Computational Aspects of Cooperative Game Theory (Synthesis Lectures on Artificial Intelligence and Machine Learning)

Computational Aspects of Cooperative Game Theory (Synthesis Lectures on Artificial Intelligence and Machine Learning)

Michael Wooldridge

Language: English

Pages: 168

ISBN: 1608456528

Format: PDF / Kindle (mobi) / ePub

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.

Table of Contents: Introduction / Basic Concepts / Representations and Algorithms / Weighted Voting Games / Beyond Characteristic Function Games / Coalition Structure Formation / Advanced Topics

"This manuscript was a pleasure to discover, and a pleasure to read -- a broad, but succinct, overview of work in computational cooperative game theory. I will certainly use this text with my own students, both within courses and to provide comprehensive background for students in my research group. The authors have made a substantial contribution to the multiagent systems and algorithmic game theory communities." --Professor Jeffrey S. Rosenschein, The Hebrew University of Jerusalem, Israel

"With the advent of the internet, the computational aspects of cooperative game theory are ever more relevant. This unique and timely book by Chalkiadakis, Elkind, and Wooldridge gives a concise and comprehensive survey of the subject, and serves at the same time as a one-stop introduction to cooperative game theory." --Professor Bernhard von Stengel, London School of Economics, UK

"In recent years, research on the computational aspects of cooperative game theory has made tremendous progress, but previous textbooks have not included more than a short introduction to this important topic. I am excited by the thorough treatment in this new book, whose authors have been and continue to be at the very forefront of this research. Newcomers to the area are well advised to read this book carefully and cover to cover." --Professor Vincent Conitzer, Duke University, USA

"Cooperative game theory has proved to be a fertile source of challenges and inspiration for computer scientists. This book will be an essential companion for everyone wanting to explore the computational aspects of cooperative game theory." --Prof Makoto Yokoo, Kyushu University, Japan

"An excellent treatise on algorithms and complexity for cooperative games. It navigates through the maze of cooperative solution concepts to the very frontiers of algorithmic game theory research.The last chapter in particular will be enormously valuable for graduate students and young researchers looking for research topics." --Professor Xiaotie Deng, University of Liverpool, UK

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VWVG-Minimality can be solved efficiently by dynamic programming. Boolean Combinations of Weighted Voting Games Vector weighted voting games can be interpreted as conjunctions of weighted voting games. One can also combine weighted voting games according to more complex Boolean formulas: for instance, to win in the game (G1 ∨ G2 ) ∧ G3 , a coalition must win in one of the games G1 and G2 as well as in G3 . This representation for simple games is studied in detail by Faliszewski et al.

computing v ∗ (N). We will now explain how to compute v ∗ (C) for all coalitions C ⊆ N, starting from coalitions of size 1 and 2, and culminating in computing v ∗ (N ). The optimal coalition structure itself can then be recovered using standard dynamic programming techniques. We will need the following simple lemma. Lemma 6.1 For any C ⊆ N we have v ∗ (C) = max{max{v ∗ (C ) + v ∗ (C ) | C ∪ C = C, C ∩ C = ∅, C , C = ∅}, v(C)}. (6.1) Proof. We will first show that the left-hand side of (6.1)

Observe that 3n < (n/4)n/2 for large enough values of n—i.e., this algorithm avoids searching all coalition structures. Moreover, its running time is polynomial in the number of coalitions 2n . However, in practice, the dynamic programming algorithm is quite slow. The next section presents an approach that, despite having weaker worst-case performance guarantees, tends to work well in practice. 6.1.2 ANYTIME ALGORITHMS Sandholm et al. [224] developed a technique for finding a coalition

more than two agents need to share the same resource at any given time period. Specifically, the agents play two different roles: one of them already has access to the resource and is using it during the negotiation process (gaining over time), while the other is waiting to use the resource (losing over time). Shehory and Kraus [237, 238] develop coalition formation algorithms which account for uncertainty regarding the capabilities of agents. Agents rely on information communicated to them by

Tuomas Sandholm, Bernhard von Stengel, Yair Zick, and the anonymous reviewers arranged by the publisher, who carefully read a preliminary draft of the book and gave us detailed and enormously helpful feedback. It goes without saying that any remaining errors and omissions are entirely the responsibility of the authors. Edith Elkind gratefully acknowledges research fellowship RF2009-08 “Algorithmic aspects of coalitional games” from the National Research Foundation (Singapore). PERSONAL PRONOUNS

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