## Computational Aspects of Cooperative Game TheoryCooperative 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. |

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### Contents

1 | |

Introduction | 5 |

Basic Concepts | 11 |

Representations and Algorithms | 37 |

Weighted Voting Games | 49 |

Beyond Characteristic Function Games | 71 |

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### Common terms and phrases

algorithmic game theory Artificial Intelligence assume Bachrach Banzhaf index Bayesian core Chalkiadakis chapter characteristic function game choice Cited on page(s coalition formation process coalition structure coalitional actions coalitional games coalitional values consider convex game cooperative game theory core of G CS-core defined Definition denote deviate dummy efficient Elkind equilibrium example feasible form coalitions fuzzy game given goal grand coalition graph hedonic games induced subgraph induced subgraph games InternationalJoint Conference linear program MC-nets multiagent systems Nash Nash equilibrium nodes non-cooperative game non-empty core NP-hard NTU games nucleolus ofa coalition ofgames ofplayers ofthis outcome ofa overlapping coalition partition function partition function games payoff allocation payoff vector permutation polynomial Prisoner’s Dilemma problem proposed rational representation satisfies separation oracle set of players Shapley value simple game solution concepts subset superadditive game Theorem utility veto player voting game G weighted voting games winning coalition Wooldridge