American Institute of Mathematical Sciences

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
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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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