Complexity of Computing the Shapley Value in Games with Externalities

We study the complexity of computing the Shapley value in games with externalities. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets) and five extensions of the Shapley value to games with externalities. Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value: two out of five extensions can be computed in polynomial time for embedded MC-nets and only one for weighted MC-nets.

[1]  Makoto Yokoo,et al.  Partition decision trees: representation for efficient computation of the Shapley value extended to games with externalities , 2019, Autonomous Agents and Multi-Agent Systems.

[2]  Oskar Skibski,et al.  Attachment centrality: Measure for connectivity in networks , 2019, Artif. Intell..

[3]  Takaaki Abe,et al.  The non-emptiness of the core of a partition function form game , 2017, Int. J. Game Theory.

[4]  Makoto Yokoo,et al.  k-Coalitional Cooperative Games , 2016, AAMAS.

[5]  Makoto Yokoo,et al.  A Graphical Representation for Games in Partition Function Form , 2015, AAAI.

[6]  Pushmeet Kohli,et al.  The cost of principles: analyzing power in compatibility weighted voting games , 2014, AAMAS.

[7]  Nicholas R. Jennings,et al.  Efficient Computation of the Shapley Value for Game-Theoretic Network Centrality , 2014, J. Artif. Intell. Res..

[8]  Nicholas R. Jennings,et al.  Computational Analysis of Connectivity Games with Applications to the Investigation of Terrorist Networks , 2013, IJCAI.

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

[10]  Michael Wooldridge,et al.  Computational Aspects of Cooperative Game Theory (Synthesis Lectures on Artificial Inetlligence and Machine Learning) , 2011 .

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

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

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

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

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

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

[17]  Istvan Mezo,et al.  The r-Bell numbers , 2009, 0909.4417.

[18]  Vincent Conitzer,et al.  Coalition Structure Generation Utilizing Compact Characteristic Function Representations , 2009, CP.

[19]  Rahul Savani,et al.  Power Indices in Spanning Connectivity Games , 2009, AAIM.

[20]  Michael Wooldridge,et al.  On the computational complexity of weighted voting games , 2009, Annals of Mathematics and Artificial Intelligence.

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

[22]  Jeffrey S. Rosenschein,et al.  Power in threshold network flow games , 2009, Autonomous Agents and Multi-Agent Systems.

[23]  Luigi Palopoli,et al.  On the complexity of core, kernel, and bargaining set , 2008, Artif. Intell..

[24]  Piotr Faliszewski,et al.  Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) Manipulating the Quota in Weighted Voting Games , 2022 .

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

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

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

[28]  Paul E. Dunne,et al.  Multiagent Resource Allocation in the Presence of Externalities , 2005, CEEMAS.

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

[30]  Sang-Seung Yi.  Endogenous Formation of Economic Coalitions: A Survey on the Partition Function Approach , 2003 .

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

[32]  T. Matsui,et al.  A SURVEY OF ALGORITHMS FOR CALCULATING POWER INDICES OF WEIGHTED MAJORITY GAMES , 2000 .

[33]  Martin Aigner,et al.  A Characterization of the bell numbers , 1999, Discret. Math..

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

[35]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[36]  Jerry S. Kelly,et al.  NP-completeness of some problems concerning voting games , 1990 .

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

[38]  L. Shapley.  A Value for n-person Games , 1988 .

[39]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[40]  Roger B. Myerson,et al.  Graphs and Cooperation in Games , 1977, Math. Oper. Res..

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

[42]  Francesco Scarcello,et al.  Coalitional games induced by matching problems: Complexity and islands of tractability for the Shapley value , 2020, Artif. Intell..

[43]  Makoto Yokoo,et al.  Signed Graph Games: Coalitional Games with Friends, Enemies and Allies , 2020, AAMAS.

[44]  J. S. Mateo.  The Shapley Value , 2012 .

[45]  Tomasz P. Michalak,et al.  Anytime coalition structure generation in multi-agent systems with positive or negative externalities , 2012, Artif. Intell..

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

[47]  Roland Bacher,et al.  Determinants of matrices related to the Pascal triangle , 2002 .