Money as Minimal Complexity

31 Pages Posted: 21 Feb 2015

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: February 19, 2015

Abstract

We consider mechanisms that provide traders the opportunity to exchange commodity i for commodity j, for certain ordered pairs ij. Given any connected graph G of opportunities, we show that there is a unique mechanism M_G that satisfies some natural conditions of "fairness" and "convenience." Let \cal{M}(m) denote the class of mechanisms M_G obtained by varying G on the commodity set {1,...,m}. We define the complexity of a mechanism M in \cal{M}(m) to be a pair of integers \tau(M), \pi(M) which represent the "time" required to exchange i for j and the "information" needed to determine the exchange ratio (each in the worst case scenario, across all i not equal to i \ne j). This induces a quasiorder \preceq on \cal{M}(m) by the rule: M \preceq M' if tau(M) \le \tau(M') and \pi(M) \le \pi(M'). We show that, for m > 3, there are precisely three \preceq-minimal mechanisms M_G in \cal{M}(m), where G corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity -- the money -- that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. Our main result is that, for any weights \lambda, \mu > 0; the star mechanism is the unique minimizer of \lambda\tau(M) \mu\pi(M) on \cal{M}(m) for large enough m.

Keywords: Exchange mechanism, Minimal complexity, Prices, money

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

Suggested Citation

Dubey, Pradeep K. and Sahi, Siddhartha and Shubik, Martin, Money as Minimal Complexity (February 19, 2015). Cowles Foundation Discussion Paper No. 1990, Available at SSRN: https://ssrn.com/abstract=2567447

Pradeep K. Dubey

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 (Contact Author)

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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