Designing economic mechanisms

A mechanism is a mathematical structure that models institutions through which economic activity is guided and coordinated. There are many such institutions; markets are the most familiar ones. Lawmakers, administrators and officers of private companies create institutions in order to achieve desired goals. They seek to do so in ways that economize on the resources needed to operate the institutions, and that provide incentives that induce the required behaviors. This book presents systematic procedures for designing mechanisms that achieve specified performance, and economize on the resources required to operate the mechanism. The systematic design procedures are algorithms for designing informationally efficient mechanisms. Most of the book deals with these procedures of design. When there are finitely many environments to be dealt with, and there is a Nash-implementing mechanism, our algorithms can be used to make that mechanism into an informationally efficient one. Informationally efficient dominant strategy implementation is also studied.

Acknowledgements	p. ix
Introduction	p. 1
1 Mechanisms and Mechanism Design	p. 14
1.0 Introduction	p. 14
1.1 Mechanisms and Design	p. 18
1.2 Environments and Goal Functions	p. 25
1.3 Mechanisms: Message Exchange Processes and Game Forms	p. 26
1.4 Initial Dispersion of Information and Privacy Preservation	p. 29
1.5 Mechanism Design	p. 30
1.6 Mechanism Design Illustrated in a Walrasian Example	p. 31
1.6.1 An Edgeworth Box Economy	p. 31
1.6.2 The Walrasian Goal Function	p. 32
1.6.3 Mechanisms: The Competitive Mechanism	p. 35
1.6.4 Competitive Equilibrium Conditions	p. 35
1.6.5 The Competitive Mechanism Is a Mechanism	p. 36
1.6.6 The Competitive Mechanism Illustrates Some Concepts Used in Mechanism Design	p. 37
1.6.7 Privacy Preservation in the Competitive Mechanism	p. 38
1.6.8 Deriving a Mechanism (Not the Competitive Mechanism) from a Covering for the Walrasian Goal Function	p. 40
1.6.9 Informational Properties of the Two Mechanisms	p. 42
1.6.10 The Rectangles Method Applied to the Walrasian Goal Function - Informal	p. 44
1.7 Introductory Discussion of Informational Efficiency Concepts	p. 46
1.8 A National Forest	p. 50
2 From Goals to Means: Constructing Mechanisms	p. 63
2.1 Phase One: Mechanism Construction	p. 74
2.1.1 Two Examples	p. 74
2.1.2 Constructing a "Universal" Method of Designing Informationally Efficient Mechanisms Realizing a Given Goal Function	p. 83
2.1.3 The Method of Rectangles (RM)	p. 86
2.2 Phase 2: Constructing Decentralized Mechanisms, from Parameter Indexed Product Structures: Transition to Message-Indexed Product Structures	p. 101
2.2.0 Introduction	p. 101
2.2.1 Basic Concepts	p. 102
2.2.2 The L-dot Example	p. 104
2.2.3 More Examples	p. 105
2.2.4 General Issues in Mechanism Construction	p. 109
2.2.5 Mechanism Construction for L-dot	p. 114
2.3 Smooth Transversal Construction for Partitions by the "Flagpole" Method	p. 117
2.3.1 Flagpoles: General Principles	p. 117
2.3.2 Flagpoles: Example 2 (Augmented Inner Product)	p. 120
2.3.3 Flagpoles: A Walrasian Example	p. 125
2.3.4 Unique Solvability Implies Partition	p. 129
2.4 Analytic Aspects	p. 130
2.4.1 Phase Two via Condensation. General Principles	p. 131
2.4.2 The Mount-Reiter Condensation Theorem (Sufficiency)	p. 136
2.4.3 Walrasian Mechanism Construction	p. 140
2.4.4 Phase Two of Mechanism Design via Condensation for the Augmented Two-Dimensional Inner Product	p. 149
2.5 Overlaps	p. 154
2.5.0 Constructing a Mechanism When the Parameter-Indexed Product Structure Is Not a Partition: An Example	p. 154
Appendix	p. 163
2.6 Informational Efficiency	p. 165
2.6.1 Main Results	p. 165
2.6.2 The Maximality of Reflexive RM-Coverings	p. 166
2.6.3 Informational Efficiency: General Considerations	p. 168
2.6.4 A Comment on Informational Efficiency Concepts	p. 171
2.6.5 Minimal Informational Size Is Achievable by an rRM Mechanism	p. 172
2.6.6 Two rRM Coverings of Different Informational Size for the Same Goal Function: An Example	p. 175
Appendix	p. 180
3 Designing Informationally Efficient Mechanisms Using the Language of Sets	p. 182
3.1 Introduction	p. 182
3.2 Mechanism Design	p. 183
3.2.1 Decentralization	p. 184
3.3 Mechanisms and Coverings	p. 186
3.4 A Systematic Process for Constructing an rRM Covering	p. 188
3.4.1 OrRM: An Algorithm for Constructing an rRM Covering of a Finite Parameter Space That Is Minimal in the Class of Rectangular, F-Contour Contained Coverings	p. 197
3.5 Constructing a Mechanism from a Covering by the Transversals Method (TM)	p. 220
3.6 Coverings and Partitions	p. 230
3.7 Informational Efficiency	p. 244
3.7.1 Introduction	p. 244
3.7.2 Observational Efficiency	p. 245
3.7.3 The Maximality of rRM-Coverings	p. 246
3.7.4 Informational Size and Coarseness	p. 250
3.8 Section 1.8 Revisited: A Graphical Presentation	p. 263
3.9 Strategic Behavior	p. 274
3.9.1 Dominant Strategy Implementation	p. 274
3.9.2 Designing Informationally Efficient Nash-Implementing Mechanisms	p. 279
Appendix Characterizations of Partitions	p. 290
4 Revelation Mechanisms	p. 296
4.1 Introduction	p. 296
4.1.1 Computational Complexity of Functions	p. 299
4.1.2 Separator Sets and Quotients	p. 303
4.1.3 Algebraic Conditions	p. 306
4.1.4 Privacy-Preserving Mechanisms	p. 307
4.2 Initial Set-Theoretic Constructions	p. 310
4.2.1 Encoded and Essential Revelation Mechanisms	p. 310
4.2.2 F-Equivalence and Encoded Revelation Mechanisms	p. 310
4.3 The Topological Case	p. 313
4.3.1 Differential Separability	p. 315
4.3.2 The Number of Variables on which F Really Depends	p. 316
4.3.3 Rank Conditions and Construction of an Essential Revelation Mechanism for F	p. 317
4.4 Proofs and Examples	p. 322
4.4.1 Leontief and Abelson Theorem	p. 322
4.4.2 Leontief's Theorem	p. 324
4.4.3 An Example of the Coordinate Construction	p. 329
4.4.4 Proof of Theorem 4.4.6	p. 331
References	p. 335
Index	p. 341

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