Assigning agents to a line

We consider the problem of assigning agents to slots on a line, where only one agent can be served at a slot and each agent prefers to be served as close as possible to his target. Our focus is on utilitarian methods, i.e., those that minimize the total gap between targets and assigned slots. We first consider deterministic assignment of agents to slots, and provide a direct method for testing if a given deterministic assignment is utilitarian. We then consider probabilistic assignment of agents to slots, and make use of the previous method to propose a utilitarian modification of the classic random priority method to solve this class of problems. We also provide some logical relations in our setting among standard axioms in the literature on assignment prob- lems.