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.
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