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A Computational Complexity Analysis of Various Types of Influence And Beyond Intractability: A Computational Complexity Analysis of Various Types of Influence and Stability in Cooperative Games Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Heinrich-Heine-Universität Düsseldorf vorgelegt von Anja Rey aus Wuppertal Düsseldorf, September 2015 aus dem Institut für Informatik der Heinrich-Heine-Universität Düsseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Heinrich-Heine-Universität Düsseldorf Referent: Prof. Dr. Jörg Rothe Koreferent: Prof. Dr. Gerhard Woeginger Koreferent: Prof. Dr. Thomas Ågotnes Tag der mündlichen Prüfung: 21. April 2016 Erklärung Ich versichere an Eides Statt, dass die vorliegende Dissertation von mir selbständig und ohne unzulässige fremde Hilfe unter Beachtung der „Grundsätze zur Sicherung guter wis- senschaftlicher Praxis an der Heinrich-Heine-Universität Düsseldorf“ erstellt worden ist. Desweiteren erkläre ich, dass ich eine Dissertation in der vorliegenden oder in ähnlicher Form noch bei keiner anderen Institution eingereicht habe. Teile dieser Arbeit wurden bereits in den folgenden Schriften veröffentlicht bzw. zur Publi- kation angenommen: [RRM16]∗, [RRSS16]∗, [RR14a], [RR16]∗, [NRR+16]∗, [LRR+15], [RRSS14], [MRR14], [RR14b], [RR12], [RR11], [RR10b], [RR10a], Düsseldorf, 16. September 2015 Anja Rey ∗ aktualisiert am 28. August 2016 iii Acknowledgements I would like to thank all the people who have inspired me throughout my progress of writing this thesis and the way towards it. In particular, my acknowledgements are directed to my advisor, Jörg Rothe, for welcoming me in his working group, for impressive support and motivation, for providing insight into the academic world, and for many original and encouraging suggestions. Moreover, I would like to thank Gerhard Woeginger, not only for agreeing to act as a referee for this thesis, but also for a challenging talk at ComSoC2012 introducing me to the world of hedonic games. Additionally, I would like to thank Thomas Ågotnes for agreeing to act as a referee. I would furthermore like to thank Dorothea Baumeister for being a mentor to me ever since the early stages of studying in Düsseldorf, for inspiring talks about research, teaching and life. My special thanks go to Lena Schend for accompanying me throughout my time at university. I am happy about the mere fact that we were able to write our theses simultaniously. Sharing an office and attitude to life alike, as well as exchanging impressions and thoughts has always been wonderful. I would like to thank my colleges at Heinrich-Heine-Universität Düsseldorf, Dorothea Baumeister, Claudia Forstinger (next to her administrative support, I always enjoy anecdotes and creative thoughts), Guido Königstein (for great technical support and never giving up on that semi- nar notebook), Daniel Neugebauer (for great discussions at coffee breaks), Nhan-Tam Nguyen (for highly valuable recommendations where to find useful literature, and a gadget shop in London), Lisa Rey, Hilmar Schadrack (for great discussions at coffee breaks), Lena Schend, and Ann-Kathrin Selker (for great discussions at coffee breaks). I would like to emphasize my recognition to Lisa, Tam, Daniel, and Lena for partly proofreading this thesis und providing helpful comments from dif- ferent points of view. For inspiring collaboration, I would like to thank my co-authors Jérôme Lang, Adrian Marple, Nhan-Tam Nguyen, Lisa Rey, Jörg Rothe, Hilmar Schadrack, and Lena Schend. Ad- ditionally, I would like to thank Judith Bock, Jens Dietze, Michael Singhof, and Harald Werner for proofreading parts of this thesis from yet other perspectives, encouraging discussions, and friendship. Nonetheless, I would like to thank the numerous anonymous reviewers for their many helpful comments. This work was supported in part by DFG grants RO 1202/12-1, RO-1202/14-1, and RO-1202/15-1 as well as by the DAAD-PPP / PHC PROCOPE programme entitled “Fair Division of Indivisible Goods: Incomplete Preferences, Communication Protocols and Computational Resis- tance to Strategic Behavior,” Naturally, I would like to thank my Mama, Barbara Rey, for her invariable loving care, the mirabelles, many telephone calls, proofreading a part of this thesis, positivity, and trust in me and my work. My thanks are also dedicated to my Papa, Günter Rey, for having taught me indefatigably the most basic necessities for this thesis, and life. I would like to express my gratitude to my sister Lisa again who managed the exceptional task of looking after me in both situations, work and social life. It is hard to put into words what your support in manyfold ways, and the mere fact of being so close mean to me. Finally, I am eternally grateful to my husband, Denis Lütke-Wiesmann, for his immense patience with me. Partly proofreading my thesis was only a tiny part of his support. I don’t know what I would have done if it weren’t for your love and being there for me in this exceptional situation and beyond. v Zusammenfassung Diese Arbeit beschäftigt sich mit der Berechnungskomplexität unterschiedlicher Problemstellungen aus der kooperativen Spieltheorie. Zum einen betrachten wir Existenz und Verifizierung von Stabili- tätskonzepten in kooperativen Spielen mit übertragbarem Nutzen und in hedonischen Spielen; zum anderen wenden wir uns der Einflussnahme auf den Ausgang eines Spiels durch Manipulation, Kon- trolle und Bestechung zu. Wir widmen uns manipulativem Verhalten im Sinne von strategischem Zusammenschluss von Spielern oder unter Angabe falscher Identitäten sowie struktureller Kontrolle durch Hinzufügen oder Entfernen von Spielern in gewichteten Wahlspielen. Betrachtete Lösungs- konzepte hier sind der probabilistische Penrose–Banzhaf-Index und der Shapley–Shubik-Index, die die Manipulatoren beabsichtigen, zu ihrem Vorteil zu verbessern. Wir zeigen unter anderem für das Problem, ob ein Zusammenschluss einer gegebenen Koalition in einem gegebenen Spiel vorteilhaft ist, Vollständigkeit für probabilistische Polynomialzeit. Außerdem verallgemeinern wir ein formales Modell für Manipulation auf beliebige Klassen von kooperativen Spielen und untersuchen allgemei- ne Eigenschaften sowie beispielhaft einzelne Klassen wie Einstimmigkeitsspiele. Darüber hinaus betrachten wir Bestechung in Pfad-Unterbrechungs-Spielen, wobei ein Gegen- spieler versucht, durch ein Netzwerk von einem Start- zu einem Zielknoten zu gelangen und dabei ausgewählte Agenten für die Freigabe ihrer Knoten zu bezahlen, sodass es sich für übrige Koalitio- nen nicht mehr lohnt oder unmöglich ist, alle Wege zu blockieren. Für mehrere Gegenspieler und Kosten zur Knotenblockierung zeigen wir, dass das Bestechungsproblem vollständig für die zweite Stufe der Polynomialzeithierarchie ist. Wir erweitern diese Spiele auf ein probabilistisches Modell, in dem das genaue Ziel des Gegenspielers unbekannt ist, und untersuchen es im Hinblick auf be- kannte Stabilitätskonzepte wie den Kern. Hier kann beobachtet werden, dass sich der allgemeinere Fall bezüglich Komplexität nicht anders verhält als der speziellere. Fragen der Stabilität analysieren wir ebenfalls in hedonischen Spielen. Dabei gehen wir zu- nächst auf wundervolle Stabilität in feind-orientierten hedonischen Spielen ein. Wir heben die untere Schranke des Problems, ob es im Graphen eines zugrunde liegenden gegebenen Spiels eine wunder- voll stabile Aufteilung gibt, auf DP an, also auf die zweite Stufe der booleschen Hierarchie. Auf dem Weg zur exakten Komplexität zeigen wir, dass coDP-Härte ausreichen würde, um Vollständigkeit für parallelen Zugriff auf NP zu beweisen. Des Weiteren führen wir ein neues Modell hedonischer Spiele mit ordinalen Präferenzen und Schwellenwerten ein, in dem Spieler ihre Mitspieler in Freunde, Fein- de und neutrale Spieler unterteilen und gleichzeitig eine schwache Ordnung über die ersten beiden Mengen angeben. Erweitert wird diese Relation auf eine Menge möglicher Präferenzen, sodass es sinnvoll ist, Begriffe der möglichen und notwendigen Stabilität zu definieren. Hierfür prüfen wir die Komplexität unterschiedlicher Konzepte wie Nash-Stabilität und ermitteln axiomatische Spieleigen- schaften. Zuletzt stellen wir eine weitere Variante hedonischer Spiele mit altruistischen Einflüssen vor. Bisher bekannte Darstellungen hedonischer Spiele gehen von eigennützigen Agenten aus, de- ren Präferenzen nur von ihrer Meinung abhängen. Basierend auf freund-orientierten Erweiterungen, lassen wir nun Einflüsse von Freunden auf die Präferenzrelation eines Spielers zu, indem wir in drei Graden der Selbstlosigkeit die durchschnittliche Meinung der Freunde in einer Koalition miteinbe- ziehen. Auch hier behandeln wir neben sinnvoll hierfür modellierten Eigenschaften die Komplexität von Stabilitätskonzepten wie strikte Popularität beim direkten Vergleich von Koalitionsstrukturen. vii Abstract This thesis deals with the computational complexity of various problems from cooperative game theory. On the one hand we examine the existence and verification of stability concepts in cooperative games with transferable utility and in hedonic games; on the other hand we look into several forms of influence on the output of a game via manipulation, control and bribery. We turn to manipulative action in the sense of strategically merging players or splitting a player into false identities as well as structural control by adding or deleting players in weighted voting games. In this setting, considered solution concepts are the probabilistic Penrose–Banzhaf index and the Shapley–Shubik
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