Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity

Abstract

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.

An axiomatic characterization of the potential decisiveness index

“…[4], [1], or modifications of them as was done for semivalues in [13]. The study of the ordinal equivalence of these extended measures, topics which have [37] and [30] as seminal works, would also merit an special attention.…”

Voting games with abstention: Linking completeness and weightedness

“…Indeed, this relation is very strong. We consider as in [18] the three separate relations that compose the influence relation defined in [24]. The completeness of each of these relations is a necessary condition for a (3, 2) game to be weighted.…”

“…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.…”

Achievable hierarchies in voting games with abstention

“…There are weighted games not being complete for the in ‡uence relation, something di¤erent to what occurs for simple games. They introduced several extensions of the desirability relation (see also Pongou et al [25]) by considering each condition in the de…nition of I-in ‡uence relation. A stronger form of I-completeness is H-completeness for which all the relations that intervene in the de…nition of completeness coincide.…”