A logic-based representation for coalitional games with externalities

We consider the issue of representing coalitional games in multiagent systems that exhibit externalities from coalition formation, i.e., systems in which the gain from forming a coalition may be affected by the formation of other co-existing coalitions. Although externalities play a key role in many real-life situations, very little attention has been given to this issue in the multi-agent system literature, especially with regard to the computational aspects involved. To this end, we propose a new representation which, in the spirit of Ieong and Shoham [9], is based on Boolean expressions. The idea behind our representation is to construct much richer expressions that allow for capturing externalities induced upon coalitions. We show that the new representation is fully expressive, at least as concise as the conventional partition function game representation and, for many games, exponentially more concise. We evaluate the efficiency of our new representation by considering the problem of computing the Extended and Generalized Shapley value, a powerful extension of the conventional Shapley value to games with externalities. We show that by using our new representation, the Extended and Generalized Shapley value, which has not been studied in the computer science literature to date, can be computed in time linear in the size of the input.

[1]  Vincent Conitzer,et al.  A Compact Representation Scheme for Coalitional Games in Open Anonymous Environments , 2006, AAAI.

[2]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[3]  Onn Shehory,et al.  Coalition structure generation with worst case guarantees , 2022 .

[4]  W. Lucas,et al.  N‐person games in partition function form , 1963 .

[5]  H. Norde,et al.  The Shapley Value for Partition Function Form Games , 2002, IGTR.

[6]  L. S. Shapley,et al.  17. A Value for n-Person Games , 1953 .

[7]  L. Shapley,et al.  The Shapley Value , 1994 .

[8]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

[9]  Martin Shubik,et al.  Game theory models and methods in political economy , 1977 .

[10]  Vincent Conitzer,et al.  Coalition Structure Generation Utilizing Compact Characteristic Function Representations , 2011 .

[11]  Tomasz P. Michalak,et al.  Optimal Coalition Structure Generation In Partition Function Games , 2008, ECAI.

[12]  Yoav Shoham,et al.  Marginal contribution nets: a compact representation scheme for coalitional games , 2005, EC '05.

[13]  Vincent Conitzer,et al.  Complexity of determining nonemptiness of the core , 2003, EC '03.

[14]  Brown,et al.  MARGINAL CONTRIBUTIONS AND EXTERNALITIES IN THE VALUE , 2008 .

[15]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[16]  Michael Wooldridge,et al.  On the computational complexity of coalitional resource games , 2006, Artif. Intell..

[17]  Robert M. Feinberg,et al.  Market Power and Incentives to Form Research Consortia , 2006 .

[18]  Michael Wooldridge,et al.  A Tractable and Expressive Class of Marginal Contribution Nets and Its Applications , 2008, Math. Log. Q..

[19]  Vincent Conitzer,et al.  Computing Shapley Values, Manipulating Value Division Schemes, and Checking Core Membership in Multi-Issue Domains , 2004, AAAI.

[20]  R. Myerson.  Values of games in partition function form , 1977 .

[21]  Nicholas R. Jennings,et al.  On representing coalitional games with externalities , 2009, EC '09.

[22]  Luigi Palopoli,et al.  On the Complexity of Compact Coalitional Games , 2009, IJCAI.

[23]  David Housman.  Values for Partition Function Form Games , 2008 .

[24]  L. Kóczy.  A recursive core for partition function form games , 2006 .

[25]  Ben McQuillin,et al.  The extended and generalized Shapley value: Simultaneous consideration of coalitional externalities and coalitional structure , 2009, J. Econ. Theory.

[26]  David Wettstein,et al.  Sharing the surplus: An extension of the Shapley value for environments with externalities , 2007, J. Econ. Theory.

[27]  Nicholas R. Jennings,et al.  Coalition Structure Generation in Multi-Agent Systems with Positive and Negative Externalities , 2009, IJCAI.

[28]  L. Lambertini.  Game Theory in the Social Sciences , 2011 .

[29]  M. Shubik.  Game theory in the social sciences: Concepts and solutions , 1982 .

[30]  J. M. Bilbao,et al.  Cooperative Games on Combinatorial Structures , 2000 .