  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 $\mathfrak{M}(m)$ denote the class of mechanisms $M_{G}$ obtained by
varying $G$ on the commodity set $\left\{1,\ldots,m\right\} $. We define the
complexity of a mechanism $M$ in $\mathfrak{M(m)}$ to be a certain 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 $i\neq j$). This induces a quasiorder $\preceq$
on $\mathfrak{M}(m)$ by the rule \[ M\preceq
M^{\prime}\text{if}\tau(M)\leq\tau(M^{\prime})\text{and}\pi(M)\leq\pi(M^{\prime}).
\] We show that, for $m>3$, there are precisely three $\preceq$-minimal
mechanisms $M_{G}$ in $\mathfrak{M}(m)$, 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 $\mathfrak{M}(m)$ for
large enough $m$.
