Ascending auctions: some impossibility results and their resolutions with final price discounts

When bidders are not substitutes, we show that there is no standard ascend-ing auction that implements a bidder-optimal competitive equilibrium under truthful bidding. Such an impossibility holds also in environments where the Vickrey payoff vector is a competitive equilibrium payoff and is thus stronger than de Vries, Schummer and Vohra s [On ascending Vickrey auctions for het-erogeneous objects, J. Econ. Theory, 132, 95-118] impossibility result with regards to the Vickrey payoff vector under general valuations. Similarly to Mishra and Parkes [Ascending price Vickrey auctions for general valuations, J. Econ. Theory, 132, 335-366], the impossibility can be circumvented by giving price discounts to the bidders from the final vector of prices. Nevertheless, the similarity is misleading: the solution we propose satisfies a minimality infor-mation revelation property that fails to be satisfied in any ascending auction that implements the Vickrey payoffs for general valuations. We investigate related issues when strictly positive increments have to be used under general continuous valuations.ascending auctions ; combinatorial auctions ; bidder-optimal competitive equilibrium ; non-linear pricing ; Vickrey payoffs ; increments

Spectrum Auction Design

Spectrum auctions are used by governments to assign and price licenses for wireless communications. The standard approach is the simultaneous ascending auction, in which many related lots are auctioned simultaneously in a sequence of rounds. I analyze the strengths and weaknesses of the approach with examples from US spectrum auctions. I then present a variation, the package clock auction, adopted by the UK, which addresses many of the problems of the simultaneous ascending auction while building on its strengths. The package clock auction is a simple dynamic auction in which bidders bid on packages of lots. Most importantly, the auction allows alternative technologies that require the spectrum to be organized in different ways to compete in a technology-neutral auction. In addition, the pricing rule and information policy are carefully tailored to mitigate gaming behavior. An activity rule based on revealed preference promotes price discovery throughout the clock stage of the auction. Truthful bidding is encouraged, which simplifies bidding and improves efficiency. Experimental tests and early auctions confirm the advantages of the approach.Auctions, spectrum auctions, market design, package auction, clock auction, combinatorial auction

An Overview of Combinatorial Auctions

An auction is combinatorial when bidders can place bids on combinations of items, called “packages,” rather than just individual items. Computer scientists are interested in combinatorial auctions because they are concerned with the expressiveness of bidding languages, as well as the algorithmic aspects of the underlying combinatorial problem. The combinatorial problem has attracted attention from operations researchers, especially those working in combinatorial optimization and mathematical programming, who are fascinated by the idea of applying these tools to auctions. Auctions have been studied extensively by economists, of course. Thus, the newly emerging field of combinatorial auctions lies at the intersection of computer science, operations research, and economics. In this article, we present a brief introduction to combinatorial auctions, based on our book, Combinatorial Auctions (MIT Press, 2006), in which we look at combinatorial auctions from all three perspectives.Auctions

On the Economic Efficiency of the Combinatorial Clock Auction

Since the 1990s spectrum auctions have been implemented world-wide. This has provided for a practical examination of an assortment of auction mechanisms and, amongst these, two simultaneous ascending price auctions have proved to be extremely successful. These are the simultaneous multiround ascending auction (SMRA) and the combinatorial clock auction (CCA). It has long been known that, for certain classes of valuation functions, the SMRA provides good theoretical guarantees on social welfare. However, no such guarantees were known for the CCA. In this paper, we show that CCA does provide strong guarantees on social welfare provided the price increment and stopping rule are well-chosen. This is very surprising in that the choice of price increment has been used primarily to adjust auction duration and the stopping rule has attracted little attention. The main result is a polylogarithmic approximation guarantee for social welfare when the maximum number of items demanded $\mathcal{C}$ by a bidder is fixed. Specifically, we show that either the revenue of the CCA is at least an $\Omega\Big(\frac{1}{\mathcal{C}^{2}\log n\log^2m}\Big)$-fraction of the optimal welfare or the welfare of the CCA is at least an $\Omega\Big(\frac{1}{\log n}\Big)$-fraction of the optimal welfare, where $n$ is the number of bidders and $m$ is the number of items. As a corollary, the welfare ratio -- the worst case ratio between the social welfare of the optimum allocation and the social welfare of the CCA allocation -- is at most $O(\mathcal{C}^2 \cdot \log n \cdot \log^2 m)$. We emphasize that this latter result requires no assumption on bidders valuation functions. Finally, we prove that such a dependence on $\mathcal{C}$ is necessary. In particular, we show that the welfare ratio of the CCA is at least $\Omega \Big(\mathcal{C} \cdot \frac{\log m}{\log \log m}\Big)$

Measuring the Efficiency of an FCC Spectrum Auction

FCC spectrum auctions sell licenses to provide mobile phone service in designated geographic territories. We propose a method to structurally estimate the deterministic component of bidder valuations and apply it to the 1995–1996 C-block auction. We base our estimation of bidder values on a pairwise stability condition, which implies that two bidders cannot exchange licenses in a way that increases total surplus. Pairwise stability holds in many theoretical models of simultaneous ascending auctions, including some models of intimidatory collusion and demand reduction. Pairwise stability is also approximately satisfied in data that we examine from economic experiments. The lack of post-auction resale also suggests pairwise stability. Using our estimates of deterministic valuations, we measure the allocative efficiency of the C-block outcome.

Quadratic Core-Selecting Payment Rules for Combinatorial Auctions

We report on the use of a quadratic programming technique in recent and upcoming spectrum auctions in Europe. Specifically, we compute a unique point in the core that minimizes the sum of squared deviations from a reference point, for example, from the Vickrey-Clarke-Groves payments. Analyzing the Karush-Kuhn-Tucker conditions, we demonstrate that the resulting payments can be decomposed into a series of economically meaningful and equitable penalties. Furthermore, we discuss the benefits of this combinatorial auction, explore the use of alternative reserve pricing approaches in this context, and indicate the results of several hundred computational runs using CATS data.Auctions, spectrum auctions, market design, package auction, clock auction, combinatorial auction

ESSAYS ON PACKAGE AUCTIONS

The recent auctions literature has devoted much attention to mechanisms that allow package bidding: all-or-nothing bids for sets of items. Introducing package bids can improve efficiency by reducing the bidders "exposure" risk of winning undesirable combinations of items. However, package bids can also create a free-rider problem for relatively small bidders since they need to compete jointly against their larger opponents, potentially reducing efficiency. The inherent asymmetry among different package bids significantly complicates an equilibrium analysis of the costs and benefits of allowing package bids in auctions. The first chapter makes progress in solving for Bayesian-Nash equilibria of the first-price package auction. We develop a new computational method which is based on a complementarity formulation of the system of equilibrium inequalities. Additionally, we establish existence of equilibrium for special cases. Our analysis shows that introducing package bidding can significantly improve efficiency when the exposure risk faced by bidders is large, but it can reduce efficiency otherwise. We also compare the first-price package auction with other leading package alternatives. Surprisingly, in the environment considered, the first-price package auction performs reasonably well, with respect to both revenue and efficiency, despite the presence of a strong free-rider problem. The second chapter studies the core-selecting auctions that were proposed recently as alternatives to the famous Vickrey-Clarke Groves (VCG) mechanism for environments with complementarities. The existing literature on core-selecting auctions is limited to only a complete-information analysis. We consider a simple incomplete-information model which allows us to do a full equilibrium analysis, including closed-form solutions for some distributions, for four different core-selecting auction formats suggested in the literature. Our model also admits correlations among bidders values. We second that the revenues and efficiency from core-selecting auctions improve as correlations among bidders values increase, while the revenues from the Vickrey auction worsen. Thus, there may be good reasons for policymakers to utilize a core-selecting auction rather than a VCG mechanism in realistic environments