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 text provides a look at select topics in economic mechanism design through the lens of approximation. It reviews the classical economic theory of mechanism design wherein a Bayesian designer looks to find the mechanism with optimal performance in expectation over the distribution from which the preferences of the participants are drawn. It then adds to this theory practical constraints such as simplicity, tractability, and robustness. The central question addressed is whether these practical mechanisms are good approximations of the optimal ones. The resulting theory of approximation in mechanism design is based on results that come mostly from the theoretical computer science literature. The results presented are the ones that are most directly compatible with the classical (Bayesian) economic theory and are not representative of the entirety of the literature. – Jason D. Hartline 6 Chapter 1 Approximation and Mechanism Design Our world is an interconnected collection of economic and computational systems wherein individuals optimize to achieve their own, perhaps selfish, goals subject to basic laws of the system. Some of these systems perform well, e.g., the national residency matching program which assigns medical students to residency programs in hospitals, e.g., auctions for online advertising on Internet search engines; and some of these systems perform poorly, e.g., financial markets during the 2008 meltdown, e.g., gridlocked transportation networks. The success and failure of these systems depends on the basic laws governing the system. Financial regulation can prevent disastrous market meltdowns, congestion protocols can prevent gridlock in transportation networks, and market and auction design can lead to mechanisms for exchanging goods or services that are good in terms of revenue or social benefit. The two sources for economic considerations are the preferences for individuals and the performance of the system. For instance, bidders in an auction would like to maximize their gains from buying; whereas, the performance of the system could (i.e., from the perspective of the seller) be measured in terms of the revenue it generates. Likewise, the two sources for computational considerations are the individuals who must optimize their strategies, and the system which must enforce its governing rules. For instance, bidders in the auction must fig- ure out how to bid, and the auctioneer must calculate the winner and payments from the bids received. While these calculations may seem easy when auctioning a painting, they both be- come quite challenging when, e.g., the Federal Communications Commission (FCC) auctions cell phone spectrum for which individual lots have a high-degree of complementarities. These economic and computational systems are complex. The space of individual strate- gies is complex and the space of possible rules for the system is complex. Optimizing among strategies or system rules in complex environments should lead to complex strategies and system rules, yet the individuals’ strategies or system rules that are successful in practice are often remarkably simple. This simplicity may be a result of computational tractability or due to desired robustness, especially when these desiderata do not significantly sacrifice performance. 7 This text focuses on a combined computational and economic theory for the study and design of mechanisms. A central theme in will be a tradeoff between optimality and other desirable properties such as simplicity, robustness, computational tractability, and practi- cality. This tradeoff will be quantified by a theory for approximation which measures the loss of a simple, robust, and practical approximation in comparison to the complicated and delicate optimal mechanism. The theory provided does not necessarily suggest mechanisms that should be deployed in practice, instead, it pinpoints salient features of good mechanisms that should be a starting point for the practitioner. In this chapter we will explore mechanism design for routing and congestion control in computer networks as an example. Our study of this example will motivate a number of questions that will be addressed in subsequent chapters of the test. We will conclude the chapter with a formal discussion of approximation and the philosophy that underpins its relevance to the theory of mechanism design. 1.1 Example: Congestion Control and Routing in Com- puter Networks We will discuss novel mechanisms for congestion control and routing in computer networks to give a preliminary illustration of the interplay between strategic incentives and approxi- mation in mechanism design. Consider a hypothetical computer network where network users reside at computers and these computers are connected together through a network of routers. Any pair of routers in this network may be connected by a network link and if such a network link exists then each router can route a message directly through the other router. We will assume that the network is completely connected, i.e., there is a path of network links