The Banzhaf-Owen Value for Fuzzy Games with a Coalition Structure
In this paper, a generalized form of the Banzhaf-Owen value for cooperative fuzzy games with a coalition structure is proposed. Its axiomatic system is given by extending crisp case. In order to better understand the Banzhaf-Owen value for fuzzy games with a coalition structure, we briefly introduce the Banzhaf-Owen values for two special kinds of fuzzy games with a coalition structure, and give their explicit forms.
[1] G. Owen, Values of games with a priori unions, Springer-Verlag, Nueva
York, 1977.
[2] G. Owen, Characterization of the Banzhaf-Coleman index, SIAM Journal
on Applied Mathematics 35 (1978) 315-327.
[3] J. M. Alonso-Meijde and M. G. Fiestras-Janeiro, Modification of the
Banzhaf value for games with a coalition structure, Annals of Operations
Research 109 (2002) 213-227.
[4] M. J. Albizuri, Axiomatizations of the Owen value without efficiency,
Mathematical Social Sciences 55 (2008) 78-89.
[5] J. M. Alonso-Meijide, F. Carreras, M. G. Fiestras-Janeiro and G. Owen,
A comparative axiomatic characterization of the Banzhaf-Owen coalitional
value, Decision Support Systems 43 (2007) 701-712.
[6] R. Amer, F. Carreras and J. M. Gimenez, The modified Banzhaf
value for games with coalition structure: an axiomatic characterization,
Mathematical Social Sciences 43 (2002) 45-54.
[7] G. Hamiache, A new axiomatization of the Owen value for games with
coalition structures, Mathematical Social Sciences 37 (1999) 281-305.
[8] A. B. Khmelnitskaya and E. B. Yanovskaya, Owen coalitional value
without additivity axiom, Mathematical Methods of Operations Research
66 (2007) 255-261.
[9] J. P. Aubin, Mathematical methods of game and economic theory, Rev.
ed., North-Holland, Amsterdam, 1982.
[10] G. Owen, Multilinear extensions of games", Management Sciences 18
(1971) 64-79.
[11] M. Tsurumi, T. Tanino and M. Inuiguchi, A Shapley function on a class
of cooperative fuzzy games, European Journal of Operational Research
129 (2001) 596-618.
[12] D. Butnariu, Stability and Shapley value for an n-persons fuzzy game,
Fuzzy Sets and Systems 4 (1980) 63-72.
[13] D. Butnariu and T. Kroupa, Shapley mappings and the cumulative
value for n-person games with fuzzy Coalitions, European Journal of
Operational Research.186 (2008) 288-299.
[14] F. Y. Meng and Q. Zhang, The Shapley value on a kind of cooperative
fuzzy games, Journal of Computational Information Systems 7
(2011)1846-1854.
[15] Y. A. Hwang, Fuzzy games: A characterization of the core, Fuzzy Sets
and Systems 158 (2007) 2480-2493.
[16] Y. A. Hwang and Y. H. Liao, Max-consistency, complement- consistency
and the core of fuzzy games, Fuzzy Sets and Systems159 (2008) 152-
163.
[17] S. Tijs, R. Branzei, S. Ishihara and S. Muto, On cores and stable sets
for fuzzy games, Fuzzy Sets and Systems 146 (2004) 285-296.
[18] M. Sakawa and I. Nishizalzi, A lexicographical solution concept in an
n-person cooperative fuzzy game, Fuzzy Sets and Systems 61 (1994)
265-275.
[19] X. H. Yu and Q. Zhang, The fuzzy core in games with fuzzy coalitions,
Journal of Computational and Applied Mathematics 230 (2009)173-186.
[20] F. Y. Meng, Q. Zhang and J. Tang, The measure of interaction among
T-fuzzy coalitions, Systems Engineering-Theory Practice 30 (2010) 73-
83. (in Chinese)
[21] F. Y. Meng and Q. Zhang, Fuzzy cooperative games with Choquet
integral form, Systems Engineering and Electronics 32 (2010) 1430-
1436. (in Chinese)
[22] F. Y. Meng and Q. Zhang, The Shapley function for fuzzy cooperative
games with multilinear extension form, Applied Mathematics Letters 23
(2010) 644-650.
[23] E. Lehrer, An axiomatization of the Banzhaf value, International Journal
of Game Theory 17 (1988) 89-99.
[1] G. Owen, Values of games with a priori unions, Springer-Verlag, Nueva
York, 1977.
[2] G. Owen, Characterization of the Banzhaf-Coleman index, SIAM Journal
on Applied Mathematics 35 (1978) 315-327.
[3] J. M. Alonso-Meijde and M. G. Fiestras-Janeiro, Modification of the
Banzhaf value for games with a coalition structure, Annals of Operations
Research 109 (2002) 213-227.
[4] M. J. Albizuri, Axiomatizations of the Owen value without efficiency,
Mathematical Social Sciences 55 (2008) 78-89.
[5] J. M. Alonso-Meijide, F. Carreras, M. G. Fiestras-Janeiro and G. Owen,
A comparative axiomatic characterization of the Banzhaf-Owen coalitional
value, Decision Support Systems 43 (2007) 701-712.
[6] R. Amer, F. Carreras and J. M. Gimenez, The modified Banzhaf
value for games with coalition structure: an axiomatic characterization,
Mathematical Social Sciences 43 (2002) 45-54.
[7] G. Hamiache, A new axiomatization of the Owen value for games with
coalition structures, Mathematical Social Sciences 37 (1999) 281-305.
[8] A. B. Khmelnitskaya and E. B. Yanovskaya, Owen coalitional value
without additivity axiom, Mathematical Methods of Operations Research
66 (2007) 255-261.
[9] J. P. Aubin, Mathematical methods of game and economic theory, Rev.
ed., North-Holland, Amsterdam, 1982.
[10] G. Owen, Multilinear extensions of games", Management Sciences 18
(1971) 64-79.
[11] M. Tsurumi, T. Tanino and M. Inuiguchi, A Shapley function on a class
of cooperative fuzzy games, European Journal of Operational Research
129 (2001) 596-618.
[12] D. Butnariu, Stability and Shapley value for an n-persons fuzzy game,
Fuzzy Sets and Systems 4 (1980) 63-72.
[13] D. Butnariu and T. Kroupa, Shapley mappings and the cumulative
value for n-person games with fuzzy Coalitions, European Journal of
Operational Research.186 (2008) 288-299.
[14] F. Y. Meng and Q. Zhang, The Shapley value on a kind of cooperative
fuzzy games, Journal of Computational Information Systems 7
(2011)1846-1854.
[15] Y. A. Hwang, Fuzzy games: A characterization of the core, Fuzzy Sets
and Systems 158 (2007) 2480-2493.
[16] Y. A. Hwang and Y. H. Liao, Max-consistency, complement- consistency
and the core of fuzzy games, Fuzzy Sets and Systems159 (2008) 152-
163.
[17] S. Tijs, R. Branzei, S. Ishihara and S. Muto, On cores and stable sets
for fuzzy games, Fuzzy Sets and Systems 146 (2004) 285-296.
[18] M. Sakawa and I. Nishizalzi, A lexicographical solution concept in an
n-person cooperative fuzzy game, Fuzzy Sets and Systems 61 (1994)
265-275.
[19] X. H. Yu and Q. Zhang, The fuzzy core in games with fuzzy coalitions,
Journal of Computational and Applied Mathematics 230 (2009)173-186.
[20] F. Y. Meng, Q. Zhang and J. Tang, The measure of interaction among
T-fuzzy coalitions, Systems Engineering-Theory Practice 30 (2010) 73-
83. (in Chinese)
[21] F. Y. Meng and Q. Zhang, Fuzzy cooperative games with Choquet
integral form, Systems Engineering and Electronics 32 (2010) 1430-
1436. (in Chinese)
[22] F. Y. Meng and Q. Zhang, The Shapley function for fuzzy cooperative
games with multilinear extension form, Applied Mathematics Letters 23
(2010) 644-650.
[23] E. Lehrer, An axiomatization of the Banzhaf value, International Journal
of Game Theory 17 (1988) 89-99.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53932", author = "Fan-Yong Meng", title = "The Banzhaf-Owen Value for Fuzzy Games with a Coalition Structure", abstract = "In this paper, a generalized form of the Banzhaf-Owen value for cooperative fuzzy games with a coalition structure is proposed. Its axiomatic system is given by extending crisp case. In order to better understand the Banzhaf-Owen value for fuzzy games with a coalition structure, we briefly introduce the Banzhaf-Owen values for two special kinds of fuzzy games with a coalition structure, and give their explicit forms.
", keywords = "Cooperative fuzzy game, Banzhaf-Owen value, multi linear extension, Choquet integral.", volume = "5", number = "8", pages = "1224-5", }
