Money as Minimal Complexity
Pradeep Dubey, Siddhartha Sahi, Martin Shubik
TL;DR
This work considers mechanisms that provide the opportunity to exchange commodity i for commodity j, for certain ordered pairs ij, and shows that there is a unique “G-mechanism” that satisfies some natural conditions of “fairness” and “convenience”.
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 $M_{G}$ 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 $τ(M),π(M)$ 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 $i\neq j$). This induces a quasiorder $\preceq$ on $\mathfrak{M}(m)$ by the rule \[ M\preceq M^{\prime}\text{if}τ(M)\leqτ(M^{\prime})\text{and}π(M)\leqπ(M^{\prime}). \] 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 $λ,μ>0,$ the star mechanism is the unique minimizer of $λτ(M)+μπ(M)$ on $\mathfrak{M}(m)$ for large enough $m$.