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

44 Pages Posted: 10 Apr 2014

See all articles by Pradeep K. Dubey

Pradeep K. Dubey

SUNY Stony Brook - Center for Game Theory in Economics

Siddhartha Sahi

Rutgers, The State University of New Jersey - New Brunswick/Piscataway

Martin Shubik

Yale University - School of Management; Yale University - Cowles Foundation

Date Written: April 9, 2014

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

JEL Classification: C70, C72, C79, D44, D63, D82

Suggested Citation

Dubey, Pradeep K. and Sahi, Siddhartha and Shubik, Martin, Minimally Complex Exchange Mechanisms: Emergence of Prices, Markets, and Money (April 9, 2014). Cowles Foundation Discussion Paper No. 1945, Available at SSRN: https://ssrn.com/abstract=2422863 or http://dx.doi.org/10.2139/ssrn.2422863

Pradeep K. Dubey (Contact Author)

SUNY Stony Brook - Center for Game Theory in Economics ( email )

Stony Brook, NY 11794
United States
631-632-7555 (Phone)
631-632-7516 (Fax)

Siddhartha Sahi

Rutgers, The State University of New Jersey - New Brunswick/Piscataway ( email )

94 Rockafeller Road
New Brunswick, NJ 08901
United States

Martin Shubik

Yale University - School of Management ( email )

Box 208200
New Haven, CT 06520-8200
United States

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States
203-432-3694 (Phone)
203-432-6167 (Fax)

HOME PAGE: http://cowles.econ.yale.edu/P/au/d_shubik.htm

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