Robust Mechanism Design: The Role of Private Information and Higher Order Beliefs (World Scientific Economic Theory)

Foreword by Eric Maskin (Nobel Laureate in Economics, 2007)This volume brings together the collected contributions on the theme of robust mechanism design and robust implementation that Dirk Bergemann and Stephen Morris have been working on for the past decade. The collection is preceded by a comprehensive introductory essay, specifically written for this volume with the aim of providing the readers with an overview of the research agenda pursued in the collected papers.The introduction selectively presents the main results of the papers, and attempts to illustrate many of them in terms of a common and canonical example, namely a single unit auction with interdependent values. It is our hope that the use of this example facilitates the presentation of the results and that it brings the main insights within the context of an important economic mechanism, namely the generalized second price auction. Contents Foreword Acknowledgments Robust Mechanism Design: An Introduction 1 Introduction 2 Leading Example: Allocating a Private Good with Interdependent Values 3 Type Spaces 4 Robust Foundations for Dominant and Ex Post Incentive Compatibility 5 Full Implementation 5.1 Ex Post Implementation 5.2 Robust Implementation in the Direct Mechanism 5.3 The Robustness of Robust Implementation 5.4 Robust Implementation in the General Mechanism 5.5 Rationalizable Implementation 5.6 The Role of the Common Prior 5.7 Dynamic Mechanisms 5.8 Virtual Implementation 6 Open Issues References Chapter 1 Robust Mechanism Design 1 Introduction 2 Setup 2.1 Payoff Environment 2.2 Type Spaces 2.3 Solution Concepts 2.4 Questions 2.5 Implicit versus Explicit Modelling of Higher Order Uncertainty and the Universal Type Space 3 Examples 3.1 F is Interim Implementable on All Type Spaces But not Ex Post Implementable 3.2 F is Interim Implementable on All Payo. Type Spaces But not Interim Implementable on All Type Spaces 4 Separable Environments 4.1 Separable Environments 4.2 Full Support Conditions 5 The Quasilinear Environment with Budget Balance 6 Discussion 6.1 A Classical Debate 6.2 Genericity 6.3 Augmented Ex Post Equivalence References Chapter 2 Ex Post Implementation 1 Introduction 2 Model 3 Monotonicity 3.1 Ex Post Monotonicity 3.2 Maskin Monotonicity 3.3 Public Good Example 4 Ex Post Implementation 4.1 Necessary Conditions 4.2 Sufficient Conditions in Economic Environments 4.3 Sufficiency Conditions in Non-economic Environments 5 Single Crossing Environment 6 Direct Mechanisms 7 Single Unit Auction 7.1 Model 7.2 Monotonicity and the VCG Mechanism 7.3 Private Versus Interdependent Values 8 Social Choice Sets 8.1 Pareto Correspondence 8.2 Functions, Sets and Correspondences 9 Mixed Strategy Implementation 10 Conclusion Acknowledgments Appendix A A.1. Private values A.2. Ex post monotonicity no veto for sets References Chapter 3 Robust Implementation in Direct Mechanisms 1 Introduction 2 Setup 3 A Public Good Example 4 Robust Implementation 5 Necessity of Contraction Property 6 The Linear Model 7 Single Unit Auction 8 Discussion 8.1 Dimensionality and Aggregation 8.2 Relation to Partial and Ex Post Implementation 8.3 Robust and Virtual Implementation in General Environments 8.4 Social Choice Correspondences and Sets 8.5 The Common Prior Assumption and Strategic Substitutes/Complements 8.6 Informational Foundation of Interdependence 9 Appendix References Chapter 4 Robust Implementation in General Mechanisms 1 Introduction 2 Setup 2.1 The Payoff Environment 2.2 Type Spaces 2.3 Mechanisms 2.4 Solution Concepts 2.5 Implementation 3 Finite Mechanisms 3.1 Ex Post Incentive Compatibility 3.2 Robust Monotonicity 3.3 Robust Measurability 3.4 Single Crossing Aggregator Environments 3.5 Robust Virtual Implementation 3.6 A Coordination Example 4 Rationalizable and Robust Implementation in In.nite Mechanisms 4.1 Best Response 4.2 Material Implementation 5 Infinite Mechanisms 6 Extensions, Variations and Discussion 6.1 Lotteries, Pure Strategies and Bayesian Implementation 6.2 Ex Post and Robust Implementation 6.3 Extensions 7 Appendix 7.1 Robust Monotonicity and Dual Robust Monotonicity 7.2 A Badly Behaved Mechanism 7.3 Coordination Example Continued References Chapter 5 The Role of the Common Prior in Robust Implementation 1 Introduction 2 Setup 3 A Public Good Example 4 Discussion References Chapter 6 An Ascending Auction for Interdependent Values: Uniqueness and Robustness to Strategic Uncertainty 1 Model 2 Static Auction 3 Dynamic Auction 4 Discussion 5 Conclusion References Chapter 7 Robust Virtual Implementation 1 Introduction 2 Setting 2.1 Environment 2.2 Mechanisms and Solution Concept 2.3 Separability 3 An Environment with Interdependent Values for a Single Good 4 Strategic Distinguishability 4.1 Main Result 4.2 The Maximally Revealing Mechanism 4.2.1 A class of maximally revealing mechanisms 4.2.2 Characterizing rationalizable behavior for small ε 4.3 Constructing a Rich Enough Test Set 5 Robust Virtual Implementation 5.1 Definitions 5.2 Necessity 5.3 Sufficiency 6 Discussion 6.1 Abreu–Matsushima Measurability 6.2 Interdependence and Pairwise Separability 6.3 Intermediate Robustness Notions 6.4 Rationalizability and All Equilibria on All Type Spaces 6.5 Iterated Deletion of Weakly Dominated Strategies 6.6 Implementation in a Direct Mechanism 6.7 Exact Implementation and Integer Games 7 Appendix References Chapter 8 Multidimensional Private Value Auctions 1 Introduction 2 TheModel 3 Seller’s Expected Revenue 3.1 Second-Price Auction 3.2 First-Price Auction 3.3 Revenue Non-Equivalence 4 Efficiency 5 Equilibrium Existence 6 Discussion: Revenue and Information Acquisition 6.1 Revenue 6.2 Information Acquisition 7 Conclusion Appendix A. Proofs Appendix B References Chapter 9 The Robustness of Robust Implementation 1 Introduction 1.1 Literature 2 Setup 3 Baseline payoff environments 3.1 Approximate Common Knowledge 3.2 Payoff Environment Solution Concept 3.3 One-Dimensional, Contractive, Supermodular Payoff Type Environments 4 Main Result 5 Discussion Appendix A References Chapter 10 Rationalizable Implementation 1 Introduction 2 Setup 3 Main Result 4 The Non-Responsive Case 5 Concluding Remarks References Chapter 11 Pricing without Priors 1 Introduction 2 Model 3 Pricing Without Priors 4 Discussion References Chapter 12 Robust Monopoly Pricing 1 Introduction 2 Model 3 Maximin Utility 4 Minimax Regret 5 Discussion Appendix A References Author Index Subject Index This volume brings together the collected contributions on the theme of robust mechanism design and robust implementation that Dirk Bergemann and Stephen Morris have been working on for the past decade. The collection is preceded by a comprehensive introductory essay, specifically written for this volume with the aim of providing the readers with an overview of the research agenda pursued in the collected papers. The introduction selectively presents the main results of the papers, and attempts to illustrate many of them in terms of a common and canonical example, namely a single unit auction with interdependent values. It is our hope that the use of this example facilitates the presentation of the results and that it brings the main insights within the context of an important economic mechanism, namely the generalized second price auction