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

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 integers 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; and 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.