PRIOR-INDEPENDENCE: A NEW LENS FOR MECHANISM DESIGN A DISSERTATION SUBMITTED TO THE DEPARTMENT OF COMPUTER SCIENCE AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Qiqi Yan June 2012 © 2012 by Qiqi Yan. All Rights Reserved. Re-distributed by Stanford University under license with the author. This dissertation is online at: http://purl.stanford.edu/cg968ch3729 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Timothy Roughgarden, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ashish Goel I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Serge Plotkin Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract We propose to study revenue-maximizing auctions in the prior-independent analysis framework. The goal is to identify a single auction mechanism for all underlying valuation distributions, so that its expected revenue approximates that of the optimal mechanism tailored for the underlying distribution, under standard weak conditions on the distribution. We use the prior-independent analysis framework to analyze natural and practical auction mechanisms such as welfare-maximization with reserve prices, limiting supply to induce artificial scarcity, sequentially posting prices, etc. Our framework allows us to argue that these simple mechanisms give near-optimal revenue guarantee in a very robust manner. iv Acknowledgments Pursuing a PhD degree is an amazing intellectual journey. This journey would not be successful or as fun without the help and support from the various people along the way. First I’d like to thank my PhD advisor Tim Roughgarden. Tim taught me a lot on almost every important aspect of research. Tim taught me to judge whether a research problem is important based on examining its interpretations carefully, helped me form the belief that there is always a simple explanation for apparently complex stuff, which is a powerful tool in problem solving, and showed me what is the right way of presenting things either in writing papers or giving talks. I benefited tremendously from learning these things from Tim. I also thank Tim for always being encouraging to students, and his stress-free advising style. I also want to thank Jason Hartline, Shaddin Dughmi, and Mukund Sundarara- jan not only for being brilliant collaborators, but also for having deeply shaped my way of thinking. In particular, I thank Jason for teaching me a whole lot about his intuition and methodology on mechanism design, plus his magic in connecting in- equalities, Shaddin for his ridiculous amount of optimism on research and for sharing his knowledge on submodularity and algorithmic mechanism design, and Mukund for giving various advice, from research, to pingpong, to life. I thank Inbal Talgam-Cohen and Peerapong Dhangwatnotai for being great col- laborators and in particular for contributing many ideas to this thesis. I also thank Mohammad Mahdian, Nitish Korula, Vahab Mirrokni for the wonderful summer in- ternship experiences, which were eye-openers to me. I thank my defense committee Yinyu (chair), Tim, Serge, Ashish, Ryan for giving me various feedback on the thesis. v I thank Stanford for providing such a great academic environment, and for sur- rounding me with an amazing group of smart people: Ananth, Aleksandra, Aneesh, Bahman, Damon, David A., Eric L., Florian, Hart, Ian, Kshipra, Martin, Michael K., Nikola, Pranav, Shayan, Shipra, Uri, Ying W., Ying X., among others. In par- ticular I thank Kshipra Bhawalker for being a great officemate, who constantly tries to improve the research/living environment around her, helping making our office, the theory wing, and even the department a better place. I also thank my first-year buddies: Aditya, Ankur, Eric K., Quoc, Steven, former roommates Marc, Jose, Wen, and Janet, pingpong pals Calvin, Chenguang, Ran, Xin, and various other friends. I also want to thank Don Knuth for the signed free books, thank Jan Vondrak for being a valuable oracle on submodularity. My interest in theory was ignited by a class on set theory and logic given by Prof. Enshao Shen at Shanghai Jiao Tong University, who later became my master thesis advisor. I thank Prof. Shen for guiding me into theory, and for being extremely kind while being stern on mathematical rigor. Finally, last but not least, I thank my parents and my wife Shanshan for the unconditional support throughout. vi Contents Abstract iv Acknowledgments v I Background and Overview 1 1 Introduction 2 1.1 Roadmap . 2 1.2 Background: Auction Mechanism Design . 3 1.3 An Algorithmic View . 5 1.4 Average-Case Analysis . 6 1.4.1 Problems with the Known Distribution Assumption . 7 1.5 Worst-Case Analysis . 8 1.5.1 Hartline-Roughgarden’s Revenue Benchmark . 9 1.6 Our Prior-Independent Analysis Framework . 9 1.6.1 Optimality vs. Robustness . 10 1.6.2 Formalizing the Optimality vs. Robustness Trade-Off . 10 1.6.3 Independent Regular Distributions . 12 1.7 Prior-Independent Mechanisms . 13 1.7.1 Mechanism with a Single Dial . 13 1.7.2 VCG with Sampled Reserve . 14 1.7.3 Supply-Limiting Mechanisms . 14 1.7.4 Sequential Posted-Price Mechanisms . 14 vii 1.7.5 Reduction to Bulow-Klemperer-Style Statements . 15 1.8 Prior-Free Mechanisms and Envy-Freeness . 15 1.9 Differences From Original Papers . 16 1.10 Related Work . 17 2 Preliminaries 20 2.1 Single-Dimensional Environments . 20 2.1.1 Matroids . 22 2.2 Mechanisms . 23 2.2.1 Objectives . 24 2.2.2 Example: VCG and Variants . 25 2.2.3 Example: Virtual Surplus Maximizer . 25 2.2.4 Example: Sequential Posted-Price Mechanisms . 26 2.3 Distributions . 26 2.3.1 Regular Distributions . 28 2.3.2 Monotone Hazard Rate Distributions . 29 2.4 Myerson’s Optimal Auction Theory . 31 2.5 Bulow-Klemperer Theorem and Its Generalizations . 32 2.6 Properties of the Optimal Revenue Function . 33 II Prior-Independent Mechanisms 36 3 Prior-Independence: Definition, Example, and Reduction 37 3.1 Prior-Independent Approximation . 37 3.2 Distribution Classes . 38 3.3 Case Study: Digital Goods with Two Bidders . 38 3.3.1 Characterizing All Truthful Mechanisms . 39 3.3.2 Revenue of the Vickrey Auction . 39 3.3.3 Interpretation I: A Single Sample is Near-Optimal . 41 3.3.4 Interpretation II: Supply-Limiting is Good for Revenue . 41 3.3.5 Interpretation III: Market Expansion is Good for Revenue . 42 viii 3.3.6 Vickrey is the Best Prior-Independent Mechanism . 42 4 VCG with Sampled Reserves 44 4.1 Introduction . 44 4.1.1 Settings . 44 4.1.2 Approach . 45 4.1.3 Main Results . 46 4.1.4 Other Results . 46 4.2 Revenue Guarantees with a Single Sample . 47 4.2.1 The Single Sample Mechanism . 47 4.2.2 Warm-Up: I.I.D. Matroid Environments . 48 4.2.3 Proof Framework . 52 4.2.4 Regular Matroid Environments . 53 4.2.5 M.H.R. Downward-Closed Environments . 55 4.2.6 Counterexample for Regular Downward-Closed Environments 60 4.2.7 Computationally Efficient Variants . 61 4.3 Revenue Guarantees with Multiple Samples . 61 4.3.1 Estimating Monopoly Reserve Prices . 62 4.3.2 The Many Samples Mechanism . 65 4.4 Prior-Independent Mechanisms Via Reductions . 68 5 Supply-Limiting Mechanisms 71 5.1 Introduction . 71 5.1.1 A Matching Problem . 71 5.1.2 A Supply-Limiting Mechanism . 72 5.1.3 Technical Approach: Reduction to Proving Bulow-Klemperer- Style Theorems . 73 5.1.4 Our Results and Organization . 73 5.1.5 Related Work . 74 5.2 Preliminaries . 75 5.3 Reduction to Bulow-Klemperer-Style Statements . 77 5.3.1 General Reduction to Bulow-Klemperer-Style Statements . 77 ix 5.3.2 Examples of Reductions . 78 5.3.3 Fractional Subadditivity of Optimal Revenue . 79 5.4 Supply-Limiting for I.I.D. k-Unit Auctions . 79 5.4.1 A Supply-Limiting Mechanism for I.I.D. Matroid Environments 81 5.5 Supply-Limiting Mechanism for Parallel Multi-Unit Auctions . 83 5.5.1 Parallel Multi-Unit Auctions . 83 5.5.2 A Bulow-Klemperer-Style Theorem . 84 5.5.3 Proof of the Bulow-Klemperer-Style Theorem for Parallel Multi-Unit Auctions . 85 5.6 I.I.D. Matching Environments: Overview of Results . 88 5.6.1 Supply-Limiting Mechanisms . 88 5.6.2 Reduction to Bulow-Klemperer-Style Theorems . 89 5.7 BK-Style Theorem for Matching with m More Bidders . 92 5.7.1 Basic Upper and Lower Bounds . 92 5.7.2 Relating the Upper and Lower Bounds via Deferred Allocation 94 5.8 Weak BK-Style Theorem for Matching with n More Bidders . 97 5.9 BK-Style Theorem for Matching with O(n) More Bidders . 98 5.9.1 Notations and Definitions . 98 5.9.2 Relating Upper- and Lower- Bounds . 100 5.9.3 Putting It All Together . 103 5.10 Extension to Multi-Unit Matching Environments . 105 5.10.1 Extending the Weak Bound . 106 5.10.2 Upper and Lower Bounds .