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On the grey Baker-Thompson rule

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  • Cost sharing problems can arise from situations in which some service is provided to a variety of different customers who differ in the amount or type of service they need. One can think of and airports computers, telephones. This paper studies an airport problem which is concerned with the cost sharing of an airstrip between airplanes assuming that one airstrip is sufficient to serve all airplanes. Each airplane needs an airstrip whose length can be different across airplanes. Also, it is important how should the cost of each airstrip be shared among airplanes. The purpose of the present paper is to give an axiomatic characterization of the Baker-Thompson rule by using grey calculus. Further, it is shown that each of our main axioms (population fairness, smallest-cost consistency and balanced population impact) together with various combina tions of our minor axioms characterizes the best-known rule for the problem, namely the Baker-Thompson rule. Finally, it is demonstrated that the grey Shapley value of airport game and the grey Baker-Thompson rule coincides.

    Mathematics Subject Classification: Primary: 91A12.

    Citation:

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  • Figure 1.  The flowchart of this study

    Table 1.  Grey marginal vectors

    $\sigma $ $m_{1}^{\sigma }\left(c^{\prime }\right) $ $m_{2}^{\sigma }\left(c^{\prime }\right) $ $m_{3}^{\sigma }\left(c^{\prime }\right) $
    $\sigma _{1} = \left(1, 2, 3\right) $ $m_{1}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[30, 36\right] $ $m_{2}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[14, 18\right] $ $m_{3}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[16, 26\right] $
    $\sigma _{2} = \left(1, 3, 2\right) $ $m_{1}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[30, 36\right] $ $m_{2}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{3}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[30, 44\right] $
    $\sigma _{3} = \left(2, 1, 3\right) $ $m_{1}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{2}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[44, 54\right] $ $m_{3}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[16, 26\right] $
    $\sigma _{4} = \left(2, 3, 1\right) $ $m_{1}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{2}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[44, 54\right] $ $m_{3}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[16, 26\right] $
    $\sigma _{5} = \left(3, 1, 2\right) $ $m_{1}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{2}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{3}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[60, 80\right] $
    $\sigma _{6} = \left(3, 2, 1\right) $ $m_{1}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{2}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[0, 0\right] $ $m_{3}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[60, 80\right] $
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