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Set-Valued Solution Concepts in Social Choice and Game Theory Axiomatic and Computational Aspects

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Set-Valued Solution Concepts in Social Choice and Game Theory Axiomatic and Computational Aspects TECHNISCHEUNIVERSITÄTMÜNCHEN Lehrstuhl für Wirtschaftsinformatik und Entscheidungstheorie Set-Valued Solution Concepts in Social Choice and Game Theory Axiomatic and Computational Aspects Markus Brill Vollständiger Abdruck der von der Fakultät für Informatik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Tobias Nipkow, Ph.D. Prüfer der Dissertation: 1. Univ.-Prof. Dr. Felix Brandt 2. Prof. Dr. Jérôme Lang, Université Paris-Dauphine/Frankreich Die Dissertation wurde am 5. Juli 2012 bei der Technischen Universität München eingereicht und durch die Fakultät für Informatik am 12. Oktober 2012 angenom- men. SET-VALUED SOLUTION CONCEPTS IN SOCIALCHOICEANDGAMETHEORY markus brill Axiomatic and Computational Aspects Markus Brill: Set-Valued Solution Concepts in Social Choice and Game Theory, Axiomatic and Computational Aspects, c July 2012 Für Papa ABSTRACT This thesis studies axiomatic and computational aspects of set-valued solution concepts in social choice and game theory. It is divided into two parts. The first part focusses on solution concepts for normal-form games that are based on varying notions of dominance. These concepts are in- tuitively appealing and admit unique minimal solutions in important subclasses of games. Examples include Shapley’s saddles, Harsanyi and Selten’s primitive formations, Basu and Weibull’s CURB sets, and Dutta and Laslier’s minimal covering sets. Two generic algorithms for computing these concepts are proposed. For each of these algorithms, properties of the underlying dominance notion are identified that en- sure the soundness and efficiency of the algorithm. Furthermore, it is shown that several solution concepts based on weak and very weak dominance are computationally intractable, even in two-player games. The second part is concerned with social choice functions (SCFs), an important subclass of which is formed by tournament solutions. The winner determination problem is shown to be computationally intractable for different variants of Dodgson’s rule, Young’s rule, and Tideman’s method of ranked pairs. For a number of tractable SCFs such as maximin and Borda’s rule, the complexity of computing pos- sible and necessary winners for partially specified tournaments is de- termined. Special emphasis is then put on tournament solutions that are defined via retentiveness. The axiomatic properties and the asymp- totic behavior of these solutions is studied in depth, and a new attrac- tive tournament solution is proposed. Finally, necessary and sufficient conditions for the strategyproofness of irresolute SCFs are presented. vii PUBLICATIONS This thesis is based on the following publications. Authors are listed in alphabetical order. 1. H. Aziz, M. Brill, F. Fischer, P. Harrenstein, J. Lang, and H. G. Seedig. Possible and necessary winners of partial tournaments. In V. Conitzer and M. Winikoff, editors, Proceedings of the 11th In- ternational Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). IFAAMAS, 2012. 2. F. Brandt and M. Brill. Necessary and sufficient conditions for the strategyproofness of irresolute social choice functions. In K. Apt, editor, Proceedings of the 13th Conference on Theoretical As- pects of Rationality and Knowledge (TARK), pages 136–142. ACM Press, 2011. 3. F. Brandt and M. Brill. Computing dominance-based solution concepts. In B. Faltings, K. Leyton-Brown, and P. Ipeirotis, ed- itors, Proceedings of the 13th ACM Conference on Electronic Commerce (ACM-EC), page 233. ACM Press, 2012. 4. F. Brandt, M. Brill, F. Fischer, and P. Harrenstein. Computational aspects of Shapley’s saddles. In Proceedings of the 8th Interna- tional Joint Conference on Autonomous Agents and Multi-Agent Sys- tems (AAMAS), pages 209–216. IFAAMAS, 2009. 5. F. Brandt, M. Brill, F. Fischer, P. Harrenstein, and J. Hoffmann. Computing Shapley’s saddles. ACM SIGecom Exchanges, 8(2), 2009. 6. F. Brandt, M. Brill, F. Fischer, and P. Harrenstein. Minimal reten- tive sets in tournaments. In W. van der Hoek, G. A. Kaminka, Y. Lespérance, and M. Luck, editors, Proceedings of the 9th Inter- national Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pages 47–54. IFAAMAS, 2010. 7. F. Brandt, M. Brill, E. Hemaspaandra, and L. Hemaspaandra. By- passing combinatorial protections: Polynomial-time algorithms for single-peaked electorates. In M. Fox and D. Poole, editors, Proceedings of the 24th AAAI Conference on Artificial Intelligence (AAAI), pages 715–722. AAAI Press, 2010. 8. F. Brandt, M. Brill, F. Fischer, and P. Harrenstein. On the com- plexity of iterated weak dominance in constant-sum games. The- ory of Computing Systems, 49(1):162–181, 2011. ix 9. F. Brandt, M. Brill, F. Fischer, and J. Hoffmann. The computa- tional complexity of weak saddles. Theory of Computing Systems, 49(1):139–161, 2011. 10. M. Brill and F. Fischer. The price of neutrality for the ranked pairs method. In J. Hoffmann and B. Selman, editors, Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI), pages 1299–1305. AAAI Press, 2012. x ACKNOWLEDGMENTS I want to say thank you to many people. To Felix Brandt, for being the best advisor I can imagine. I well remember the day when I first met Felix, and his support, guidance, and brilliance have been outstanding ever since. I truly hope that we will have fruitful collaborations for decades to come. To all former and current members of the PAMAS research group. Haris Aziz, Felix Brandt, Felix Fischer, Paul Harrenstein, Keyvan Kardel, Evangelia Pyrga, Hans Georg Seedig, and Troels Bjerre Sørensen: You have been amazing colleagues and friends, and I have been very lucky to be in the right place at the right time. To all colleagues at LMU and TUM, for creating a stimulating work- ing environment. To all my coauthors and collaborators, for teach- ing me everything I know about research. Besides PAMAS members, the set of my coauthors includes Edith and Lane Hemaspaandra, Jan Hoffmann, and Jérôme Lang. Jérôme has also provided extensive and very helpful feedback to my thesis and my work in general. My work has further benefitted from helpful discussions with Craig Boutilier, Vincent Conitzer, Edith Elkind, Ulle Endriss, Piotr Faliszewski, Chris- tian Klamler, Jean-François Laslier, Michel Le Breton, Vincent Mer- lin, Rolf Niedermeier, Noam Nisan, Ariel Procaccia, Jörg Rothe, Tim Roughgarden, Tuomas Sandholm, Arkadii Slinko, Toby Walsh, Ger- hard Woeginger, Lirong Xia, and Bill Zwicker, among others. To Jean-François Laslier and his research group at École Polytech- nique, for hosting me in the summer of 2011. To José Rui Figueira, the special one, for organizing an unforgettable summer school in Estoril. To the A-Team Aarhus consisting of Matúš Mihal’ák, Leo Rüst, Rahul Savani, and Alex Souza, for the fun we had in Denmark. To the writ- ing group under the guidance of Aniko Balazs, for helpful tips and feedback. To Tobias Nipkow, for chairing my thesis committee. To TopMath and the M9 research group, for allowing me to dive into research at an early stage. In particular, I want to thank René Brandenberg, Peter Gritzmann, and Anusch Taraz from M9, and An- drea Echtler, Ralf Franken, Christian Kredler, and Denise Lichtig from TopMath. To my TopMath football team, for all the great memories, including winning the Elite-Cup trophy twice. Let me write them in formation: Stefan Jerg—Péter Koltai, Stephan Ritscher—Thorsten Knott, Lukas Höhndorf, Suyang Pu—Sascha Böhm. To all my friends in academia, including Doro Baumeister, Nadja Betzler, Paul Dütting, Gábor Erdélyi, Britta Dorn, and Umberto Grandi, for making conferences so much fun. xi To all my friends outside academia, including Maria Eltsova, Melanie Götz, Eva Langhauser, Nico Noppenberger, Nóra Reg˝os,and Tea Todorova, for supporting me throughout. To Kriszta and Wolf- gang Endreß, for providing me with Hungarian wines to keep me going. To my parents and family, for all they have done for me. Finally, to all those great people who deserve a thank you but whom I forgot. funding sources My work was supported by the Deutsche Forschungsgemeinschaft under grants BR-2312/6-1 (within the Eu- ropean Science Foundation’s EUROCORES program LogICCC) and BR-2312/10-1, by the TUM Graduate School, and by ParisTech. xii CONTENTS 1 introduction1 1.1 Illustrative Examples 3 1.1.1 Example 1: Choice of Action 3 1.1.2 Example 2: Choice Based on Pairwise Comparisons 6 1.1.3 Example 3: Choice Based on Preferences 7 1.2 The Value of Set-Valued Solution Concepts 9 1.2.1 A Classification of Solution Concepts 9 1.2.2 Set-Valued Solution Concepts in Game Theory 11 1.2.3 Set-Valued Solution Concepts in Social Choice Theory 12 1.3 Overview of This Thesis 13 1.3.1 Publications on Which This Thesis is Based 13 1.3.2 Work Excluded From This Thesis 14 1.4 Prerequisites 14 i game theory 15 2 games, dominance, and solutions 17 2.1 Strategic Games 17 2.2 Dominance 19 2.2.1 Dominance Structures 19 2.2.2 Properties of Dominance Structures 22 2.3 Solution Concepts 23 2.3.1 Nash Equilibrium 23 2.3.2 Iterated Dominance 25 2.4 Summary 26 3 dominance-based solution concepts 27 3.1 Motivation 27 3.2 D-solutions and D-sets 28 3.3 Uniqueness Results for D-Sets 31 3.4 Games with an Exponential Number of D-Sets 36 3.5 Summary 39 4 algorithms for dominance-based solutions 41 4.1 Generic Greedy Algorithm 41 4.2 Generic Sophisticated Algorithm 43 4.3 Greedy Algorithms 45 4.4 Sophisticated Algorithms 47 4.5 Summary 48 xv xvi Contents
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