Matching Markets: Design and Analysis David John Abraham CMU-CS-09-167 September, 2009 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee: R. Ravi, Chair Alan Frieze David Manlove (University of Glasgow) Luis von Ahn Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Copyright c 2009 David John Abraham This research was sponsored by the National Science Foundation under grant numbers IIS-0121678, CCF- 0514922, CCR-0313148 and IIS-0427858. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. Keywords: Mechanism Design, Algorithms, Matching, Stable Marriage, Kidney Exchange Abstract A market consists of buyers and sellers of some commodity, say a DVD movie. In this thesis, we assume the role of market operator. Our goal is to ensure that the market has certain desirable properties, including truthfulness, fairness and stability. We explore how to achieve these properties by designing rules for how the participants (buyers and sellers) can interact. We also explore how to efficiently compute the outcome of large numbers of participants interacting at once. The main type of market we study is called a matching market. We study several particular matching markets, including keyword auction, kidney exchange and stable roommate mar- kets. In each of these cases, the aim is to match the participants to each other, somehow taking into account their preferences for one another. Our results focus on properties of the matching process and the design of efficient algorithms for finding various types of match- ings. In particular, we present new polynomial time algorithms for finding matchings that have one of the following properties: popularity, rank-maximality and fairness. We also give an efficient algorithm for clearing large swap markets such as kidney exchanges. Finally, we present a new decomposition technique for designing keyword auctions. iii iv Acknowledgements I would like to thank my advisor R. Ravi. Ravi has a well-deserved reputation as an out- standing advisor. He is patient, energetic, insightful and always able to give good advice. In short, I’ve learned a lot and had a great time working with him. I would like to give Ravi special thanks for guiding me through the most difficult period of my PhD, and also giving me the freedom to follow my interests. Thanks also to Luis von Ahn for giving me the opportunity to join his startup, reCAPTCHA. It has been an amazing ride so far, something that few people get to experience. I hope reCAPTCHA’s success continues and that one day, we achieve its goal of digitizing all the books, newspapers and important historical documents in the world. I would also like to thank David Manlove, my MSc advisor and main collaborator. David trained me in the field of matching markets and research in general. Without David’s influence, I would not have come across the two areas of research I enjoyed the most: popular matching and kidney exchange markets. Also, I would like to give him special thanks for his understanding and help in getting me into CMU, to be with my partner, Liz. Thanks also to my committee in general. I appreciate your guidance, and the work involved in providing it. Finally, thanks to my family for their love and support. v vi Table of Contents 1 Introduction 1 1.1 Organization ................................... 2 1.2 MatchingAgentswithItems . .. 3 1.2.1 PopularMatching............................. 3 1.2.2 KeywordAuctions ............................ 4 1.3 MatchingAgentswithotherAgents . .... 5 1.3.1 Finding a Maximum Exchange . 6 1.3.2 Stable Roommates with Globally-Ranked Pairs . ..... 6 1.3.3 EgalitarianMatchings . 7 2 Popular Matching 9 2.1 Introduction.................................... 10 2.1.1 PreviousWork .............................. 11 2.1.2 Preliminaries ............................... 12 vii 2.1.3 ChapterOutline.............................. 12 2.2 Strictly-ordered Preference Lists . ........ 12 2.2.1 Characterizing Popular Matchings . .... 13 2.2.2 AlgorithmicResults. 16 2.3 PreferenceListswithTies . ... 19 2.3.1 Characterizing Popular Matchings . .... 19 2.3.2 AlgorithmicResults. 25 2.4 EmpiricalResults................................ 27 2.5 RecentWork ................................... 29 3 Layerable Mechanisms for Keyword Auctions 33 3.1 Introduction.................................... 34 3.1.1 Preliminaries ............................... 34 3.1.2 PreviousWork .............................. 36 3.1.3 ChapterOutline.............................. 38 3.2 LayerableMechanisms . .. .. 38 3.2.1 Designing Layerable Mechanisms . .. 40 3.2.2 Example Layerable Mechanisms . 43 3.3 Selling Items to the Auctioneer . .... 45 3.3.1 Setting and Preliminary Discussion . .... 45 viii 3.3.2 Generalized Laddered Auction . .. 46 3.4 Conclusion..................................... 51 4 Clearing Algorithms for Barter Exchange Markets 53 4.1 Introduction.................................... 55 4.1.1 PreviousWork .............................. 57 4.1.2 ChapterOutline.............................. 59 4.2 Market Characteristics and Instance Generator . .......... 59 4.3 ProblemComplexity ............................... 60 4.4 Solution Approaches Based on an Edge Formulation . ........ 62 4.4.1 ConstraintSeeder............................. 64 4.4.2 ConstraintGeneration . 65 4.4.3 Experimentalperformance . 65 4.5 Solution Approaches Based on a Cycle Formulation . ........ 65 4.5.1 EdgevsCycleFormulation. 66 4.5.2 ColumnGenerationfortheLP . 67 4.5.3 Branch-and-PriceSearchfortheILP . ... 72 4.6 ExperimentalResults. .. .. .. 74 4.7 Conclusion..................................... 75 5 The Stable Roommates Problem with Globally-Ranked Pairs 79 ix 5.1 Introduction.................................... 80 5.1.1 Motivation................................. 81 5.1.2 PreliminaryResults. 82 5.1.3 Previouswork............................... 84 5.1.4 ChapterOutline.............................. 85 5.2 Rank-MaximalMatching . 85 5.3 HardnessResults ................................. 91 5.4 Conclusion..................................... 93 5.5 RecentWork ................................... 93 6 Egalitarian Matching 95 6.1 Introduction.................................... 96 6.1.1 Egalitarianism: Lorenz Dominance . ... 96 6.1.2 Motivation................................. 97 6.1.3 PreviousWork .............................. 99 6.1.4 ChapterOutline.............................. 100 6.2 StructuralCharacterization . ..... 100 6.2.1 Gallai-EdmondsDecomposition . 101 6.2.2 Removing Isolated and Perfectly Demanded Vertices . ....... 101 6.2.3 LocalSearchApproach. 102 x 6.2.4 ShrinkingOddComponents . 103 6.2.5 Upper and Lower Bounds on Vertex Probabilities . ..... 105 6.2.6 From Edge Probabilities to a Feasible Distribution over Maximum Matchings105 6.2.7 LocalSearchAlgorithm . 106 6.3 FlowNetworkApproach ............................. 110 6.4 SamplingAlgorithm ............................... 112 7 Conclusion 115 7.1 SummaryofContributions . 115 7.1.1 PopularMatching............................. 115 7.1.2 Layerable Mechanisms for Keyword Auctions . ..... 116 7.1.3 Clearing Algorithms for Barter Exchange Markets . ....... 117 7.1.4 The Stable Roommates Problem with Globally-Ranked Pairs. 118 7.1.5 EgalitarianMatching . 118 7.2 OpenProblems .................................. 119 7.2.1 EgalitarianMatching . 119 7.2.2 OnlineMarkets .............................. 120 7.2.3 Query-CommitModel . .. .. 120 7.3 ConcludingRemarks ............................... 121 xi xii List of Figures 2.1 An instance for which there is no popular matching. ......... 10 2.2 Anillustrativeexample. ... 13 2.3 The f-posts and s-postsfortheexampleinstance. 14 2.4 The reduced graph G′ fortheexampleinstance . 15 2.5 Linear-time popular matching algorithm for instances with strictly-ordered preference lists 17 2.6 Linear-time algorithm for finding an applicant-complete matching in G′ ... 17 2.7 An example with ties in the preference lists. ........ 19 2.8 The graph G1 fortheexampleinstancewithties . 20 2.9 An example with ties in the preference lists. ........ 22 2.10 The reduced graph G′ for the example instance with ties . 24 2.11 O(√nm) popular matching algorithm for preference lists with ties....... 26 3.1 Deterministic allocation written with telescoping sums ............ 39 3.2 Truthful Layer-j Mechanism........................... 47 3.3 Prices and Click-through rate allocations for generalized laddered auction . 49 xiii 3.4 Allocation function for Slots 1 ...a in the generalized laddered auction . 50 3.5 Allocation function for Slots (a + 1) ...k in the generalized laddered auction 50 4.1 Examplebarterexchangemarket. .... 56 4.2 NP-completeness gadget for triple ti and maximum cycle length L....... 62 4.3 Perfect matching encoding of the market in Figure 4.1. .......... 63 4.4 Maximum-weight matching encoding of the market in Figure4.1. ...... 66 4.5 Cycleformulation. ............................... 68 4.6 Experimental results: average runtime with standard deviationbars.. 76 5.1 Exampleofshrinkingoperation . .... 88 5.2 Non-bipartite rank-maximal matching algorithm . .......... 88 5.3 Preference lists for the constructed instance of min-bp-sm-grp ........ 93 6.1 Example Lorenz dominant distribution. ....... 97 6.2 Sample execution of the local search algorithm . ......... 108 xiv