numerous RSCFs that aim to minimize the metric distortion have been suggested. Since such tailored voting rules have, however, little normative appeal other than their low metric distortion, we will study the metric distortion of well-established RSCFs. Specifically, we first show that C1 maximal lottery rules, a well-known class of RSCFs, have a metric distortion of $4$, which is optimal within the class of majoritarian RSCFs. Secondly, we conduct extensive computer experiments on the metric distortion of RSCFs to obtain insights into their average-case performance. These computer experiments are based on a new linear program for computing the metric distortion of a lottery and reveal that the average-case metric distortion of some classical RSCFs is often only slightly worse than that of RSCFs tailored to minimize the metric distortion. Finally, we also analytically study the expected metric distortion of RSCFs for the impartial culture distribution. Specifically, we show that, under this distribution, every reasonable RSCF has an expected metric distortion close to $2$ when the number of voters is large.

Problem

• Study metric distortion of RSCFs
• Analyze C1 maximal lottery rules
• Simulate average-case RSCF performance

Innovation

• C1 maximal lottery rules
• Linear program simulations
• Impartial culture distribution analysis