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Auction Theory

by Vijay Krishna    
Table of Contents
1. Introduction 1
Part I.        Single Object Auctions 10
2. Private Value Auctions: A First Look                                       13
     2.1 The Symmetric Model                       

     2.2 Second-Price Auctions    

     2.3 First-Price Auctions

     2.4 Revenue Comparison         

     2.5 Reserve Prices
3. The Revenue Equivalence Principle 29
     3.1 Main Result
     3.2 Some Applications of the Revenue Equivalence Principle 
4. Qualifications and Extensions 37
     4.1 Risk-Averse Bidders
     4.2 Budget Constraints
     4.3 Asymmetries among Bidders
     4.4 Resale and Efficiency
5. Mechanism Design 61
     5.1 Mechanisms
     5.2 Optimal Mechanisms
     5.3 Efficient Mechanisms
6. Auctions with Interdependent Values 83
     6.1 The Symmetric Model
     6.2 Second-Price Auctions
     6.3 English Auctions
     6.4 First-Price Auctions
     6.5 Revenue Comparisons
     6.6 Efficiency
7. The Revenue Ranking (“Linkage”) Principle                                      103
     7.1 The Main Result
     7.2 Public Information
     7.3 An Alternative Linkage Principle
8. Asymmetries and Other Complications                                                111
     8.1 Failures of the Linkage Principle
     8.2 Asymmetric Equilibria in Symmetric Second-Price Auctions
     8.3 Asymmetrically Informed Bidders
     8.4 Reserve Prices and Entry Fees
9. Efficiency and the English Auction                                                     125
     9.1 The Single Crossing Condition
     9.2 Two-Bidder Auctions
     9.3 The Average Crossing Condition
     9.4 Three or More Bidders
     9.5 Proof of Proposition 9.2
     9.6 Miscellany
10. Mechanism Design with Interdependent Values 141
     10.1 Efficient Mechanisms
     10.2 Optimal Mechanisms
11. Bidding Rings                                                                                       151
     11.1 Collusion in Second-Price Auctions
     11.2 Collusion in First-Price Auctions
Part II.      Multiple Object Auctions                                                     163
12. An Introduction to Multiple Object Auctions 165
     12.1 Sealed-Bid Auctions for Selling Multiple Units
     12.2 Some Open Auctions
13. Equilibrium and Efficiency with Private Values 179
     13.1 The Basic Model
     13.2 Vickrey Auctions
     13.3 Efficiency in Multiunit Auctions
     13.4 Uniform Price Auctions
     13.5 Discriminatory Auctions
14. Some Revenue Considerations 199
     14.1 Revenue Equivalence in Multiunit Auctions
     14.2 Revenue Equivalence with Multiunit Demand: An Example
15. Sequential Sales 209
     15.1 Sequential First-Price Auctions
     15.2 Sequential Second-Price Auctions
16. Nonidentical Objects 223
     16.1 The Model
     16.2 Efficient Allocations
     16.3 Substitutes and Complements
     16.4 Bundling
     16.5 Some Computational Issues
     16.6 Budget Constraints
17. Multiple Objects and Interdependent Values 235
     17.1 One Dimensional Signals
     17.2 Multidimensional Signals
Part 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