Computational Aspects of Cooperative Game Theory

Abstract

Part of: Synthesis digital library of engineering and computer science. Cooperative game theory is a branch of (micro- )economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this book is to present a survey of work on the computational aspects of cooperative game theory. We begin by formally defining transferable utility games in characteristic function form, and introducing key solution concepts such as the core and the Shapley value. We then discuss two major issues that arise when considering such games from a computational perspective: identifying compact representations for games, and the closely related problem of efficiently computing solution concepts for games. We survey several formalisms for cooperative games that have been proposed in the literature, including, for example, cooperative games defined on networks, as well as general compact representation schemes such as MC-nets and skill games. As a detailed case study, we consider weighted voting games: a widely-used and practically important class of cooperative games that inherently have a natural compact representation. We investigate the complexity of solution concepts for such games, and generalizations of them. We briefly discuss games with non-transferable utility and partition function games. We then overview algorithms for identifying welfare-maximizing coalition structures and methods used by rational agents to form coalitions (even under uncertainty), including bargaining algorithms. We conclude by considering some developing topics, applications, and future research directions. Basic concepts -- Representations and algorithms -- Weighted voting games -- Beyond characteristic function games -- Coalition structure formation -- Advanced topics. 1. Introduction -- 1.1 Why are non-cooperative games non-cooperative? -- 1.2 Computational problems in game theory -- 1.3 The remainder of this book -- 1.4 Further reading. 2. Basic concepts -- 2.1 Characteristic function games -- 2.1.1 Outcomes -- 2.1.2 Subclasses of characteristic function games -- 2.2 Solution concepts -- 2.2.1 Shapley value -- 2.2.2 Banzhaf index -- 2.2.3 Core and core-related concepts -- 2.2.4 Nucleolus -- 2.2.5 Kernel -- 2.2.6 Bargaining set -- 2.2.7 Stable set. 3. Representations and algorithms -- 3.1 Combinatorial optimization games -- 3.1.1 Induced subgraph games -- 3.1.2 Network flow games -- 3.1.3 Assignment and matching games -- 3.1.4 Minimum cost spanning tree games -- 3.1.5 Facility location games -- 3.2 Complete representations -- 3.2.1 Marginal contribution nets -- 3.2.2 Synergy coalition groups -- 3.2.3 Skill-based representations -- 3.2.4 Algebraic decision diagrams -- 3.3 Oracle representation. 4. Weighted voting games -- 4.1 Definition and examples -- 4.2 Dummies and veto players -- 4.2.1 Power and weight -- 4.2.2 Computing the power indices -- 4.2.3 Paradoxes of power -- 4.3 Stability in weighted voting games -- 4.3.1 The least core, the cost of stability, and the nucleolus -- 4.4 Vector weighted voting games -- 4.4.1 Computing the dimension of a simple game. 5. Beyond characteristic function games -- 5.1 Non-transferable utility games -- 5.1.1 Formal model -- 5.1.2 Hedonic games -- 5.1.3 Qualitative games -- 5.2 Partition function games. 6. Coalition structure formation -- 6.1 Coalition structure generation -- 6.1.1 Dynamic programming -- 6.1.2 Anytime algorithms -- 6.2 Coalition formation by selfish rational agents -- 6.2.1 Coalition formation via bargaining -- 6.2.2 Dynamic coalition formation -- 6.2.3 Coalition formation under uncertainty -- 6.3 Coalition formation and learning. 7. Advanced topics -- 7.1 Links between cooperative and non-cooperative games -- 7.1.1 Cooperation in normal-form games -- 7.1.2 Non-cooperative justifications of cooperative solution concepts -- 7.1.3 Program equilibrium -- 7.2 Using mechanism design for coalition formation -- 7.2.1 Anonymity-proof solution concepts -- 7.3 Overlapping and fuzzy coalition formation -- 7.4 Logical approaches to cooperative game theory -- 7.5 Applications -- 7.5.1 Coalitions in communication networks -- 7.5.2 Coalitions in the electricity grid -- 7.5.3 Core-selecting auctions -- 7.6 Research directions.