Allocation Problems, by the Method of Alternative Representation of the Inverse Set, for Values of Cooperative TU Games

In earlier works, we introduced the Inverse Problem, relative to the Shapley Value, as follows: for a given n-dimensional vector L, find out the transferable utilities’ games , such that The same problem has been discussed further for Semivalues. A connected problem has been considered more recently: find out TU-games for which the Shapley Value equals L, and this value is coalitional rational, that is belongs to the Core of the game . Then, the same problem was discussed for other two linear values: the Egalitarian Allocation and the Egalitarian Nonseparable Contribution, even though these are not Semivalues. To solve such problems, we tried to find a solution in the family of so called Almost Null Games of the Inverse Set, relative to the Shapley Value, by imposing to games in the family, the coalitional rationality conditions. In the present paper, we use the same idea, but a new tool, an Alternative Representation of Semivalues. To get such a representation, the definition of the Binomial Semivalues due to A. Puente was extended to all Semivalues. Then, we looked for a coalitional rational solution in the Family of Almost Null games of the Inverse Set, relative to the Shapley Value. In each case, such games depend on a unique parameter, so that the coalitional rationality will be expressed by a simple inequality, determined by a number, the coalitional rationality threshold. The relationships between the three numbers corresponding to the above three efficient values have been found. Some numerical examples of the method are given.

Some Recursive Definitions for Linear Values of Cooperative TU Games

We give recursive definitions for the Banzhaf Value and the Semivalues of cooperative TU games. These definitions were suggested by the concept of potential for the Shapley Value due to Hart and Mas-Colell and by some results of the author who introduced the potentials of these values and the Power Game of a given game.

AVERAGE RULES FOR COOPERATIVE TU GAMES

In this paper, we propose a new single valued rule based on the concept of fair division for all cooperative transferable utility (TU) games. In any cooperative TU game, primarily the coalitions that are likely to form are identified and each such coalition is fixed with a payoff vector based on the notion of fairness. The value of the single valued rule is obtained from the collection of all coalition structures consisting of the coalitions that are likely to form. The uniqueness of the new rule is followed by its existence and computational simplicity for all TU games. Finally, a linear average rule is defined, and some of its properties are discussed.

A NOTE ON CHARACTERIZING CORE STABILITY WITH FUZZY GAMES

This paper investigates core stability of cooperative (TU) games via a fuzzy extension of the totally balanced cover of a cooperative game. The stability of the core of the fuzzy extension of a game, the concave extension, is shown to reflect the core stability of the original game and vice versa. Stability of the core is then shown to be equivalent to the existence of an equilibrium of a certain correspondence.