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Algorithmic Modeling of Decision Making Over Networks

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Algorithmic Modeling of Decision Making Over Networks UNIVERSITY OF CALIFORNIA, SAN DIEGO Algorithmic Modeling of Decision Making over Networks A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Computer Science by Andrea Vattani Committee in charge: Professor Ramamohan Paturi, Chair Professor Sanjoy Dasgupta Professor Massimo Franceschetti Professor Alex Snoeren Professor Joel Sobel 2012 Copyright Andrea Vattani, 2012 All rights reserved. The dissertation of Andrea Vattani is approved, and it is acceptable in quality and form for publication on micro- film and electronically: Chair University of California, San Diego 2012 iii DEDICATION To my family iv EPIGRAPH Don't be such a hush-hush. v TABLE OF CONTENTS Signature Page.................................. iii Dedication..................................... iv Epigraph.....................................v Table of Contents................................. vi List of Figures.................................. ix List of Algorithms................................ xii Acknowledgements................................ xiii Vita........................................ xvi Abstract of the Dissertation........................... xvii Chapter 1 Introduction............................1 1.1 Central fields........................4 1.2 Central topics........................7 1.2.1 Part I: Models for Networked Experiments....8 1.2.2 Part II: Models for Other Phenomena....... 13 I Models for Networked Experiments 16 Chapter 2 Coloring Experiments....................... 17 2.1 Results............................ 19 2.2 Related work........................ 21 2.3 Model............................ 22 2.4 Ring networks........................ 24 2.4.1 A natural algorithm................. 24 2.4.2 An optimal algorithm................ 29 2.4.3 Slow-down with asymmetric incentives...... 36 2.5 General bipartite graphs.................. 37 2.5.1 A natural algorithm................. 37 2.5.2 An optimal algorithm................ 39 2.6 Discussion: Leader election vs 2-coloring......... 41 2.7 Preferential attachment graphs............... 42 2.7.1 Analysis of the algorithm.............. 43 2.8 Conclusions......................... 46 vi Chapter 3 Matching Experiments...................... 48 3.1 Results............................ 50 3.2 Related work........................ 54 3.3 The matching games.................... 55 3.4 The algorithmic model................... 56 3.5 Analysis........................... 58 3.6 Prediction and validation of the model.......... 66 3.7 Conclusions......................... 68 3.8 Appendix: analysis for Theorems 20{21.......... 69 3.8.1 Properties of matchings in M1........... 70 ∗ 3.8.2 The tree Tn ...................... 72 3.8.3 Proof of Theorem 20................ 74 3.8.4 Proof of Theorem 21................ 76 Chapter 4 Finding Red Balloons....................... 79 4.1 The query incentive network model............ 81 4.2 Results............................ 83 4.3 Additional related work................... 85 4.4 Preliminaries........................ 86 4.4.1 Split contracts.................... 87 4.4.2 Propagation of the payment............ 88 4.4.3 Difference with respect to previous work..... 89 4.4.4 Roadmap...................... 90 4.5 Properties of Nash equilibria................ 91 4.6 The Nash equilibrium.................... 95 4.7 Guaranteeing h-consistency................. 102 4.8 Efficiency.......................... 105 4.9 Discussion: non-uniqueness of the Nash equilibrium... 109 4.10 Simulations......................... 111 II Models for Other Phenomena 114 Chapter 5 Models for Aggregation...................... 115 5.1 Results............................ 116 5.2 Related work........................ 118 5.3 Preliminaries........................ 120 5.3.1 Notation....................... 121 5.4 Homogeneous Populations................. 122 5.4.1 A Population of Followers............. 122 5.4.2 A Population of Leaders.............. 124 5.4.3 Lower Bounds for Homogeneous Populations... 125 5.5 Heterogeneous Populations................. 127 vii 5.5.1 Achieving Low Price of Anarchy.......... 127 5.5.2 Price of Stability and Relation to Price of Anarchy 130 5.6 Extensions.......................... 132 5.6.1 Generalized β-leaders................ 133 5.6.2 The effects of information............. 135 5.7 Conclusions......................... 138 Chapter 6 The Secretary Problem...................... 140 6.1 Results............................ 141 6.2 Related work........................ 143 6.3 Preliminaries........................ 144 6.4 Algorithm and Analysis................... 144 6.4.1 Warmup: Analysis for a single secretary..... 145 6.4.2 Analysis for general posets............. 146 6.5 Upper bounds on success.................. 153 6.6 Tightness of the algorithm................. 157 6.7 Conclusions......................... 160 Bibliography................................... 162 viii LIST OF FIGURES Figure 3.1: Computer interface. The subject is matched with the node on the right and is being requested by three unmatched nodes.. 49 Figure 3.2: Approximate and maximum matching. Left: an approxi- mate maximum matching of size 5 on a network with 12 nodes (matching edges are represented in bold red, matched nodes are colored, unmatched nodes are white). Right: a maximum matching of size 6 on the same network (note that the maximum matching is also a perfect matching, as all nodes are matched). 50 Figure 3.3: Affinity between humans' and algorithm's performance, 16-node networks. The performance of the human subjects (red points joined by continuous line) and of the algorithm (blue points) over eight bipartite 16-node networks (triangles) and eight non-bipartite 16-node networks (circles) are plotted. The experiment was run multiple times on each network and the average behavior is reported. The x-axis shows the indexes of the networks sorted by increasing average time required to reach a maximum matching. Bipartite networks are labeled from 1 to 8, while non-bipartite networks are labeled from 9 to 16. The y-axis shows the average time (in seconds) required to reach a maximum matching for humans, while the average number of rounds of the algorithm is scaled by a constant factor...... 51 Figure 3.4: Affinity between humans' and algorithm's performance, 24-node networks. The performance of the human subjects (red points joined by continuous line) and of the algorithm (blue points) over different 24-node networks are plotted. In particular, small-world networks (triangles), a ring network (di- amonds), and preferential attachment networks (circles) were tested. The experiment was run multiple times on each network and the average behavior is reported. The x-axis shows the in- dexes of the networks sorted by increasing average time required to reach a maximum matching. The y-axis shows the average time (in seconds) required to reach a maximum matching for humans, while the average number of rounds of the algorithm is scaled by a constant factor.................... 52 ix Figure 3.5: Algorithm's asymptotic performance. Prudence algo- rithm's performance with respect to the network's size for the \bad" graph Gn (black diamonds), for preferential attachment model (green squares), small-world model (red triangles). For each generative model and network size we generated 100 net- works and run the algorithm 1000 times on each. The average behavior is reported. The x-axis shows the network size, and the y-axis shows the average number of rounds required by the algorithm to converge to a maximum matching.......... 52 Figure 3.6: Experimental performance, 24-node networks. Perfor- mance of the experimental subjects on networks of 24 nodes. The plot shows the time to reach a perfect matching of size 12 (red), an approximate matching of size 11 (a 0:92{approximate matching, in blue) and a matching of size 6 (a 1=2{approximate matching, in green). Results for single games are reported. The x-axis shows the indexes of the games sorted by increasing solv- ing time, while the y-axis shows the time in seconds. The right- most four games on the red plot did not converge to a maximum matching and correspond to three instances of the \bad" graph Gn and to one instance of the preferential attachment network. 53 Figure 3.7: The bad graph. The \bad" graph Gn for n = 3. One of the \bad" matchings of Theorem 20 is highlighted in red....... 65 ∗ ∗ Figure 3.8: Tree Tn . Tree Tn with labels, for n = 6. This is used in the proof of Theorem 20......................... 70 Figure 4.1: Investment as a function of the rarity n for b = 1:95 with split- contracts (in red triangles and circles, for = 0:2 and = 0:05, respectively) and with fixed-payment contracts (in blue triangles and circles, for = 0:2 and = 0:05, respectively)... 83 Figure 4.2: Investment as a function of the rarity n for b = 4 with split- contracts (in red triangles and circles, for = 0:2 and = 0:05, respectively) and with fixed-payment contracts (in blue triangles and circles, for = 0:2 and = 0:05, respectively)... 84 Figure 4.3: As a function of b > 1: investment with split-contracts (in red circles) and 12logarithm of the investment with fixed-payment 1 contracts (in blue circles). In green, the function b−1 scaled by a constant factor........................... 112 Figure 5.1: 2-stable placement with high price of anarchy. This is used in Observation 45............................ 124 Figure 5.2: Graphs for lower bounds. They are used in the proof of Theo- rem 47................................ 126 x Figure 5.3: Graphs for
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