Strategy-Proof Facility Location for Concave Cost Functions

TL;DR

This paper introduces a randomized, strategyproof facility location mechanism that guarantees bounded approximation ratios for social and maximum costs across any number of facilities and agents, for concave cost functions.

Contribution

It presents the Equal Cost mechanism, the first to achieve bounded approximation ratios for any number of facilities and agents with concave costs, and analyzes its properties.

Findings

Equal Cost is group strategyproof with approximation ratio at most 2 for Max Cost.

Equal Cost achieves an approximation ratio at most n for Social Cost.

A negative result excludes similar mechanisms for strictly convex costs.

Abstract

We consider k-Facility Location games, where n strategic agents report their locations on the real line, and a mechanism maps them to k facilities. Each agent seeks to minimize his connection cost, given by a nonnegative increasing function of his distance to the nearest facility. Departing from previous work, that mostly considers the identity cost function, we are interested in mechanisms without payments that are (group) strategyproof for any given cost function, and achieve a good approximation ratio for the social cost and/or the maximum cost of the agents. We present a randomized mechanism, called Equal Cost, which is group strategyproof and achieves a bounded approximation ratio for all k and n, for any given concave cost function. The approximation ratio is at most 2 for Max Cost and at most n for Social Cost. To the best of our knowledge, this is the first mechanism with a bounded approximation ratio for instances with k > 2 facilities and any number of agents. Our result implies an interesting separation between deterministic mechanisms, whose approximation ratio for Max Cost jumps from 2 to unbounded when k increases from 2 to 3, and randomized mechanisms, whose approximation ratio remains at most 2 for all k. On the negative side, we exclude the possibility of a mechanism with the properties of Equal Cost for strictly convex cost functions. We also present a randomized mechanism, called Pick the Loser, which applies to instances with k facilities and n = k+1 agents, and for any given concave cost function, is strongly group strategyproof and achieves an approximation ratio of 2 for Social Cost.