Mechanisms and Games for Dynamic Spectrum Allocationmarx/bio/papers/... · 10.7 Capacity scaling with generalized fading distributions 318 10.8 Capacity scaling with reduced CSI 321

Mechanisms and Games forDynamic SpectrumAllocation

Edited byTansu Alpcan, Holger Boche, Michael Honig, H. Vincent Poor

Contents

List of contributors page xiiiForeword xvii

Part I Theoretical Fundamentals 1

1 Games and Mechanisms for Networked Systems: Incentives and Algo-rithms Chorppath, Alpcan, Boche 3

1.1 Introduction 31.2 System Model 61.3 Interference and Utility Function Models 81.4 Pricing Mechanisms for Multi-Carrier Wireless Systems 10

1.4.1 Net Utility Maximization 111.4.2 Alternative Designer Objectives 15

1.5 Learning in Pricing Mechanisms 171.6 Auction-Based Mechanisms 181.7 Discussion and Open Problems 27

2 Competition in Wireless Systems via Bayesian Interference Games S.Adlakha, R. Johari and A. Goldsmith 31

2.1 Introduction 312.2 Static Gaussian Interference Games 34

2.2.1 Preliminaries 342.2.2 The Gaussian Interference Game with Unknown Channel Gains 352.2.3 Bayesian Gaussian Interference Game 37

2.3 Sequential Interference Games with Incomplete Information 392.3.1 A Two-Stage Sequential Game 402.3.2 A Sequential Game with Entry 42

2.4 Repeated Games with Entry: The Reputation Effect 442.4.1 A Repeated SBGI-E Game 452.4.2 Sequential Equilibrium of the Repeated Game 46

2.5 Conclusion 482.6 Sequential Interference Games with Two-Sided Uncertainty 49

2.6.1 A Two Stage Sequential Game 492.6.2 A Sequential Game with Entry 50

iv Contents

2.7 Sequential Equilibrium for a Two Period Repeated Game 52

3 Reacting to the Interference Field M. Debbah and H. Tembine 573.1 Introduction 57

3.1.1 Spectrum Access as a Game 573.1.2 Cognitive Access Game 573.1.3 Mean-Field Game Approach 583.1.4 Interference Management in Large-Scale Networks 583.1.5 Objectives 593.1.6 Structure of the Chapter 593.1.7 Notations 59

3.2 Wireless Model 603.2.1 Channel Model 613.2.2 Mobility Model 623.2.3 Path-Loss Model 623.2.4 Remaining Energy Dynamics 623.2.5 Queue Dynamics 633.2.6 SINR Model 63

3.3 Game-Theoretic Formulations 643.4 Reaction to the Interference Field 64

3.4.1 Introduction to Mean Field Games 643.4.2 The Interference Field 66

3.5 Mean Field Stochastic Game 673.5.1 On a Game with One-and-Half Player 683.5.2 Strategies and Payoffs 683.5.3 Mean Field Equilibrium 683.5.4 Structure of the Optimal Strategy 693.5.5 Performance 693.5.6 Mean Field Deterministic Game 703.5.7 Hierarchical Mean Field Game 71

3.6 Discussions 713.7 Conclusions 713.8 Open Issues 72

4 Walrasian Model for Resource Allocation and Transceiver Design in In-terference Networks Jorswieck and Mochaourab 75

4.1 Consumer Theory 764.1.1 Standard Consumer Theory 774.1.2 Consumer Theory for utility α − βx1 + γx1x2 794.1.3 Example 1: Protected and Shared Bands 804.1.4 Example 2: Two-user MISO Interference Channel 854.1.5 Example 3: Multi-carrier Interference Channel 884.1.6 Discussion and comparison of consumer models 90

4.2 Walrasian Market Model 91

Contents v

4.2.1 Existence of a Walrasian Equilibrium 924.2.2 Uniqueness of a Walrasian Equilibrium 934.2.3 Convergence of a Tatonnement Process 944.2.4 Efficiency of a Walrasian Equilibrium 944.2.5 Example 1: Two-user Protected and Shared Bands 954.2.6 Example 2: Two-user MISO Interference Channel 994.2.7 Example 3: MC Interference Channel 102

5 Power Allocation and Spectrum Sharing in Wireless Networks: An Im-plementation Theory Approach A. Kakhbod, A. Nayyar, S.Sharma, and D. Teneketzis 107

5.1 Introduction 1075.1.1 Chapter Organization 108

5.2 What is Implementation Theory? 1085.2.1 Game Forms/Mechanisms 1095.2.2 Implementation in Different Types of Equilibria 1105.2.3 Desirable Properties of Game Forms 1135.2.4 Key Results on Implementation Theory 114

5.3 Nash implementation for social welfare maximization and weak Paretooptimality 1175.3.1 The model (MPS A) 1175.3.2 The power allocation and spectrum sharing problem 1205.3.3 Constructing a game form for the decentralized power and

spectrum allocation problem 1215.3.4 Social welfare maximizing power allocation in a single

frequency band 1235.3.5 Weakly Pareto optimal power and spectrum allocation 1265.3.6 Interpreting Nash Equilibrium 1285.3.7 Other approaches to power allocation and spectrum sharing 129

5.4 Revenue maximization 1305.4.1 The Model 1305.4.2 Impossibility Result from Implementation Theory 1325.4.3 Purely Spectrum Allocation Problem 1325.4.4 Purely Power Allocation Problem 1385.4.5 Other Models and Approaches on Revenue Maximization 139

5.5 Conclusion and reflections 139

6 Performance and convergence of multiuser online learning Tekin andLiu 145

6.1 Introduction 1456.2 Related work 1466.3 Problem formulation and preliminaries 149

6.3.1 Factors determining the channel quality/reward 1496.3.2 Channel models 150

vi Contents

6.3.3 The set of optimal allocations 1516.3.4 Performance measure 1526.3.5 Degree of decentralization 153

6.4 Main results 1546.5 Achievable performance with no feedback and IID channels 1556.6 Achievable performance with partial feedback and IID channels 1596.7 Achievable performance with partial feedback and synchronization for

IID and Markovian channels 1676.7.1 Analysis of the regret of DLOE 1706.7.2 Regret analysis for IID channels 1716.7.3 Regret analysis for Markovian channels 175

6.8 Discussion 1786.8.1 Strategic considerations 1786.8.2 Multiple optimal allocations 1796.8.3 Unknown sub-optimality gap 181

7 Game-Theoretic Solution Concepts and Learning Algorithms Perlazaand Lasaulce 185

7.1 Introduction 1857.2 A General Dynamic Spectrum Access Game 1867.3 Solutions Concepts and Dynamic Spectrum Access 187

7.3.1 Nash Equilibrium 1877.3.2 Epsilon-Nash Equilibrium 1937.3.3 Satisfaction Equilibrium and Efficient Satisfaction Equilibrium 1957.3.4 Generalized Nash Equilibrium 1977.3.5 Coarse Correlated Equilibrium and Correlated Equilibrium 2007.3.6 Robust Equilibrium 2027.3.7 Bayesian Equilibrium and Augmented Equilibrium 2047.3.8 Evolutionary Stable Solutions 2057.3.9 Pareto Optimal Action Profiles and Social Optimal Action

Profiles 2097.3.10 Other Equilibrium Concepts 210

7.4 Learning Equilibria 2107.4.1 Learning Nash Equilibria 2117.4.2 Learning Epsilon-Equilibrium 2147.4.3 Learning Coarse Correlated Equilibrium 2167.4.4 Learning Satisfaction Equilibrium 2177.4.5 Discussion 219

7.5 Conclusion 221

Part II Cognitive Radio and Sharing of Unlicensed Spectrum 227

8 Cooperation in Cognitive Radio Networks: From Access to MonitoringW. Saad and H.V. Poor 229

Contents vii

8.1 Introduction 2298.1.1 Cooperation in Cognitive Radio: Mutual Benefits and Costs 229

8.2 An Overview on Coalitional Game Theory 2318.3 Cooperative Spectrum Exploration and Exploitation 233

8.3.1 Motivation 2338.3.2 Basic Problem 2348.3.3 Joint Sensing and Access as a Cooperative Game 2378.3.4 Coalition Formation Algorithm for Joint Sensing and Access 2408.3.5 Numerical Results 242

8.4 Cooperative Primary User Activity Monitoring 2448.4.1 Motivation 2448.4.2 Primary User Activity Monitoring: Basic Model 2458.4.3 Cooperative Primary User Monitoring 2478.4.4 Numerical Results 254

8.5 Summary 257

9 Cooperative Cognitive Radios with Diffusion Networks Cavalcante, Stanczak,Yamada 263

9.1 Introduction 2639.2 Preliminaries 264

9.2.1 Basic tools in convex and matrix analysis 2649.2.2 Graphs 266

9.3 Distributed spectrum sensing 2669.4 Iterative consensus-based approaches 268

9.4.1 Average consensus algorithms 2689.4.2 Acceleration techniques for iterative consensus algorithms 2729.4.3 Empirical Evaluation 276

9.5 Consensus techniques based on CoMAC CoMAC 2809.6 Adaptive distributed spectrum sensing based on adaptive subgradient

method adaptive subgradient techniques 2849.6.1 Distributed detection with adaptive filter adaptive filters 2849.6.2 Set-theoretic adaptive filter Set-theoretic adaptive filters for

distributed detection 2859.6.3 Empirical evaluation 291

9.7 channel probingChannel probing 2949.7.1 Introduction 2949.7.2 admissibility Admissibility problem 2959.7.3 Power and admission control algorithms 2969.7.4 Channel probing for admission control admission control 2969.7.5 Conclusions 298

10 Capacity Scaling Limits of Cognitive Multiple Access Networks E. Nek-ouei, H. Inaltekin and S. Dey 305

10.1 Introduction 305

viii Contents

10.2 Organization and notation 30610.3 Three main cognitive radio paradigms 30710.4 Power allocation in cognitive radio networks 309

10.4.1 Point-to-point time-invariant cognitive radio channels 30910.4.2 Point-to-point time-varying cognitive radio channels 30910.4.3 Fading multiple access cognitive radio channels 311

10.5 Capacity scaling with full CSI: Homogeneous CoEs 31310.6 Capacity scaling with full CSI: Heterogeneous CoEs 31710.7 Capacity scaling with generalized fading distributions 31810.8 Capacity scaling with reduced CSI 32110.9 Capacity scaling in distributed cognitive multiple access networks 32410.10 Summary and conclusions 329

11 Dynamic Resource Allocation in Cognitive Radio Relay Networks UsingSequential Auctions T. Wang, L. Song, and Z. Han 335

11.1 Introduction 33511.1.1 Cognitive Radio Relay Network 33511.1.2 Sequential Auctions 33611.1.3 Chapter Outline 337

11.2 System Model and Problem Formulation 33811.2.1 System Model of Cognitive Radio Relay Network 33811.2.2 Bandwidth Allocation Problem and Optimal Solution 338

11.3 Auction Formulation and Sequential Auctions 34011.3.1 Auction Formulation 34111.3.2 Sequential First Price Auction 34111.3.3 Sequential Second Price Auction 34411.3.4 Example 345

11.4 Simulation Results 34711.4.1 Total Transmission Rate 34711.4.2 Feedback and Complexity 34911.4.3 Fairness 352

11.5 Conclusions 352

12 Incentivized Secondary Coexistence D. Zhang and N. B. Mandayam 35512.1 Introduction 35512.2 System Model and Bandwidth Exchange 356

12.2.1 System Model 35612.2.2 Bandwidth Exchange 357

12.3 Database Assisted Nash Bargaining for Bandwidth Exchange 35812.3.1 Using Database to Obtain Bargaining Parameters 35912.3.2 Effect of Existence of Other Users 36212.3.3 Pairwise Nash Bargaining Solution 36212.3.4 Convergence 36412.3.5 Complexity Analysis 365

Contents ix

12.4 Performance Improvement 36512.5 Implementation in a Dynamic Environment 36612.6 Extension to Other Access Methods 36812.7 Numerical Results 369

12.7.1 Simulation model 36912.7.2 Simulation Results 370

12.8 Conclusion and Discussions 370

Part III Management and Allocation of Licensed Spectrum 373

13 Self-Organizing Context-Aware Small Cell Networks: Challenges and Fu-ture Opportunities Khanafer, Saad, and Basar 375

13.1 Introduction 37513.2 Strategic access polices in the uplink of femtocell networks 378

13.2.1 System model 37913.2.2 Game formulation and best response algorithm 38113.2.3 Numerical results 384

13.3 Context-aware resource allocation 38713.3.1 Frequent and occasional users 38813.3.2 Context-aware power and frequency allocation game 38913.3.3 Numerical results 391

13.4 Summary 392

14 Economic Viability of Dynamic Spectrum Management Jianwei Huang 39914.1 Background 39914.2 Taxonomy and a brief literature review 40014.3 Incomplete network information 402

14.3.1 Case study: Dynamic spectrum bargaining with incompleteinformation 402

14.3.2 Further research directions 40914.4 Primary-secondary decision coupling 410

14.4.1 Case study: Revenue maximization based on interferenceelasticity 410

14.4.2 Further research directions 41414.5 Interaction mechanisms 415

14.5.1 Case study: Cognitive mobile virtual network operator 41514.5.2 Further research directions 424

14.6 Dynamic decision processes 42414.6.1 Case study: Admission control and resource allocation delay-

sensitive communications 42414.6.2 Further research directions 430

14.7 Conclusion 431

x Contents

15 Auction driven market mechanisms for dynamic spectrum managementG. Iosifidis, and I. Koutsopoulos 435

15.1 Introduction 43615.2 Auction theory fundamentals 439

15.2.1 Auction design objectives 43915.2.2 Multiple-item auctions 44315.2.3 Sponsored search auctions 447

15.3 Hierarchical spectrum auctions 44915.3.1 Background 44915.3.2 Examples of inefficient hierarchical spectrum allocation 45015.3.3 Related work 45315.3.4 Mechanisms for efficient hierarchical spectrum allocation 454

15.4 Double auction mechanism for secondary spectrum markets 45515.4.1 Background 45515.4.2 System model 45715.4.3 The double auction mechanism 459


16 Enabling Sharing in Auctions for Short-term Spectrum Licenses I. A.Kash, R. Murty, and D. C. Parkes 467

16.1 Introduction 46716.1.1 Related Work 469

16.2 Challenges in Auction Design 47016.3 The Model of Shared Spectrum and Externalities 472

16.3.1 User Model 47216.3.2 Allocation Model 474

16.4 Auction Algorithm 47716.4.1 Externalities and Monotonicity 47716.4.2 High level approach 47816.4.3 The SATYA Algorithm 47916.4.4 Pricing Algorithm 48216.4.5 Running time 48316.4.6 Extensions 48316.4.7 SATYA’s use of a MAC 484

16.5 Evaluation 48416.5.1 Varying the Number of Users 48616.5.2 Varying the Number of Channels 48816.5.3 Measuring Revenue 49016.5.4 SATYA’s Performance with Multiple Channels 492


17 Economic Models for Secondary Spectrum Lease: A Spatio-TemporalPerspective A. Al Daoud, M. Alanyali, and D. Starobinski 497

17.1 Introduction 497

Contents xi

17.2 Spatio-temporal model for spectrum access 50017.3 Economic framework for spectrum leasing 503

17.3.1 Pricing of spectrum lease 50417.3.2 Computation of optimal prices 506

17.4 Economic model for private commons 50817.4.1 Reservation policies 50917.4.2 EFPA for reservation policies 51017.4.3 Implied cost 51217.4.4 Revenue maximization via adaptive reservation 514

17.5 Conclusion 516

18 How to use a strategic game to optimize the performance of CDMA wire-less network synchronization G Bacci and M Luise 521

18.1 Introduction 52218.2 CDMA power control as a two-player game 523

18.2.1 The near-far effect game 52318.2.2 The need for power control 52718.2.3 The impact of initial code synchronization 531

18.3 CDMA power control as a multiple-player game 53518.3.1 System model 53518.3.2 Formulation of the game 538

18.4 Energy-efficient resource allocation 54418.4.1 Implementation of the distributed algorithm 54418.4.2 A numerical example 546

18.5 Discussion and perspectives 54918.A Serial-search code acquisition 550

19 Economics and the efficient allocation of spectrum licenses S. Loertscherand L. M. Marx 555

19.1 Introduction 55519.2 Basic model 557

19.2.1 Setup 55719.2.2 Mechanisms and strategies 559

19.3 Results 56019.3.1 Efficient benchmark for complete information 56019.3.2 Results for private information and strategic interaction 56319.3.3 Implications for the design of primary and secondary markets 566

19.4 Generalization 56719.4.1 Model 56719.4.2 Results 57019.4.3 Implications flowing from the general case 573

19.5 Practical implementation 57319.5.1 FCC approach 57519.5.2 Experimental approach 577

xii Contents


Author index 583Subject index 584

List of contributors

Anil Kumar ChorppathTechnical University of Munich

Tansu AlpcanThe University of Melbourne

Holger BocheTechnical University of Munich

Sachin AdlakhaCalifornia Institute of Technology

Ramesh JohariStanford University

Andrea GoldsmithStanford University

Merouane DebbahEcole superieure d’electricite (SUPELEC)

Hamidou TembineEcole superieure d’electricite (SUPELEC)

Eduard A. JorswieckTechnische Universitat Dresden

Rami MochaourabFraunhofer Heinrich Hertz Institute

Ali KakhbodUniversity of Pennsylvania

xiv List of contributors

Ashutosh NayyarUniversity of California Berkeley

Shrutivandana SharmaUniversity of TorontoSingapore University of Technology and Design

Demosthenis TeneketzisUniversity of Michigan, Ann Arbor

Cem TekinUniversity of California, Los Angeles

Mingyan LiuUniversity of Michigan

Samir M. PerlazaDepartment of Electrical Engineering, Princeton University

Samson LasaulceLaboratoire des Signaux et Systemes, Ecole superieure d’electricite (SUPELEC)

Walid SaadUniversity of Miami

H. Vincent PoorPrinceton University

Renato Luis Garrido CavalcanteFraunhofer Heinrich Hertz Institute

Slawomir StanczakFraunhofer Heinrich Hertz Institute and Technische Universitat Berlin

Isao YamadaTokyo Institute of Technology

Ehsan NekoueiThe University of Melbourne

Hazer InaltekinAntalya International University

List of contributors xv

Subhrakanti DeyThe University of Melbourne

Tianyu WangPeking University

Lingyang SongPeking University

Zhu HanUniversity of Houston

Dan ZhangWINLAB, Rutgers University

Narayan B. MandayamWINLAB, Rutgers University

Ali KhanaferCoordinated Science Laboratory, Electrical and Computer Engineering Department,University of Illinois at Urbana-Champaign

Walid SaadElectrical and Computer Engineering Department, University of Miami

Tamer BasarCoordinated Science Laboratory, Electrical and Computer Engineering Department,University of Illinois at Urbana-Champaign

Jianwei HuangNetwork Communications and Economics Lab., Department of InformationEngineering, The Chinese University of Hong Kong.

George IosifidisUniversity of Thessaly, and CERTH

Iordanis KoutsopoulosAthens University of Economics and Business, and CERTH

Ian A. KashMicrosoft Research Cambridge

Rohan MurtySociety of Fellows, Harvard University

xvi List of contributors

David C. ParkesSchool of Engineering and Applied Sciences, Harvard University

Ashraf Al DaoudGerman Jordanian University

Murat AlanyaliBoston University

David StarobinskiBoston University

Giacomo BacciUniversity of Pisa

Marco LuiseUniversity of Pisa

Simon LoertscherDepartment of Economics, University of Melbourne

Leslie M. MarxDuke University

19 Economics and the efficientallocation of spectrum licensesSimon Loertscher and Leslie M. Marx

We discuss the economics literature underpinning the development of a market designapproach for both primary and secondary markets for spectrum licenses and considerthe practical implications for implementation.

19.1 Introduction

The development of mobile wireless technologies for voice and data, and of the mech-anisms used to allocate electromagnetic spectrum for those uses, provides an insightfulcase study into how markets work and why market design matters for social outcomes.This chapter introduces the key concepts and theorems from economic theory and illus-trates them based on the historical development of mobile wireless services. Althoughthere is much to be learned from the experiences of countries around the globe, given theprominent role that the U.S. Federal Communications Commission (FCC) has playedhistorically, we focus on the U.S. experience.

We show that under the assumption that buyers and sellers are privately informedabout their valuations and costs, the distinction between primary markets and secondarymarkets is critical for what can be achieved with carefully designed allocation mecha-nisms, where by a primary market we mean a situation in which the seller (or possiblythe buyer) of the assets also chooses the mechanism, and by a secondary market wemean a situation in which an entity other than a party to the transaction chooses thetrading mechanism and organizes the exchange. The economics literature on mecha-nism design and auction theory has primarily focused on designing primary markets,notwithstanding a few notable exceptions such as [52], [42], [21] and [39]. We reviewthe literature on primary market design and show that for the primary market an efficientallocation mechanism that does not run a deficit exists. This is in stark contrast to theknown results for secondary markets, according to which such mechanisms do not exist.Moreover, we derive a new result that generalizes these impossibility results to the caseof heterogeneous objects and arbitrary quasilinear utility and profit functions.

This chapter provides background on the underlying economics, including possibilityand impossibility results, relevant to the design of dynamic spectrum allocation mech-anisms. The basic framework is provided in Section 19.2 and theoretical foundationsare provided in Sections 19.3.1 for primary markets and in 19.3.2 for secondary mar-kets. The corresponding implications for market design, which may be of key interest to

556 S. Loertscher and L. M. Marx

those approaching the problem from an engineering perspective, are provided in Section19.3.3. We provide a generalization of the foundational results from the economics liter-ature in Section 19.4, with associated implications for market design in Section 19.4.3.Finally, Section 19.5 reviews the FCC’s approach to the issues raised in this chapter,including the role of results from experimental economics. Readers focused on an en-gineering approach to secondary market design may find the results of Sections 19.3.3,19.4.3, and 19.5 to be of the greatest interest; however, we hope that the economicfoundations presented in this chapter will allow one to better understand the underlyingbases for the discussions in these sections.

The main part of this chapter focuses on the model with a homogenous product inwhich each buyer has demand for one unit and each seller has the capacity to produceone unit. With one exception, the focus of this chapter is on dominant strategy mech-anisms. The exception is Proposition 19.4. This is the most general statement of theimpossibility theorem (due to [42]) that in the domain of Bayesian mechanisms, ex postefficient trade is not possible without running a deficit. Because the set of Bayesianmechanisms contains the set of dominant strategy mechanisms, this is a remarkablygeneral result. Together with the fact that an efficient allocation mechanism exists forthe primary market, it provides both an important rationale for taking the primary mar-ket allocation problem seriously and an explanation for why the economics literaturehas primarily focused on the primary market problem.

That said, there are, of course, instances in which reliance on secondary markets be-comes inevitable. Perhaps the most important reason for this is technological change.For example, the increasing demand for mobile wireless services and the developmentof digital television make it appear highly desirable that TV broadcasters offer some oftheir spectrum licenses for sale to providers of mobile wireless services. This develop-ment has, for example, been the rationale for the U.S. Congress to mandate that the FCCset up and run a secondary market for spectrum licenses. Another important reason forthe desirability of secondary markets is dispersed ownership of the assets, which makesit impossible for a single seller to design the trading mechanism. For example, this isthe case for kidney exchanges (see e.g. [50]) and for problems of providing servicessuch as the provision of container port drop-off and pick-up slots by terminal operatorsto trucking companies and the provision of child care, kindergarten and school seats tofamilies and their children. Finally, even without dispersed ownership, it may not bein society’s best interest to have the seller design the trading mechanism if the seller’sinterest is something other than efficiency. This is a particularly salient issue when theseller is a private entity that aims to maximize profits.

This provides motivation to examine the extent to which efficiency can be achievedin well-designed secondary markets without running a deficit. Although the Myerson-Satterthwaite Theorem and its generalizations are important and remarkable, it isequally important to note that these theorems make qualitative statements. In partic-ular, the efficiency loss of a well-designed, centrally run exchange decreases quicklyas the number of buyers and sellers increases. This result has been shown by [21] forBayesian mechanisms and by [39] within the domain of dominant strategy mechanisms.We illustrate it using a version of [39]’s second price double auction.

Economics and the efficient allocation of spectrum licenses 557

This chapter is organized as follows. It starts in Section 19.2 by defining the basicsetup and notation. Then in Section 19.3 we present the benchmark case described bythe Coase Theorem (due to [11]), according to which private transactions achieve thebest social outcomes, provided only that the government defines and protects propertyrights, and transaction costs are negligible. An immediate implication of the Coase The-orem is that, whenever it applies, the initial allocation of property rights is irrelevant forthe efficiency of the final allocation. We illustrate the Coase Theorem in an environmentin which perfectly competitive markets maximize social welfare. The Coase Theoremand its implications contrast with the development of mobile wireless services, whichwas dormant as long as lotteries (or other inefficient procedures) were used for allo-cating spectrum and overwhelmingly quick as soon as auctions were used to allocatespectrum in the primary market, suggesting the existence of substantial frictions. Wethen extend the analysis to the cases of one-sided and then two-sided private informa-tion.

Relaxing the assumptions of homogenous goods and unit demands and capacities, weintroduce and analyze in Section 19.4 the general environment with heterogenous ob-jects and, essentially, no assumptions on utility and profit functions other than quasilin-earity. In this setup we study the celebrated Vickrey-Clarke-Groves (VCG) mechanism,which is named after the authors – [52], [9] and [22] – whose independent contributionsled to it. The VCG mechanism is a dominant strategy mechanism that allocates effi-ciently any finite number of possibly heterogenous goods in the primary market withoutrunning a deficit. The VCG mechanism is a generalization of the second-price auctionand provides an important theoretical benchmark for what can be achieved under thesemore complicated and often more realistic conditions. We extend the standard VCGmechanism to the two-sided problem that one faces in the secondary market when sell-ers can produce different packages of goods and buyers have heterogenous demandsfor these packages. This is the problem the U.S. Federal Communications Commission(FCC) faces for the forthcoming so-called incentive auctions. We show that the two-sided VCG mechanism always runs a deficit if it achieves efficiency, which is the spiritof the Myerson-Satterthwaite Theorem and an extension of Vickrey’s result to heteroge-nous objects.

Section 19.5 discusses issues of practical implementation as embraced by the FCCand the role for an experimental economics approach to addressing practical implemen-tation issues. Section 19.6 offers concluding comments.

19.2 Basic model

19.2.1 Setup

There is a single homogeneous product, as well as a numeraire, which we interpretas money. There are n risk-neutral buyers indexed by i ∈ {1, .., n}, where each buyeri has value vi ∈ [v, v] for a single unit of the product and, for simplicity, unlimitedbudget. There are m risk-neutral sellers indexed j ∈ {1, ..,m}, where each seller j has


cost c j ∈ [c, c] to produce a single unit of the product and unlimited budget. A buyer’svaluation vi and a seller’s cost c j are sometimes also referred to as their types. Thesellers’ costs capture both the case where sellers have to produce the goods at some costand the case of a pure exchange economy, in which each seller is endowed with one unitof the product and c j is his private valuation and hence his (opportunity) cost of sellingthe unit.

In line with the assumptions of the Bayesian mechanism design approach, we mayassume that buyer i’s valuation vi is an independent random draw from some distributionF with support [v, v] and positive density on this support for all i = 1, .., n. Similarly,seller j’s cost c j may be thought of as the realization of an independent random variablewith distribution G, which has support [c, c] and positive density everywhere on thesupport. LetV = [v, v]n and C = [c, c]m be the product sets of buyers’ and sellers’ types.We refer to the model as one with complete information if both realized valuationsv = [v1, .., vn] and realized costs c = [c1, .., cm] are known by all the agents and themechanism designer. We say that the model is one with incomplete information ifevery agent only knows the realization of his or her own type and the commonly knowndistributions F and G (as well as n and m), while the mechanism designer only has theinformation about distributions and numbers of buyers and sellers. Lastly, the model issaid to be of one-sided private information if only the sellers’ costs (and n,m and F)are commonly known, while each buyer’s type is her private information.1 We assumethat

c < v and v ≤ c.

The first part of the assumption makes sure that trade is sometimes ex post efficient andthe second one is sufficient for trade not to be always ex post efficient. [34, 35] pro-vide conditions under which ex post efficient trade with private information is possiblewithout running a deficit.

We order (and relabel) the buyers and sellers so that v1 > v2 > ... > vn and c1 < c2 <

... < cm, where we simplify by assuming no tied values or tied costs. The assumptionthat there are no ties is satisfied almost surely when types are drawn from continuousdistribution functions.

An allocation is a (n + m)-dimensional vector Γ = [β, σ] consisting of 0’s and 1’s.For i ∈ {1, ..., n}, element βi specifies whether buyer i receives the good, and for j ∈{1, ...,m}, element σ j specifies whether seller j produces a unit of the good, with 1meaning receive/produce. An allocation Γ is said to be feasible if

∑ni=1 βi ≤

∑mj=1 σ j.

This implicitly assumes that goods can be freely disposed off, which for all intents andpurposes is without loss of generality. Let G be the set of feasible allocations.

When buying a unit at price pi, buyer i’s utility when of type vi is u(vi, pi) = vi − pi.Seller’s j’s profit with cost c j when selling at price p j is π(c j, p j) = p j − c j. Theseutility and profit functions are sometimes called quasilinear because they are linear in

1 Of course, one could also study the converse problem with one-sided private information that is held bysellers. However, this problem is isomorphic to the one with one-sided private information held by buyersand is skipped in the interest of space.


monetary payments.2 We assume that agents’ outside options are 0, that is, regardless ofhis or her type, an agent who does not trade and does not make or receive any paymentshas a payoff of 0.

Embodied in these definitions are further the assumptions of private values, i.e., anagent’s payoff is not affected by the values or costs of other agents; and no externalities,i.e., an agent’s payoff is not affected by the allocations, transfers, or utility of otheragents.

19.2.2 Mechanisms and strategies

Let p = [p1, .., pn+m] be the vector of prices of dimension (n + m). A mechanismspecifies agents’ set of actions (and the order in which they choose their actions) and afeasible allocation Γ together with a price vector p. A direct mechanism is a mechanismthat asks agents to report their types and makes the allocation and prices contingent onthese reports. Formally, a direct mechanism 〈Γ, p〉 is a function: V × C → G × Rn+m.The focus on direct mechanisms is without loss of generality because of the revelationprinciple, which states that whatever allocation and expected payments can be obtainedas the (Bayes Nash) equilibrium outcome of some mechanism can be obtained as the(Bayes Nash) equilibrium outcome of a direct mechanism (see, for example, [28]).3 Adominant strategy mechanism is a mechanism that makes it a dominant strategy forevery player to report his or her type truthfully. A Bayesian mechanism, on the otherhand, is a mechanism whose allocation and payment rule are defined with respect tosome Bayes Nash equilibrium under the mechanism.

Two kinds of constraints are important in mechanism design. The first one is thatagents act in their own best interest. That is, a mechanism needs to make sure thatagents do what they are supposed to do. For direct mechanisms, these are referred toas incentive compatibility constraints. Second, the mechanism should satisfy the (in-terim) individual rationality constraints that agents, once they know their types andthe expected payments and allocations conditional on this information, given the mech-anism, are better off participating in the mechanism than walking away. Knowing one’sown type but not the types of the other players is referred to as the interim stage. Thestage (which need never be reached) in which all players’ types are known is called theex post stage. Accordingly, if individual rationality constraints are satisfied given anyrealization of allocations and payments, the individual rationality constraints are said tobe satisfied ex post. Similarly, a feasible allocation Γ∗ is said to be ex post efficient if itis efficient given the realized types, that is, Γ∗ ∈ arg maxΓ∈G

∑ni=1 βivi −

∑mj=1 σ jc j.

2 They could be non-linear in the good that is being traded, whence the name.3 Recall that a Bayes Nash equilibrium of a game is a strategy profile such that every type of every player

maximizes his or her expected payoff, keeping fixed the strategy profile of every other player-type, theexpectation being taken with respect to the player-type’s beliefs that are updated using Bayes’ rule.


19.3 Results

19.3.1 Efficient benchmark for complete information

In seminal work in the economics literature, Ronald Coase ( [11]) put forward the ideathat in a theoretical environment with complete information and no transactions costs,the initial allocation of properties rights (as long as these property rights are well de-fined and protected) is irrelevant because, Coase argued, agents will continue to engagein transactions as long as the allocation remains inefficient. In Coase’s theoretical envir-onment, any inefficiency in the initial allocation is eliminated through exchange in thesecondary market. The general idea that, in environments with complete information,well-defined and protected property rights, and no transaction costs, efficient outcomescan be achieved through private transactions has come to be known as the Coase Theo-rem.4

We begin with an illustration of an instance of the Coase Theorem. For this purpose,we assume in this subsection that all values and costs are common knowledge amongall the agents and also known by the mechanism designer. If c1 ≥ v1, then the lowestproduction cost is equal to or exceeds the highest value and so there are no gains fromtrade. If there are at least as many buyers as sellers (n ≥ m) and cm < vm, then efficiencyrequires that all sellers produce and that the m units produced be allocated to buyers1, ...,m.5 Similarly, if there are more sellers than buyers (m > n) and cn < vn, thenefficiency requires that all buyers be allocated a unit and that the n units required beproduced by sellers 1, ..., n. (More generally, these last two cases occur when cmin{m,n} <

vmin{m,n}.)In all other cases, c1 < v1 and there exists k ∈ {2, ...,min {m, n}} such that ck−1 < vk−1

and ck ≥ vk, implying that the efficient outcome involves production by sellers 1, ..., k−1and an allocation of products to buyers 1, ..., k − 1. Using this definition of k, we cannow define

κ :=

0, if c1 ≥ v1

min {m, n} , if cmin{m,n} < vmin{m,n}

k − 1, otherwise,

so that the efficient outcome is for sellers j ≤ κ to produce and for buyers i ≤ κ toreceive one unit each.

The Coase TheoremOne way to view the market in this environment is to assume the existence of a “two-sided” Walrasian auctioneer accepting bids and asks for the product. The Walrasian

4 There is no single formal statement of the Coase Theorem. For example, see [37].5 In slight abuse of everyday language, we mean that sellers sell their units when we say that “sellers

produce”, regardless of whether they actually physically produce the units or only sell units of goods orassets they are endowed with.


equilibrium in this model involves a price

p ∈

[v1, c1], if κ = 0[cmin{m,n}, vmin{m,n}], if κ = min {m, n}[max {cκ, vκ+1} ,min{cκ+1, vκ}], otherwise.

First, note that by the definition of κ and our ordering of the costs and values, the pricep is well defined. If κ = 0, then v1 ≤ c1; if κ = min {m, n} , then cmin{m,n} < vmin{m,n}; andotherwise max {cκ, vκ+1} ≤ min{cκ+1, vκ}, where we define cm+1 to be c and vn+1 to be v.Second, given such a price p, there is an equilibrium in which only buyers i ≤ κ wishto purchase, implying demand of κ. (If the price is set at its lower bound and that lowerbound is vκ+1, then buyer κ+1 is indifferent between purchasing and not.) Third, there isan equilibrium in which only sellers j ≤ κ wish to produce, implying supply of κ. (If theprice is set at its upper bound and that upper bound is cκ+1, then seller κ+1 is indifferentbetween producing and not.) Thus, a market price p as defined above implements theefficient outcome.

In the sense described here, the competitive market equilibrium delivers the efficientoutcome.

Proposition 19.1 The competitive equilibrium for the environment with unit demandand supply and complete information is efficient.

The efficiency of the competitive equilibrium does not rely on which of the first κsellers trades with which of the first κ buyers, only that it is the first κ sellers whoproduce and the first κ buyers who are ultimately allocated the products.

The so-called “efficient rationing rule” allocates the product of the least-cost sellerto the highest-valuing buyer, and so on, so that for i ∈ {1, ..., κ}, seller i trades withbuyer i; see [49], [4], [31], [27], and [16] for more on rationing rules. For example, ifwe assume sellers charge differential prices for their products, then the demand for alow-priced seller’s product may be greater than 1. Once that seller’s product has beenpurchased by a buyer, the residual demand facing the remaining sellers will depend onwhich buyer made that first purchase. Under the efficient rationing rule, the residualdemand at any price is simply the original demand minus 1 (as long as that quantity isnonnegative). Efficient rationing obtains under the assumption that buyers are able tocostlessly resell products to each other. In a game in which sellers first set prices andthen buyers state whether they are willing to purchase from each seller, then efficientrationing delivers the efficient outcome as a Nash equilibrium of the game.

Under the alternative rationing rule of “random rationing” each buyer is equallylikely to be given the opportunity to trade with the least-cost seller. That is, all buyerswith values greater than the price of the least-cost seller are equally likely to trade withthat seller. Under the random rationing rule, the allocation is not necessarily efficientbecause a buyer with value less than vκ might be allocated a unit.

One can define games in which efficient rationing arises in a game involving bilat-eral bargaining. For example, suppose the market is organized as a series of bilateral


negotiations, each delivering the Nash bargaining outcome.6 First, buyer 1 and seller 1engage in bilateral bargaining. If they do not come to agreement, then neither trades.Then buyer 2 and seller 2 do the same, and this continues until there are either no morebuyers or no more sellers. In this game, if v1 > c1, the outcome of the first Nash bar-gaining game is for seller 1 to produce the good and sell it to buyer 1 for a payment ofv1+c1

2 . The outcome of this sequence of Nash bargaining games is for the first κ buyersand sellers to trade, but no others. Thus, efficient rationing and the efficient outcome isachieved.

Other sequences of negotiations are possible as well, including potentially those thatinvolve buyers other than the first κ acquiring product initially, but then later selling tothe higher valuing buyers.

The Walrasian equilibrium and the differential pricing game with efficient rationingproduce outcomes that are in the “core” in the sense that no subset of agents can prof-itably deviate from the specified outcome. Furthermore, only efficient allocations arein the core; see [47]. For example, if an outcome involved production by a seller withcost greater than cκ and no production by seller κ, then a coalition involving the sellerwith cost greater than cκ, seller κ, and the purchaser of the product from the seller withcost greater than cκ could profitably deviate by sharing in the efficiency gains associatedwith moving production to seller κ. Finally, every core outcome has the property that alltraders receive payoffs equal to those in one and the same Walrasian equilibrium p forevery p ∈ [max{cκ, vκ+1},min{cκ+1, vκ}], assuming for simplicity that the efficient quan-tity κ is neither 0 nor min{n,m}. This can easily be seen by noting that if two buyers paydifferential prices, then the buyer paying the higher price can form a profitable deviatingcoalition with the lower priced seller. Thus payoffs must be as if all trades occurred atthe same price, and any such price must be a Walrasian equilibrium price.

The result of the Coase Theorem is that in an environment with complete information,as long as property rights are well defined and protected, and as long as there are notransactions costs, it should not matter precisely what the market processes are. As longas any inefficiency remains, there are mutually beneficial trades that can be made, andone would expect those to be realized.

Implications for market designAs described by [41] (pp.19-20), critics of an auction approach to allocating spectrumlicenses have argued based on the Coase Theorem, saying:

[O]nce the licenses are issued, parties will naturally buy, sell, and swap them to correct anyinefficiencies in the initial allocation. Regardless of how license rights are distributed initially, thefinal allocation of rights will take care of itself. Some critics have gone even farther, arguing onthis basis that the only proper object of the government is to raise as much money as possible inthe sale, because it should not and cannot control the final allocation.

However, the evidence suggests that the market for spectrum licenses does not satisfyall the requirements of the Coase Theorem. As described by [41], p.40:

6 Under Nash bargaining a buyer of type v and a seller of type c trade at the price p(v, c) that maximizes(v − p)(p − c) over p, which is p(v, c) = (v + c)/2 provided v ≥ c. Otherwise, they do not trade.


The history of the US wireless telephone service offers direct evidence that the fragmented andinefficient initial distribution of rights was not quickly correctable by market transactions. Despitedemands from consumers for nationwide networks and the demonstrated successes of similarlywide networks in Europe, such networks were slow to develop in the United States.

In order to understand the market design issues for spectrum licenses, we must beginby understanding the ways in which deviations from the perfect world of the CoaseTheorem potentially affect the efficiency of market outcomes. We begin in the nextsubsection by considering the simple adjustment to the environment just considered toallow the values of buyers and costs of sellers to be the agents’ own private information.

19.3.2 Results for private information and strategic interaction

We begin by providing an example of an efficient mechanism for the case in whichthere is a single seller that acts as the mechanism designer with the goal of maximizingefficiency. Then we examine the case where sellers must be incentivized to participateand reveal their privately held information, in which case the Myerson-SatterthwaiteTheorem implies that the outcome is generally not efficient. We then examine the im-plications for the design of primary market versus secondary market institutions.

Efficient mechanisms generate a surplus for one-sided private informationIn order to illustrate the existence of an efficient mechanism for a primary market, wesimplify the above environment by assuming there is a single seller that can producem units at increasing marginal cost c j for j ∈ {1, ...,m}. But we adjust the above en-vironment so that the values of the buyers v1, ..., vn are the private information of theindividual buyers. In this environment, there exists an efficient mechanism that neverruns a deficit.

To see this, consider a mechanism in which buyers submit reports of their values andorder the reports as r1 ≥ r2 ≥ ... ≥ rn. Let the mechanism identify the set of efficienttrades as described above based on the vector of reports r (rather than the true valuesas above) and the seller’s marginal costs. The mechanism identifies the number κ ofunits to allocated based on reports and costs, and those units are allocated to buyerswith reports r1, ..., rκ. Each buyer receiving a unit of the good pays the same amountp = max {cκ, rκ+1} to the seller.

It follows from the analysis above that if buyers truthfully report their values, thismechanism is efficient. The result that no buyer has an incentive to misreport her truevalue follows from the usual second-price auction logic. Suppose that buyer i considersreporting ri > vi rather than vi. This change only affects the outcome for buyer i if buyeri does not receive a unit when she reports vi but does receive a unit when she reportsri, in which case the change results in buyer i being allocated a unit at a price greaterthan her value for the unit. Thus, the buyer prefers to report truthfully rather than anyamount greater than her value.

To see that no buyer has an incentive to underreport her value, suppose that buyeri considers reporting ri < vi rather than vi. This change only affects the outcome forbuyer i if buyer i receives a unit when she reports vi but does not receive a unit when


she reports ri, in which case the change results in buyer i not receiving a unit when shewould have acquired a unit at a price less than her value, giving her positive surplus.Thus, it is a weakly dominant strategy for buyers to report truthfully.7

Proposition 19.2 In an environment with a single multi-unit seller and unit demandbuyers with private information, there exists an ex post efficient, incentive compatible,and individually rational mechanism.

Proposition 19.2 establishes that efficiency can be achieved when there are no incen-tive issues for the seller, such as might be the case in the primary market for spectrumlicenses where the government is the seller. However, as we show next, this result doesnot extend to the case of secondary markets in which sellers must be incentivized aswell.

Efficient mechanisms run a deficit for two-sided private informationIn order to examine the effects of having private information for both buyers and sellers,consider the environment above, but assume that each buyer’s value and each seller’scost is the agent’s own private information. As we show, incentives for buyers and sellersto engage in strategic behavior are unavoidable whenever values are private.

[52] deserves credit for having first presented an impossibility result for market mak-ing in the domain of dominant strategy mechanisms. Vickrey states, “When it comes tomarkets where the amounts which each trader might buy or sell are not predeterminedbut are to be determined by the negotiating procedure along with the amount to be paid,the prospects for achieving an optimum allocation of resources become much dimmer.A theoretical method exists, to be sure, which involves essentially paying each sellerfor his supply an amount equal to what he could extract as a perfectly discriminatingmonopolist faced with a demand curve constructed by subtracting the total supply ofhis competing suppliers from the total demand, and symmetrically for purchase. But . . .the method is far too expensive in terms of the inflow of public funds that would berequired . . . .” ( [52] p.29). We present and prove a more general version of this resultbelow.

In the domain of Bayesian mechanism, the seminal impossibility result of [42], whichwas originally stated for one buyer and one seller, extends to this environment and estab-lishes that under weak conditions there does not exist a mechanism that is ex post effi-cient, incentive compatible, interim individually rational and that does not run a deficit.

In what follows, we assume v < c, so that we avoid the trivial case in which it isalways efficient for all buyers and sellers to trade.

Consider the VCG mechanism, whose operation in this context requires that the sell-ers report their costs and that the buyers report their values.8 Given the reported costand values, the mechanism would implement the efficient allocation given the reports.Assuming truthful reporting, each buyer i ≤ κ receives a unit and pays max {cκ, vκ+1} ,

and each seller j ≤ κ produces a unit and receives min {cκ+1, vκ}. For reasons that are7 It is also worth noting that nothing in this argument hinges on the assumption that buyers are risk neutral

because it extends straightforwardly to the case where buyers’ utility functions u(vi, p) are increasingfunctions of the monetary payoff vi − p, regardless of the sign of the second derivative of these functions.

8 Section 19.4 contains the general description of the two-sided VCG mechanism.


analogous to those in the setup with one-sided private information, truthful reporting isa weakly dominant strategy for every buyer and every seller irrespective of their types.

Participation in this mechanism is clearly individually rational. A buyer with value vhas expected surplus of zero and a buyer’s expected surplus is increasing in her value.A seller with cost c has expected surplus of zero and a seller’s expected surplus isdecreasing in his cost.

If there exists some other efficient mechanism that is also incentive compatible, thenby the revenue equivalence theorem9 there is a constant ξb such that the expected pay-ment for any buyer b with value v under this mechanism differs from the expectedpayment under the VCG mechanism by ξb. Similarly, there is a constant ξs such thatthe expected receipts for any seller s with cost c under this mechanism differs from thatunder the VCG mechanism by ξs. If this other mechanism is also individually rational,it must be that ξb ≤ 0 because otherwise the buyer with value v would have negativeexpected surplus, and it must be that ξs ≥ 0 because otherwise a seller with cost c wouldhave negative expected surplus, contradicting individual rationality. Thus, in this othermechanism, the sum of buyers’ payments minus the sum of sellers’ receipts must beweakly lower than in the VCG mechanism.

This gives us the following lemma, which is a straightforward extension of Proposi-tion 5.5 in [28].

lemma 19.3 Consider the model with incomplete information. Among all allocationmechanisms that are efficient, incentive compatible, and individually rational, the VCGmechanism maximizes the sum of buyers’ payments minus the sum of sellers’ receipts.

Using Lemma 19.3, we can prove a generalized version of the Myerson-SatterthwaiteTheorem.

Proposition 19.4 In the model with incomplete information, there is no mechanismthat is efficient, incentive compatible, individually rational, and at the same time doesnot run a deficit.

Proof. Consider the VCG mechanism described above. The total amount received bysellers, κmin {cκ+1, vκ} , is weakly greater than the total payments made by the buyers,κmax {cκ, vκ+1} , and strictly greater in general given our assumption that v < c. Thus,in general, the VCG mechanism runs a deficit. Using Lemma 19.3, if the VCG runsa deficit, then every other incentive compatible and individually rational mechanismalso runs a deficit. Thus, there does not exist an efficient mechanism that is incentivecompatible, individually rational, and balances the budget for all realizations of valuesand costs. Q.E.D.

For large markets approximately efficient mechanisms generate a surplusfor two-sided private informationAs a contrast to Proposition 19.4, [21] show that in the limit as a market becomes large,the expected inefficiency associated with the optimally designed mechanism decreases;see also [46], [51], and [15] for limit results with large double auctions. Intuitively, as

9 See Proposition 14.1 in [28] for a statement of the revenue equivalence theorem for the multi-unit case.


the market becomes large, each trader has no effect on the market clearing price andso no incentive to disguise private information, allowing the efficient allocation to beobtained.

To provide a specific example of an almost efficient two-sided mechanism that doesnot run a deficit, we provide a slightly simplified version of the dominant strategy mech-anism proposed by [39].

Buyers and sellers simultaneously submit bids bi and s j to the clearinghouse. Givensubmitted bids, the clearinghouse induces the Walrasian quantity minus one, i.e. κ − 1,to be traded, κ being defined with respect to the submitted bids. Buyers who trade paythe price pB = max{bκ, sκ−1} to the clearinghouse. Sellers who trade receive the pricepS = min{bκ−1, sκ}. All other buyers and sellers pay and receive nothing.

By the usual arguments, each buyer and seller has a weakly dominant strategy to bidtruthfully, i.e. bi = vi and s j = c j are weakly dominant strategies, because no agent canever affect the price he or she pays or receives. The bids only affect whether or not anagent trades. By bidding truthfully, agents can make sure that they trade in exactly thoseinstances in which it is in their best interest. For a buyer i that is to trade if and only ifvi ≥ pB and for a seller j this is to trade if and only if c j ≤ pS .

Given truthful bidding, pB = max{vκ, cκ−1} = vκ and pS = min{vκ−1, cκ} = cκ. Bythe definition of κ, vκ > cκ, whence it follows that pB > pS , and so the mechanismgenerates a surplus.

For given v and c, the efficiency loss is small in the sense that the only loss is thefailure to trade by the buyer-seller pair whose contribution to overall welfare is smallestamongst the κ pairs that would trade under an efficient allocation. Moreover, as thenumber of buyers and sellers becomes large, this efficiency loss becomes smaller inexpectation by a law of large numbers argument.

19.3.3 Implications for the design of primary and secondary markets

As shown above, the existence of second-price auctions suggests that efficiently func-tioning primary markets for spectrum licenses are possible. The efficiency result holdsin an environment in which all available units for sale and all potential buyers of thoseunits participate in a centralized mechanism. This suggests a role for a market designerin the primary market.

Turning to secondary markets, the Myerson-Satterthwaite Theorem implies that effi-ciency without running a deficit is not possible for secondary markets. As stated by [41]p.21:

[E]ven in the simplest case with just a single license for sale, there exists no mechanism that willreliably untangle an initial misallocation. ... According to a famous result in mechanism designtheory – the Myerson-Satterthwaite theorem – there is no way to design a bargaining protocolthat avoids this problem: delays or failures are inevitable in private bargaining if the good startsout in the wrong hands. (emphasis in the original)

The key implication of these results is that we cannot rely on competitive markets toproduce efficient outcomes in secondary markets.


Nonetheless, there is potentially an important role for market design approaches toimprove the efficiency of secondary markets. This is clear from the results of [39] andothers showing that approximate efficiency can be achieved in large secondary markets.Because one might not expect large secondary markets to arise in a typical environmentwith dispersed ownership, these results highlight the potentially valuable role a marketdesigner can play by centralizing secondary markets to achieve a larger market size.

Given the pace of technological change in our use of spectrum, it is inevitable thatsecondary markets must play an important role. The FCC and Congress recognize this intheir current work to design “incentive auctions” to facilitate a move to a more efficientallocation of spectrum licenses. As stated by the FCC:

“Commercial uses of spectrum change over time with the development of new technologies. ...The use of incentive auctions is one of the ways we can help meet the dramatic rise in demandfor mobile broadband. Current licensees – like over-the-air-broadcasters – would have the optionto contribute spectrum for auction in exchange for a portion of the proceeds. ... The use of in-centive auctions to repurchase spectrum is a complex and important undertaking that will helpensure America’s continued leadership in wireless innovation. The best economists, engineersand other experts are hard at work at the Commission to get the job done right – with opennessand transparency.”10

19.4 Generalization

In this section, we introduce the setup for the general two-sided VCG mechanism,whose standard one-sided form is due to the independent contributions of [52], [9]and [22]. We show that the two-sided VCG mechanism provides agents with domi-nant strategies, allocates goods efficiently when the agents play these strategies, andalways runs a deficit when the agents play their dominant strategies. Lastly, we look ata number of simple mechanisms the two-sided VCG mechanism specializes to underappropriate restrictions.

19.4.1 Model

For the general model, we move away from the assumption of a single homogeneousproduct and assume there are K ≥ 1 potentially heterogenous objects that the sellerscan produce. These K objects can be bundled into 2K − 1 different packages (excludingthe nil package). All buyers i have valuations vk

i for each possible package k and allsellers j have costs ck

j for producing any package k with k = 1, .., 2K − 1, i = 1, .., n andj = 1, ..,m. So buyer i’s valuations can be summarized as a 2K − 1 dimensional vectorvi = [v1

i , .., v2K−1i ], whose k-th element is i’s valuation for package k, and seller j’s costs

are a 2K − 1 dimensional vector c j = [c1j , .., c

2K−1j ], whose k-th element is j’s cost for

producing package k. All agents (i.e. buyers and sellers) have quasilinear preferences,linear in payments. So if buyer i is allocated package k and pays the price p, her net

10 Federal Communications Commission website, http://www.fcc.gov/topic/incentive-auctions, accessedJune 27, 2012.


payoff is vki − p. Similarly, if seller j produces package k and is paid p, his net payoff is

p − ckj. We set the valuation of every buyer for receiving the nil package and the cost of

every seller of producing the nil package to 0.We refer to vi and c j as buyer i’s and seller j ’s type and assume that each agent’s

type is his or her private information, that types are independent across agents, thatvk

i ∈ [vki , v

ki ] for all k = 1, .., 2K − 1, so that vi ∈ ×

2K−1k=1 [vk

i , vki ] := Vi for all i = 1, .., n,

and that ckj ∈ [ck

j, ckj] for all k = 1, .., 2K − 1, so that c j ∈ ×

2K−1k=1 [c j, c j] := C j for all

j = 1, ..,m. Let vi = [v1i , .., v

2K−1i ] and c j = [c1

j , .., c2K−1j ] denote buyer i’s lowest and

seller j’s highest types, respectively.The assumption that types are drawn from rectangular sets allows us to speak of an

agent’s highest and lowest possible type without any ambiguity. It is, for example, anappropriate assumption if the vk

i ’s are independent draws from some distributions Fki , or,

perhaps more realistically, if vi is drawn from a distribution Fi with full support onVi,and analogously for sellers. Clearly, this assumption comes at the cost of some loss ofgenerality because it rules out the case where a buyer may never have the lowest possible(or highest possible) valuations for all objects. This assumption is usually not made inthe literature on one-sided VCG mechanisms because there the (implicit or explicit)assumption is that the seller’s cost is less than every buyer’s valuation for every package,which is typically normalized to 0. This structure also does not need to be imposed in thetwo-sided setup where each buyer i has a maximal willingness to pay vi ∈ [vi, vi] for oneunit only and each seller j has a capacity to produce one unit only at cost c j ∈ [c j, c j],which has been analyzed in a variety of setups (see e.g. [42], [21], [28]). It is necessaryhere because the VCG mechanism requires that we can unambiguously identify a lowesttype for every buyer and a highest (or least efficient) type for every seller.11 We assumethat types are private information of the agents but that Vi and C j are known by themechanism designer for i = 1, .., n and j = n + 1, .., n + m.

In the case of spectrum licenses, and in many other instances, it is sensible to assumethat packages can be re-packaged at zero cost to society if trade is centralized. There-fore, no congruence between packages produced and packages delivered is required.12

SetupLet v = (v1, .., vn) and c = (c1, .., cm). An allocation is a K × (n + m)-matrix Γ = [β, σ]consisting of 0’s and 1’s. The h-th column of Γ, denoted Γh, specifies which goods buyerh receives for h = 1, .., n and which goods seller h produces for h = n + 1, .., n + m (with1 meaning receive/produce). That is, βhi = 1(0) means that buyer i gets (does not get)good h and σh j = 1(0) means that seller j produces (does not produce) good h. An

11 To be more precise, VCG mechanism does not exactly require this since we could also define paymentsof agent h by excluding h (see below); however, if we did so, then the mechanism would necessarilyalways run a deficit. So the assumption is necessary only if we want to allow for the possibility that themechanism does run a surplus under some conditions.

12 If, on the other hand, sellers and buyers trade directly, then efficiency may dictate that the packagesproduced be identical to the packages delivered or, at least, that the packages buyers receive are unions ofpackages produced.


allocation Γ is said to be feasible if for every h = 1, ..,Kn∑

i=1

βhi ≤

n+m∑

j=n+1

σh j. (19.1)

Observe that there are 2K possible Γh’s, so every non-zero Γh corresponds to some spe-cific package. Letting Γh correspond to package h, we let vi(Γh) ≡ vh

i and c j(Γh) ≡ chj and

vi(Γh) ≡ vhi and c j(Γh) ≡ ch

j in slight abuse of notation. Letting G be the set of feasibleallocations and W(Γ, v, c) :=

∑ni=1 vi(Γh) −

∑mj=1 c j(Γh) be social welfare at allocation

Γ given v and c, the (or a) welfare maximizing allocation for given v and c is denotedΓ∗(v, c) and satisfies

γ∗(v, c) ∈ arg maxΓ∈G

W(Γ, v, c). (19.2)

Mechanisms and strategiesLike the standard (one-sided) VCG mechanism, the two-sided VCG mechanism is adirect mechanism that asks all agents to submit bids on all possible packages, where thespace of an agent’s possible bids is identical to the space of his or her possible types. Inany direct mechanism, an allocation rule is a matrix valued function Γ : R(n+m)(2K−1)

+ →

G that maps agents’ bids into feasible allocations. An allocation rule Γ(v, c) is said to beefficient if Γ(v, c) = Γ∗(v, c) for all (v, c) ∈ V × C, whereV = ×n

i=1Vi and C = ×mj=1C j.

And like the one-sided VCG mechanism, the two-sided VCG mechanism uses anefficient allocation rule. That is, if the bids are v and c, the allocation is Γ∗(v, c). 13

Letting v−i be the bids of buyers other than i , the payment pBi (v, c) that buyer i has to

make given bids (v, c) is

pBi (v, c) = W(Γ∗(vi, v−i, c), vi, v−i, c) −W−i(Γ∗(v, c), v, c), (19.3)

where W−i(Γ∗(v, c), v, c) = W(Γ∗(v, c), v, c) − vi(Γ∗i (v, c)). Notice that these paymentslook somewhat different from the way in which they are usually described, where buyeri’s payment would be: maximal social welfare excluding buyer i (i.e. where the welfaremaximizing allocation is chosen as if i did not exist) minus the social welfare exclud-ing buyer i at the welfare maximizing allocation that includes buyer i (see e.g. [41],p.49). The difference is due to the fact that the lowest types vi are not necessarily 0here, whereas if they are 0 (as mentioned, this is the standard assumption in one-sidedsetups), then there is obviously no difference between the two. However, in two-sidedsetups, the distinction matters with regards to revenue. Intuitively, this makes sense: Themechanism does have to provide the incentives to all agents of revealing their types, andthey could always pretend to be the least efficient type.

Similarly, the payment seller j gets when the bids are v and c is

pSj (v, c) = W− j(Γ∗(v, c), v, c) −W(Γ∗(c j, v, c− j), c j, v, c− j), (19.4)

where c− j is the collection of the bids of all sellers other than j and W− j(Γ∗(v, c), v, c) =

W(Γ∗(v, c), v, c) + c j(Γ∗j(v, c)).

13 If there are multiple welfare maximizing allocations, the mechanism can pick any of those.


For reasons mirroring those in one-sided setups, the two-sided VCG mechanism pro-vides each agent with dominant strategies. To see this, notice that buyer i’s payoff whenher type is vi and her bid is bi while every one else bids v−i and c is

vi(Γ∗(bi, v−i, c))−pBi (bi, v−i, c)

= W(Γ∗(bi, v−i, c), v, c)) −W(Γ∗(vi, v−i, c), vi, v−i, c), (19.5)

which is maximized at bi = vi since Γ∗(bi, v−i, c) is the maximizer of W(Γ, bi, v−i, c)rather than of W(Γ, v, c) and since the last term is independent of bi. Analogously, sellerj’s payoff when his type is c j and his bid is s j while every one else bids v and c− j is

pSj (v, s j, c− j)−c j(Γ∗j(v, s j, c− j))

= W(Γ∗(v, s j, c− j), v, c)) −W(Γ∗(v, c j, c− j), v, c j, c− j), (19.6)

which is maximal at s j = c j.

19.4.2 Results

In the results presented below, we find it useful in terms of establishing notation to firststate results for two-sided private information and then specialize to one-sided privateinformation.

Efficient mechanisms run a deficit for two-sided private informationDenote, respectively, buyer i’s and seller j’s marginal contribution to welfare at (v, c) by

MWi(v, c) := W(Γ∗(v, c), v, c) −W(Γ∗(vi, v−i, c), vi, v−i, c)

and

MW j(v, c) := W(Γ∗(v, c), v, c) −W(Γ∗(v, c j, c− j), v, c j, c− j).

We now assume thatn∑

i=1

MWi(v, c) +

m∑

j=1

MW j(v, c) > W(Γ∗(v, c), v, c). (19.7)

Condition (19.7) typically holds in two-sided setups with unit demand and unit ca-pacities. For example, in the classical setup with one buyer and one seller and asingle unit with v > c and v = c and v = c, W(Γ∗(v, c), v, c) = v − c andMWi(v, c) = v − c = MW j(v, c), so MWi(v, c) + MW j(v, c) = 2W(Γ∗(v, c), v, c).14

More generally, in the setup with M sellers with unit capacities and N buyers withunit demand with v = c and v = c, we have

∑Ni=1 MWi(v, c) +

∑Mj=1 MW j(v, c) =

W(Γ∗(v, c), v, c)+κ[min{cκ+1, vκ}−max{cκ, vκ+1}], where κ is the efficient quantity and c j

the j-lowest cost and vi the i-th highest valuation. Assuming continuous distributions,

14 This is due to the two-sided nature of the market making problem. In a second price sales auction of asingle object with n buyers and MWi(v) := W(Γ∗(v), v) −W(Γ∗(vi, v−i), vi, v−i),∑n

i=1 MWi(v) = v1 − v2 < W(Γ∗(v), v) = v1, where vi is the i-th highest draw. Accordingly, revenue R(v)is R(v) = W(Γ∗(v), v) −

∑ni=1 MWi(v) = v1 − (v1 − v2) = v2.


this is strictly larger than W(Γ∗(v, c), c, c) with probability 1 whenever κ > 0. Note thatin a one-sided setup such as a single unit auction, the sum of the marginal contribu-tions to welfare of the buyers,

∑ni=1 MWi(v), is less than maximum welfare W(Γ∗(v), v)

because W(Γ∗(v), v) = v1 and∑n

i=1 MWi(v) = v1 − v2. An analogous statement willhold for one-sided allocation problems that only involve sellers. Intuitively, as observedby [47] agents on the same side of the market are substitutes to each other while agentsfrom different sides are complements for each other. Based on a result of [48], [32]show that the inequality in (19.7) is never reversed in the assignment game of [47],where buyers perceive sellers as heterogenous and where all buyers have unit demandand all sellers have unit capacities, provided only the least efficient type of a buyerand the least efficient seller type optimally never trade. To see that condition (19.7) canhold for very general preferences and costs in two-sided setups, consider the case withM = 1 seller and assume that c ≥ vi for all i, where c is the highest possible cost forthe seller S . Then MWS (v, c) = W(Γ∗(v, c), v, c) because if the seller had the highestpossible cost draw the optimal allocation would involve no production at all. On top ofthat,

∑ni=1 MWi(v, c) > 0 will hold with probability 1 whenever W(Γ∗(v, c), v, c) > 0,

and so condition (19.7) will be satisfied.We can now state a generalization of [52]’s observation that for homogenous goods

efficient market making with a dominant strategy mechanism is only possible by run-ning a deficit.

Proposition 19.5 Under condition (19.7), the two-sided VCG mechanism runs a deficiteach time it is run and some trade is ex post efficient.

Proof. Given bids (v, c) the revenue under the mechanism is

R(v, c) =

n∑

i=1

pBi (v, c) −

m∑

j=1

pSj (v, c) = W(Γ∗(v, c), v, c)

−

n∑

i=1

{W(Γ∗(v, c), v, c) −W(Γ∗(vi, v−i, c), vi, v−i, c)

}

−

m∑

j=1

{W(Γ∗(v, c), v, c) −W(Γ∗(v, c j, c− j), v, c j, c− j)

}

= W(Γ∗(v, c), v, c) −

n∑

i=1

MWi(v, c) +

m∑

j=1

MW j(v, c)

< 0,

where the inequality is due to (19.7). Q.E.D.To fix ideas, let us revisit the case where buyers have unit demand, sellers have unit

capacities, and all goods are homogenous. Under the Walrasian allocation rule, we have

W(Γ∗(v, c), v, c)) =

κ∑

h=1

vh −

κ∑

j=1

c j


and, for vκ+1 ≤ cκ,

W(Γ∗(vi, v−i, c), vi, v−i, c) =

κ∑

h,i

vh −

κ−1∑

j=1

c j

for any i ≤ κ. Accordingly,

pBi (v, c) = W(Γ∗(vi, v−i, c), vi, v−i, c)) −W(Γ∗(v, c), v, c) + vi = cκ. (19.8)

On the other hand, if vκ+1 > cκ,

W(Γ∗(vi, v−i, c), vi, v−i, c) =

κ+1∑

h,i

vh −

κ∑

j=1

c j

and hence

pBi (v, c) = W(Γ∗(vi, v−i, c), vi, v−i, c)) −W(Γ∗(v, c), v, c) + vi = vκ+1. (19.9)

Summarizing,

pBi (v, c) = max{cκ, vκ+1}, (19.10)

and, analogously,

pSj (v, c) = min{cκ+1, vκ} (19.11)

for any i and j that trade.This double-auction has uniform prices for all buyers and for all sellers who trade. Let

pB(v, c) = max{cκ, vκ+1} be the price trading buyers pay and pS (v, c) = min{cκ+1, vκ} bethe price all sellers who produce pay. Observe that by construction, pS (v, c) > pB(v, c).Therefore, the mechanism runs a deficit whenever trade is efficient.

Efficient mechanisms generate a surplus for one-sided private informationLet us next assume that sellers’ costs are known, so that there is no need to give in-centives to sellers to reveal their information. This does not affect buyers’ incentiveproblem, and so each buyer who buys is still asked to pay pB

i (v, c). However, sellerswho trade do not have to be paid more than their cost, and so the mechanism never runsa deficit.

Again to fix ideas, let us revisit the case where buyers have unit demand, sellers haveunit capacities, and all goods are homogenous. In this case, each buyer who buys isasked to pay pB(v, c) = max{cκ, vκ+1} and sellers who produce are not paid more than cκ,and so the mechanism never runs a deficit. In the case with one seller, one object, and acost c1 = 0, the object is sold to the buyer who values it the most at the second-highestbid. Given our convention of relabeling agents, the object is sold to buyer 1 at pricev2. Notice that W(Γ∗(v, c), v, c) = v1 and W(Γ∗(vi, v−i, c), v, c) = v1 for all i ≥ 2, whileW(Γ∗(v1, v−1, c), v, c) = v2. Therefore, revenue R(v, c) is R(v, c) = v1 − (v1 − v2) = v2 asit should be for a second-price auction.


19.4.3 Implications flowing from the general case

It is remarkable that there exists a mechanism, namely the generalized VCG mechanismdescribed above, that generates efficient outcomes in primary markets (with a surplus)and in secondary markets (with a deficit).

The efficiency of the VCG mechanism is achieved at a potentially significant cost interms of complexity because, in general, efficiency requires full combinatorial bidding,with each bidder submitting 2K − 1 bids. Even in moderately complex environments,say, with K = 10 objects, evaluating every single package accurately and placing anappropriate bid on it will in general impose too heavy a burden on bidders whose ra-tionality is inevitably bounded. An additional concern is the computational burden theVCG mechanism imposes on the mechanism designer, who has to determine the win-ners and the bidder-specific payments. However, in some cases, a designer might haveinformation about the likely structure of buyer preferences, in which case it might bepossible with little or no reduction in efficiency to reduce the dimensionality of the bidsby limiting the set of packages on which bids may be submitted. (See, for example, thediscussion of hierarchical package bidding in Section 19.5.1.)

Even if practical consideration limit the real-world applicability of the VCG mecha-nism, it continues to be useful as a theoretical benchmark and goal. It provides a bench-mark for evaluating proposed designs that might have more straightforward implemen-tation or, for secondary markets, that overcome the issue that the VCG mechanism runsa deficit. A natural and open question is how [39]’s almost efficient dominant strategymechanism that never runs a deficit can be extended to this setup.

In the next section we address these issues further by considering the issue of practicalimplementation.

19.5 Practical implementation

As described above, even when it is theoretically possible to construct an efficient mech-anism, there can be practical hurdles to implementation.

An early advocate of the use of auctions by the FCC was [10], who made the decid-edly Coasian argument:

There is no reason why users of radio frequencies should not be in the same position as other busi-nessmen. There would not appear, for example, to be any need to regulate the relations betweenusers of the same frequency. Once the rights of potential users have been determined initially,the rearrangement of rights could be left to the market. The simplest way of doing this wouldundoubtedly be to dispose of the use of a frequency to the highest bidder, thus leaving the subdi-vision of the use of the frequency to subsequent market transactions. ( [10], p.30)

The U.S. Communications Act of 1934 (as amended by the Telecom Act of 1996)gives the FCC the legal authority to auction spectrum licenses. The language of theAct suggests that efficiency concerns should dominate revenue concerns in the FCC’sauction design choices. Section 309(j) of the Act states that one objective of the auctionsis “recovery for the public of a portion of the value of the public spectrum,” but it also


states that “the Commission may not base a finding of public interest, convenience, andnecessity solely or predominantly on the expectation of Federal revenues.” Thus, fromthe beginning, the FCC faced the conflicting objectives of revenue versus efficiency,15

although efficiency concerns have been predominant in the design concerns of the FCC.In this section we discuss practical implementation considerations, with a focus on

primary markets, which is where the economics literature, including theoretical and ex-perimental literature, and practice have been most developed and advanced. Issues willbe similar (and possibly exacerbated) for the market making problem for a secondarymarket with two-sided private information.

In a theoretical sense, the problem of designing an efficient mechanism for the pri-mary market is solved in the concept of the VCG mechanism. However, the VCG mech-anism has proved challenging in terms of its practical implementation; see [1] and [44].For efficiency to obtain, VCG mechanisms require bidders to reveal their true values,but bidders may have concerns that doing so would disadvantage them in future compe-tition. The VCG mechanism is vulnerable to collusion by a coalition of losing bidders,and history has shown that bidder collusion in spectrum license auctions is a realisticconcern. The VCG mechanism is vulnerable to the use of multiple bidding identitiesby a single bidder, which is something that is currently permitted in FCC auctions, andgiven the possibility for complex contractual relationships among bidders, is somethingthat might be difficult to prohibit. There are potential political repercussions for theseller if submitted bids in a VCG mechanism reveal that the winner’s willingness to paywas greatly more than the actual payment. In addition, the FCC’s revenue from a VCGmechanism could be low or even zero. Finally, the efficiency of the VCG mechanism isobtained by allowing combinatorial bidding, but given the number of different spectrumlicenses, the computational complexity associated with winner determination may beinsurmountable, and, perhaps more importantly, it is unrealistic to expect bidders to beable to provide reports of their values for all of the potentially very large number ofpossible packages.16

In practice, the FCC has had to develop more suitable designs that accommodatethese concerns, at least to some extent, while keeping in mind the objectives of effi-ciency and, to a lesser degree, revenue. The use of open ascending auctions reducesthe complexity concerns for both the auctioneer and bidders, and is not problematic forsubstitute preferences, but creates “exposure risk” for bidders who perceive differentobjects as complements because these bidders risk receiving only parts of a packageand overpaying for what they receive because the package would be of more value tothem than the sum of its components. It seems very plausible that bidders may havesuch preferences, for example, for various collections of spectrum licenses.

Concerns regarding the exposure risk problem have focused attention on the needfor some kind of package bidding. Such combinatorial auction designs create issues

15 [24] show that there is a conflict between efficiency and revenue maximization in multi-object auctionseven with symmetric bidders.

16 [33] mentions that, even in the absence of complexity, second price auctions the seller may face theproblem that the bidders do not trust him to reveal the second highest bid truthfully (which was an issuewith stamp collection auctions in the middle of the 20th century).


of their own, such as bid shading or demand reduction (that is, bidders strategicallybidding less than their valuations), leaving designers once again in the world of thesecond best. There is little theoretical guidance available associated with designs such ascombinatorial clock auctions and hierarchical package bidding because they are rarelyaddressed in the theoretical literature, presumably based on the perception that they aretypically not theoretically tractable. Instead, design has relied heavily on experiments,economic intuition, and practical experience.

It does not seem very plausible that secondary markets can or should be relied upon tocorrect for deficiencies associated with primary market allocations; indeed, it has beenargued that combining primary and secondary markets may be a source of inefficiencyby inducing speculative bidding ( [18]) and facilitating collusion ( [19]).

In Section 19.5.1 we describe some of the choices made by the FCC in the face ofthese challenges. In Section 19.5.2 we describe the role that experimental economicscan play in guiding practical responses to resolving tradeoffs in market design.

19.5.1 FCC approach

The FCC acted as a primary market designer for spectrum licenses, holding auctions forthose licenses since 1994; see [40], [38], [12], [30], and [36]. When initially developingits auction design, the FCC had to address the issues that it needed to auction a largenumber of heterogeneous licenses where there were potentially strong complementari-ties as well as substitutability among various licenses. Existing auction mechanisms ofthe time did not appear well suited to address these challenges.

The FCC needed a mechanism that would allow bidders to shift bids between licensesthey viewed as substitutes as prices changed and to bid simultaneously on licenses theyviewed as complementary. As described above, a buyer’s willingness to pay might bedefined over packages of licenses and not independently over individual licenses.

Even in cases where the potential complexity of combinatorial bidding is not a con-cern and the efficient outcome can be induced by a VCG mechanism, as pointed outby [1] and [44], there are practical implementation problems associated with the VCGmechanism. For example, VCG mechanisms require that bidders reveal their true val-ues. Bidders may not want to do this if, for example, it weakens their bargaining positionin future transactions, and the FCC may not be able to maintain the secrecy of bids be-cause of the Freedom of Information Act. In addition, as stated in [44] p.195, “[i]ngovernment sales of extremely valuable assets, the political repercussions of revealingthe gap between large offers and small revenue could be a dominant concern.”

Ultimately, based on substantial input from FCC and academic economists, the FCCdeveloped a simultaneous multiple round (SMR) auction, also known as a simultaneousascending auction. This basic auction format, with various modifications and exten-sions,17 continues to be used today. As described by the FCC:

In a simultaneous multiple-round (SMR) auction, all licenses are available for bidding throughout

17 A number of relatively minor modifications to the original design have been made to addresssusceptibility to collusion by bidders. See, e.g., [7], [13, 14], [29], [38], and [36]. Regarding the use of acontingent re-auction format, see [8].


the entire auction, thus the term “simultaneous.” Unlike most auctions in which bidding is contin-uous, SMR auctions have discrete, successive rounds, with the length of each round announced inadvance by the Commission. After each round closes, round results are processed and made pub-lic. Only then do bidders learn about the bids placed by other bidders. This provides informationabout the value of the licenses to all bidders and increases the likelihood that the licenses will beassigned to the bidders who value them the most. The period between auction rounds also allowsbidders to take stock of, and perhaps adjust, their bidding strategies. In an SMR auction, there isno preset number of rounds. Bidding continues, round after round, until a round occurs in whichall bidder activity ceases. That round becomes the closing round of the auction.18

The FCC has allowed limited combinatorial bidding at certain auctions. The FCCrecognizes the benefits of what it refers to as package bidding: “This approach allowsbidders to better express the value of any synergies (benefits from combining comple-mentary items) that may exist among licenses and to avoid the risk of winning only partof a desired set.”19 But the FCC has balanced these benefits with the potential costs ofcomplexity by limiting the set of packages on which bidders may submit bids, gener-ally to a set of packages with a hierarchical structure (“hierarchical package bidding”or HPB).20 The FCC used the SMR-HPB format in Auction 73 for C-block licenses, al-lowing bidding on three packages (“50 States,” “Atlantic,” and “Pacific”) as well as theindividual licenses.21 Under some reasonable restrictions on preferences, it may be pos-sible to maintain efficiency with only restricted package bidding, such as hierarchicalpackage bidding.

The FCC has also used an auction design whereby the auction mechanism itself se-lected among multiple competing band plans. This differs from the FCC’s standardapproach of predetermining the single band plan under which licenses will be offered atauction.22

[53] criticizes the mechanism design approach for focusing on the development ofoptimal designs for particular environments, often involving parameters that are un-likely to be known by the designer in practice, rather than identifying simple mech-anisms that perform reasonably well across a variety of environments that might beencountered. Overall, the FCC’s approach has been to do the best possible to design aprimary market that balances efficiency and complexity concerns, while also recoveringsome of the value of the spectrum resource for taxpayers. The FCC has relied on sec-ondary market transactions to address remaining inefficiencies in the initial allocationand to address the dynamic nature of the efficient allocation as demand, technology, andthe set of market participants evolve over time.

18 FCC website, http://wireless.fcc.gov/auctions/default.htm?job=about auctions&page=2, accessed June28, 2012.

19 FCC website, http://wireless.fcc.gov/auctions/default.htm?job=about auctions&page=2, accessed June28, 2012.

20 The hierarchical structure of HPB was suggested by [45]. The pricing mechanism for HPB was proposedby [20].

21 FCC website, Auction 73 Procedures Public Notice, available athttp://hraunfoss.fcc.gov/edocs public/attachmatch/DA-07-4171A1.pdf, accessed June 28, 2012.

22 FCC website, Auction 65 Procedures Public Notice, available athttp://hraunfoss.fcc.gov/edocs public/attachmatch/DA-06-299A1.pdf, accessed June 28, 2012.


19.5.2 Experimental approach

Given the inability of theory to provide an efficient, easily implementable mechanismthat does not run a deficit, the FCC has requested, commissioned, and relied upon eco-nomic experiments to guide its decision making in spectrum license market design.

Economic experiments typically involve the use of cash to motivate subjects (oftenuniversity students) to participate in an exercise that is designed to reflect some aspectof real-world markets under study. The details of the exercise would typically be trans-parent and carefully explained to the subjects. Subjects would make choices and berewarded in a way designed to elicit the type of behavior one might expect from marketparticipants motivated by profit. Data collected from these experiments, which are gen-erally repeated many times, sometimes with the same subjects and sometimes with newsubjects, are used to generate insights as to the behavior one would expect in real-worldsettings.23

For example, the FCC relied on experimental work by [20] in determining whetherand in what form to allow limited package bidding in FCC Auction 73 (700 MHz Auc-tion).24 [20] describe their work as providing a “wind tunnel” test of three alternativeauction formats under consideration.

Experimental work by [6] raises cautions regarding a package bidding format pro-posed by the FCC, referred to as SMRPB: “We find clear differences among the packageformats ... both in terms of efficiency and seller revenue. ... The SMRPB auction per-formed worse than the other combinatorial formats, which is one of the main reasonswhy the FCC has decided not to implement SMRPB procedure for package bidding.”( [6] p.1)

Experimental work by [2] allows those researchers to suggest improvements to thedetails of the FCC’s standard SMR auction format. They also find that when license val-ues are superadditive, the combinatorial auction outperforms the FCC’s standard auc-tion format in terms of efficiency,25 but at a cost to bidders in terms of the time requiredto complete the auction. In addition, they report results suggesting the combinatorialauction format considered might not perform well in certain “stress test” scenarios.Other economic experimental work that is relevant to spectrum license market designincludes [3], [43], [29], and [17].

19.6 Conclusions

In this chapter, we provide the economic underpinnings for a government role in allocat-ing spectrum licenses in the primary market and the issues that arise when one considersa role for a market maker in the secondary market. We examine when economic theorysuggests market design could usefully play a role and what that role might be.

23 See [26] for an introduction to the methodology of experimental economics.24 See the FCC’s Procedures Public Notice for Auction 73, available at

http://hraunfoss.fcc.gov/edocs public/attachmatch/DA-07-4171A1.pdf), accessed July 11, 2012.25 [6] also find efficiency benefits from combinatorial bidding.

578 References

The results presented here are limited in that we focus on the case of private val-ues.26 In addition, we focus on dominant strategy mechanisms rather than consideringthe broader class of Bayesian mechanisms. However, the impossibility result of Myer-son and Satterthwaite for efficient secondary market design that does not run a deficitalso holds for Bayesian mechanisms, and so the result is stronger than what we havepresented here.

Finally, given the swift pace of technological innovation in mobile wireless technolo-gies, it is clear that dynamic issues are a key concern. We have focused on static mecha-nism design, although there is a growing literature on dynamic mechanism design; see,e.g., [5]. As our understanding of the issues and possibilities associated with dynamicmechanism design advances, there may be new possibilities to improve spectrum li-cense allocation mechanisms, although one might expect that issues of complexity andother weaknesses of theoretically desirable designs will mean a continued reliance onbasic economic foundations together with results of experiments, economic intuition,and practical experience.

Acknowledgement

Simon Loertscher gratefully acknowledges The Department of Treasury and Financeof Victoria and the Australian Communications and Media Authority for helpful dis-cussions that led to the development of the ideas for this chapter and the comments byYuelan Chen and two anonymous reviewers that have improved the exposition. Thiswork was financially supported by the Centre for Market Design at the University ofMelbourne.

Leslie Marx thanks the U.S. Federal Communications Commission and especiallyEvan Kwerel and Martha Stancill for helpful discussions related to FCC spectrum li-cense auctions.

The views expressed here are those of the authors and do not necessarily reflect thoseof the agencies and individuals acknowledged above.

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