Approximation in Economic Design

Jason D. Hartline1

Draft: April 15, 2012

1Northwestern University, Evanston, IL 60208. Email: [email protected]. 2 Contents

1 Approximation and Mechanism Design 7 1.1 Example: Congestion Control and Routing in Computer Networks...... 8 1.1.1 Non-monetarypayments ...... 14 1.1.2 PostedPricing...... 15 1.1.3 GeneralRoutingMechanisms ...... 16 1.2 MechanismDesign ...... 18 1.3 Approximation ...... 19 1.3.1 Philosophy of Approximation ...... 19 1.3.2 ApproximationFactors ...... 22

2 Equilibrium 25 2.1 CompleteInformationGames ...... 25 2.2 IncompleteInformationGames ...... 27 2.3 Bayes-Nash Equilibrium ...... 27 2.4 Single-dimensionalGames ...... 29 2.5 Characterization of Bayes-Nash equilibrium ...... 30 2.6 Characterization of Dominant Strategy Equilibrium ...... 34 2.7 RevenueEquivalence ...... 35 2.8 Solving for Bayes-Nash Equilibrium ...... 36 2.9 TheRevelationPrinciple ...... 39

3 Optimal Mechanisms 43 3.1 Single-dimensional Environments ...... 44 3.2 SocialSurplus...... 45 3.3 Profit ...... 48 3.3.1 QuantileSpace ...... 48 3.3.2 RevenueCurves...... 49 3.3.3 Expected Revenue and Virtual Values ...... 50 3.3.4 Optimal Mechanisms and Regular Distributions ...... 51 3.3.5 Single-itemAuctions ...... 53 3.4 IrregularDistributions ...... 54 3.4.1 IronedRevenueCurves...... 54 3.4.2 OptimalMechanisms ...... 57

3 3.4.3 Single-itemAuctions ...... 58

4 Bayesian Approximation 63 4.1 Single-itemAuctions ...... 64 4.1.1 RegularDistributions...... 64 4.1.2 IrregularDistributions ...... 66 4.1.3 AnonymousReserves ...... 72 4.2 General Feasibility Settings ...... 73 4.2.1 Monotone-hazard-rate Distributions (and Downward-closed Feasibility) 73 4.2.2 Matroid Feasibility (and Regular Distributions) ...... 76

5 Prior-independent Approximation 83 5.1 Motivation...... 83 5.2 “Resource”Augmentation ...... 85 5.2.1 Single-itemAuctions ...... 85 5.2.2 MatroidEnvironments ...... 86 5.3 Single-sample Mechanisms ...... 87 5.3.1 TheGeometricInterpretation ...... 87 5.3.2 RandomversusMonopolyReserves ...... 88 5.3.3 Single-sample versus Optimal ...... 89 5.4 Prior-independent Mechanisms ...... 91 5.4.1 DigitalGoodEnvironments ...... 91 5.4.2 GeneralEnvironments ...... 92

6 Prior-free Mechanisms 95 6.1 TheDigitalGoodEnvironment ...... 96 6.1.1 DeterministicAuctions ...... 97 6.1.2 RandomSampling ...... 98 6.1.3 DecisionProblems ...... 101 6.1.4 Lowerbounds ...... 104 6.2 TheEnvy-freeBenchmark ...... 106 6.3 Multi-unitEnvironments ...... 110 6.4 Matroid Permutation and Position Environments ...... 113 6.5 Downward-closed Permutation Environments ...... 117

7 Multi-dimensional Approximation 123 7.1 ItemPricing...... 124 7.2 Reduction: Unit-demand to Single-dimensional Preferences ...... 125 7.2.1 Single-dimensional Analogy ...... 126 7.2.2 Upperbound ...... 126 7.2.3 Reduction ...... 127 7.2.4 Instantiation ...... 128 7.3 Lottery Pricing and Randomized Mechanisms ...... 130

4 7.4 Beyond Independent Unit-demand Environments ...... 133 7.5 Optimal Lottery-pricing via Linear Programming ...... 133

8 Computational Tractability 137 8.1 Tractability ...... 137 8.2 Single-minded Combinatorial Auctions ...... 139 8.2.1 ApproximationAlgorithms...... 139 8.2.2 Approximation Mechanisms ...... 142 8.3 Bayesian Algorithm and Mechanism Design ...... 144 8.3.1 Monotonization ...... 145 8.3.2 BlackboxComputation...... 149 8.3.3 PaymentComputation ...... 149 8.4 Computational Overhead of Payments ...... 151 8.4.1 Communication Complexity Lower Bound ...... 152 8.4.2 ImplicitPayments ...... 154

A Mathematical Reference 161 A.1 Big-ohNotation...... 161 A.2 Common Probability Distributions ...... 161 A.3 ExpectationandOrderStatistics ...... 162 A.4 IntegrationbyParts ...... 162 A.5 HazardRates ...... 163

5 Author’s Note

This text is suitable for advanced undergraduate or graduate courses; it has been developed at Northwestern U. as the primary text for such a course since 2008. This