Egalitarian Allocations and the Inverse Problem for the Shapley Value

Irinel Dragan

doi:10.4236/ajor.2018.86025

Abstract

In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one. In the Inverse Set, we determined a family of games for which the Shapley Value is also a coalitional rational value. The Egalitarian Allocation of the game is efficient, so that in the set called the Inverse Set relative to the Shapley Value, the allocation is the same as the initial one, but may not be coalitional rational. In this paper, we shall find out in the same family of the Inverse Set, a subfamily of games with the Egalitarian Allocation is also a coalitional rational value. We show some relationship between the two sets of games, where our values are coalitional rational. Finally, we shall discuss the possibility that our procedure may be used for solving a very similar problem for other efficient values. Numerical examples show the procedure to get solutions for the efficient values.

Highlights

A cooperative transferable utilities game (TU game), is a pair ( N, v), where N is a finite set, the set of players, and v : P ( N ) → R, the characteristic function, is defined on P ( N ), the set of subsets of N, with v (∅) =0
In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players
The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one

Summary

Introduction

A cooperative transferable utilities game (TU game), is a pair ( N, v) , where N is a finite set, the set of players, and v : P ( N ) → R , the characteristic function, is defined on P ( N ) , the set of subsets of N, with v (∅) =0. Simpler value is the Egalitarian Allocation, EAi ( N= , v) v(N) Both values have the property of efficiency, that is the sum of individual wins makes v ( N ). We shall see later the difference between these two games This means that in both cases there is a small chance that the grand coalition will be formed, taking into account that the coalitions with two players give some better wins. In an earlier work, (see [1]), we introduced and solved what we called the Inverse Problem, relative to the Shapley Value: for a game with a computed Shapley Value, finding out the set of all games with the same Shapley Value. In a more recent work, (see [3]), we intro-

Dragan DOI

Egalitarian Allocations for Three Person Games

Egalitarian Allocations and Coalitional Rationality

Discussions and Remarks