Soft cooperation systems and games

Abstract A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. In order to obtain this allocation, a characteristic function is known. It establishes the profit of each coalition of agents if this coalition decides to act alone. Originally players are considered symmetric and then the allocation only depends on the characteristic function; this paper is about cooperative games with an asymmetric set of agents. We introduced cooperative games with a soft set of agents which explains those parameters determining the asymmetry among them in the cooperation. Now the characteristic function is defined not over the coalitions but over the soft coalitions, namely the profit depends not only on the formed coalition but also on the attributes considered for the players in the coalition. The best known of the allocation rules for cooperative games is the Shapley value. We propose a Shapley kind solution for soft games.

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

[2]  Lei Wei,et al.  Caristi Type Coincidence Point Theorem in Topological Spaces , 2013, J. Appl. Math..

[3]  D. Molodtsov.  Soft set theory—First results , 1999 .

[4]  Aslihan Sezgin Sezer,et al.  On operations of soft sets , 2011, Comput. Math. Appl..

[5]  Surajit Borkotokey,et al.  The Shapley value of cooperative games under fuzzy settings: a survey , 2014, Int. J. Gen. Syst..

[6]  Cheng-Fu Yang,et al.  A note on "Soft Set Theory" [Comput. Math. Appl 45 (4-5) (2003) 555-562] , 2008, Comput. Math. Appl..

[7]  Andrés Jiménez-Losada,et al.  Fuzzy restrictions and an application to cooperative games with restricted cooperation , 2017, Int. J. Gen. Syst..

[8]  Andrés Jiménez-Losada,et al.  Duality on combinatorial structures. An application to cooperative games , 2017, Int. J. Gen. Syst..

[9]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[10]  Jesús Mario Bilbao,et al.  Cooperative games on antimatroids , 2004, Discret. Math..

[11]  Qiaoyan Wen,et al.  Operations on Soft Sets Revisited , 2012, J. Appl. Math..

[12]  Theo S. H. Driessen,et al.  The Shapley value for games on matroids: The static model , 2001, Math. Methods Oper. Res..

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

[14]  Xun Ge,et al.  Investigations on Some Operations of Soft Sets , 2011 .

[15]  Michel Grabisch,et al.  Games on fuzzy communication structures with Choquet players , 2010, Eur. J. Oper. Res..

[16]  Andrés Jiménez-Losada,et al.  Cooperation among agents with a proximity relation , 2016, Eur. J. Oper. Res..

[17]  Jesús Mario Bilbao,et al.  The Shapley value on convex geometries , 2000, Discret. Appl. Math..

[18]  Andrés Jiménez-Losada,et al.  Games with fuzzy authorization structure: A Shapley value , 2015, Fuzzy Sets Syst..

[19]  M. Gloria Fiestras-Janeiro,et al.  Some structural properties of a lattice of embedded coalitions , 2016, Int. J. Gen. Syst..

[20]  Theo S. H. Driessen,et al.  The Shapley value for games on matroids: The dynamic model , 2001, Math. Methods Oper. Res..

[21]  R. J. Aumann,et al.  Cooperative games with coalition structures , 1974 .

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

[23]  Michel Grabisch,et al.  Games on lattices, multichoice games and the shapley value: a new approach , 2007, Math. Methods Oper. Res..

[24]  Xiaoyan Liu,et al.  On some new operations in soft set theory , 2009, Comput. Math. Appl..

[25]  A. R. Roy,et al.  Soft set theory , 2003 .

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

[27]  Rudolf Henn,et al.  Mathematical Economics and Game Theory , 1977 .

[28]  G. Owen,et al.  Games with permission structures: The conjunctive approach , 1992 .