Resumen de Positive and normative assessment of voting situations

Annick Laruelle , Federico Valenciano Llovera

• This paper is concerned with the analysis of voting situations. By a voting situation, we mean a situation in which a set of voters faces collective decisionmaking by means of a procedure that specifies when a proposal is to be accepted and when is to be rejected after a vote is cast. More specifically we are interested in the accurate formulation and quantification of several features involved in these situations. Traditionally voting situations have been modelled by simple superadditive games. In this context several measures or �power indices� have been proposed in order to assess the voters� a priori �decisiveness� or �power�.

  This classical approach can be criticized for several reasons. First, the gametheoretical framework and terminology is inadequate and misleading. There is no transferable utility involved, there are not really �players� nor �voters�, for the voting rule only specifies when a proposal will pass: to describe a voting rule one should more properly speak just of �seats�. There are no �coalitions� -a term that suggests cooperation with a purpose where there is only coincidence of vote.

  There are no �marginal contributions� where there is no cake to share. Moreover, as a rule the axiomatic characterizations of power indices either do not fit the specificity of the context, or lack compellingness. Second, the existence and misuse of several indices without a clear interpretation is confusing and does not contribute to their credit. Finally, power indices are often criticized on the basis that the only information they take into account is the voting rule, while the voters� preferences, which clearly influence their capacity of being successful, decisive or lucky, are ignored.

  In this paper we propose a more general model which includes the two separate ingredients in any real-world voting situation: the voting rule and the voters. The voting rule, specifies for a given set of seats when a proposal is to be accepted or rejected depending on the resulting vote configuration. Voters, the second ingredient in a voting situation, are included via their voting behavior, which is summarized by a distribution of probability over the vote configurations. This distribution of probability depends on the preferences of the actual voters over the issues they will have to decide upon, the likelihood of these issues being proposed, etc. In each real-world voting situation these probabilities must be approximated from the available data. Thus this general model, unlike the traditional one, is apt for positive or descriptive purposes.

  Within this general framework we re-examine the concepts of �succes�, �decisiveness� and �luck� that in a more or less clear formulation can be traced a long way back in the literature. This more general setting allows a simple and precise reformulation of these concepts as probabilities which depend on the voting rule and the voters� voting behavior. Moreover, our formulations extend previous purely normative notions to more general positive/descriptive concepts, which can be particularized into some familiar but not always well-understood notions, shedding new light on their meaning and relations.