The Stochastic Shapley Value for coalitional games with externalities

A long debated but still open question in the game theory literature is that of how to extend the Shapley Value to coalitional games with externalities. While previous work predominantly focused on developing alternative axiomatizations, in this article we propose a novel approach which centers around the coalition formation process and the underlying probability distribution from which a suitable axiomatization naturally follows. Specifically, we view coalition formation in games with externalities as a discrete-time stochastic process. We focus, in particular, on the Chinese Restaurant Process – a well-known stochastic process from probability theory. We show that reformulating Shapley's coalition formation process as the Chinese Restaurant Process yields in games with externalities a unique value with various desirable properties. We then generalize this result by proving that all values that satisfy the direct translation of Shapley's axioms to games with externalities can be obtained using our approach based on stochastic processes.

[1]  Franz Lehner.  Cumulants in noncommutative probability theory IV. Noncrossing cumulants: De Finetti's theorem and Lp-inequalities , 2006 .

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

[3]  Isa Emin Hafalir,et al.  Efficiency in coalition games with externalities , 2007, Games Econ. Behav..

[4]  Roded Sharan,et al.  Bayesian haplo-type inference via the dirichlet process , 2004, ICML.

[5]  Michel Grabisch,et al.  A coalition formation value for games with externalities , 2011 .

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

[7]  Michael Finus,et al.  New developments in coalition theory : An application to the case of global pollution , 2003 .

[8]  G. Owen.  VALUES OF GAMES WITH A PRIORI UNIONS , 1977 .

[9]  R. Myerson.  Conference structures and fair allocation rules , 1978 .

[10]  Zhaohui S. Qin,et al.  Clustering microarray gene expression data using weighted Chinese restaurant process , 2006, Bioinform..

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

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

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

[14]  Radford M. Neal.  Markov Chain Sampling Methods for Dirichlet Process Mixture Models , 2000 .

[15]  Thomas L. Griffiths,et al.  Hierarchical Topic Models and the Nested Chinese Restaurant Process , 2003, NIPS.

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

[17]  Javier Arin,et al.  An Axiom System for a Value for Games in Partition Function Form , 2005, IGTR.

[18]  Geoffroy de Clippel.  Marginal Contributions and Externalities in the Value , 2007 .

[19]  W. Ewens.  The sampling theory of selectively neutral alleles. , 1972, Theoretical population biology.

[20]  H. Young.  Monotonic solutions of cooperative games , 1985 .

[21]  Peter McCullagh,et al.  Random Permutations and Partition Models , 2011, International Encyclopedia of Statistical Science.

[22]  R. Serrano,et al.  Marginal Contributions and Externalities in the Value , 2005 .

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

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

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