The Shapley value in positional queueing problems

A group of agents are waiting to be served in a facility. Each server in the facility can serve only one agent at a time and agents differ in their cost-types. For this queueing problem, we are interested in finding the order in which to serve agents and the corresponding monetary transfers for the agents. In the standard queueing problem, each agent’s waiting cost is assumed to be constant per unit of time. In this paper, we allow the waiting cost of each agent to depend on the cost-type of each agent and the position assigned to be served. Furthermore, this function is assumed to be supermodular with respect to the cost-type and the position, and non-decreasing with respect to each argument. Our “positional queueing problem” generalizes the queueing problem with multiple parallel servers (Chun and Heo in Int J Econ Theory 4:299–315, 2008) as well as the position allocation problem (Essen and Wooders in J Econ Theory 196:105315, 2021). By applying the Shapley value to the problem, we obtain the optimistic and the pessimistic Shapley rules which are extensions of the minimal (Maniquet in J Econ Theory 109:90–103, 2003) and the maximal (Chun in Math Soc Scie 51:171–181, 2006) transfer rules of the standard queueing problem. We also present axiomatic characterizations of the two rules. The optimistic Shapley rule is the only rule satisfying efficiency and Pareto indifference together with (1) equal treatment of equals and independence of larger cost-types or (2) the identical cost-types lower bound, negative cost-type monotonicity, and last-agent equal responsibility. On the other hand, the pessimistic Shapley rule is the only rule satisfying efficiency and Pareto indifference together with (1) equal treatment of equals and independence of smaller cost-types or (2) the identical cost-types lower bound, positive cost-type monotonicity, and first-agent equal responsibility under constant completion time.