Auction theory

318 Pages·2002·1.926 MB·English

by  Vijay Krishna

Auction Theory
Vijay Krishna
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Contents
Preface ix
1 Introduction 1
I Single Object Auctions 11
2 Private Value Auctions: A First Look 13
2.1 The Symmetric Model . . . . . . . . . . . . . . . . . . . . . 14
2.2 Second-Price Auctions . . . . . . . . . . . . . . . . . . . . . 15
2.3 First-Price Auctions . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Revenue Comparison . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Reserve Prices . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 The Revenue Equivalence Principle 29
3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Some Applications of the Revenue Equivalence Principle . . 31
3.2.1 Unusual Auctions. . . . . . . . . . . . . . . . . . . . 31
3.2.2 Uncertain Number of Bidders . . . . . . . . . . . . . 34
4 Qualifications and Extensions 37
4.1 Risk-Averse Bidders . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Budget Constraints . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Second-Price Auctions . . . . . . . . . . . . . . . . . 43
vi Contents
4.2.2 First-Price Auctions . . . . . . . . . . . . . . . . . . 45
4.2.3 Revenue Comparison . . . . . . . . . . . . . . . . . . 45
4.3 Asymmetries among Bidders . . . . . . . . . . . . . . . . . 46
4.3.1 Asymmetric First-Price Auctions with Two Bidders 46
4.3.2 Revenue Comparison . . . . . . . . . . . . . . . . . . 51
4.3.3 Efficiency Comparison . . . . . . . . . . . . . . . . . 54
4.4 Resale and Efficiency . . . . . . . . . . . . . . . . . . . . . . 54
5 Mechanism Design 61
5.1 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.1 The Revelation Principle . . . . . . . . . . . . . . . 62
5.1.2 Incentive Compatibility . . . . . . . . . . . . . . . . 63
5.1.3 Individual Rationality . . . . . . . . . . . . . . . . . 67
5.2 Optimal Mechanisms . . . . . . . . . . . . . . . . . . . . . . 67
5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.3 Discussion and Interpretation . . . . . . . . . . . . . 71
5.2.4 Auctions versus Mechanisms . . . . . . . . . . . . . 75
5.3 Efficient Mechanisms . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 The VCG Mechanism . . . . . . . . . . . . . . . . . 76
5.3.2 Budget Balance . . . . . . . . . . . . . . . . . . . . . 78
5.3.3 An Application to Bilateral Trade . . . . . . . . . . 80
6 Auctions with Interdependent Values 83
6.1 The Symmetric Model . . . . . . . . . . . . . . . . . . . . . 86
6.2 Second-Price Auctions . . . . . . . . . . . . . . . . . . . . . 87
6.3 English Auctions . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 First-Price Auctions . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Revenue Comparisons . . . . . . . . . . . . . . . . . . . . . 96
6.5.1 English versus Second-Price Auctions . . . . . . . . 96
6.5.2 Second-Price versus First-Price Auctions. . . . . . . 97
6.6 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 The Revenue Ranking (“Linkage”) Principle 103
7.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Public Information . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 An Alternative Linkage Principle . . . . . . . . . . . . . . . 108
8 Asymmetries and Other Complications 111
8.1 Failures of the Linkage Principle . . . . . . . . . . . . . . . 111
8.2 Asymmetric Equilibria in Symmetric Second-Price Auctions 116
8.3 Asymmetrically Informed Bidders. . . . . . . . . . . . . . . 119
8.4 Reserve Prices and Entry Fees. . . . . . . . . . . . . . . . . 121
Contents vii
9 Efficiency and the English Auction 125
9.1 The Single Crossing Condition . . . . . . . . . . . . . . . . 126
9.2 Two-Bidder Auctions. . . . . . . . . . . . . . . . . . . . . . 127
9.3 The Average Crossing Condition . . . . . . . . . . . . . . . 130
9.4 Three or More Bidders . . . . . . . . . . . . . . . . . . . . . 131
9.5 Proof of Proposition 9.2 . . . . . . . . . . . . . . . . . . . . 133
9.6 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10 Mechanism Design with Interdependent Values 141
10.1 Efficient Mechanisms . . . . . . . . . . . . . . . . . . . . . . 142
10.2 Optimal Mechanisms . . . . . . . . . . . . . . . . . . . . . . 146
11 Bidding Rings 151
11.1 Collusion in Second-Price Auctions . . . . . . . . . . . . . . 152
11.1.1 Efficient Collusion . . . . . . . . . . . . . . . . . . . 154
11.1.2 Reserve Prices in the Face of Collusion . . . . . . . . 158
11.2 Collusion in First-Price Auctions . . . . . . . . . . . . . . . 160
II Multiple Object Auctions 163
12 An Introduction to Multiple Object Auctions 165
12.1 Sealed-Bid Auctions for Selling Identical Units . . . . . . . 166
12.1.1 Discriminatory Auctions . . . . . . . . . . . . . . . . 168
12.1.2 Uniform-Price Auctions . . . . . . . . . . . . . . . . 169
12.1.3 Vickrey Auctions . . . . . . . . . . . . . . . . . . . . 171
12.2 Some Open Auctions . . . . . . . . . . . . . . . . . . . . . . 172
12.2.1 Dutch Auctions . . . . . . . . . . . . . . . . . . . . . 172
12.2.2 English Auctions . . . . . . . . . . . . . . . . . . . . 172
12.2.3 Ausubel Auctions. . . . . . . . . . . . . . . . . . . . 173
13 Equilibrium and Efficiency with Private Values 179
13.1 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . 179
13.2 Vickrey Auctions . . . . . . . . . . . . . . . . . . . . . . . . 181
13.3 Efficiency in Multiunit Auctions . . . . . . . . . . . . . . . 183
13.4 Uniform-Price Auctions . . . . . . . . . . . . . . . . . . . . 185
13.4.1 Demand Reduction . . . . . . . . . . . . . . . . . . . 187
13.4.2 Single-Unit Demand . . . . . . . . . . . . . . . . . . 190
13.5 Discriminatory Auctions . . . . . . . . . . . . . . . . . . . . 191
13.5.1 Structure of Equilibria . . . . . . . . . . . . . . . . . 192
13.5.2 Single-Unit Demand . . . . . . . . . . . . . . . . . . 195
14 Some Revenue Considerations 199
14.1 Revenue Equivalence in Multiunit Auctions . . . . . . . . . 200
14.2 Revenue Equivalence with Multiunit Demand: An Example 202
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15 Sequential Sales 209
15.1 Sequential First-Price Auctions . . . . . . . . . . . . . . . . 209
15.1.1 Two Units. . . . . . . . . . . . . . . . . . . . . . . . 210
15.1.2 More than Two Units . . . . . . . . . . . . . . . . . 213
15.1.3 Equilibrium Bids and Prices. . . . . . . . . . . . . . 216
15.2 Sequential Second-Price Auctions . . . . . . . . . . . . . . . 217
15.2.1 Revenue Equivalence . . . . . . . . . . . . . . . . . . 217
15.2.2 Equilibrium Bids . . . . . . . . . . . . . . . . . . . . 218
16 Nonidentical Objects 223
16.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
16.2 Efficient Allocations . . . . . . . . . . . . . . . . . . . . . . 225
16.3 Substitutes and Complements . . . . . . . . . . . . . . . . . 226
16.4 Bundling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
16.5 Some Computational Issues . . . . . . . . . . . . . . . . . . 230
16.6 Budget Constraints . . . . . . . . . . . . . . . . . . . . . . . 232
17 Multiple Objects and Interdependent Values 235
17.1 One-Dimensional Signals . . . . . . . . . . . . . . . . . . . . 235
17.1.1 An Efficient Direct Mechanism . . . . . . . . . . . . 236
17.1.2 Efficiency via Open Auctions . . . . . . . . . . . . . 240
17.2 Multidimensional Signals . . . . . . . . . . . . . . . . . . . 243
17.2.1 Single Object . . . . . . . . . . . . . . . . . . . . . . 244
17.2.2 Multiple Objects . . . . . . . . . . . . . . . . . . . . 248
III Appendices 251
A Continuous Distributions 253
B Stochastic Orders 259
C Order Statistics 265
D Affiliated Random Variables 269
E Some Linear Algebra 273
F Games of Incomplete Information 279
G Existence of Equilibrium in First-Price Auctions 283
References 289
Index 299
Preface
The analysis of auctions as games of incomplete information originates in
the work of William Vickrey (1961). In this book I discuss the theory of
auctions in this tradition. The goal is to give an account of developments
in the field in the 40 years since Vickrey’s pioneering paper.
I do not attempt to provide a comprehensive survey of auction theory.
The field has burgeoned, especially in the past couple of decades, and a
comprehensive survey would be nearly impossible–the Econ Lit database
has more than a thousand entries with the word “auction” or “auctions”
in their titles, and about one-half of these papers are theoretical. Instead I
have opted to concentrate on selected themes that I consider to be central
to the theory. I adopt the point of view that a detailed consideration of a
few basic models is more fruitful than a perfunctory discussion of a large
number of variations. I can only hope that my choice of themes is not too
arbitrary.
The models that are considered are discussed in some detail and, with a
few minor exceptions, complete proofs of all propositions are provided. It
is my contention that the strengths and weaknesses of the theory can be
appreciatedonlyby examiningthe inner workingsof thepropositionswith
some care.
Game Theory. The theory of games, especially concerning games of in-
complete information, constitutes the basic apparatus of the book. Most
moderngraduatetextsinmicroeconomicsnowhaveasubstantialemphasis
on game theory (see Kreps, 1990 or Mas-Colell et al., 1995); any of these