A dynamic model of barter exchange

We consider the problem of efficient operation of a barter exchange platform for indivisible goods. We introduce a dynamic model of barter exchange where in each period one agent arrives with a single item she wants to exchange for a different item. We study a homogeneous and stochastic environment: an agent is interested in the item possessed by another agent with probability p, independently for all pairs of agents. We consider two settings with respect to the types of allowed exchanges: a) Only two-way cycles, in which two agents swap their items, b) Two or three-way cycles. The goal of the platform is to minimize the average waiting time of an agent. Somewhat surprisingly, we find that in each of these settings, a policy that conducts exchanges in a greedy fashion is near optimal, among a large class of policies that includes batching policies. Further, we find that for small p, allowing three-cycles can greatly improve the waiting time over the two-cycles only setting. Specifically, we find that a greedy policy achieves an average waiting time of Θ(1/p2) in setting a), and Θ(1/p3/2) in setting b). Thus, a platform can achieve the smallest waiting times by using a greedy policy, and by facilitating three cycles, if possible. Our findings are consistent with empirical and computational observations which compare batching policies in the context of kidney exchange programs.