Metric Distortion under Group-Fair Objectives

TL;DR

The paper studies metric voting where agents are partitioned into groups and aims to minimize group-aware costs under two objectives, $Max{-}of{-}Avg$ and $Avg{-}of{-}Max$. It analyzes three classes of mechanisms with varying information: full-information group-oblivious, ordinal-information group-oblivious, and group-aware mechanisms. The authors prove tight distortion bounds across scenarios: $3$ for Max{-}of{-}Avg under full-information, and $3$ (symmetric) or $k$ (general) for Avg{-}of{-}Max; ordinal-information group-oblivious mechanisms yield $5$-distortion for both objectives in the symmetric case and $2k+1$ in the general case. When groups are known, group-aware ordinal mechanisms achieve $3$-distortion for two alternatives, and with known distances the same bound can be achieved via Virtual-MiniMax/-MiniAvg mechanisms; the paper also outlines open problems for more-than-two alternatives. This work advances understanding of fairness-aware efficiency trade-offs in distributed voting and points toward further exploration of group-aware ordinal mechanisms and randomized approaches.

Abstract

We consider a voting problem in which a set of agents have metric preferences over a set of alternatives, and are also partitioned into disjoint groups. Given information about the preferences of the agents and their groups, our goal is to decide an alternative to approximately minimize an objective function that takes the groups of agents into account. We consider two natural group-fair objectives known as Max-of-Avg and Avg-of-Max which are different combinations of the max and the average cost in and out of the groups. We show tight bounds on the best possible distortion that can be achieved by various classes of mechanisms depending on the amount of information they have access to. In particular, we consider group-oblivious full-information mechanisms that do not know the groups but have access to the exact distances between agents and alternatives in the metric space, group-oblivious ordinal-information mechanisms that again do not know the groups but are given the ordinal preferences of the agents, and group-aware mechanisms that have full knowledge of the structure of the agent groups and also ordinal information about the metric space.

Figures (1)

• Figure 1: An illustration of the worst-case instance given by Lemma \ref{['lem:reduction_lemma']}.

Theorems & Definitions (38)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof