Knightian self uncertainty in the vcg mechanism for unrestricted combinatorial auctions

Abstract

We study the social welfare performance of the VCG mechanism in the well-known and challenging model of self uncertainty initially put forward by Frank H. Knight and later formalized by Truman F. Bewley. Namely, the only information that each player i has about his own true valuation consists of a set of distributions, from one of which i's valuation has been drawn. We assume that each player knows his true valuation up to an additive inaccuracy δ, and study the social welfare performance of the VCG mechanism relative to δ > 0. Denoting by MSW the maximum social welfare, we have already shown in [Chiesa, Micali and Zhu 2012] that, even in single-good auctions, no mechanism can guarantee any social welfare greater than MSW / n in dominant strategies or ex-post Nash equilibrium strategies, where n is the number of players. In a separate paper [CMZ14], we have proved that for multi-unit auctions, where it coincides with the Vickrey mechanism, the VCG mechanism performs very well in (Knightian) undominated strategies. Namely, in an n-player m-unit auction, the Vickrey mechanism guarantees a social welfare ≥ - MSW - 2mδ, when each Knightian player chooses an arbitrary undominated strategy to bid in the auction. In this paper we focus on the social welfare performance of the VCG mechanism in unrestricted combinatorial auctions, both in undominated strategies and regret-minimizing strategies. (Indeed, both solution concepts naturally extend to the Knightian setting with player self uncertainty.) Our first theorem proves that, in an n-player m-good combinatorial auction, the VCG mechanism may produce outcomes whose social welfare is ≤ - MSW - ω(2m δ), even when n=2 and each player chooses an undominated strategy. We also geometrically characterize the set of undominated strategies in this setting. Our second theorem shows that the VCG mechanism performs well in regret-minimizing strategies: the guaranteed social welfare is ≥-MSW - 2min{m,n}δ if each player chooses a pure regret-minimizing strategy, and ≥- MSW - O(n2 δ) if mixed strategies are allowed. Finally, we prove a lemma bridging two standard models of rationality: utility maximization and regret minimization. A special case of our lemma implies that, in any game (Knightian or not), every implementation for regret-minimizing players also applies to utility-maximizing players who use regret ONLY to break ties among their undominated strategies. This bridging lemma thus implies that the VCG mechanism continues to perform very well also for the latter players.