Facility location with double-peaked preferences

Abstract

We study the problem of locating a single facility on a real line based on the reports of self-interested agents, when agents have double-peaked preferences, with the peaks being on opposite sides of their locations. We observe that double-peaked preferences capture reallife scenarios and thus complement the well-studied notion of single-peaked preferences. As a motivating example, assume that the government plans to build a primary school along a street; an agent with single-peaked preferences would prefer having the school built exactly next to her house. However, while that would make it very easy for her children to go to school, it would also introduce several problems, such as noise or parking congestion in the morning. A 5-min walking distance would be sufficiently far for such problems to no longer be much of a factor and at the same time sufficiently close for the school to be easily accessible by the children on foot. There are two positions (symmetrically) in each direction and those would be the agent's two peaks of her double-peaked preference. Motivated by natural scenarios like the one described above, we mainly focus on the case where peaks are equidistant from the agents' locations and discuss how our results extend to more general settings. We show that most of the results for single-peaked preferences do not directly apply to this setting, which makes the problem more challenging. As our main contribution, we present a simple truthfulin-expectation mechanism that achieves an approximation ratio of 1 + b/c for both the social and the maximum cost, where b is the distance of the agent from the peak and c is the minimum cost of an agent. For the latter case, we provide a 3/2 lower bound on the approximation ratio of any truthful-in-expectation mechanism. We also study deterministic mechanisms under some natural conditions, proving lower bounds and approximation guarantees. We prove that among a large class of reasonable strategyproof mechanisms, there is no deterministic mechanism that outperforms our truthful-in-expectation mechanism. In order to obtain this result, we first characterize mechanisms for two agents that satisfy two simple properties; we use the same characterization to prove that no mechanism in this class can be groupstrategyproof.

Strategyproof Mechanism for Two Heterogeneous Facilities with Constant Approximation Ratio

“…This is the pioneering work in the field. Later on, more characterizations are studied on networks [Schummer and Vohra, 2002], for two facilities [Fotakis and Tzamos, 2014] and for double-peaked preference [Filos-Ratsikas et al, 2017]. There are also characterizations for other variants of facility location games.…”

“…[Zou and Li, 2015] and [Feigenbaum and Sethuraman, 2015] studied dual preference where agents have different preferences towards the single facility. [Filos-Ratsikas et al, 2017] studied double-peaked preference.…”

The Surprising Power of Hiding Information in Facility Location

“…All rights reserved. many variants have been studied such as: locating multiple facilities (Escoffier et al 2011), locating a facility in multiple dimensions (Sui, Boutilier, and Sandholm 2013), exploring different types of agent preferences (Filos-Ratsikas et al 2017) and objectives (Feldman and Wilf 2013), and strategic opening of facilities (Chen et al 2019).…”

“…In the aforementioned example, residents may report both their home and work address, or a school district may report the addresses of multiple schools under its purview. This has been somewhat explored in facility location (Dekel, Fischer, and Procaccia 2010;Filos-Ratsikas et al 2017). We also note additional motivation from a different line of literature.…”

Strategyproof Facility Location for Three Agents on a Circle

“…For deterministic mechanisms even the existence of a single cycle in a graph entails that any strategyproof mechanism must be dictatorial on a subdomain, and thus has an approximation ratio that increases linearly with the number of agents n [Dokow et al, 2012, Schummer andVohra, 2004]. Many variations of the problem have since been explored in the AI and multiagent systems community, including multiple facilities [Anastasiadis and Deligkas, 2018, Escoffier et al, 2011, Serafino and Ventre, 2015, complex incentives and forms of strategic behavior [Filos-Ratsikas et al, 2017, Sui and Boutilier, 2015, Todo et al, 2011, Zou and Li, 2015, and alternative design goals [Alon et al, 2010a, Feldman and Wilf, 2013, Mei et al, 2016. The circle in particular has received much attention in the facility location literature [Alon et al, 2010a,b, Cai et al, 2016, Dokow et al, 2012, Schummer and Vohra, 2004, both because it is the simplest graph for which median-like mechanisms cannot work, and because it is an abstraction of actual problems like selecting a time-of-the-day or a server in a ring of computers.…”