Minimally complex exchange mechanisms: Emergence of prices, markets, and money(R)∗

We consider abstract exchange mechanisms wherein individuals submit “diversified” offers in m commodities, which are then redistributed to them. Our first result is that if the mechanism satisfies certain natural conditions embodying “fairness”and “convenience”then it admits unique prices, in the sense of consistent exchange-rates across commodity pairs ij that equalize the valuation of offers and returns for each individual. We next define integers τij, πij and ki which represent the “time”required to exchange i for j, the “diffi culty” in determining the exchange ratio, and the “dimension”of the offer space in i; and refer to these as time, priceand messagecomplexity of the mechanism. Our second result is that there are only a finite number of minimally complex mechanisms, which moreover correspond to certain directed graphs G in a precise sense. The edges of G can be regarded as markets for commodity pairs, and prices play a stronger role in that the return to a trader depends only on his own offer and the prices. ∗In honor of Lloyd Shapley. (This is a slightly revised version of an earlier Stony Brook Working Paper, as well as Cowles Foundation Discussion Paper 1945, both dated 8 April 2014.) †Center for Game Theory, Department of Economics, Stony Brook University; and Cowles Foundation for Research in Economics, Yale University ‡Department of Mathematics, Rutgers University, New Brunswick, New Jersey §Cowles Foundation for Research in Economics, Yale University; and Santa Fe Institute, New Mexico.