On the Coalitional Rationality and the Egalitarian Nonseparable Contribution

Irinel Dragan

doi:10.4236/am.2019.105026

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.

Highlights

In [1], we introduced a new problem, connected to the Shapley Value, that was called the Inverse Problem, relative to the Shapley Value: Let L ∈ R+n be the Shapley Value of a given TU game
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
We proved the following result: Theorem: The family of almost null games in the Inverse Set, relative to the Shapley Value, is providing a family of TU games in which the ENSC value will be unchanged and coalitional rational, if the parameter satisfies the inequality (13), providing a new coalitional rationality threshold

Summary

Introduction

In [1], we introduced a new problem, connected to the Shapley Value, that was called the Inverse Problem, relative to the Shapley Value: Let L ∈ R+n be the Shapley Value of a given TU game.

Dragan DOI

Dragan

The ENSC Value for the Games in the Almost Null Family

Discussion and Remarks