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Robust Mechanism Design： An Introduction1

1 Introduction1

2 Leading Example： Allocating a Private Good with Interdependent Values4

3 Type Spaces7

4 Robust Foundations for Dominant and Ex Post Incentive Compatibility15

5 Full Implementation21

5.1 Ex Post Implementation22

5.2 Robust Implementation in the Direct Mechanism23

5.3 The Robustness of Robust Implementation32

5.4 Robust Implementation in the General Mechanism33

5.5 Rationalizable Implementation34

5.6 The Role of the Common Prior36

5.7 Dynamic Mechanisms37

5.8 Virtual Implementation38

6 Open Issues42

References43

Chapter 1 Robust Mechanism Design49

1 Introduction50

2 Setup54

2.1 Payoff Environment54

2.2 Type Spaces55

2.3 Solution Concepts56

2.4 Questions58

2.5 Implicit versus Explicit Modelling of Higher Order Uncertainty and the Universal Type Space59

3 Examples62

3.1 F is Interim Implementable on All Type Spaces But not Ex Post Implementable63

3.2 F is Interim Implementable on All Payoff Type Spaces But not Interim Implementable on All Type Spaces67

4 Separable Environments72

4.1 Separable Environments73

4.2 Full Support Conditions77

5 The Quasilinear Environment with Budget Balance81

6 Discussion90

6.1 A Classical Debate90

6.2 Genericity92

6.3 Augmented Ex Post Equivalence92

References93

Chapter 2 Ex Post Implementation97

1 Introduction97

2 Model103

3 Monotonicity105

3.1 Ex Post Monotonicity105

3.2 Maskin Monotonicity107

3.3 Public Good Example109

4 Ex Post Implementation111

4.1 Necessary Conditions112

4.2 Sufficient Conditions in Economic Environments114

4.3 Sufficiency Conditions in Non-economic Environments117

5 Single Crossing Environment121

6 Direct Mechanisms126

7 Single Unit Auction129

7.1 Model130

7.2 Monotonicity and the VCG Mechanism130

7.3 Private Versus Interdependent Values135

8 Social Choice Sets137

8.1 Pareto Correspondence137

8.2 Functions， Sets and Correspondences141

9 Mixed Strategy Implementation142

10 Conclusion147

Appendix A148

A.1.Private values148

A.2.Ex post monotonicity no veto for sets148

References150

Chapter 3 Robust Implementation in Direct Mechanisms153

1 Introduction154

2 Setup159

3 A Public Good Example161

4 Robust Implementation164

5 Necessity of Contraction Property170

6 The Linear Model175

7 Single Unit Auction178

8 Discussion179

8.1 Dimensionality and Aggregation179

8.2 Relation to Partial and Ex Post Implementation183

8.3 Robust and Virtual Implementation in General Environments183

8.4 Social Choice Correspondences and Sets185

8.5 The Common Prior Assumption and Strategic Substitutes/ Complements186

8.6 Informational Foundation of Interdependence187

9 Appendix187

References192

Chapter 4 Robust Implementation in General Mechanisms195

1 Introduction195

2 Setup200

2.1 The Payoff Environment200

2.2 Type Spaces201

2.3 Mechanisms201

2.4 Solution Concepts201

2.5 Implementation204

3 Finite Mechanisms206

3.1 Ex Post Incentive Compatibility207

3.2 Robust Monotonicity208

3.3 Robust Measurability212

3.4 Single Crossing Aggregator Environments214

3.5 Robust Virtual Implementation216

3.6 A Coordination Example216

4 Rationalizable and Robust Implementation in Infinite Mechanisms218

4.1 Best Response218

4.2 Material Implementation221

5 Infinite Mechanisms223

6 Extensions， Variations and Discussion230

6.1 Lotteries， Pure Strategies and Bayesian Implementation230

6.2 Ex Post and Robust Implementation232

6.3 Extensions234

7 Appendix234

7.1 Robust Monotonicity and Dual Robust Monotonicity234

7.2 A Badly Behaved Mechanism235

7.3 Coordination Example Continued236

References238

Chapter 5 The Role of the Common Prior in Robust Implementation241

1 Introduction241

2 Setup243

3 A Public Good Example245

4 Discussion249

References250

Chapter 6 An Ascending Auction for Interdependent Values： Uniqueness and Robustness to Strategic Uncertainty253

1 Model255

2 Static Auction256

3 Dynamic Auction258

4 Discussion260

5 Conclusion261

References262

Chapter 7 Robust Virtual Implementation263

1 Introduction264

2 Setting270

2.1 Environment270

2.2 Mechanisms and Solution Concept271

2.3 Separability272

3 An Environment with Interdependent Values for a Single Good274

4 Strategic Distinguishability277

4.1 Main Result277

4.2 The Maximally Revealing Mechanism279

4.2.1 A class of maximally revealing mechanisms279

4.2.2 Characterizing rationalizable behavior for small ε280

4.3 Constructing a Rich Enough Test Set285

5 Robust Virtual Implementation286

5.1 Definitions286

5.2 Necessity287

5.3 Sufficiency289

6 Discussion293

6.1 Abreu-Matsushima Measurability293

6.2 Interdependence and Pairwise Separability295

6.3 Intermediate Robustness Notions296

6.4 Rationalizability and All Equilibria on All Type Spaces298

6.5 Iterated Deletion of Weakly Dominated Strategies299

6.6 Implementation in a Direct Mechanism301

6.7 Exact Implementation and Integer Games302

7 Appendix303

References315

Chapter 8 Multidimensional Private Value Auctions319

1 Introduction319

2 The Model324

3 Seller’s Expected Revenue326

3.1 Second-Price Auction326

3.2 First-Price Auction327

3.3 Revenue Non-Equivalence328

4 Efficiency331

5 Equilibrium Existence333

6 Discussion： Revenue and Information Acquisition334

6.1 Revenue334

6.2 Information Acquisition336

7 Conclusion339

Appendix A…Proofs340

Appendix B353

References354

Chapter 9 The Robustness of Robust Implementation357

1 Introduction357

1.1 Literature359

2 Setup360

3 Baseline payoff environments362

3.1 Approximate Common Knowledge362

3.2 Payoff Environment Solution Concept363

3.3 One-Dimensional， Contractive， Supermodular PayoffType Environments364

4 Main Result366

5 Discussion368

Appendix A369

References372

Chapter 10 Rationalizable Implementation375

1 Introduction375

2 Setup377

3 Main Result380

4 The Non-Responsive Case391

5 Concluding Remarks397

References403

Chapter 11 Pricing without Priors405

1 Introduction405

2 Model408

3 Pricing Without Priors410

4 Discussion413

References415

Chapter 12 Robust Monopoly Pricing417

1 Introduction417

2 Model421

3 Maximin Utility425

4 Minimax Regret428

5 Discussion434

Appendix A436

References440

Author Index443

Subject Index447