2015/12/15 by Dubey, Pradeep, Sahi, Siddhartha, Shubik, Martin
#91B64 #Combinatorics (math.CO) #Computer Science and Game Theory (cs.GT) #FOS: Computer and information sciences #FOS: Economics and business #FOS: Mathematics #Theoretical Economics (econ.TH)
paper · doi:10.48550/arxiv.1512.04637
Consider an exchange mechanism which accepts diversified offers of various commodities and redistributes everything it receives. We impose certain conditions of fairness and convenience on such a mechanism and show that it admits unique prices, which equalize the value of offers and returns for each individual. We next define the complexity of a mechanism in terms of certain integers τij,πij and ki that represent the time required to exchange i for j, the difficulty in determining the exchange ratio, and the dimension of the message space. We show that there are a finite number of minimally complex mechanisms, in each of which all trade is conducted through markets for commodity pairs. Finally we consider minimal mechanisms with smallest worst-case complexities τ=maxτij and π=maxπij. For m>3 commodities, there are precisely three such mechanisms, one of which has a distinguished commodity -- the money -- that serves as the sole medium of exchange. As m→ ∞ the money mechanism is the only one with bounded ( π,τ) .