VCG Auction Mechanism Cost Expectations and Variances

TL;DR

This paper analyzes the expected costs and variances of VCG auction mechanisms in combinatorial settings with uniform random item costs, revealing key relationships and asymptotic behaviors.

Contribution

It provides new theoretical insights into the expected costs and variances of VCG auctions, especially in matroid structures and for minimum spanning trees.

Findings

Expected VCG cost is at least double the nominal cost.

Exact double expectation in bridgeless matroids.

Asymptotic variance of VCG cost for minimum spanning trees.

Abstract

We consider Vickrey-Clarke-Groves (VCG) auctions for a very general combinatorial structure, in an average-case setting where item costs are independent, identically distributed uniform random variables. We prove that the expected VCG cost is at least double the expected nominal cost, and exactly double when the desired structure is a basis of a bridgeless matroid. In the matroid case we further show that, conditioned upon the VCG cost, the expectation of the nominal cost is exactly half the VCG cost, and we show several results on variances and covariances among the nominal cost, the VCG cost, and related quantities. As an application, we find the asymptotic variance of the VCG cost of the minimum spanning tree in a complete graph with random edge costs.