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 mechanismsMG obtained by varying G on the commodity set {1, …, m}. We define the complexity of a mechanism M in M(m) to be a 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 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.