CORC Technical Report TR - 2005 - 3 Cooperation in Queues ∗

A set of agents arrive simultaneously to a service facility. Each agent requires the facility for a certain length of time and incurs a cost for each unit of time spent in queue. Attached to each agent is an index, which is the ratio of her waiting cost rate to processing time. Efficiency dictates that the agents be served in decreasing order of their indices. If monetary compensations are disallowed, fairness suggests that agents be served in a random order, each ordering of the agents being equally likely. The efficient ordering is unfair to agents with low indices; the random service order is typically extremely inefficient. It is well-known that this gap can be bridged by using monetary compensations. This paper is motivated by the need to design compensation schemes that are fair to all the agents. Assuming quasi-linear preferences, we find money transfers for the efficient ordering of the agents so that every coalition of agents is at least as well off in the proposed solution as they are in the random service order. To that end, we propose two solution concepts (RP and CRP core), which serve to place upper bounds on the cost share of any coalition of agents. The solutions differ in the definition of the expected worth of a coalition, when the agents are served in a random order. A detailed study of these two concepts as well as their compatibility with other fairness criteria are the main focus of this paper. We show that the RP core is not convex, and that the CRP core is. Furthermore, we show that standard solution concepts like the Shapley solution (of a related game) and the equal gains solution are also in the CRP core. We describe an efficient algorithm to find the equal costs solution in this core. Finally, we axiomatically characterize the Shapley solution. This research was supported by an NSF grant DMI-0093981. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY; email: [email protected] Department of Industrial Engineering and Operations Research, Columbia University, New York, NY; email: [email protected]