Characterization of Group-Strategyproof Mechanisms for Facility Location in Strictly Convex Space

TL;DR

This paper characterizes group-strategyproof mechanisms for facility location in strictly convex spaces, showing that deterministic mechanisms are dictatorial and randomized ones are 2-dictatorial, with implications for approximation bounds.

Contribution

It provides a complete characterization of group-strategyproof mechanisms in strictly convex spaces, revealing their dictatorial nature and establishing tight approximation bounds.

Findings

Deterministic group-strategyproof mechanisms are dictatorial.

Randomized group-strategyproof mechanisms are 2-dictatorial.

The paper establishes tight approximation bounds for these mechanisms.

Abstract

We characterize the class of group-strategyproof mechanisms for the single facility location game in any unconstrained strictly convex space. A mechanism is \emph{group-strategyproof}, if no group of agents can misreport so that all its members are \emph{strictly} better off. A strictly convex space is a normed vector space where $\|x+y\|<2$ holds for any pair of different unit vectors $x \neq y$, e.g., any $L_p$ space with $p\in (1,\infty)$. We show that any deterministic, unanimous, group-strategyproof mechanism must be dictatorial, and that any randomized, unanimous, translation-invariant, group-strategyproof mechanism must be \emph{2-dictatorial}. Here a randomized mechanism is 2-dictatorial if the lottery output of the mechanism must be distributed on the line segment between two dictators' inputs. A mechanism is translation-invariant if the output of the mechanism follows the same translation of the input. Our characterization directly implies that any (randomized) translation-invariant approximation algorithm satisfying the group-strategyproofness property has a lower bound of $2$-approximation for maximum cost (whenever $n \geq 3$), and $n/2 - 1$ for social cost. We also find an algorithm that $2$-approximates the maximum cost and $n/2$-approximates the social cost, proving the bounds to be (almost) tight.