No abstract

Weighted games for several levels of approval in input and output were introduced in [9]. An extension of the desirability relation for simple games, called the influence relation, was introduced for games with several levels of approval in input in [24] (see also [18]). However, there are weighted games not being complete for the influence relation, something different to what occurs for simple games. In this paper we introduce several extensions of the desirability relation for simple games and from the completeness of them it follows the consistent link with weighted games, which solves the existing gap. Moreover, we prove that the influence relation is consistent with a known subclass of weighted games: strongly weighted games.Keywords: Decision making process, Voting systems in democratic organizations, Multiple levels of approval, Weightedness and completeness, Desirability relations 2000 MSC: 91A12, 90B50, 91A35, 05C65, 94C10

We study political in ‡uence in institutions where members choose from among several options their levels of support to a collective goal, these individual choices determining the degree to which the goal is reached. In ‡uence is assessed by newly de…ned binary relations, each of which compares any two individuals on the basis of their relative performance at a corresponding level of participation. For institutions with three levels of support (e.g., voting games in which each voter may vote "yes", "abstain", or vote "no"), we obtain three in ‡uence relations, and show that the strict component of each of them may be cyclical. The cyclicity of these relations contrasts with the transitivity of the unique in ‡uence relation of binary voting games. Weak conditions of anonymity are su¢ cient for each of them to be transitive. We also obtain a necessary and su¢ cient condition for each of them to be complete. Further, we characterize institutions for which the rankings induced by these relations, and the Banzhaf-Coleman and Shapley-Shubik power indices coincide. We argue that the extension of these relations to …rms would be useful in e¢ ciently allocating workers to di¤erent units of production.Applications to various forms of political and economic organizations are provided.

This work focuses on multi-type games in which there are a number of nonordered types in the input, while the output consists of a single real value. When considering the dichotomous case, we extend the Shapley-Shubik power index and provide a full characterization of this extension. Our results generalize the literature on classical cooperative games.