# American Institute of Mathematical Sciences

April  2021, 8(2): 139-150. doi: 10.3934/jdg.2020023

## On the equal surplus sharing interval solutions and an application

 1 Süleyman Demirel University, Faculty of Economics and Administrative Sciences, Department of Business Administration, Isparta, 32260, Turkey 2 Usak University, Faculty of Education, Department of Mathematics and Science Education, Usak, 64000, Turkey 3 Süleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, Isparta, 32260, Turkey

* Corresponding author: zeynepalparslan@yahoo.com

Received  November 2019 Revised  February 2020 Published  April 2021 Early access  July 2020

In this paper, we focus on the equal surplus sharing interval solutions for cooperative games, where the set of players are finite and the coalition values are interval numbers. We consider the properties of a class of equal surplus sharing interval solutions consisting of all convex combinations of them. Moreover, an application based on transportation interval situations is given. Finally, we propose three solution concepts, namely the interval Shapley value, ICIS-value and IENSC-value, for this application and these solution concepts are compared.

Citation: Osman Palanci, Mustafa Ekici, Sirma Zeynep Alparslan Gök. On the equal surplus sharing interval solutions and an application. Journal of Dynamics and Games, 2021, 8 (2) : 139-150. doi: 10.3934/jdg.2020023
##### References:
 [1] S. Z. Alparslan Gök, R. Branzei and S. Tijs, Cores and Stable Sets for Interval-Valued Games, CentER Discussion Paper No. 2018-17, (2008), 15 pp. doi: 10.2139/ssrn.1094653. [2] S. Z. Alparslan Gök, R. Branzei and S. Tijs, Convex interval games, Journal of Applied Mathematics and Decision Sciences, (2009) Art. ID 342089, 14 pp. doi: 10.1155/2009/342089. [3] S. Z. Alparslan Gök, S. Miquel and S. Tijs, Cooperation under interval uncertainty, Mathematical Methods of Operations Research, 69 (2009), 99-109.  doi: 10.1007/s00186-008-0211-3. [4] P. Borm, H. Hamers and R. Hendrickx, Operations Research games: A survey, TOP, 9 (2001), 139-216.  doi: 10.1007/BF02579075. [5] R. Branzei, D. Dimitrov and S. Tijs, Models in Cooperative Game Theory, Springer-Verlag, Berlin, 2008. [6] T. S. H. Driessen, Properties of 1-convex n-person games, OR Spektrum, 7 (1985), 19-26.  doi: 10.1007/BF01719757. [7] T. S. H. Driessen, Cooperative Games, Solutions, and Applications, Kluwer Academic Publishers, Dordrecht, 1988. doi: 10.1007/978-94-015-7787-8. [8] T. S. H. Driessen and Y. Funaki, Coincidence of and collinearity between game-theoretic solutions, OR Spektrum, 13 (1991), 15-30.  doi: 10.1007/BF01719767. [9] T. S. H. Driessen and Y. Funaki, Reduced game properties of egalitarian division rules for cooperative games, In Operations Research '93, Physica, Heidelberg, 1994, 126–129. doi: 10.1007/978-3-642-46955-8_33. [10] Y. Funaki, Upper and lower bounds of the kernel and nucleolus, International Journal of Game Theory, 15 (1986), 121-129.  doi: 10.1007/BF01770980. [11] C. Kiekintveld, T. Islam and V. Kreinovich, Security games with interval uncertainty, AAMAS Conference: Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems, (2013), 231–238. [12] P. Legros, Allocating joint costs by means of the nucleolus, International Journal of Game Theory, 15 (1986), 109-119.  doi: 10.1007/BF01770979. [13] H. Moulin, The separability axiom and equal-sharing methods, Journal of Economic Theory, 36 (1985), 120-148.  doi: 10.1016/0022-0531(85)90082-1. [14] O. Palancı, S. Z. Alparslan Gök, 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. [15] J. Sánchez-Soriano, M. A. López and I. García-Jurado, On the core of transportation games, Mathematical Social Sciences, 41 (2001), 215-225.  doi: 10.1016/S0165-4896(00)00057-3. [16] L. S. Shapley, A value for n-person games,Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, NJ, 1953. [17] R. van den Brink and Y. Funaki, Axiomatizations of a class of equal surplus sharing solutions for TU-games, Theory and Decision, 67 (2009), 303-340.  doi: 10.1007/s11238-007-9083-x.

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##### References:
 [1] S. Z. Alparslan Gök, R. Branzei and S. Tijs, Cores and Stable Sets for Interval-Valued Games, CentER Discussion Paper No. 2018-17, (2008), 15 pp. doi: 10.2139/ssrn.1094653. [2] S. Z. Alparslan Gök, R. Branzei and S. Tijs, Convex interval games, Journal of Applied Mathematics and Decision Sciences, (2009) Art. ID 342089, 14 pp. doi: 10.1155/2009/342089. [3] S. Z. Alparslan Gök, S. Miquel and S. Tijs, Cooperation under interval uncertainty, Mathematical Methods of Operations Research, 69 (2009), 99-109.  doi: 10.1007/s00186-008-0211-3. [4] P. Borm, H. Hamers and R. Hendrickx, Operations Research games: A survey, TOP, 9 (2001), 139-216.  doi: 10.1007/BF02579075. [5] R. Branzei, D. Dimitrov and S. Tijs, Models in Cooperative Game Theory, Springer-Verlag, Berlin, 2008. [6] T. S. H. Driessen, Properties of 1-convex n-person games, OR Spektrum, 7 (1985), 19-26.  doi: 10.1007/BF01719757. [7] T. S. H. Driessen, Cooperative Games, Solutions, and Applications, Kluwer Academic Publishers, Dordrecht, 1988. doi: 10.1007/978-94-015-7787-8. [8] T. S. H. Driessen and Y. Funaki, Coincidence of and collinearity between game-theoretic solutions, OR Spektrum, 13 (1991), 15-30.  doi: 10.1007/BF01719767. [9] T. S. H. Driessen and Y. Funaki, Reduced game properties of egalitarian division rules for cooperative games, In Operations Research '93, Physica, Heidelberg, 1994, 126–129. doi: 10.1007/978-3-642-46955-8_33. [10] Y. Funaki, Upper and lower bounds of the kernel and nucleolus, International Journal of Game Theory, 15 (1986), 121-129.  doi: 10.1007/BF01770980. [11] C. Kiekintveld, T. Islam and V. Kreinovich, Security games with interval uncertainty, AAMAS Conference: Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems, (2013), 231–238. [12] P. Legros, Allocating joint costs by means of the nucleolus, International Journal of Game Theory, 15 (1986), 109-119.  doi: 10.1007/BF01770979. [13] H. Moulin, The separability axiom and equal-sharing methods, Journal of Economic Theory, 36 (1985), 120-148.  doi: 10.1016/0022-0531(85)90082-1. [14] O. Palancı, S. Z. Alparslan Gök, 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. [15] J. Sánchez-Soriano, M. A. López and I. García-Jurado, On the core of transportation games, Mathematical Social Sciences, 41 (2001), 215-225.  doi: 10.1016/S0165-4896(00)00057-3. [16] L. S. Shapley, A value for n-person games,Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, NJ, 1953. [17] R. van den Brink and Y. Funaki, Axiomatizations of a class of equal surplus sharing solutions for TU-games, Theory and Decision, 67 (2009), 303-340.  doi: 10.1007/s11238-007-9083-x.
Interval marginal vectors
 $\sigma$ $m_{1}^{\sigma}\left( w\right)$ $m_{2}^{\sigma}\left( w\right)$ $m_{3}^{\sigma}\left( w\right)$ $\sigma_{1} = \left( 1,2,3\right)$ $\left[ 0,0\right]$ $\left[ 6,20\right]$ $\left[ 5,8\right]$ $\sigma_{2} = \left( 1,3,2\right)$ $\left[ 0,0\right]$ $\left[ 6,10\right]$ $\left[ 5,18\right]$ $\sigma_{3} = \left( 2,1,3\right)$ $\left[ 6,20\right]$ $\left[ 0,0\right]$ $\left[ 5,8\right]$ $\sigma_{4} = \left( 2,3,1\right)$ $\left[ 11,28\right]$ $\left[ 0,0\right]$ $\left[ 0,0\right]$ $\sigma_{5} = \left( 3,1,2\right)$ $\left[ 5,18\right]$ $\left[ 6,10\right]$ $\left[ 0,0\right]$ $\sigma_{6} = \left( 3,2,1\right)$ $\left[ 11,28\right]$ $\left[ 0,0\right]$ $\left[ 0,0\right]$
 $\sigma$ $m_{1}^{\sigma}\left( w\right)$ $m_{2}^{\sigma}\left( w\right)$ $m_{3}^{\sigma}\left( w\right)$ $\sigma_{1} = \left( 1,2,3\right)$ $\left[ 0,0\right]$ $\left[ 6,20\right]$ $\left[ 5,8\right]$ $\sigma_{2} = \left( 1,3,2\right)$ $\left[ 0,0\right]$ $\left[ 6,10\right]$ $\left[ 5,18\right]$ $\sigma_{3} = \left( 2,1,3\right)$ $\left[ 6,20\right]$ $\left[ 0,0\right]$ $\left[ 5,8\right]$ $\sigma_{4} = \left( 2,3,1\right)$ $\left[ 11,28\right]$ $\left[ 0,0\right]$ $\left[ 0,0\right]$ $\sigma_{5} = \left( 3,1,2\right)$ $\left[ 5,18\right]$ $\left[ 6,10\right]$ $\left[ 0,0\right]$ $\sigma_{6} = \left( 3,2,1\right)$ $\left[ 11,28\right]$ $\left[ 0,0\right]$ $\left[ 0,0\right]$
The equal surplus sharing interval solutions of Example 5.4
 Interval Solutions Player 1 Player 2 Player 3 Interval Shapley value $\left[ 5\tfrac{1}{2},15\tfrac{2}{3}\right]$ $\left[ 3,6\tfrac{2}{3}\right]$ $\left[ 2\tfrac{1}{2},5\tfrac{2}{3}\right]$ ICIS-value $\left[ 3\tfrac{2}{3},9\tfrac{1}{3}\right]$ $\left[ 3\tfrac{2}{3},9\tfrac{1}{3}\right]$ $\left[ 3\tfrac{2}{3},9\tfrac{1}{3}\right]$ IENSC-value $\left[ 7\tfrac{1}{3},22\right]$ $\left[ 2\tfrac {1}{3},4\right]$ $\left[ 1\tfrac{1}{3},2\right]$
 Interval Solutions Player 1 Player 2 Player 3 Interval Shapley value $\left[ 5\tfrac{1}{2},15\tfrac{2}{3}\right]$ $\left[ 3,6\tfrac{2}{3}\right]$ $\left[ 2\tfrac{1}{2},5\tfrac{2}{3}\right]$ ICIS-value $\left[ 3\tfrac{2}{3},9\tfrac{1}{3}\right]$ $\left[ 3\tfrac{2}{3},9\tfrac{1}{3}\right]$ $\left[ 3\tfrac{2}{3},9\tfrac{1}{3}\right]$ IENSC-value $\left[ 7\tfrac{1}{3},22\right]$ $\left[ 2\tfrac {1}{3},4\right]$ $\left[ 1\tfrac{1}{3},2\right]$
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