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 certain integers τ ij , π ij , and k i which represent the “time” required to exchange i for j , the “diﬀiculty” in determining the exchange ratio, and the “dimension” of the oﬀer space in i ; we refer to these as time- , price- and message- complexity 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. Finally we consider “strongly” minimal mechanisms, with smallest “worst case” complexities τ = max τ ij and pi = max π ij . Our third main result is that for m > 3 commodities that there are precisely three such mechanisms, which correspond to the star, cycle, and complete graphs, and have complexities ( π,τ ) = (4,2), (2, m − 1), ( m 2 − m , 1) respectively. Unlike the other two mechanisms, the star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange. As m approaches inﬁnity it is the only mechanism with bounded ( π,τ ).
Dubey, Pradeep; Sahi, Siddhartha; and Shubik, Martin, "Minimally Complex Exchange Mechanisms: Emergence of Prices, Markets, and Money" (2014). Cowles Foundation Discussion Papers. 2346.