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. Artikeln behandlar hur man kan räkna ut en marknadsjämvikt genom att betrakta uppdatering av allokeringar istället för, som traditionellt, betrakta uppdatering av priser. Bidraget i det sammanhanget är att visa att detta är möjligt även på en marknad med många olika varor (priser). Fördelar och nackdelar med denna metod jämfört med traditionella metoder diskuteras. Artikeln behandlar också hur man kan ta fram en marknadsjämvikt även då beräkningen sker under begränsad tid.