Envy Freedom and Prior-free Mechanism Design

We consider the provision of an abstract service to single-dimensional agents. Our model includes position auctions, single-minded combinatorial auctions, and constrained matching markets. When the agents' values are drawn from a distribution, the Bayesian optimal mechanism is given by Myerson (1981) as a virtual-surplus optimizer. We develop a framework for prior-free mechanism design and analysis. A good mechanism in our framework approximates the optimal mechanism for the distribution if there is a distribution; moreover, when there is no distribution this mechanism still performs well. We define and characterize optimal envy-free outcomes in symmetric single-dimensional environments. Our characterization mirrors Myerson's theory. Furthermore, unlike in mechanism design where there is no point-wise optimal mechanism, there is always a point-wise optimal envy-free outcome. Envy-free outcomes and incentive-compatible mechanisms are similar in structure and performance. We therefore use the optimal envy-free revenue as a benchmark for measuring the performance of a prior-free mechanism. A good mechanism is one that approximates the envy free benchmark on any profile of agent values. We show that good mechanisms exist, and in particular, a natural generalization of the random sampling auction of Goldberg et al. (2001) is a constant approximation

Truthful approximation mechanisms for restricted combinatorial auctions

When attempting to design a truthful mechanism for a computationally hard problem such as combinatorial auctions, one is faced with the problem that most efficiently computable heuristics can not be embedded in any truthful mechanism (e.g. VCG-like payment rules will not ensure truthfulness). We develop a set of techniques that allow constructing efficiently computable truthful mechanisms for combinatorial auctions in the special case where each bidder desires a specific known subset of items and only the valuation is unknown by the mechanism (the single parameter case). For this case we extend the work of Lehmann, O'Callaghan, and Shoham, who presented greedy heuristics. We show how to use If-Then-Else constructs, perform a partial search, and use the LP relaxation. We apply these techniques for several canonical types of combinatorial auctions, obtaining truthful mechanisms with provable approximation ratios