Ascending Price Vickrey Auctions Using Primal-Dual Algorithms∗

The direct Vickrey-Clarke-Groves (VCG) mechanism requires solving underlying optimization problems of the main economy and of every “marginal” economy. We show that VCG outcome can be implemented using a competitive equilibrium (CE) of the main economy if and only if that CE price is also a CE price of every marginal economy. We call such a CE price a universal competitive equilibrium (UCE) price. We design an ascending price auction that implements the VCG outcome under any valuation profile of buyers using this UCE price concept. The auction searches for a UCE price, and calculates payments of buyers from the final UCE price and allocation. This approach is particularly useful in settings where a CE price supporting VCG payments may be absent (for example, when substitutes condition on valuations does not hold) and answers the question whether VCG outcome can be implemented using iterative ascending price auctions for generic valuation profiles. We relate our work to the recent work of de Vries et al. [13] and show that when buyers are substitutes, their auction converges to a UCE price. When buyers are not substitutes, their auction gives a starting price for our auction.