# American Institute of Mathematical Sciences

March  2020, 16(2): 919-931. doi: 10.3934/jimo.2018185

## Extension of generalized solidarity values to interval-valued cooperative games

 a. School of Economics and Management, Fuzhou University, Fuzhou, Fujian 350108, China b. School of Architecture, Fuzhou University, Fuzhou, Fujian 350108, China

* Corresponding author: School of Economics and Management, Fuzhou University, No. 2, Xueyuan Road, Daxue New District, Fuzhou District, Fuzhou, Fujian 350108, China. Tel./Fax: +86-0591-83768427; E-mail: lidengfeng@fzu.edu.cn, feiwei@fzu.edu.cn

Received  March 2018 Revised  August 2018 Published  December 2018

The main purpose of this paper is to extend the concept of generalized solidarity values to interval-valued cooperative games and hereby develop a simplified and fast approach for solving a subclass of interval-valued cooperative games. In this paper, we find some weaker coalition monotonicity-like conditions so that the generalized solidarity values of the $\alpha$-cooperative games associated with interval-valued cooperative games are always monotonic and non-decreasing functions of any parameter $\alpha \in [0,1]$. Thereby the interval-valued generalized solidarity values can be directly and explicitly obtained by computing their lower and upper bounds through only using the lower and upper bounds of the interval-valued coalitions' values, respectively. The developed method does not use the interval subtraction and hereby can effectively avoid the issues resulted from it. Furthermore, we discuss the effect of the parameter $\xi$ on the interval-valued generalized solidarity values of interval-valued cooperative games and some significant properties of interval-valued generalized solidarity values.

Citation: Deng-Feng Li, Yin-Fang Ye, Wei Fei. Extension of generalized solidarity values to interval-valued cooperative games. Journal of Industrial & Management Optimization, 2020, 16 (2) : 919-931. doi: 10.3934/jimo.2018185
##### References:
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Alparslan G$\rm{\ddot{o}}$k and G. W. Weber, An axiomatization of the interval Shapley value and on some interval solution concepts, Contributions to Game Theory and Management, 8 (2015), 243-251.   Google Scholar [28] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.  Google Scholar [29] S. K. Roy and P. Mula, Solving matrix game with rough payoffs using genetic algorithm, Operational Research: An International Journal, 16 (2016), 117-130.   Google Scholar [30] S. K. Roy, G. Maity and G. W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, Central European Journal of Operations Research, 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5.  Google Scholar [31] E. Savku and G. W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, Journal of Optimization Theory and Applications, 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.  Google Scholar [32] G. J. Xu, H. Dai, D. S. Hou and H. Sun, A-potential function and a non-cooperative foundation for the Solidarity value, Operations Research Letters, 44 (2016), 86-91.  doi: 10.1016/j.orl.2015.12.002.  Google Scholar

show all references

##### References:
 [1] S. Z. Alparslan G$\rm{\ddot{o}}$k, On the interval Shapley value, Optimization, 63 (2014), 747-755.  doi: 10.1080/02331934.2012.686999.  Google Scholar [2] S. Z. Alparslan G$\rm{\ddot{o}}$k, O. Branzei, R. Branzei and S. Tijs, Set-valued solution concepts using interval-type payoffs for interval games, Journal of Mathematical Economics, 47 (2011), 621-626.  doi: 10.1016/j.jmateco.2011.08.008.  Google Scholar [3] S. B$\rm{\acute{e}}$al, E. R$\rm{\acute{e}}$mila and P. Solal, Axiomatization and implementation of a class of solidarity values for TU-games, Theory and Decision, 83 (2017), 61-94.  doi: 10.1007/s11238-017-9586-z.  Google Scholar [4] A. Bhaumik, S. K. Roy and D.-F. Li, Analysis of triangular intuitionistic fuzzy matrix games using robust ranking, Fuzzy Systems, 33 (2017), 327-336.   Google Scholar [5] R. Branzei, O. Branzei, S. Z. Alparslan G$\rm{\ddot{o}}$k and S. Tijs, Cooperative interval games: A survey, Central European Journal of Operations Research, 18 (2010), 397-411.  doi: 10.1007/s10100-009-0116-0.  Google Scholar [6] R. Branzei, D. Dimitrov and S. Tijs, Shapley-like values for interval bankruptcy games, Economics Bulletin, 3 (2003), 1-8.   Google Scholar [7] A. Calik, T. Paksoy, A. Yildizbasi and N. Y. Pehlivan, A decentralized model for allied closed-loop supply chains: Comparative analysis of interactive fuzzy programming approaches, International Journal of Fuzzy Systems, 19 (2017), 367-382.   Google Scholar [8] E. Calvo and E. Guti$\rm{\acute{e}}$rrez-L$\rm{\acute{o}}$pez, Axiomatic characterizations of the weighted solidarity values, Mathematical Social Sciences, 71 (2014), 6-11.  doi: 10.1016/j.mathsocsci.2014.03.005.  Google Scholar [9] A. Casajus and F. Huettner, On a class of solidarity values, European Journal of Operational Research, 236 (2014), 583-591.  doi: 10.1016/j.ejor.2013.12.015.  Google Scholar [10] F. Guan, D. Y. Xie and Q. Zhang, Solutions for generalized interval cooperative games, Fuzzy Systems, 28 (2015), 1553-1564.   Google Scholar [11] W. B. Han, H. Sun and G. J. Xu, A new approach of cooperative interval games: The interval core and Shapley value revisited, Operations Research Letters, 40 (2012), 462-468.  doi: 10.1016/j.orl.2012.08.002.  Google Scholar [12] F. X. Hong and D.-F. Li, Nonlinear programming approach for interval-valued n-person cooperative games, Operational Research: An International Journal, 17 (2017), 479-497.   Google Scholar [13] X. F. Hu and D.-F. Li, A new axiomatization of the Shapley-solidarity value for games with a coalition structure, Operations Research Letters, 46 (2018), 163-167.  doi: 10.1016/j.orl.2017.12.006.  Google Scholar [14] Y. Kamijo and T. Kongo, Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value, European Journal of Operational Research, 216 (2012), 638-646.  doi: 10.1016/j.ejor.2011.08.011.  Google Scholar [15] G. Kara, A. $\rm{\ddot{O}}$zmen and G. W. Weber, Stability advances in robust portfolio optimization under parallelepiped uncertainty, Central European Journal of Operations Research, (2017), 1--21. doi: 10.1007/s10100-017-0508-5.  Google Scholar [16] B. B. Kirlar, S. Erg$\rm{\ddot{u}}$n, S. Z. Alparslan G$\rm{\ddot{o}}$k and G. W. Weber, A game-theoretical and cryptographical approach to crypto-cloud computing and its economical and financial aspects, Annals of Operations Research, 260 (2018), 217-231.  doi: 10.1007/s10479-016-2139-y.  Google Scholar [17] D.-F. Li, Linear programming approach to solve interval-valued matrix games, Omega: The International Journal of management Science, 39 (2011), 655-666.   Google Scholar [18] D.-F. Li and J. C. Liu, A parameterized nonlinear programming approach to solve matrix games with payoffs of I-fuzzy numbers, IEEE Transactions on Fuzzy Systems, 23 (2015), 885-896.   Google Scholar [19] F. Y. Meng, X. H. Chen and C. Q. Tan, Cooperative fuzzy games with interval characteristic functions, Operational Research: An International Journal, 16 (2016), 1-24.   Google Scholar [20] R. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia, 1979.  Google Scholar [21] A. S. Nowak and T. Radzik, A solidarity value for n-person transferable utility games, International Journal of Game Theory, 23 (1994), 43-48.  doi: 10.1007/BF01242845.  Google Scholar [22] B. Oksendal, L. Sandal and J. Uboe, Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models, Control, 37 (2013), 1284-1299.  doi: 10.1016/j.jedc.2013.02.010.  Google Scholar [23] A. $\rm{\ddot{O}}$zmen, E. Kropat and G. W. Weber, Robust optimization in spline regression models for multi-model regulatory networks under polyhedral uncertainty, Optimization, 66 (2017), 2135-2155.  doi: 10.1080/02331934.2016.1209672.  Google Scholar [24] A. $\rm{\ddot{O}}$zmen, G. W. Weber, I. Batmaz and E. Kropat, RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 4780-4787.  doi: 10.1016/j.cnsns.2011.04.001.  Google Scholar [25] O. Palanci, S. Z. Alparslan G$\rm{\ddot{o}}$k, S. Erg$\rm{\ddot{u}}$n and G. W. Weber, Cooperative grey games and the grey Shapley value, Optimization, 64 (2015), 1657-1668.  doi: 10.1080/02331934.2014.956743.  Google Scholar [26] O. Palanci, S. Z. Alparslan G$\rm{\ddot{o}}$k, M. O. Olgun and G. W. Weber, Transportation interval situations and related games, OR Spectrum, 38 (2016), 119-136.  doi: 10.1007/s00291-015-0422-y.  Google Scholar [27] O. Palanci, S. Z. Alparslan G$\rm{\ddot{o}}$k and G. W. Weber, An axiomatization of the interval Shapley value and on some interval solution concepts, Contributions to Game Theory and Management, 8 (2015), 243-251.   Google Scholar [28] M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.  Google Scholar [29] S. K. Roy and P. Mula, Solving matrix game with rough payoffs using genetic algorithm, Operational Research: An International Journal, 16 (2016), 117-130.   Google Scholar [30] S. K. Roy, G. Maity and G. W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, Central European Journal of Operations Research, 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5.  Google Scholar [31] E. Savku and G. W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, Journal of Optimization Theory and Applications, 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.  Google Scholar [32] G. J. Xu, H. Dai, D. S. Hou and H. Sun, A-potential function and a non-cooperative foundation for the Solidarity value, Operations Research Letters, 44 (2016), 86-91.  doi: 10.1016/j.orl.2015.12.002.  Google Scholar
Some interval-valued generalized solidarity values
 ${{{\boldsymbol{\bar \rho }}}^{{\rm{GSV}}\xi }}(\bar \upsilon ')$ $\xi = 0$ $\xi = 0.25$ $\xi = 0.5$ $\xi = 0.75$ $\xi = 1$ $\bar \rho _1^{{\rm{GSV}}\xi }(\bar \upsilon ')$ $[2.14,2.69]$ $[2.31,2.85]$ $[2.45,2.99]$ $[2.56,3.10]$ $[8/3,3.2]$ $\bar \rho _2^{{\rm{GSV}}\xi }(\bar \upsilon ')$ $[2.73,3.23]$ $[2.71,3.23]$ $[2.69,3.22]$ $[2.68,3.21]$ $[8/3,3.2]$ $\bar \rho _3^{{\rm{GSV}}\xi }(\bar \upsilon ')$ $[3.13,3.68]$ $[2.98,3.52]$ $[2.86,3.39]$ $[2.76,3.29]$ $[8/3,3.2]$
 ${{{\boldsymbol{\bar \rho }}}^{{\rm{GSV}}\xi }}(\bar \upsilon ')$ $\xi = 0$ $\xi = 0.25$ $\xi = 0.5$ $\xi = 0.75$ $\xi = 1$ $\bar \rho _1^{{\rm{GSV}}\xi }(\bar \upsilon ')$ $[2.14,2.69]$ $[2.31,2.85]$ $[2.45,2.99]$ $[2.56,3.10]$ $[8/3,3.2]$ $\bar \rho _2^{{\rm{GSV}}\xi }(\bar \upsilon ')$ $[2.73,3.23]$ $[2.71,3.23]$ $[2.69,3.22]$ $[2.68,3.21]$ $[8/3,3.2]$ $\bar \rho _3^{{\rm{GSV}}\xi }(\bar \upsilon ')$ $[3.13,3.68]$ $[2.98,3.52]$ $[2.86,3.39]$ $[2.76,3.29]$ $[8/3,3.2]$
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