D ec 2 01 5 Money as Minimal Complexity ∗

We consider mechanisms that provide traders the opportunity to exchange commodity i for commodity j, for certain ordered pairs ij. Given any connected graph G of opportunities, we show that there is a unique mechanism MG that satisfies some natural conditions of “fairness” and “convenience”. Let M(m) denote the class of mechanisms MG obtained by varying G on the commodity set {1, . . . , m}. We define the complexity of a mechanism M in M(m) to be a certain pair of integers τ(M), π(M) which represent the time required to exchange i for j and the information needed to determine the exchange ratio (each in the worst case scenario, across all i 6= j). This induces a quasiorder on M(m) by the rule M M ′ if τ(M) ≤ τ(M ) and π(M) ≤ π(M ). We show that, form > 3, there are precisely three -minimal mechanisms MG in M(m), where G corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity – the money – that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. In honor of Lloyd Shapley. Stony Brook Center for Game Theory, Dept. of Economics; and Cowles Foundation for Research in Economics, Yale University Department of Mathematics, Rutgers University, New Brunswick, New Jersey Cowles Foundation for Research in Economics, Yale University; and Santa Fe Institute, New Mexico.