On the Competition Complexity of Dynamic Mechanism Design

TL;DR

This paper investigates the competition complexity in dynamic auctions, establishing bounds and providing the first prior-independent results for correlated stages, extending duality frameworks to dynamic settings.

Contribution

It proves bounds on the competition complexity of dynamic auctions and introduces the first prior-independent dynamic auction results for correlated stages.

Findings

Competition complexity of dynamic auctions is at most 3n.

Competition complexity is at least linear in n.

First non-trivial guarantees for simple ex-post IR dynamic auctions with correlated stages.

Abstract

The Competition Complexity of an auction measures how much competition is needed for the revenue of a simple auction to surpass the optimal revenue. A classic result from auction theory by Bulow and Klemperer [9], states that the Competition Complexity of VCG, in the case of n i.i.d. buyers and a single item, is 1, i.e., it is better to recruit one extra buyer and run a second price auction than to learn exactly the buyers' underlying distribution and run the revenue-maximizing auction tailored to this distribution. In this paper we study the Competition Complexity of dynamic auctions. Consider the following setting: a monopolist is auctioning off m items in m consecutive stages to n interested buyers. A buyer realizes her value for item k in the beginning of stage k. We prove that the Competition Complexity of dynamic auctions is at most 3n, and at least linear in n, even when the buyers' values are correlated across stages, under a monotone hazard rate assumption on the stage (marginal) distributions. We also prove results on the number of additional buyers necessary for VCG at every stage to be an {\alpha}-approximation of the optimal revenue; we term this number the {\alpha}-approximate Competition Complexity. As a corollary we provide the first results on prior-independent dynamic auctions. This is, to the best of our knowledge, the first non-trivial positive guarantees for simple ex-post IR dynamic auctions for correlated stages. A key step towards proving bounds on the Competition Complexity is getting a good benchmark/upper bound to the optimal revenue. To this end, we extend the recent duality framework of Cai et al. [12] to dynamic settings. As an aside to our approach we obtain a revenue non-monotonicity lemma for dynamic auctions, which may be of independent interest.