Mechanism Design and Approximation Jason D. Hartline Manuscript date: September 2, 2014 Copyright c 2011–2014 http://jasonhartline.com/MDnA/ Contents 1 Mechanism Design and Approximation page 1 1.1 AnExample:CongestionControlandRouting 2 1.1.1 Non-monetary payments 9 1.1.2 Posted Pricing 11 1.1.3 General Routing Mechanisms 14 1.2 Mechanism Design 15 1.3 Approximation 17 1.3.1 Philosophy of Approximation 18 1.3.2 Approximation Factors 20 2 Equilibrium 24 2.1 Complete Information Games 24 2.2 Incomplete Information Games 26 2.3 Bayes-Nash Equilibrium 27 2.4 Single-dimensional Games 29 2.5 Characterization of Bayes-Nash Equilibrium 31 2.6 Characterization of Dominant Strategy Equilibrium 36 2.7 Revenue Equivalence 37 2.8 Solving for Bayes-Nash Equilibrium 38 2.9 Uniqueness of Equilibria 42 2.10 The Revelation Principle 46 Exercises 48 3 Optimal Mechanisms 53 3.1 Single-dimensional Environments 54 3.2 Social Surplus 56 3.3 Profit 59 3.3.1 Highlevel Approach: Amortized Analysis 61 Copyright c 2011–2014 by Jason D. Hartline. Source: http://jasonhartline.com/MDnA/ Manuscript Date: September 2, 2014. iv Contents 3.3.2 The Virtual Surplus Maximization Mecha- nism 64 3.3.3 Single-item Environments 66 3.3.4 Quantile Space, Price-posting Revenue, and Derivation of Virtual Values 71 3.3.5 Virtual Surplus Maximization Subject to Monotonicity 75 3.4 Multi- to Single-agent Reduction 79 3.4.1 Revenue Curves 80 3.4.2 Optimal and Marginal Revenue 81 3.4.3 Downward Closure and Pricing 82 3.4.4 Revenue Linearity 83 3.4.5 Optimal Ex Ante Pricings, Revisited 85 3.4.6 Optimal Interim Pricings, Revisited 87 3.5 Social Surplus with a Balanced Budget 89 3.5.1 Lagrangian Relaxation 90 3.5.2 Monotone Lagrangian Virtual Values 92 3.5.3 Non-monotone Lagrangian Virtual Values and Partial Ironing 94 Exercises 96 4 Bayesian Approximation 100 4.1 Monopoly Reserve Pricing 102 4.1.1 Approximation for Regular Distributions 104 4.1.2 Inapproximability Irregular Distributions 107 4.2 Oblivious Posted Pricings and the Prophet Inequality 110 4.2.1 The Prophet Inequality 111 4.2.2 Oblivious Posted Pricing 114 4.3 Sequential Posted Pricings and Correlation Gap 116 4.3.1 The Ex Ante Relaxation 117 4.3.2 The Correlation Gap 119 4.3.3 Sequential Posted Pricings 121 4.4 AnonymousReservesandPricings 123 4.4.1 Identical Distributions 124 4.4.2 Non-identical Distributions 126 4.5 Multi-unit Environments 127 4.6 Ordinal Environments and Matroids 129 4.6.1 Matroid Set Systems 131 4.6.2 Monopoly Reserve Pricing 135 4.6.3 Oblivious and Adaptive Posted Pricings 137 Contents v 4.6.4 Sequential Posted Pricings 137 4.6.5 AnonymousReserves 141 4.6.6 Beyond Ordinal Environments 141 4.7 Monotone-hazard-rate Distributions 144 Exercises 148 5 Prior-independent Approximation 155 5.1 Motivation 156 5.2 “Resource”Augmentation 158 5.2.1 Single-item Environments 158 5.2.2 Multi-unit and Matroid Environments 159 5.3 Single-sample Mechanisms 160 5.3.1 The Geometric Interpretation 161 5.3.2 Monopoly versus Single-sample Reserves 163 5.3.3 Optimal versus Lazy Single-sample-reserve Mechanism 164 5.4 Prior-independent Mechanisms 165 5.4.1 Digital Good Environments 166 5.4.2 General Environments 167 Exercises 169 6 Prior-free Mechanisms 172 6.1 The Framework for Prior-free Mechanism Design 174 6.2 The Digital-good Environment 177 6.2.1 TheEnvy-freeBenchmark 178 6.2.2 Deterministic Auctions 181 6.2.3 The Random Sampling Auction 183 6.2.4 Decision Problems for Mechanism Design 188 6.3 TheEnvy-freeBenchmark 192 6.3.1 Envy-free Pricing 193 6.3.2 Envy-freeOptimalRevenue 195 6.3.3 Envy freedom versus Incentive Compatibility 197 6.3.4 Permutation Environments 200 6.3.5 TheEnvy-freeBenchmark 201 6.4 Multi-unit Environments 203 6.4.1 Reduction to Digital Goods 204 6.4.2 Combination of Benchmarks and Auctions 207 6.4.3 The Random Sampling Auction 208 6.5 Matroid Permutation and Position Environments 209 6.6 Downward-closed Permutation Environments 214 Exercises 218 vi Contents Appendix Mathematical Reference 221 References 225 Index 230 Contents vii 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 de- sign 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 distri- bution 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 prac- tical mechanisms are good approximations of the optimal ones. The re- sulting 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 1 Mechanism Design and Approximation Our world is an interconnected collection of economic and computational systems. Within such a system, individuals optimize their actions to achieve their own, perhaps selfish, goals; and the system combines these actions with its basic laws to produce an outcome. Some of these systems perform well, e.g., the national residency matching program which as- signs medical students to residency programs in hospitals, e.g., auctions for online advertising on Internet search engines; and some of these sys- tems 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 reg- ulation can prevent disastrous market meltdowns, congestion protocols can prevent gridlock in transportation networks, and market and auc- tion design can lead to mechanisms for allocating and exchanging goods or services that yield higher profits or increased value to society. The two sources for economic considerations are the preferences of 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 govern- ing rules. For instance, bidders in the auction must figure 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 become 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 Copyright c 2011–2014 by Jason D. Hartline. Source: http://jasonhartline.com/MDnA/ Manuscript Date: September 2, 2014. 2 Mechanism Design and Approximation of individual strategies 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 conse- quence of individuals and designers preference for ease of understanding and optimization (i.e., tractability) or robustness to variations in the scenario, especially when these desiderata do not significantly sacrifice performance. This text focuses on a combined computational and economic the- ory for the study and design of mechanisms. A central theme will be the tradeoff between optimality and other desirable properties such as simplicity, robustness, computational tractability, and practicality. This tradeoff will be quantified by a theory of approximation which measures the loss of performance of a simple, robust, and practical approxima- tion mechanism in comparison to the complicated and delicate optimal mechanism. The theory provided does not necessarily suggest mecha- nisms 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 con- gestion 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 text. 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 An Example: Congestion Control and Routing in Computer 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, approximation, and computation in mech- anism design. In this discussion, we will introduce basic questions that will be answered in the subsequent chapters of this text. 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 1.1 An Example: Congestion Control and Routing 3 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 between all pairs of users.