Duality in Multi-Commodity Market Computations

In search for general equilibrium in multi-commodity markets, price-oriented schemes are normally used. That is, a set of prices (one price for each commodity) is updated until supply meets demand for each commodity. It is well known that in a two-commodity market resource-oriented schemes are conceivable. In this paper we demonstrate the duality between price- and resource-oriented schemes in the general multi-commodity case. We also discuss important properties of the two approaches. In resource-oriented schemes the resource constraint, which says that supply must equal demand, is always fulfilled, implying that at any time the auctioneer can provide a feasible allocation. This is not the case in price-oriented schemes outside market equilibrium. In this paper we introduce a novel any-time algorithm, Proportion, for the price-oriented scheme as well, that allows the auctioneer to deliver a suitable allocation at some deadline (possibly unknown in advance) also before market equilibrium is reached. We also show how the findings for the any-time algorithms can enable more efficient price-oriented markets