Facility Location Games with Thresholds

In classic facility location games, a facility is to be placed based on the reported locations from agents. Each agent wants to minimize the cost (distance) between her location and the facility. In real life, the cost of an agent may not strictly increase with the distance. In this paper, we introduce two types of thresholds to the agent's cost. For the model with lower thresholds, the agent's cost is 0 if the distance is within the threshold, otherwise it increases linearly until the value 1. Similarly, for the model with upper thresholds, the cost is 1 if the distance is beyond the threshold, otherwise it is a linear function with the value from 0 to 1. We aim to prevent the agent from misreporting her location while optimizing social objectives in both models. For the first model, we design a strategyproof mechanism optimal for the social cost objective and a strategyproof mechanism with an approximation ratio of 3 for the maximum cost objective. For the second model, we use the median mechanism for the social cost with a threshold-based approximation ratio and design a new mechanism for the maximum cost with tight bounds. We also show lower bounds for both models. Finally, we derive results for the scenario where each agent has both thresholds.