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 M_G that satisfies some natural conditions of "fairness" and "convenience." Let \cal{M}(m) denote the class of mechanisms M_G obtained by varying G on the commodity set {1,...,m}. We define the complexity of a mechanism M in \cal{M}(m) to be a pair of integers \tau(M), \pi(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 not equal to i \ne j). This induces a quasiorder \preceq on \cal{M}(m) by the rule: M \preceq M' if tau(M) \le \tau(M') and \pi(M) \le \pi(M'). We show that, for m > 3, there are precisely three \preceq-minimal mechanisms M_G in \cal{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. Our main result is that, for any weights \lambda, \mu > 0; the star mechanism is the unique minimizer of \lambda\tau(M) \mu\pi(M) on \cal{M}(m) for large enough m