On Voting and Facility Location

Abstract

We study mechanisms for candidate selection that seek to minimize the social cost, where voters and candidates are associated with points in some underlying metric space. The social cost of a candidate is the sum of its distances to each voter. Some of our work assumes that these points can be modeled on a real line, but other results of ours are more general.A question closely related to candidate selection is that of minimizing the sum of distances for facility location. The difference is that in our setting there is a fixed set of candidates, whereas the large body of work on facility location seems to consider every point in the metric space to be a possible candidate. This gives rise to three types of mechanisms which differ in the granularity of their input space (voting, ranking and location mechanisms). We study the relationships between these three classes of mechanisms.While it may seem that Black's 1948 median algorithm is optimal for candidate selection on the line, this is not the case. We give matching upper and lower bounds for a variety of settings. In particular, when candidates and voters are on the line, our universally truthful spike mechanism gives a [tight] approximation of two. When assessing candidate selection mechanisms, we seek several desirable properties: (a) efficiency (minimizing the social cost) (b) truthfulness (dominant strategy incentive compatibility) and (c) simplicity (a smaller input space). We quantify the effect that truthfulness and simplicity impose on the efficiency.

Improved Metric Distortion for Deterministic Social Choice Rules

“…In addition to deterministic social choices rules, randomized rules have been studied in the metric distortion setting. It is possible that they can achieve better distortion than deterministic ones: Randomly selecting a voter to be the dictator achieves distortion of 3 [2,12]. On the other hand, no randomized rule can get distortion better than 2 [2,12], no truthful randomized rule can beat 3 [12] and no weighted tournament rule can beat 3 [16].…”

Awareness of Voter Passion Greatly Improves the Distortion of Metric Social Choice

“…Randomized vs Deterministic Mechanisms In this paper we restrict our attention to deterministic social choice rules, instead of randomized ones as in e.g., [2,11,17,22], for several reasons. First, consider looking at our mechanisms from a social choice perspective, i.e., as voting rules that need to be adopted by organizations and used in practice.…”

Strategyproof Facility Location for Three Agents on a Circle

“…In 2009, the agenda of approximation mechanisms without money was made explicit in a paper by Tennenholtz [2009, 2013], who used facility location as their primary domain of demonstration due to its simplicity. Moreover, facility location is often a bridge between mechanism design and social choice [Caragiannis et al, 2010, Feldman et al, 2016 and has applications to transport [Moujahed et al, 2006], disaster relief [Florez et al, 2015] and more. Facility location is thus often used as a testbed for ideas and techniques in mechanism design and noncooperative multiagent systems.…”

“…In the constrained variant, the RD mechanism obtains 3− 2 n approximation and this is known to be tight for all strategyproof mechanisms. The upper bound holds for any metric space , whereas the lower bound requires specific constructions on the n-dimensional binary cube [Feldman et al, 2016, Meir et al, 2011. Anshelevich and Postl [2017] show a smooth transition of the RD approximation ratio from 2 − 2 n to 3 − 2 n as the location of the facility becomes more constrained.…”