Rutgers Discrete Mathematics Seminar

Title: Minimally complex exchange mechanisms: Emergence of prices, markets, and money

Speaker: Siddhartha Sahi, Rutgers University

Date: Tuesday, April 29, 2014 2:00pm

Location: Hill Center, Room 525, Rutgers University, Busch Campus, Piscataway, NJ


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, and these 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 π = 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 which have complexities (π,τ) = (4,2), (2,m-1), (mē-m,1) respectively. The star mechanism, unlike the other two, has a distinguished commodity -- the -- -- money -- that serves as the sole medium of exchange. As m→∞ it is the only one with bounded (π,τ).