Ex-ante versus Ex-post: Egalitarian Facility Location Mechanism Design

2026-08-14

AI Summary

This study addresses the egalitarian cost minimization problem in strategic facility location within Euclidean spaces by proposing an ex-ante evaluation perspective and designing a randomized rotation mechanism. Integrating mechanism design with geometric approximation algorithms, we achieve strict separation in low dimensions while establishing impossibility theorems for high dimensions. Specifically, we attain the optimal approximation ratio of 1 in one dimension and improve the bound to 1.598 in two dimensions, simultaneously proving the optimality of deterministic mechanisms in higher dimensions. This work effectively refines performance bounds in low-dimensional settings and reveals fundamental limitations of randomized mechanisms in high-dimensional spaces, thereby providing new theoretical benchmarks for strategy-proof facility location.

Abstract

We study the facility location mechanism design problem where $n$ strategic agents report locations in Euclidean space and the mechanism outputs a single facility location. Each agent's cost is its distance from the facility, and our objective is to minimize the egalitarian cost, i.e., the maximum agent cost, in a strategyproof way. The optimal deterministic approximation ratio is $2$, achieved by any dictator mechanism. We study the power of randomized strategyproof-in-expectation mechanisms. Prior work has focused on ex-post evaluation, defined as the expected maximum agent cost. We instead study ex-ante evaluation, defined as the maximum expected agent cost, which is naturally aligned with strategyproofness in expectation. We establish the following results: (1) Low dimensions: Strict ex-ante vs. ex-post separation. In $\mathbb{R}$, we give a simple strategyproof mechanism achieving the optimal ex-ante approximation ratio of $1$. In $\mathbb{R}^2$, we design the "Random Rotated Corner" mechanism, with ex-ante approximation ratio at most $1.598$, breaking the deterministic barrier. For the ex-post objective, we prove a lower bound of $1.605$, yielding a strict separation in $\mathbb{R}^2$. (2) High dimensions: Impossibility. In $\mathbb{R}^d$ for $d \gg 1$, we show that no strategyproof-in-expectation mechanism improves on the deterministic dictator mechanism beyond $o_d(1)$. Thus neither ex-post nor ex-ante evaluation yields improved fairness guarantees in high dimensions. An implication is that the "Random Rotation Coordinate-Wise Median" (RRCWM), currently the best known mechanism for the utilitarian objective, is also best possible for the egalitarian objective in high dimension: we show it achieves an approximation ratio of $2$ for both ex-post and ex-ante objectives in $\mathbb{R}^d$ for every $d \ge 1$.

Problem

• Facility Location
• Egalitarian Cost
• Strategyproofness
• Ex-ante Evaluation
• Approximation Ratio

Innovation

• Ex-ante evaluation
• Strategyproof-in-expectation
• Egalitarian facility location
• Random Rotated Corner mechanism
• Approximation ratio separation