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# On the equal surplus sharing interval solutions and an application

• * Corresponding author: zeynepalparslan@yahoo.com
• 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.

Mathematics Subject Classification: Primary:91A12.

 Citation: • • Table 1.  Interval marginal vectors

 $\sigma$ $m_{1}^{\sigma}\left( w\right)$ $m_{2}^{\sigma}\left( w\right)$ $m_{3}^{\sigma}\left( w\right)$ $\sigma_{1} = \left( 1,2,3\right)$ $\left[ 0,0\right]$ $\left[ 6,20\right]$ $\left[ 5,8\right]$ $\sigma_{2} = \left( 1,3,2\right)$ $\left[ 0,0\right]$ $\left[ 6,10\right]$ $\left[ 5,18\right]$ $\sigma_{3} = \left( 2,1,3\right)$ $\left[ 6,20\right]$ $\left[ 0,0\right]$ $\left[ 5,8\right]$ $\sigma_{4} = \left( 2,3,1\right)$ $\left[ 11,28\right]$ $\left[ 0,0\right]$ $\left[ 0,0\right]$ $\sigma_{5} = \left( 3,1,2\right)$ $\left[ 5,18\right]$ $\left[ 6,10\right]$ $\left[ 0,0\right]$ $\sigma_{6} = \left( 3,2,1\right)$ $\left[ 11,28\right]$ $\left[ 0,0\right]$ $\left[ 0,0\right]$

Table 2.  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 $\left[ 7\tfrac{1}{3},22\right]$ $\left[ 2\tfrac {1}{3},4\right]$ $\left[ 1\tfrac{1}{3},2\right]$
• Tables(2)

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