Mew mathematical properties of the semivalues of cooperative tu games^*

Abstract

The Semivalues were introduced by Dubey, Neiman and Weber (1981). as thc values of TU games satisfying a set of axioms, precisely: linearity, symmetry, monotonicity and projection axioms. In thin paper, a potential approach is used to prove a characterization of Semivalues as the unique values satisfying a new type of consistency relative to a new implicitly defined reduced game of HartIMas CoIelI type and weighted standardness for all two person games. The definition of the reduced game requires some combinatorial results on the auxiliary game of the given game. As a byproduct, we derive from these results two other combinatorial properties of the Semivalues: (i) any Semivalue is the Shapley value of the auxiliary game of the given game, and (ii) any Semivalue satisfies the fairness principle introduced by Myerson (1977) as the principle of balanced contributions.

On the Coalitional Rationality and the Inverse Problem for Shapley Value and the Semivalues

“…As in the previous section, any TU game may get a representation in terms of a potential basis of the vector space, as shown in [1]; the Inverse Set relative to a Semivalue is given by a similar formula. The difference is that the basic vectors are given by more general expressions, depending on the weight vectors of the Semivalues.…”

“…The solution of the Inverse Problem can be also obtained from [1]. As the weight vectors for the Banzhaf Value of a three person game are…”

“…Now, the scalar form of the Inverse Set relative to the Banzhaf Value (see [1], Example 2), is Note that in general the efficiency is missing; therefore we shall define the Coalitional rationality by means of the Power Game in which the efficiency holds. Recall that the Power Game, for this game (see [4], 3.1), is given by…”

“…Then, as shown in [1], (Th.2), the coefficients of the expansion of the given game in the potential basis of the vector space of three person TU games are 2 2 3 2 3 2 3 2 3 2 2 3 2 3 2 3 2 3 1 1 1 2 1 3 2 1 1 2,1 1, …”

“…In an earlier paper of the author, ( [1]), it was introduced and solved what was called the Inverse Problem for Semivalues of cooperative TU games. The present paper is starting with two examples of Semivalues (see [2] …”