Graphical exchange mechanisms

Consider an exchange mechanism which accepts “diversified” offers of various commodities and then redistributes them. Under some natural conditions of “fairness” and “convenience”, such a mechanism admits unique prices, which equalize the value of offers and returns for every individual. Next define the complexity of a mechanism via certain integers τij, πij and ki that represent the time required to exchange i for j, the difficulty in determining the exchange ratio, and the dimension of the offers. There are finitely many minimally complex mechanisms, in each of which all trade occurs through markets for commodity pairs. Finally consider minimal mechanisms with smallest worst-case complexities τ=max⁡τij and π=max⁡πij. For m>3 commodities, there are precisely three such mechanisms, one of which has a distinguished commodity - the money - as the sole medium of exchange. As m→∞ the money mechanism is the only one with bounded (π,τ).