# Strategyproof Facility Location for Three Agents on a Circle

## TL;DR

This paper introduces a new strategyproof randomized mechanism for three agents on a circle that improves social welfare and establishes new bounds for such mechanisms in metric spaces.

## Contribution

It proposes a novel strategyproof mechanism based on distance-proportional probabilities and provides the first lower bounds for randomized strategyproof facility location in metric spaces.

## Key findings

- Mechanism is strategyproof in expectation.
- Mechanism outperforms the random dictator in social welfare.
- New upper bound of 7/6 for three agents on a circle.

## Abstract

We consider the facility location problem in a metric space, focusing on the case of three agents. We show that selecting the reported location of each agent with probability proportional to the distance between the other two agents results in a mechanism that is strategyproof in expectation, and dominates the random dictator mechanism in terms of utilitarian social welfare. We further improve the upper bound for three agents on a circle to 7/6 (whereas random dictator obtains 4/3); and provide the first lower bounds for randomized strategyproof facility location in any metric space, using linear programming.