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Minimally Complex Exchange Mechanisms: Emergence of Prices, Markets, and Money

Pradeep Dubey, Siddhartha Sahi and Martin Shubik
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Siddhartha Sahi: Dept. of Mathematics, Rutgers University

No 1945, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

Abstract: 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 tau_{ij}, pi_{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 tau = max tau_{ij} and pi = max pi_{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 (pi,tau) = (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 (pi,tau).

Keywords: Exchange mechanism; Minimal complexity; Prices; Markets; Money (search for similar items in EconPapers)
JEL-codes: C70 C72 C79 D44 D63 D82 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-mic
Date: 2014-04
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