Separability by semivalues modified for games with coalition structure

Abstract

Abstract. Two games are inseparable by semivalues if both games obtain the same allocation whatever semivalue is considered. The problem of separability by semivalues reduces to separability from the null game. For four or more players, the vector subspace of games inseparable from the null game by semivalues contains games different to zero-game. Now, for five or more players, the consideration of a priori coalition blocks in the player set allows us to reduce in a significant way the dimension of the vector subspace of games inseparable from the null game. For these subspaces we provide basis formed by games of a particular type.

Symmetric Coalitional Binomial Semivalues

“…Since the Banzhaf value β is a particular p-binomial semivalue (p = 1/2), this example also shows that the coalitional p-binomial semivalues, which can be obtained from the work by Albizuri and Zarzuelo [3] or Amer and Giménez [8] by applying Owen's scheme to any p-binomial semivalue, satisfy, in general, none of both properties. That's why we will generalize Alonso and Fiestras' procedure.…”

“…., 1/2) give β[v] (Owen [40]. This latter procedure extends well to any p-binomial semivalue (see Puente [48], Freixas and Puente [29] or Amer and Giménez [8]) by evaluating the derivatives at point (p, p, . .…”

“…Taking into account that players 2 and 3, on one hand, and players 4 and 5 on the other, are symmetric in v, the computation method for binomial semivalues stated by Puente [48] (see also Amer and Giménez [8]) gives…”

“…These authors provide axiomatic characterizations in both cases: the homogeneous one, when a common semivalue is used by unions in the quotient game and by players within each one of them (see also Giménez [30] and Amer and Giménez [8] for this case), and the heterogeneous one, where different semivalues apply in the quotient game and (uniformly) within all unions.…”

“…In Puente [48] it is shown that the binomial semivalues can be computed in a way very close to that of the Banzhaf value. Giménez [30] and Amer and Giménez [8] prove that (a) any other semivalue requires using a geometrical reference system of the semivalue simplex, given by any n different binomial semivalues, and a linear map whose matrix depends on (the partial derivatives of the multilinear extension of) the game and the reference system -it also applies, of course, to the Shapley value, with no integration step-, and (b) the homogeneous coalitional semivalues can be computed by means of a bilinear form whose matrix depends, again, on the game and the reference system. It is worthy mentioning here that in the case of coalitional binomial semivalues Carreras and Magaña's [18] procedure applies as well.…”