Sensitivity of the Performance of a Simple Exchange Model to its Topology

Abstract

We study a simple exchange model in which price is fixed and the amount of a good transferred between actors depends only on the actors' respective budgets and the existence of a link between transacting actors. The model induces a simply-connected but possibly multi-component bipartite graph. A trading session on a fixed graph consists of a sequence of exchanges between connected buyers and sellers until no more exchanges are possible. We deem a trading session "feasible" if all of the buyers satisfy their respective demands. If all trading sessions are feasible the graph is deemed "successful", otherwise the feasibility of a trading session depends on the order of the sequence of exchanges. We demonstrate that topology is important for the success of trading sessions on graphs. In particular, for the case that supply equals demand for each component of the graph, we prove that the graph is successful if and only if the graph consists of components each of which are complete bipartite. For the case that supply exceeds demand, we prove that the other topologies also can be made successful but with finite reserve (i.e., excess supply) requirements that may grow proportional to the number of buyers. Finally, with computations for a small instance of the model, we provide an example of the wide range of performance in which only the connectivity varies. These results taken together place limits on the improvements in performance that can be expected from proposals to increase the connectivity of sparse exchange networks.