Majorities with a quorum

Based on a general model of "quaternary" voting rule, sensitive to voters' choices between four different options (abstaining, voting "yes", voting "no" and staying home), we systematically study different types of majority and quorum. The model allows for a precise formulation of majority rules and quorum constraints. For such rules four types of majority can be defined. We also consider four types of quorum. Then we study the possible combinations of a majority system with a type of quorum and provide examples from rules actually used in parliaments.

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley-Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation. (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author’s final draft

Political Influence in Multi-Choice Institutions: Cyclicity, Anonymity and Transitivity

We study political influence 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. Influence is assessed by newly defined 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 influence 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 influence relation of binary voting games. Weak conditions of anonymity are sufficient for each of them to be transitive. We also obtain a necessary and sufficient 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 firms would be useful in efficiently allocating workers to different units of production. Applications to various forms of political and economic organizations are provided.Level-based influence relations, Multi-choice institutions, cyclicity, anonymity, transitivity

Quaternary dichotomous voting rules

In this paper we provide a general model of "quaternary" dichotomous voting rules (QVRs), namely, voting rules for making collective dichotomous decisions (to accept or reject a proposal), based on vote profiles in which four options are available to each voter: voting ("yes", "no" or "abstaining") or staying home and not turning out. The model covers most of actual real-world dichotomus rules, where quorums are often required, and some of the extensions considered in the literature. In particular, we address and solve the question of the representability of QVRs by means of weighted rules and extend the notion of "dimension" of a rule.

Preferences, actions and voting rules

In this paper we address several issues related to collective dichotomous decision-making by means of quaternary voting rules, i.e., when voters may choose between four actions: voting yes, voting no, abstaining and not turning up - which are aggregated by a voting rule into a dichotomous decision: acceptance or rejection of a proposal. In particular we study the links between the actions and preferences of the actors. We show that quaternary rules (unlike binary rules, where only two actions -yes or no- are possible) leave room for manipulability (i.e., strategic behaviour). Thus a preference profile does not in general determine an action profile. We also deal with the notions of success and decisiveness and their ex ante assessment for quaternary voting rules, and discuss the role of information and coordination in this context.

Probabilistic power indices for games with abstention

S'introdueixen vuit índexs de poder que admeten una interpretació probabilística per les normes de votació amb abstenció o amb tres nivells d'aprovació en l'entrada. S'analitzen les semblances i diferències entre els índexs estàndards coneguts pels jocs simples i per les extensions per aquest context més general. Es conclou la feina proporcionant procediments basats en la generació de funcions per jocs(3,2) extensibles a jocs (j,k).Preprin

Majorities with a quorum

Based on a general model of "quaternary" voting rule, sensitive to voters' choices between four different options (abstaining, voting "yes", voting "no" and staying home), we systematically study different types of majority and quorum. The model allows for a precise formulation of majority rules and quorum constraints. For such rules four types of majority can be defined. We also consider four types of quorum. Then we study the possible combinations of a majority system with a type of quorum and provide examples from rules actually used in parliaments.This research is supported by the Spanish Ministerio de Ciencia e Innovación under projects SEJ2006-05455 and ECO2009-11213, co-funded by ERDF, and from the Basque Government's funding to Grupo Consolidado GIC07/146-IT-377-07

Generalized roll-call model for the Shapley-Shubik index

In 1996 Dan Felsenthal and Mosh\'e Machover considered the following model. An assembly consisting of $n$ voters exercises roll-call. All $n!$ possible orders in which the voters may be called are assumed to be equiprobable. The votes of each voter are independent with expectation $0<p<1$ for an individual vote {\lq\lq}yea{\rq\rq}. For a given decision rule $v$ the \emph{pivotal} voter in a roll-call is the one whose vote finally decides the aggregated outcome. It turned out that the probability to be pivotal is equivalent to the Shapley-Shubik index. Here we give an easy combinatorial proof of this coincidence, further weaken the assumptions of the underlying model, and study generalizations to the case of more than two alternatives.Comment: 19 pages; we added a reference to an earlier proof of our main resul

Referendum Design, Quorum Rules and Turnout

In this article, we focus on the consequences of quorum requirements for turnout in referendums. We use a rational choice, decision theoretic voting model to demonstrate that participation quorums change the incentives some electors face, inducing those who oppose changes in the status quo and expect to be in the minority to abstain. As a result, paradoxically, participation quorums decrease electoral participation. We test our model’s predictions using data for all referendums held in current European Union countries from 1970 until 2007, and show that the existence of a participation quorums increases abstention by more than ten percentage points.Referendum Design; Voter turnout

Measuring Power and Satisfaction in Societies with Opinion Leaders: Dictator and Opinion Leader Properties

A well known and established model in communication policy in sociology and marketing is that of opinion leadership. Opinion leaders are actors in a society who are able to affect the behavior of other members of the society called followers. Hence, opinion leaders might have a considerable impact on the behavior of markets and other social agglomerations being made up of individual actors choosing among a number of alternatives. For marketing or policy purposes it appears to be interesting to investigate the effect of different opinion leader-follower structures in markets or any other collective decision-making situations in a society. We study a two-action model in which the members of a society are to choose one action, for instance, to buy or not to buy a certain joint product, or to vote yes or no on a specific proposal. Each of the actors has an inclination to choose one of the actions. By definition opinion leaders have some power over their followers, and they exercise this power by influencing the behavior of their followers, i.e. their choice of action. After all actors have chosen their actions, a decision-making mechanism determines the collective choice resulting out of the individual choices. Making use of bipartite digraphs we introduce novel satisfaction and power scores which allow us to analyze the actors' satisfaction and power with respect to the collective choice for societies with different opinion leader-follower structures. Moreover, we study common dictator and opinion leader properties of the above scores and illustrate our findings for a society with five members.Bipartite digraph ; influence ; inclination ; collective choice ; opinion leader ; follower ; satisfaction ; power ; dictator properties ; opinion leader properties