Mechanism Design: A Linear Programming

Mechanism design is an analytical framework for thinking clearly and carefully about what exactly a given institution can achieve when the information necessary to make decisions is dispersed and privately held. This analysis provides an account of the underlying mathematics of mechanism design based on linear programming. Three advantages characterize the approach. The first is simplicity: arguments based on linear programming are both elementary and transparent. The second is unity: the machinery of linear programming provides a way to unify results from disparate areas of mechanism design. The third is reach: the technique offers the ability to solve problems that appear to be beyond solutions offered by traditional methods. No claim is made that the approach advocated should supplant traditional mathematical machinery. Rather, the approach represents an addition to the tools of the economic theorist who proposes to understand economic phenomena through the lens of mechanism design.

Rakesh V. Vohra is the John L. and Helen Kellogg Professor of Managerial Economics and Decision Sciences at the Kellogg School of Management, Northwestern University.

Introduction	p. 1
Outline	p. 5
Arrow's Theorem and Its Consequences	p. 7
The Integer Program	p. 8
General Domains	p. 13
Social Choice Functions	p. 14
Strategic Candidacy	p. 17
Mechanisms and Revelation	p. 20
Network Flow Problem	p. 24
Graphs	p. 24
Network Flow Problem	p. 26
Flow Decomposition	p. 30
The Shortest-Path Polyhedron	p. 33
Interpreting the Dual	p. 35
Infinite Networks	p. 36
Incentive Compatibility	p. 38
Notation	p. 38
Dominant Strategy Incentive Compatibility	p. 40
2-Cycle Condition	p. 44
Convex Type Spaces	p. 45
Convex Valuations	p. 51
Roberts's Theorem	p. 54
Revenue Equivalence	p. 59
A Demand-Rationing Example	p. 65
The Classical Approach	p. 69
Interdependent Values	p. 73
Bayesian Incentive Compatibility	p. 77
Efficiency	p. 79
Vickrey-Clarke-Groves Mechanism	p. 79
Combinatorial Auctions	p. 82
The Core	p. 87
Ascending Auctions	p. 89
Primal-Dual Algorithm	p. 89
Incentives	p. 98
Subgradient Algorithm	p. 99
Gross Substitutes	p. 105
An Impossibility	p. 107
A Recipe	p. 109
Revenue Maximization	p. 110
What Is a Solution?	p. 112
One-Dimensional Types	p. 114
A Formulation	p. 117
Optimal Mechanism for Sale of a Single Object	p. 118
Polyhedral Approach	p. 121
Ironing and Extreme Points	p. 126
From Expected Allocations to the Allocation Rule	p. 129
Correlated Types	p. 130
The Classical Approach	p. 133
Budget Constraints	p. 135
The Continuous Type Case	p. 139
Asymmetric Types	p. 140
Bargaining	p. 140
Multidimensional Types	p. 141
Wilson's Example	p. 143
Capacity-Constrained Bidders	p. 150
Rationalizability	p. 160
The Quasilinear Case	p. 160
The General Case	p. 161
References	p. 165
Index	p. 171
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Mechanism Design: A Linear Programming Approach

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