Combination in Multi-Unit 

Estelle Cantillonyand Martin Pesendorferz

First Version: July 2001 This Draft: November 2006

Abstract

This paper considers the problem of identi…cation and estimation in the …rst-price multi- unit . It is motivated by the auctions of bus routes held in London where bidders submit bids on combinations of routes as well as on individual routes. We show that submitting a combination bid lower than the sum of the bids on the constituent routes does not require cost synergies and can instead serve as a tool to leverage market power across the di¤erent routes. As a result, the welfare consequences of allowing combina- tion bidding in the …rst price auction are ambiguous, and depend on the importance of the cost synergies. We provide conditions for non-parametric identi…cation of the mul- tidimensional private information in the multi-unit …rst price auction and derive partial identi…cation results when they are not satis…ed. We propose an estimation method consisting of two stages: In the …rst stage, the distribution of bids is estimated para- metrically. In the second stage, the (set of) costs and distribution(s) of costs consistent with the observed behavior are inferred based on the …rst order conditions for optimally chosen bids. We apply the estimation method to data from the London bus routes mar- ket. We quantify the magnitude of cost synergies and assess possible e¢ ciency losses arising in this market.

We are grateful to several o¢ cials at London Transport Buses and at the London Omnibus Traction Society for their help in collecting the data, and to Ioannis Ioannou, Bryan Lincoln, Bill Simpson, Qinan Tang and Jay Cox for their excellent research assistance. We are indebted to many seminar and conference audiences and to many individuals, in particular Mireia Jofre-Bonet, Ariel Pakes, Wolfgang Pesendorfer and Bill Simpson, for their helpful suggestions. Costas Meghir and four anonymous referees made comments that substantially improved this draft. We thank the NSF under grant SES 0214222 and the Division of Research at Harvard Business School for …nancial support.

yFNRS, ECARES (Université Libre de Bruxelles) and CEPR. Email: [email protected]

zDepartment of Economics, London School Economics, CEPR and NBER. Email: [email protected].

1 1 Introduction

This paper considers the problem of identi…cation and estimation in the …rst-price multi- unit auction. It is motivated by the auctions held by the London Transportation authority to award contracts to service bus routes. Two special features of these auctions are that several bus routes are auctioned at the same time, and that bidders may submit combination bids in addition to stand-alone bids. In other words, the London bus routes market is an example of a combinatorial auction. Combinatorial auctions allow bidders to transmit rich information regarding the value they attach to the objects for sale. When bidders value bundles of objects di¤erently than the sum of the constituent parts, allowing bids for a combination of routes is a necessary condition for e¢ ciency and optimality. This was well understood by the London Trans- portation authority. Indeed, two of the motivations for allowing combination bids in the London bus routes market were: (i) that they would allow bidders to pass on some of the cost savings resulting from cost synergies between routes through lower bids, and (ii) that they would enhance the e¢ ciency of the allocation of routes across bidders. However, allowing combination bids in the …rst price auction may also hurt e¢ ciency and costs. Section 2 introduces a model of a private value multi-unit …rst price procure- ment auction that allows for cost synergies between routes. Two distinct motivations for combination bidding are illustrated. First, combination bidding gives r