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

Abstract

We characterize the class of group-strategyproof mechanisms for the single facility location game in any unconstrained strictly convex space. A mechanism is group-strategyproof, if no group of agents can misreport so that all its members are strictly better off. A strictly convex space is a normed vector space where ∥ + ∥ < 2 holds for any pair of different unit vectors ≠ , e.g., any space with ∈ (1, ∞). We show that any deterministic, unanimous, group-strategyproof mechanism must be dictatorial, and that any randomized, unanimous, translation-invariant, group-strategyproof mechanism must be 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 ≥ 3), and /2 − 1 for social cost. We also find an algorithm that 2-approximates the maximum cost and /2-approximates the social cost, proving the bounds to be (almost) tight.CCS Concepts: • Theory of computation → Algorithmic mechanism design; Algorithmic game theory.

Heterogeneous Facility Location with Limited Resources

“…In their setting, the facilities are strategic and need to be compensated monetarily in order for them to be built; the goal is to maximize an aggregate objective given that the total payment is below a predefined budget. Besides these works, there is long literature of (homogeneous) facility location, studying different objectives (Alon et al 2010;Cai, Filos-Ratsikas, and Tang 2016;Feigenbaum, Sethuraman, and Ye 2013;Feldman and Wilf 2013), multiple facilities (Escoffier et al 2011;Fotakis and Tzamos 2013;Lu, Wang, and Zhou 2009;Lu et al 2010), different domains (Schummer and Vohra 2002;Tang, Yu, and Zhao 2020;Sui, Boutilier, and Sandholm 2013;Sui and Boutilier 2015), different cost functions (Filos-Ratsikas et al 2015;Fotakis and Tzamos 2016), and several interesting variants (Golomb and Tzamos 2017;Kyropoulou, Ventre, and Zhang 2019;Zhang and Li 2014;Filos-Ratsikas and Voudouris 2021). The very recent survey of Chan et al (2021) provides an excellent overview of the literature on mechanism design for facility location problems and the survey of Anshelevich et al (2021) provides an overview of the literature on distortion, which has been applied for analyzing facility location settings.…”

Mechanism Design for Facility Location Problems: A Survey

“…Dokow et al [26] presented similar characterizations to that of [68] for discrete lines and circles. Recently, Tang et al [75] provided a characterization of deterministic and randomized group strategyproof unanimous mechanisms in convex spaces (generalizing and extending some classic results of [12]), and showed approximation ratio bounds that can be obtained as a consequence of their characterization. Filimonov and Meir [33] gave a full characterization of onto and strategyproof mechanisms in discrete trees.…”

Approximate mechanism design for distributed facility location

“…e problem became extremely popular in the economics and computation community a er Procaccia and Tennenholtz [56] used it to put forward their agenda of approximate mechanism design without money, following the similar agenda of Nisan and Ronen [52] for se ings with money. Since then, the facility location problem has been studied extensively, for different objectives [3,16,27,28], multiple facilities [25,36,46,47], different domains [58,65,64,63], different cost functions [32,37], and several variants of the problem [5,20,22,21,23,34,35,41,45,61,60,66,59].…”