Two-facility Location Games with Minimum Distance Requirement

Abstract

We study the mechanism design problem of a social planner for locating two facilities on a line interval [0, 1], where a set of n strategic agents report their locations and a mechanism determines the locations of the two facilities. We consider the requirement of a minimum distance 0 ≤ d ≤ 1 between the two facilities. Given the two facilities are heterogeneous, we model the cost/utility of an agent as the sum of his distances to both facilities. In the heterogeneous two-facility location game to minimize the social cost, we show that the optimal solution can be computed in polynomial time and prove that carefully choosing one optimal solution as output is strategyproof. We also design a strategyproof mechanism minimizing the maximum cost. Given the two facilities are homogeneous, we model the cost/utility of an agent as his distance to the closer facility. In the homogeneous two-facility location game for minimizing the social cost, we show that any deterministic strategyproof mechanism has unbounded approximation ratio. Moreover, in the obnoxious heterogeneous two-facility location game for maximizing the social utility, we propose new deterministic group strategyproof mechanisms with provable approximation ratios and establish a lower bound (7 − d)/6 for any deterministic strategyproof mechanism. We also design a strategyproof mechanism maximizing the minimum utility. In the obnoxious homogeneous two-facility location game for maximizing the social utility, we propose deterministic group strategyproof mechanisms with provable approximation ratios and establish a lower bound 4/3. Besides, in the two-facility location game with triple-preference, where each facility may be favorable, obnoxious, indifferent for any agent, we further motivate agents to report both their locations and preferences towards the two facilities truthfully, and design a deterministic group strategyproof mechanism with an approximation ratio 4.

Constrained heterogeneous facility location games with max-variant cost

“…Since then, a plethora of different versions have been studied in the literature making different assumptions, always aiming to design strategyproof mechanisms with an as low approximation ratio in terms of some social objective as possible. Indicatively, different variants have been considered depending on the number of facilities to locate [Lu et al, 2010, Fotakis andTzamos, 2014], whether the facilities are desirable or obnoxious [Cheng et al, 2013], whether the preferences of the agents are homogeneous or heterogeneous [Feigenbaum and Sethuraman, 2015, Serafino and Ventre, 2016, Kanellopoulos et al, 2023a, Deligkas et al, 2023, Chen et al, 2020, and whether there are constraints about where the facilities can be placed [Feldman et al, 2016, Kanellopoulos et al, 2023b, Xu et al, 2021 in combination with min or sum individual costs. For a detailed exposition of the truthful facility location literature, we refer the interest reader to the survey of Chan et al [2021].…”

Two-Winner Election Using Favorite-Candidate Voting Rule

“…Fotakis and Tzamos (2014) show that for instance with n ≥ 5 agents, any strategy-proof mechanism either admits a unique dictator, or places the facilities at the leftmost and rightmost. Xu et al (2021) consider the two-facility games with minimum distance requirement.…”

Facility Reallocation on the Line

“…The papers [6,31] comprise a recent study on strategy-proofness for a facility location problem on a line, where there are two facilities to be placed, and agents aim to minimise their total distance to both these facilities. Another recent work is [3], where there are again two facilities, and the utility functions of the agents are heterogeneous, where an agent may either maximise or minimise over the distances to the facilities.…”