#### Dubey, P; Sahi, S; Shubik, M

We consider mechanisms that provide 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 "G-mechanism" that satisfies some natural conditions of "fairness" and "convenience". Next we define time and price complexity for any G-mechanism as (respectively) the time required to exchange i for j, and the information needed to determine the exchange ratio (each for the worst pair ij). If the number of commodities exceeds three, there are precisely three minimally complex G -mechanisms, 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. Furthermore, for any weighted sum of complexities, the star mechanism is the unique minimizer of the sum for large enough m.