Voting games with abstention: Linking completeness and weightedness

•We study voting systems in democratic institutions in which abstention is allowed.•Some weighted systems are not complete for the influence relation.•We consider three new relations, each of them weaker than the influence relation.•We determine the desired notion of completeness being coherent with weightedness.•The research carried out is related with the determination of importance rankings.

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.