A Graphical Representation for Games in Partition Function Form

We propose a novel representation for coalitional games with externalities, called Partition Decision Trees. This representation is based on rooted directed trees, where non-leaf nodes are labelled with agents' names, leaf nodes are labelled with payoff vectors, and edges indicate membership of agents in coalitions. We show that this representation is fully expressive, and for certain classes of games significantly more concise than an extensive representation. Most importantly, Partition Decision Trees are the first formalism in the literature under which most of the direct extensions of the Shapley value to games with externalities can be computed in polynomial time.

[1]  Makoto Yokoo,et al.  Extension of MC-net-based coalition structure generation: handling negative rules and externalities , 2011, AAMAS.

[2]  Michael Wooldridge,et al.  Computational Aspects of Cooperative Game Theory , 2011, KES-AMSTA.

[3]  S. Vajda,et al.  Contributions to the Theory of Games. Volume III. Annals of Mathematics Studies Number 39. Edited by M. Dresher, A. W. Tucker and P. Wolfe. (Princeton University Press) , 1954, The Mathematical Gazette.

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

[5]  Haris Aziz,et al.  Complexity of coalition structure generation , 2011, AAMAS.

[6]  Nicholas R. Jennings,et al.  Representation of coalitional games with algebraic decision diagrams , 2011, AAMAS.

[7]  Oskar Skibski.  Steady Marginality: A Uniform Approach to Shapley Value for Games with Externalities , 2011, SAGT.

[8]  Nicholas R. Jennings,et al.  Computational Aspects of Extending the Shapley Value to Coalitional Games with Externalities , 2010, ECAI.

[9]  Nicholas R. Jennings,et al.  A logic-based representation for coalitional games with externalities , 2010, AAMAS.

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

[11]  Vincent Conitzer,et al.  Coalitional Games in Open Anonymous Environments , 2005, IJCAI.

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

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

[14]  Nicholas R. Jennings,et al.  An Efficient Vector-Based Representation for Coalitional Games , 2013, IJCAI.

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

[16]  Cheng-Cheng Hu,et al.  An axiomatic characterization of a value for games in partition function form , 2010 .

[17]  Oskar Skibski,et al.  Marginality Approach to Shapley Value in Games with Externalities , 2013 .

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

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

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

[21]  Jeffrey S. Rosenschein,et al.  Coalitional skill games , 2008, AAMAS.

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

[23]  Nicholas R. Jennings,et al.  Anytime coalition structure generation in multi-agent systems with positive or negative externalities , 2012, Artif. Intell..

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

[25]  E. M. Bolger.  A set of axioms for a value for partition function games , 1989 .

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

[27]  Makoto Yokoo,et al.  A Compact Representation Scheme of Coalitional Games Based on Multi-Terminal Zero-Suppressed Binary Decision Diagrams , 2011, PRIMA.

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

[29]  Vincent Conitzer,et al.  Complexity of constructing solutions in the core based on synergies among coalitions , 2006, Artif. Intell..

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

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