    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  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 & Games, doi: 10.3934/jdg.2020023
##### References:
  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  P. Borm, H. Hamers and R. Hendrickx, Operations Research games: A survey, TOP, 9 (2001), 139-216.  doi: 10.1007/BF02579075.  Google Scholar  R. Branzei, D. Dimitrov and S. Tijs, Models in Cooperative Game Theory, Springer-Verlag, Berlin, 2008. Google Scholar  T. S. H. Driessen, Properties of 1-convex n-person games, OR Spektrum, 7 (1985), 19-26.  doi: 10.1007/BF01719757.  Google Scholar  T. S. H. Driessen, Cooperative Games, Solutions, and Applications, Kluwer Academic Publishers, Dordrecht, 1988. doi: 10.1007/978-94-015-7787-8. Google Scholar  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.  Google Scholar  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. Google Scholar  Y. Funaki, Upper and lower bounds of the kernel and nucleolus, International Journal of Game Theory, 15 (1986), 121-129.  doi: 10.1007/BF01770980.  Google Scholar  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. Google Scholar  P. Legros, Allocating joint costs by means of the nucleolus, International Journal of Game Theory, 15 (1986), 109-119.  doi: 10.1007/BF01770979.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  L. S. Shapley, A value for n-person games. Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, NJ, 1953, 307–317. Google Scholar  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.  Google Scholar

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##### References:
  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  P. Borm, H. Hamers and R. Hendrickx, Operations Research games: A survey, TOP, 9 (2001), 139-216.  doi: 10.1007/BF02579075.  Google Scholar  R. Branzei, D. Dimitrov and S. Tijs, Models in Cooperative Game Theory, Springer-Verlag, Berlin, 2008. Google Scholar  T. S. H. Driessen, Properties of 1-convex n-person games, OR Spektrum, 7 (1985), 19-26.  doi: 10.1007/BF01719757.  Google Scholar  T. S. H. Driessen, Cooperative Games, Solutions, and Applications, Kluwer Academic Publishers, Dordrecht, 1988. doi: 10.1007/978-94-015-7787-8. Google Scholar  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.  Google Scholar  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. Google Scholar  Y. Funaki, Upper and lower bounds of the kernel and nucleolus, International Journal of Game Theory, 15 (1986), 121-129.  doi: 10.1007/BF01770980.  Google Scholar  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. Google Scholar  P. Legros, Allocating joint costs by means of the nucleolus, International Journal of Game Theory, 15 (1986), 109-119.  doi: 10.1007/BF01770979.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  L. S. Shapley, A value for n-person games. Contributions to the Theory of Games, Vol. 2, Princeton University Press, Princeton, NJ, 1953, 307–317. Google Scholar  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.  Google Scholar
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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