## Abstract

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) to be a pair of integers τ(**

*M**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*not equal to

*i*≠

*j*). This induces a quasiorder \preceq on

**(**

*M**m*) by the rule

*M* \preceq *M*’ if τ(*M*) ≤ τ(*M*’) and π(*M*) ≤ π(*M*’).

We show that, for *m* > 3, there are precisely three \preceq-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.

Our main result is that, for *any* weights λ, μ > 0; the star mechanism is the unique minimizer of λτ(*M*) + μπ (*M*) on **M**(*m*) for large enough *m*.