# American Institute of Mathematical Sciences

March  2022, 12(1): 1-14. doi: 10.3934/naco.2021047

## Axiomatic results and dynamic processes for two weighted indexes under fuzzy transferable-utility behavior

 Department of Applied Mathematics, National Pingtung University, Pingtung 900, Taiwan

Received  March 2020 Revised  December 2020 Published  March 2022 Early access  November 2021

By considering the supreme-utilities and the weights simultaneously under fuzzy behavior, we propose two indexes on fuzzy transferable-utility games. In order to present the rationality for these two indexes, we define extended reductions to offer several axiomatic results and dynamics processes. Based on different consideration, we also adopt excess functions to propose alternative formulations and related dynamic processes for these two indexes respectively.

Citation: Yu-Hsien Liao. Axiomatic results and dynamic processes for two weighted indexes under fuzzy transferable-utility behavior. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 1-14. doi: 10.3934/naco.2021047
##### References:
 [1] J. P. Aubin, Coeur et valeur des jeux flous á paiements latéraux, Comptes Rendus de l'Académie des Sciences, 279 (1974), 891-894. [2] J. P. Aubin, Cooperative fuzzy games, Mathematics of Operations Research, 6 (1981), 1-13.  doi: 10.1287/moor.6.1.1. [3] J. F. Banzhaf, Weighted voting doesn't work: a mathematical analysis, Rutgers Law Rev., 19 (1965), 317-343. [4] S. Borkotokey and R. Neog, Dynamic resource allocation in fuzzy coalitions: a game theoretic model, Fuzzy Optimization and Decision Making, 13 (2014), 211-230.  doi: 10.1007/s10700-013-9172-y. [5] R. Branzei, D. Dimitrov and S. Tijs, Egalitarianism in convex fuzzy games, Mathematical Social Sciences, 47 (2004), 313-325.  doi: 10.1016/j.mathsocsci.2003.09.003. [6] D. Butnariu, Fuzzy games: a description of the concept, Fuzzy Sets and Systems, 1 (1978), 181-192.  doi: 10.1016/0165-0114(78)90003-9. [7] S. Hart and A. Mas-Colell, Potential, value and consistency, Econometrica, 57 (1989), 589-614.  doi: 10.2307/1911054. [8] Y. A. Hwang, Fuzzy games: a characterization of the core, Fuzzy Sets and Systems, 158 (2007), 2480-2493.  doi: 10.1016/j.fss.2007.03.009. [9] Y. A. Hwang and Y. H. Liao, Consistency and dynamic approach of indexes, Social Choice and Welfare, 34 (2010), 679-694.  doi: 10.1007/s00355-009-0423-3. [10] S. Li and Q. Zhang, A simplified expression of the Shapley function for fuzzy game, Eur. J. Oper. Res., 196 (2009), 234-245.  doi: 10.1016/j.ejor.2008.02.034. [11] Y. H. Liao, Fuzzy games: a complement-consistent solution, axiomatizations and dynamic approaches, Fuzzy Optimization and Decision Making, 16 (2017), 257-268.  doi: 10.1007/s10700-016-9248-6. [12] Y. H. Liao and L. Y. Chung, Power allocation rules under fuzzy behavior and multicriteria situations, Iranian Journal of Fuzzy Systems, 17 (2020), 187-198. [13] Y. H. Liao, P. H. Wu and L. Y. Chung, The EANSC: a weighted extension and axiomatization, Economics Bulletin, 35 (2015), 475-480. [14] M. Maschler and G. Owen, The consistent Shapley value for hyperplane games, International Journal of Game Theory, 18 (1989), 389-407.  doi: 10.1007/BF01358800. [15] E. Molina and J. Tejada, The equalizer and the lexicographical solutions for cooperative fuzzy games: characterizations and properties, Fuzzy Sets and Systems, 125 (2002), 369-387.  doi: 10.1016/S0165-0114(01)00023-9. [16] H. Moulin, On additive methods to share joint costs, The Japanese Economic Review, 46 (1995), 303-332. [17] S. Muto, S. Ishihara, S. Fukuda, S. Tijs and R. Branzei, Generalized cores and stable sets for fuzzy games, International Game Theory Review, 8 (2006), 95-109.  doi: 10.1142/S0219198906000801. [18] M. Sakawa and I. Nishizaki, A lexicographical concept in an n-person cooperative fuzzy games, Fuzzy Sets and Systems, 61 (1994), 265-275.  doi: 10.1016/0165-0114(94)90169-4. [19] J. S. Ransmeier, The Tennessee Valley Authority, Vanderbilt University Press in Nashville, 1942. [20] L. S. Shapley, Discussant's comment, In Moriarity S (ed) Joint Cost Allocation, University of Oklahoma Press in Tulsa, 1982. [21] 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.  doi: 10.1016/S0165-0114(03)00329-4. [22] M. Tsurumi, T. Tanino and M. Inuiguchi, A Shapley function on a class of cooperative fuzzy games, Eur. J. Oper. Res., 129 (2001), 596-618.  doi: 10.1016/S0377-2217(99)00471-3. [23] H. C. Wei, P. T. Liu and Y. H. Liao, Two optimal allocations under management systems: Game-theoretical approaches, International Journal of Information and Management Sciences, 30 (2019), 99-112.

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##### References:
 [1] J. P. Aubin, Coeur et valeur des jeux flous á paiements latéraux, Comptes Rendus de l'Académie des Sciences, 279 (1974), 891-894. [2] J. P. Aubin, Cooperative fuzzy games, Mathematics of Operations Research, 6 (1981), 1-13.  doi: 10.1287/moor.6.1.1. [3] J. F. Banzhaf, Weighted voting doesn't work: a mathematical analysis, Rutgers Law Rev., 19 (1965), 317-343. [4] S. Borkotokey and R. Neog, Dynamic resource allocation in fuzzy coalitions: a game theoretic model, Fuzzy Optimization and Decision Making, 13 (2014), 211-230.  doi: 10.1007/s10700-013-9172-y. [5] R. Branzei, D. Dimitrov and S. Tijs, Egalitarianism in convex fuzzy games, Mathematical Social Sciences, 47 (2004), 313-325.  doi: 10.1016/j.mathsocsci.2003.09.003. [6] D. Butnariu, Fuzzy games: a description of the concept, Fuzzy Sets and Systems, 1 (1978), 181-192.  doi: 10.1016/0165-0114(78)90003-9. [7] S. Hart and A. Mas-Colell, Potential, value and consistency, Econometrica, 57 (1989), 589-614.  doi: 10.2307/1911054. [8] Y. A. Hwang, Fuzzy games: a characterization of the core, Fuzzy Sets and Systems, 158 (2007), 2480-2493.  doi: 10.1016/j.fss.2007.03.009. [9] Y. A. Hwang and Y. H. Liao, Consistency and dynamic approach of indexes, Social Choice and Welfare, 34 (2010), 679-694.  doi: 10.1007/s00355-009-0423-3. [10] S. Li and Q. Zhang, A simplified expression of the Shapley function for fuzzy game, Eur. J. Oper. Res., 196 (2009), 234-245.  doi: 10.1016/j.ejor.2008.02.034. [11] Y. H. Liao, Fuzzy games: a complement-consistent solution, axiomatizations and dynamic approaches, Fuzzy Optimization and Decision Making, 16 (2017), 257-268.  doi: 10.1007/s10700-016-9248-6. [12] Y. H. Liao and L. Y. Chung, Power allocation rules under fuzzy behavior and multicriteria situations, Iranian Journal of Fuzzy Systems, 17 (2020), 187-198. [13] Y. H. Liao, P. H. Wu and L. Y. Chung, The EANSC: a weighted extension and axiomatization, Economics Bulletin, 35 (2015), 475-480. [14] M. Maschler and G. Owen, The consistent Shapley value for hyperplane games, International Journal of Game Theory, 18 (1989), 389-407.  doi: 10.1007/BF01358800. [15] E. Molina and J. Tejada, The equalizer and the lexicographical solutions for cooperative fuzzy games: characterizations and properties, Fuzzy Sets and Systems, 125 (2002), 369-387.  doi: 10.1016/S0165-0114(01)00023-9. [16] H. Moulin, On additive methods to share joint costs, The Japanese Economic Review, 46 (1995), 303-332. [17] S. Muto, S. Ishihara, S. Fukuda, S. Tijs and R. Branzei, Generalized cores and stable sets for fuzzy games, International Game Theory Review, 8 (2006), 95-109.  doi: 10.1142/S0219198906000801. [18] M. Sakawa and I. Nishizaki, A lexicographical concept in an n-person cooperative fuzzy games, Fuzzy Sets and Systems, 61 (1994), 265-275.  doi: 10.1016/0165-0114(94)90169-4. [19] J. S. Ransmeier, The Tennessee Valley Authority, Vanderbilt University Press in Nashville, 1942. [20] L. S. Shapley, Discussant's comment, In Moriarity S (ed) Joint Cost Allocation, University of Oklahoma Press in Tulsa, 1982. [21] 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.  doi: 10.1016/S0165-0114(03)00329-4. [22] M. Tsurumi, T. Tanino and M. Inuiguchi, A Shapley function on a class of cooperative fuzzy games, Eur. J. Oper. Res., 129 (2001), 596-618.  doi: 10.1016/S0377-2217(99)00471-3. [23] H. C. Wei, P. T. Liu and Y. H. Liao, Two optimal allocations under management systems: Game-theoretical approaches, International Journal of Information and Management Sciences, 30 (2019), 99-112.
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