Facility Location Games with Optional Preferences: A Revisit

We study the k-facility location games with optional preferences on the line. In the games, each strategic agent has a public location preference on the k facility locations and a private optional preference on the preferred/acceptable set of facilities out of the k facilities. Our goal is to design strategyproof mechanisms to elicit agents’ optional preferences and locate k facilities to minimize the social or maximum cost of agents based on their facility preferences and public agent locations. We consider two variants of the facility location games with optional preferences: the Min variant and the Max variant where the agent’s cost is defined as their distance to the closest acceptable facility and the farthest acceptable facility, respectively. For the Min variant, we present two deterministic strategyproof mechanisms to minimize the maximum cost and social cost with k ≥ 3 facilities and well-separated n agents, achieving approximation ratios of 3 and 2n + 1 respectively. We complement the results by establishing lower bounds of ₂³ and ⁿ₄ for the approximation ratios achievable by any deterministic strategyproof mechanisms for the maximum cost and social cost, respectively. We then improve our results in a special setting of the Min variant where there are exactly three facilities and present two deterministic strategyproof mechanisms to minimize the maximum cost and social cost. For the Max variant, we present an optimal deterministic strategyproof mechanism for the maximum cost and a k-approximation deterministic strategyproof mechanism for the social cost.