Strategyproof mechanisms for 2-facility location games with minimax envy

We study a fairness-based model for 2-facility location games on the real line where the social objective is to minimize the maximum envy over all agents. All the agents seek to minimize their personal costs, and the envy between any two of them is the difference in their personal costs. We consider two cases of personal costs, called min-dist cost and sum-dist cost. We are interested in pursuing strategyproof mechanisms for 2-facility location games in both cases. For the min-dist personal cost, we first show that a lower bound of the additive approximation for any deterministic strategyproof mechanism is 1/4, then devise a deterministic group strategyproof mechanism with additive approximation of 1/2 and two randomized strategyproof mechanisms with additive approximation of 1/4. For the sum-dist personal cost, we devise a group strategyproof deterministic mechanism which is also optimal.