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 AM  Vol.10 No.5 , May 2019
On the Coalitional Rationality and the Egalitarian Nonseparable Contribution
Abstract: In earlier works we introduced the Inverse Problem, relative to the Shapley Value, then relative to Semivalues. In the explicit representation of the Inverse Set, the solution set of the Inverse Problem, we built a family of games, called the almost null family, in which we determined more recently a game where the Shapley Value and the Egalitarian Allocations are colalitional rational. The Egalitarian Nonseparable Contribution is another value for cooperative transferable utilities games (TU games), showing how to allocate fairly the win of the grand coalition, in case that this has been formed. In the present paper, we solve the similar problem for this new value: given a nonnegative vector representing the Egalitarian Nonseparable Contribution of a TU game, find out a game in which the Egalitarian Nonseparable Contribution is kept the same, but it is colalitional rational. The new game will belong to the family of almost null games in the Inverse Set, relative to the Shapley Value, and it is proved that the threshold of coalitional rationality will be higher than the one for the Shapley Value. The needed previous results are shown in the introduction, the second section is devoted to the main results, while in the last section are discussed remarks and connected problems. Some numerical examples are illustrating the procedure of finding the new game.
Cite this paper: Dragan, I. (2019) On the Coalitional Rationality and the Egalitarian Nonseparable Contribution. Applied Mathematics, 10, 363-370. doi: 10.4236/am.2019.105026.
References

[1]   Dragan, I. (1991) The Potential Basis and the Weighted Shapley Value. Libertas Mathematica, 11, 139-150.

[2]   Dragan, I. (2014) On the Coalitional Rationality of the Shapley Value and Other Efficient Values of Cooperative TU Games. American Journal of Operations Research, 4, 228-234.
https://doi.org/10.4236/ajor.2014.44022

[3]   Dragan, I. (2015) Coalitional Rationality of the Banzhaf Value and Other Non-Efficient Values of Cooperative TU Games. Applied Mathematics, 6, 2068-2076.
https://doi.org/10.4236/am.2015.612182

[4]   Dragan, I. (2018) Egalitarian Allocations and the Inverse Problem for the Shapley Value. American Journal of Operations Research, 8, 448-456.
https://doi.org/10.4236/ajor.2018.86025

[5]   Driessen, T. and Funaki, Y. (1997) The Egalitarian Nonpairwise Averaged Contribution Value for TU Games. In: Parthasarathy, T., et al., Eds., Game Theoretical Applications to Economics and Operations Research, Kluwer Academic Publishers, Amsterdam, 51-66.
https://doi.org/10.1007/978-1-4757-2640-4_6

 
 
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