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### Summary

### Summary

Providing a comprehensive introduction to modern auction theory and its important new applications, this book is written by a leading economic theorist whose suggestions guided the creation of the new spectrum auction designs. Aimed at graduate students and professionals in economics, the volume provides the most up-to-date analysis of traditional theories of "optimal auctions" as well as newer theories of multi-unit auctions and package auctions, and shows by example how these theories are used. It explores the limitations of prominent older designs, such as the Vickrey auction design, and evaluates the practical responses to those limitations. Paul Milgrom is the Leonard and Shirley Ely Professor of Humanities and Sciences and Professor of Economics, Stanford University. He is the author of more than sixty articles and co-author of the influential textbook, Economics, Organization and Management (Prentice Hall, 1992). Professor Milgrom is a pioneer in the economic theory of auctions and co-designer of the simultaneous, multiple round auction that the FCC adopted for selling radio spectrum licenses.

Visit the author's website for instructor resources.

### Table of Contents

Preface | p. xi |

Foreword | p. xv |

1 Getting to Work | p. 1 |

1.1 Politics Sets the Stage | p. 3 |

1.2 Designing for Multiple Goals | p. 3 |

1.2.1 Substitutes and Complements | p. 6 |

1.2.2 New Zealand's Rights Auction | p. 9 |

1.2.3 Better Auction Designs | p. 13 |

1.2.4 The FCC Design and Its Progeny | p. 13 |

1.3 Comparing Seller Revenues | p. 16 |

1.4 The Academic Critics | p. 19 |

1.4.1 Resale and the Coase Theorem | p. 19 |

1.4.2 Mechanism Design Theory | p. 21 |

1.4.3 Theory and Experiment | p. 25 |

1.4.4 Practical Concerns | p. 26 |

1.5 Plan for This Book | p. 31 |

Part 1 The Mechanism Design Approach | p. 35 |

2 Vickrey-Clarke-Groves Mechanisms | p. 45 |

2.1 Formulation | p. 45 |

2.2 Always Optimal and Weakly Dominant Strategies | p. 49 |

2.3 Balancing the Budget | p. 53 |

2.4 Uniqueness | p. 55 |

2.5 Disadvantages of the Vickrey Auction | p. 56 |

2.5.1 Practical Disadvantages | p. 56 |

2.5.2 Monotonicity Problems | p. 57 |

2.5.3 The Merger-Investment Disadvantage | p. 60 |

2.6 Conclusion | p. 61 |

3 The Envelope Theorem and Payoff Equivalence | p. 64 |

3.1 Hotelling's Lemma | p. 65 |

3.2 The Envelope Theorem in Integral Form | p. 66 |

3.3 Quasi-linear Payoffs | p. 69 |

3.3.1 Holmstrom's Lemma | p. 70 |

3.3.2 The Green-Laffont-Holmstrom Theorem | p. 71 |

3.3.3 Myerson's Lemma | p. 73 |

3.3.4 Revenue Equivalence Theorems | p. 75 |

3.3.5 The Myerson-Satterthwaite Theorem | p. 77 |

3.3.6 The Jehiel-Moldovanu Impossibility Theorems | p. 80 |

3.3.7 Myerson and Riley-Samuelson Revenue-Maximizing Auctions | p. 84 |

3.3.8 The McAfee-McMillan Weak-Cartels Theorem | p. 87 |

3.3.9 Sequential Auctions and Weber's Martingale Theorem | p. 90 |

3.3.10 Matthews Theorem: Risk Averse Payoff Equivalence | p. 91 |

3.4 Conclusion | p. 94 |

4 Bidding Equilibrium and Revenue Differences | p. 98 |

4.1 The Single Crossing Conditions | p. 99 |

4.1.1 The Monotonic Selection Theorem | p. 101 |

4.1.2 The Sufficiency Theorem | p. 102 |

4.1.3 The Constraint Simplification Theorem | p. 105 |

4.1.4 The Mirrlees-Spence Representation Theorem | p. 106 |

4.2 Deriving and Verifying Equilibrium Strategies | p. 110 |

4.2.1 The Second-Price Auction with a Reserve Price | p. 111 |

4.2.2 The Sealed Tender, or First-Price, Auction | p. 112 |

4.2.3 The War of Attrition Auction | p. 117 |

4.2.4 The All-Pay Auction | p. 119 |

4.3 Revenue Comparisons in the Benchmark Model | p. 119 |

4.3.1 Payoff Equivalence without Revenue Equivalence | p. 121 |

4.3.2 Budget Constraints | p. 132 |

4.3.3 Endogenous Quantities | p. 135 |

4.3.4 Correlated Types | p. 137 |

4.4 Expected-Revenue-Maximizing Auctions | p. 140 |

4.4.1 Myerson's Theorem | p. 144 |

4.4.2 Bulow-Klemperer Theorem | p. 148 |

4.4.3 The Irregular Case | p. 148 |

4.5 Auctions with Weak and Strong Bidders | p. 149 |

4.6 Conclusion | p. 154 |

5 Interdependence of Types and Values | p. 157 |

5.1 Which Models and Assumptions are "Useful"? | p. 158 |

5.1.1 Payoffs Depend Only on Bids and Types | p. 158 |

5.1.2 Types Are One-Dimensional and Values Are Private | p. 159 |

5.1.3 Types Are Statistically Independent | p. 161 |

5.2 Statistical Dependence and Revenue-Maximizing Auctions | p. 162 |

5.3 Wilson's Drainage Tract Model | p. 166 |

5.3.1 Equilibrium | p. 167 |

5.3.2 Profits and Revenues | p. 173 |

5.3.3 Bidder Information Policy | p. 175 |

5.3.4 Seller Information Policy | p. 177 |

5.4 Correlated Types and Interdependent Values | p. 181 |

5.4.1 Affiliation | p. 182 |

5.4.2 The Milgrom-Weber Ascending Auction Models | p. 187 |

5.4.2.1 The (Second-Price) Button Auction with Minimal Information | p. 188 |

5.4.2.2 The Button Auction with Maximal Information | p. 195 |

5.4.2.3 Some Revenue Comparisons | p. 198 |

5.4.3 First-Price Auctions | p. 200 |

5.5 Conclusion | p. 204 |

6 Auctions in Context | p. 208 |

6.1 The Profit and Surplus Contribution of an Entrant | p. 214 |

6.2 Symmetric Models with Costly Entry | p. 216 |

6.2.1 Symmetric Bidders and Uncoordinated Entry | p. 218 |

6.2.1.1 Equilibrium in Entry and Bidding Decisions | p. 218 |

6.2.1.2 Setting the Reserve Price | p. 222 |

6.2.2 Coordinating Entry among Symmetric Competitors | p. 225 |

6.2.2.1 Pre-qualifying Bidders | p. 227 |

6.2.2.2 Auctions, Negotiations, and Posted Prices | p. 230 |

6.2.2.3 Buy Prices | p. 232 |

6.3 Asymmetric Models: Devices to Promote Competition | p. 234 |

6.3.1 Example of Set-asides | p. 235 |

6.3.2 Example of Bidding Credits | p. 237 |

6.3.3 Example of Lot Structure and Consolation Prizes | p. 238 |

6.3.4 Premium Auctions | p. 239 |

6.3.5 Dutch vs. English Auctions and the Anglo-Dutch Design | p. 241 |

6.4 After the Bidding Ends | p. 243 |

6.4.1 Bankruptcy and Non-performance | p. 243 |

6.4.2 Scoring Rules vs. Price-Only Bids | p. 245 |

6.5 Conclusion | p. 247 |

Part II Multi-Unit Auctions | p. 251 |

7 Uniform Price Auctions | p. 255 |

7.1 Uniform Price Sealed-Bid Auctions | p. 257 |

7.1.1 Demand Reduction | p. 258 |

7.1.2 Low-Price Equilibria | p. 262 |

7.2 Simultaneous Ascending Auctions | p. 265 |

7.2.1 The Simultaneous Ascending Auction and the Walrasian Tatonnement | p. 268 |

7.2.2 Clock Auctions | p. 279 |

7.2.3 Strategic Incentives in Uniform Price Auctions | p. 284 |

7.2.3.1 The Basic Clock Auction Model | p. 284 |

7.2.3.2 The Alternating-Move Clock Auction | p. 287 |

7.2.3.3 Strategic Incentives with Elastic Supply | p. 290 |

7.3 Conclusion | p. 293 |

8 Package Auctions and Combinatorial Bidding | p. 296 |

8.1 Vickrey Auctions and the Monotonicity Problems | p. 302 |

8.1.1 Bidders' Vickrey Payoffs Bound Their Core Payoffs | p. 305 |

8.1.2 Vickrey Auctions and the Entry Puzzle | p. 305 |

8.1.3 When Are Vickrey Outcomes in the Core? | p. 307 |

8.1.4 Substitute Goods and Core Outcomes | p. 308 |

8.1.5 Substitute Goods and Vickrey Outcomes | p. 312 |

8.2 Bernheim-Whinston First-Price Package Auctions | p. 315 |

8.2.1 Formulation | p. 316 |

8.2.2 Profit-Target Strategies | p. 318 |

8.2.3 Equilibrium and the Core | p. 319 |

8.3 Ausubel-Milgrom Ascending Proxy Auctions | p. 324 |

8.3.1 The Proxy Auction with Unlimited Budgets | p. 325 |

8.3.1.1 Proxy Outcomes Are Core Outcomes | p. 326 |

8.3.1.2 Profit-Target Strategies and Equilibrium | p. 327 |

8.3.1.3 The Proxy Auction When Goods Are Substitutes | p. 329 |

8.3.2 The Non-transferable-Utility Proxy Auction | p. 330 |

8.4 Conclusion | p. 333 |

Bibliography | p. 339 |

Author Index | p. 347 |

Subject Index | p. 351 |