A preliminary simulative assessment of disproportionality indices

West European Politics, 2011

This article proposes a new way to measure proportionality using aggregated threshold functions. Electoral systems can be summarised by a single value that shows the necessary share of the total vote to win either one seat or half of the seats in parliament. The article calculates aggregate threshold values for 142 different electoral systems that were used in 525 democratic elections between 1946 and 2000. These results are also contrasted with the most commonly used indices of proportionality and turn out to be both substantively and empirically richer. Aggregated threshold functions provide both students and reformers of electoral systems with a measure based purely on institutional variables that offers an exhaustive summary of the functioning of many electoral systems.

Palgrave Communications

This paper deals with the fundamental issue of quantifying the degree of competitiveness underlying an election in a society. That is, given the vote shares of competing political parties in the election one needs to arrive at a representative metric of electoral competition. The vote share of a party represents its influence on the outcome of the election. A summary measure of differences of such influences across parties, aggregated in an unambiguous way, is an index of electoral competitiveness. The literature contains several such indices. Implicit under the choice of an index is a set of properties that the index satisfies. A highly attractive feature of the probability ratio index we suggest in this article, as a summary measure of electoral competition, is its suitability for making comparisons of competitiveness across elections with different number of parties. The probability ratio index is simply the ratio between the probabilities that the two voters drawn at random with...

British Journal of Political Science, 2001

Comparative studies of electoral institutions have largely neglected a fundamental characteristic of most of the world's electoral systems: malapportionment. This article provides a method for measuring malapportionment in different types of electoral systems, calculates levels of malapportionment in seventy-eight countries, and employs statistical analysis to explore the correlates of malapportionment in both upper and lower chambers. The analysis shows that the use of single-member districts is associated with higher levels of malapportionment in lower chambers and that federalism and country size account for variation in malapportionment in upper chambers. Furthermore, African and especially Latin American countries tend to have electoral systems that are highly malapportioned. The article concludes by proposing a broad, comparative research agenda that focuses on the origins, evolution and consequences of malapportionment.

Theory and Decision, 1988

Three alternative methods are proposed to determine a normative standard concerning the fair proportion of seats a party ought to receive in a representative assembly as a function of the voters' preference orderings. The methods differ from one another in their treatment of indifference relations and the assumptions they make about the type of scale underlying voters' preferences. Common to all three methods is the basic idea that the ratio between the number of voters preferring party i over j to the number of voters preferring party j over i can be tested for consistency, in a precisely defined sense, and if sufficiently consistent, can be appropriately scaled to determine the proportion of seats each party ought to receive. The proposed solutions are shown to satisfy several desiderata when the matrix of preference ratios is consistent. When there are cyclical majorities of equal size, the matrix of preference ratios is inconsistent. The main application of the proposed scheme is as a normative benchmark against which actual or proposed voting procedures can be evaluated in proportional representation systems. The theoretical implications of these solutions are briefly discussed.

aecpa.es

Using Gallagher's least squares index, we examine the determinants of disproportionality in 656 elections held in 59 new and established democracies between 1945 and 2010. We aim at understanding the fluctuations of the indices of disproportionality in countries with and without electoral reform; in this draft of the paper, however, we will only address the former case. Our findings show substantial evidence of the impact of institutional engineering on electoral disproportionality. After a permissive change in the rules of the game, the levels of deviation from perfect proportionality will tend toward zero. In contrast, in cases where a restrictive electoral reform is introduced, levels of disproportionality will likely increase. We also aim to review the conditions under which the impact of reforms on electoral disproportionality is larger and tends to vanish over time.

Group Decision and Negotiation, 2015

I use power indices to assess the level of representative efficiency of different electoral systems. A representative democracy should "give voice" to as many people's preferences as possible. This paper evaluates how close a given electoral system is to mirroring the distribution of votes cast in an election. To this goal both power indices and differences between shares of votes received and seats gained are used. The distribution of power is also compared with both the share of seats of parties in the assembly and the share of votes obtained in the election. The results show that proportional systems and runoff majority are more efficient in terms of representativeness than first-past-the-post methods. Moreover, as the total number of seats in a parliament decreases, representative efficiency tends to increase. The analysis is conducted through simulations using conditional and real (Dutch and Italian) data.

Electoral Studies, 2007

This article introduces a set of functions that measures the mechanical performance of an electoral system. Aggregated threshold functions offer the necessary and sufficient share of the vote nationwide to win a given number of seats. Traditionally, electoral systems have been measured taking into account the share of the vote required to win one seat given a district. In the approach used here, the values obtained are calculated taking into account all districts in which a country is divided and for any number of seats. This article offers the definition and formalization of these functions. Once the aggregated threshold functions are defined in all their terms, I show some data validation to test their capacity of prediction. The main goal of the article is to provide with a tool that can be used, for example, to develop a measure that summarizes in a single value the functioning of an electoral system. This value can be used by electoral reformers as well as by students of electoral systems to test the consequences of electoral systems as a whole.

IVIE working papers o®er in advance the results of economic research under way in order to encourage a discussion process before sending them to scienti¯c journals for their¯nal publication. * We gratefully acknowlegde the¯nancial support from the DGES of the Spanish Ministerio de Educaci ¶ on y Cultura, under project PB96-0247, from the Basque Government under project PI95-101 and from the Training and Mobility of Researchers programme initiated by the European Commission. Support from the Universit ¶ e catholique de Louvain, where the¯rst author completed her Ph. D. dissertation, is also gratefully acknowledged.

Italian Institutional Reforms: A Public Choice Perspective, 2008

It is self-evident that the type of electoral system used will have an effect on citizen participation in the electoral process – the question of which system is more inclusive of the citizenry has been the centre of much debate between political scientists and commentators and is the focus of this essay. In particular, this essay will look at two broad types of electoral systems – proportional representation systems (hereinafter referred to as PR systems) and majoritarian systems – to determine which of the two are more inclusive of the citizenry. To do so, each electoral system will be explained and the ways in which each system makes the citizenry more inclusive (or otherwise) will be explored. Case studies from countries that use either system will be presented to support the argument that PR systems are more inclusive of the citizenry than their majoritarian counterparts.

We develop and apply a new conceptual framework and measure for evaluating electoral systems, focusing on (in)equality in parliamentary representation. Our main arena of interest is proportional representation with districts, an electoral system employed by more than half of democratic states, and we draw on an almost entirely overlooked fact: Electoral regimes vary substantially within countries, with some voters casting their ballot in semi-majoritarian districts of few representatives and others in large and proportional ones. This within-country institutional variation, we contend, affects representational (in)equality. Evaluating equality in parliamentary representation, we demonstrate that districted proportional representation often leads to overrepresentation of voters supporting right-leaning parties. Utilizing district-level data from 20 Western parliamentary democracies and complementing our within-country approach with a cross-country analysis, we further show that where parliaments are elected by large and small districts, representational inequality among voters is greater compared with countries in which parliament is elected by even-magnitude districts.

IndElec is a software addressed to compute a wide range of indices from electoral data, which are intended to analyze both party systems and electoral systems in political studies. Further, IndElec can calculate such indices from electoral data at several levels of aggregation, even when the acronyms of some political parties change across districts. As the amount of information provided by IndElec may be considerable, this software also aids the user in the analysis of electoral data through three capabilities. First, IndElec automatically elaborates preliminary descriptive statistical reports of computed indices. Second, IndElec saves the computed information into text files in data matrix format, which can be directly loaded by any statistical software to facilitate more sophisticated statistical studies. Third, IndElec provides results in several file formats (text, CSV, HTML, R) to facilitate their visualization and management by using a wide range of application softwares (word processors, spreadsheets, web browsers, etc.). Finally, a graphical user interface is provided for IndElec to manage calculation processes, but no visualization facility is available in this environment. In fact, both the inputs and outputs for IndElec are arranged in files with the aforementioned formats.

2016

Disproportionality indices try to measure to what extent the composition of a parliament differs from the distribution of the votes among parties. Malapportionment indices measure to what extent the number of parliament seats attached to each district difffers from the distribution of the population among districts. Political scientists have noticed the similarity between both problems. Since there exist many different such indices, some axioms have recently been proposed for assessing the merits of the various indices. In this paper, we propose the first axiomatic characterization of some disproportionality indices.

European Journal of Political Economy, 1988