Computationally Feasible VCG Mechanisms

TL;DR

This paper investigates the computational infeasibility of traditional VCG mechanisms in complex problems and proposes a modified approach that encourages truthful behavior while maintaining welfare guarantees.

Contribution

It demonstrates the near universality of non-truthfulness in approximate VCG mechanisms and introduces a new modification to promote truthfulness and individual rationality.

Findings

Most approximation algorithms for combinatorial auctions are non-truthful VCG mechanisms.

The proposed modification ensures agents can improve outcomes and encourages truthful behavior.

The mechanism guarantees welfare at least as high as the underlying algorithm when agents are truthful.

Abstract

A major achievement of mechanism design theory is a general method for the construction of truthful mechanisms called VCG (Vickrey, Clarke, Groves). When applying this method to complex problems such as combinatorial auctions, a difficulty arises: VCG mechanisms are required to compute optimal outcomes and are, therefore, computationally infeasible. However, if the optimal outcome is replaced by the results of a sub-optimal algorithm, the resulting mechanism (termed VCG-based) is no longer necessarily truthful. The first part of this paper studies this phenomenon in depth and shows that it is near universal. Specifically, we prove that essentially all reasonable approximations or heuristics for combinatorial auctions as well as a wide class of cost minimization problems yield non-truthful VCG-based mechanisms. We generalize these results for affine maximizers. The second part of this paper proposes a general method for circumventing the above problem. We introduce a modification of VCG-based mechanisms in which the agents are given a chance to improve the output of the underlying algorithm. When the agents behave truthfully, the welfare obtained by the mechanism is at least as good as the one obtained by the algorithms output. We provide a strong rationale for truth-telling behavior. Our method satisfies individual rationality as well.