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Seats and Votes: A Generalization of the Cube Law of Elections

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Abstract

The empirical “cube law” applies to parliamentary elections in Anglo-Saxon countries. It says that the ratio of assembly seats of two major parties is approximately the cube of the ratio of votes. This paper presents a more general semi-empirical “seat-vote equation” which includes the cube law as a special case but which also applies to the U.S. Electoral College, labor union, direct presidential, and proportional representation elections. The paper defines “constituency” as the smallest unit within which the party with plurality wins all the seats. The smaller the number of such constituencies is, the more dramatic is the attrition of minority party representation. Thus changes in the number of constituencies can be used to bring about a desired degree of minority representation. The prediction of the average long-range effects of such changes could be an important practical application of the seat-vote equation.

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... Most of the early entries in the bias and responsiveness literature can be classified as discussing index methods. The first of these asserted, or examined qualitatively the assertion, that a cube law was an appropriate descriptor of the seats-votes relationship for the two major parties in Britain, or elsewhere (e.g., Smith 1909 3 , crediting McMahon;The Economist 1950;Stuart 1950, 1952;McHenry 1955;Rydon 1957;Butler 1963;Qualter 1968;March 1957-8;Brookes 1959;Taagepera 1973;Theil 1970). The cube law is the relationship (1) ...
... which can then be charted over time or averaged for a pooled estimate (Eldersveld 1951;Brookes 1954;March 1957-8;Qualter 1968;Theil 1969;Casstevens and Morris 1972;Taagepera 1973Taagepera , 1986Laakso 1979). 4 ...
... These can also be averaged across elections for a pooled estimate (Taagepera 1973(Taagepera , 1986 is close to this approach). Parties with zero seats or zero votes (not usually a problem) have their votes lumped with other parties. ...
Article
There is an extensive and expanding literature that examines methods for estimating the responsiveness and partisan bias of two-party electoral systems. Attempts to extend these methods into the multiparty domain appropriate for the vast majority of electoral systems, or to the analysis of the representation of other types of groups (e.g., regions, ethnic groups), have been limited. I describe index, multiyear, uniform swing, and variable swing methods -- along with novel graphical displays -- for analyzing seats-votes curves, bias, and responsiveness in multiparty systems. The variable swing method is a multiparty generalization of Gelman and King's "JudgeIt" model. Examples discussed include elections in the UK, Mauritius, and Costa Rica, and geographic representation worldwide. In comparing the various methods it is argued that variable swing is ideal for most applications, that uniform swing and index methods provide useful answers to a limited set of questions despite faulty assum...
... We begin by formalizing partisan bias and responsiveness. The two-party case (King and Browning 1987, Taagepera 1973, Tufte 1973) extends in a straightforward manner to multiparty competition. In the two-party case, partisan bias and responsiveness are typically conceptualized as a generalization of the cube law stipulating that: ...
... At the limit, when ρ tends to infinity, every district is a microcosm of the national electorate, such that the party receiving 51% of the vote wins all districts and receives 100% delegates. ρ = 3, the dotted line, characterizes the classic cube law that many have associated with plurality rule in singlemember districts (Taagepera 1973). With cube responsiveness, a party with 55% of the vote wins two-thirds of the seats, but with 33% it wins only one-tenth of the seats. ...
... We now turn to estimating overall bias and its components. We fit equation 2 (Taagepera 1973). ...
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We extend the estimation of the components of partisan bias—i.e., undue advantage conferred to some party in the conversion of votes into legislative seats—to single-member district systems in the presence of multiple parties. Extant methods to estimate the contributions to partisan bias from malapportionment, boundary delimitations, and turnout are limited to two-party competition. In order to assess the spatial dimension of multi-party elections, we propose an empirical procedure combining three existing approaches: a separation method (Grofman et al. 1997), a multi-party estimation method (King 1990), and Monte Carlo simulations of national elections (Linzer, 2012). We apply the proposed method to the study of recent national lower chamber elections in Mexico. Analysis uncovers systematic turnout-based bias in favor of the former hegemonic ruling party that has been offset by district geography substantively helping one or both other major parties.
... Cependant, des recherches approfondies sur le terrainà propos de ces systèmeś electoraux ont montré que la loi de cube ne permet ni d'approximer correctement la relation entre les voix et les sièges ni de fournir une justification normative convaincante de la façon avec laquelle les voix sont transférées en sièges. Rein Taagepera [53] a montré que cette amplification observée peutêtre inférieure ou supérieureà la loi du cube et que cette dernière n'est qu'un cas singulier d'uneéquation plus générale. Il s'agit de " la loi de puissance" (Taagepera [53]) qui est formulée comme suit : ...
... Rein Taagepera [53] a montré que cette amplification observée peutêtre inférieure ou supérieureà la loi du cube et que cette dernière n'est qu'un cas singulier d'uneéquation plus générale. Il s'agit de " la loi de puissance" (Taagepera [53]) qui est formulée comme suit : ...
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Cet articleetudie les sources du paradoxe du referendum. Celles-ci ne sont que les biais resultant du transfert des voix en sieges. Ces biais se resument principalement dans les trois facteurs suivants: le gerrymandering, le malap- portionment et le taux de participation auxquels s'ajoutent parfois la question de l'influence d'un tiers parti (victorieux ou non). En employant les donnees deselections cantonales de la France metropolitaine de 1985 `a 2004, nous cal- culons l'impact de chaque facteurs en adaptant la methode de Broockes mod- ifiee par Johnston et al. ((24)). Nous determinons par la suite la source prin- cipale du paradoxe de referendum au niveau de chaque departement touche par ce paradoxe.
... Although there is no dearth of legal briefs filed in courts involving gerrymandering over many years in the past, it is only more recently that mathematicians and applied computational researchers have started to investigate this topic, perhaps due to the tremendous progress in high-speed computation in the last two decades. For example, researchers in [1,4,5,15,16,22,23,31,32] have made conceptual or empirical attempts at quantifying gerrymandering and devising redistricting methods to optimize such quantifications using well-known notions such as compactness and symmetry, whereas researchers in [1, 6-8, 19, 31] have investigated designing efficient heuristic approach and other computer simulation approaches for this purpose. Two recent research directions deserve specific mentions here. ...
... Seat-vote equation: For the decision version of this problem, we are required to produce a re-districting plan that exactly satisfies a relationship between between normalized seat counts and normalized vote counts between the two parties. The relationship was stated by [32] as ...
Preprint
The topic of this paper is "gerrymandering", namely the curse of deliberate creations of district maps with highly asymmetric electoral outcomes to disenfranchise voters, and it has a long legal history. Measuring and eliminating gerrymandering has enormous implications to sustain the backbone of democratic principles of a society. Although there is no dearth of legal briefs involving gerrymandering over many years, it is only more recently that mathematicians and applied computational researchers have started to investigate this topic. However, it has received relatively little attention so far from the computational complexity researchers dealing with theoretical analysis of computational complexity issues, such as computational hardness, approximability issues, etc. There could be many reasons for this, such as descriptions of these problem non-CS non-math (often legal or political) journals that theoretical CS (TCS) people usually do not follow, or the lack of coverage of these topics in TCS publication venues. One of our modest goals in writing this article is to improve upon this situation by stimulating further interactions between the gerrymandering and TCS researchers. To this effect, our main contributions are twofold: (1) we provide formalization of several models, related concepts, and corresponding problem statements using TCS frameworks from the descriptions of these problems as available in existing non-TCS (perhaps legal) venues, and (2) we also provide computational complexity analysis of some versions of these problems, leaving other versions for future research. The goal of writing this article is not to have the final word on gerrymandering, but to introduce a series of concepts, models and problems to the TCS community and to show that science of gerrymandering involves an intriguing set of partitioning problems involving geometric and combinatorial optimization.
... This relationship is not as evident in plurality elections, he says, because plurality systems do not aim at proportionality (Lijphart, 1994). Notwithstanding, by referring to Taagepera (1973), Lijphart argues that there is a connection between assembly size and disproportionality in plurality systems as well. He thus finds it theoretically justified to regard assembly size as one of the important elements of electoral systems (Lijphart, 1994). ...
... That being said, assembly size has an effect on the degree of disproportionality and the number of parties. A theoretical explanation of the impact of assembly size on the degree of proportionality is provided by means of the cube law, which holds that votes divided in a ratio of a:b between two parties in a plurality election results in a seat allocation in the ratio of a³:b³ (Taagepera, 1973). Disproportionality increases as the number of votes increases and/or assembly size decreases. ...
Article
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The article examines the impact of assembly size on the degree of disproportionality and party system fragmentation. The hypothesis is as follows: assembly size has a negative effect on the degree of dispro-portionality and a positive effect on the effective number of parties in systems with single-member dis-tricts—in proportional electoral systems, by contrast, such a pattern does not exist. In PR systems, notably the average effective threshold supersedes assembly size in explaining the degree of disproportionality and the effective number of parties. Electoral thresholds, ordinal ballots and apparentement, which also have some impact on disproportionality and party system fragmentation in proportional elections, are ab-sent in systems with single-member districts (with the exception of ordinal ballots in alternative vote sys-tems). Moreover, the district magnitude does not vary between electoral districts and countries. Therefore, assembly size is a significant factor in majoritarian systems. The empirical analysis of 550 elections in democratic countries provides support for the hypothesis.
... Theil's S i = SP n i /  P n k (1969) also applies to seats and votes in single seat plurality elections: Taagepera (1973) showed that the total numbers of seats (S) and votes (V ) determine the disproportionality exponent n. When these elections go through an intervening stage with E electors, then internal consistency requires the format n = f (V )/f (S). ...
... Some time after its inception, investigation with actual election results indicated that values for p other than 3 were better descriptions of many electoral systems (see Taagepera 1973;Tufte 1973;and the citations in Grofman 1983, 317). While this mathematical relationship is straightforward, it is difficult to interpret in a theoretically meaningful way wikhout either knowledge of analytical geometry or specific applications. ...
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Government T h e translation of citizen votes into legislative seats is of central importance in democratic electoral systems. It has been a longstanding concern among scholars in political science and in numerous other disciplines. Throughout this literature, two fundamental tenets of democratic theory, partisan bias and democratic representation, have often been confused. We develop a general statistical model of the relationship between votes and seats and separate these two important concepts theoretically and empirically. In so doing, we also solve several methodological problems with the study of seats, votes, and the cube law. An application to U.S. congressional districts provides estimates of bias and representation for each state and demonstrates the model's utility. Results of this application show distinct types of representation coexisting in U.S. states. Although most states have small partisan biases, there are some with a substantial degree of bias.
... Partisan bias in electoral systems has a long history in geographic and political literatures (Brookes 1960;Gudgin and Taylor 1976;Katz 1980). From analysis and attempts to understand the relationship between seats and votes in electoral systems (Taagepera 1973;Tufte 1973) and the later proliferation and refinement of measures of political bias (Grofman 1983; King and Browning 1987;Grofman and Selb 2009), courts in the United States have largely settled on the validity of partisan bias and responsiveness as the two central dimensions of partisan symmetry (Grofman and King 2007). Partisan symmetry occurs when electoral systems provide no unfair advantages to one party over another. ...
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Compactness of a congressional district is a traditional principle in adjudicating gerrymandering claims in political redistricting. During the last decade, many states have used compactness as an important criterion to constrain the presence of gerrymandering in the redistricting process. In this study, we conducted an array of spatiotemporal analyses aiming to evaluate the changes in compactness between the 112th and 113th Congressional districting plans in California and North Carolina, two states that have been well known for their heavy gerrymandering for years. We employed classic shape-based compactness measures, moment-of-inertia-based measures, and measures of partisan bias to assess the districting plans from multiple angles, including irregularity of district boundaries, spatial dispersion, population-weighted shape dispersion, and partisan symmetry. This new and combined use of spatial measures evidenced remarkable increases on the average compactness scores for California's Congress, suggesting general alleviation of the bipartisan gerrymandering in the previous plan. On the contrary, the partisan gerrymandering in North Carolina intensified in the current map, indicated by the substantial decline in the compactness scores for a majority of the districts. Analysis of partisan bias in the districting plans suggested a very slight bias toward Democrats in California in both districting plans. In North Carolina, the partisan advantage shifted from Democrats to Republicans during redistricting. Comparative analysis between the two families of spatial measures revealed the superiority of the moment of inertia family to the classic shape-based indexes for measuring compactness of congressional districts.
... Much of this literature builds from Kendall and Stuart's (1950) statement of the cube law and searches for better and more general ways of characterizing the relationship between votes and seats (e.g. Tufte 1973, Grofman 1983, Taagepera 1973, King and Browning 1987, King 1990). This literature has given surprisingly little attention to the geography of party support. ...
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... Kendall and Stuart (1950)) that in two-party electoral systems where each district elects a single representative by simple plurality voting ("First Past the Post", or FPP), there is a relationship between the fraction x of all the votes and the fraction y of all the districts won by a party along the lines of When k = 3 this is sometimes known as the "cube law", or "law of the cubic proportion". For a more comprehensive discussion of this "law" and some alternatives to it, see Tufte (1973), Taagepera (1973Taagepera ( , 1986, Gudgin and Taylor (1979). Taking (1) as an equality gives ...
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One of the most important concerns in democratic political systems is the translation of citizens' votes into seats in representative bodies. Drawing on the literature involving the cube law, the article examines the votes-seats relationship for elections to the lower houses of U.S. state legislatures over the period 1976-1984. The model permitted an examination of the concepts of electoral bias and representation (majoritarianism). The analysis indicated that some degree of bias exists in the states, and they tend more toward a proportional votes-seats relationship than has generally been found for Congress and comparative legislative bodies.
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The uninominal election on a majority basis with two rounds has a mechanical effect of increasing electoral movements among the seats allocated to each political forces. The 2002 legislative elections are one more illustration : whereas the right/left balance could be estimated at 55/45 at the end of the second round, the moderate right had 399 out of 577 seats. When the FN doesn't skew the game, as in 1997, the votes/seats equation follows, not the cube law of elections, but the « four » law. The mechanical effects of the type of election do not explain, however, how the seats are divided among political forces within each coalition. The number of seats allocated to little parties is mainly based on how they are established territorially: the CP managed to keep a parliamentary group by safekeeping its strongholds, thanks to the way the incumbents resisted. The ecology vote, on the other hand, was more spread out which resulted in only three Green deputies.
Article
The single-member district plurality system for legislative elections, also known as first-past-the-post (FPTP), usually results in disproportional seat distributions among parties. The Cube Law, which stipulates that the ratio of seats won by the parties in a two-party system is a cubic function of the corresponding ratio of total votes, attests to the degree of disproportionaty in representation under FPTP. This so-called law, however, is really just a benchmark, and the performance of FPTP can vary from country to country and from time to time. It is well known that the validity of the Cube Law, and hence the proportionality of representation, depends on the distribution of vote share across constituencies. Scholars have pointed to contagion, heterogeneity, and the size of constituencies as factors that may affect the conditions under which the Cube Law can be sustained. In this article, we propose a spatial regression model which implies all these factors. Empirically, we investigate Taiwan's recent legislative elections to test our theory. Our findings show that, in this case, low spatial autocorrelation at the district level is associated with vastly disproportional election outcomes.
Article
Representative democracy ideally requires that the strength of every opinion in the representative assembly should be proportional to its share of popular support. Various electoral procedures have been devised to achieve near-proportionality between popular votes and the number of assembly seats of political parties. The most widely used method methods are d’Hondt, the Sainte Laguë and quota (the largest remainder) procedures. For a description of calculations involved see, e.g., Rae [1971].
Article
Two elections to the Legislative Yuan have been held under a mixed parallel system. While there have been criticisms that this new set of rules leads to a considerable disparity between parties ' vote and seat shares in the district tier, in so far as the new electoral system has been accepted and therefore treated as given by both parties and voters, its fairness should be assessed not by the degree of proportionality, but rather by examining various sources of potential bias. These include differences in electorate sizes and turnout rates across districts, and the efficiency by which votes for the main parties 'candidates are distributed. The present study investigates how "fair " the functioning of Taiwan 's new mixed parallel system was in the 2008 and 2012 legislative elections by simulating equal and reverse vote scenarios at the district level, and measuring the magnitude of each component of electoral bias. The results show that the operation of the electoral system entails no marked partisan bias, since it does not consistently confer an advantage to either of the main parties or camps. © Institute of International Relations, National Chengchi University, Taipei, Taiwan (ROC).
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The chapter surveys and summarizes key findings of various chapters and highlights several distinctive features of Taiwanese politics in the 2016 regime change. There is a recurring pattern with candidates and parties making bloated campaign promises and then failing to fulfill them. Identity politics has been a constant theme in policy deliberation and partisan competition. In any analysis of Taiwan’s national security and foreign policy, cross-Strait relations with China remain an inescapable variable for consideration. Finally, the Tsai government has decided to realign with the US, Japan, and the Southeast Asia region to mitigate China’s pressure and threats.
Article
One of the most important concerns in democratic political systems is the translation of citizens' votes into seats in representative bodies. Drawing on the literature involving the cube law, the article examines the votes-seats relationship for elections to the lower houses of U.S. state legislatures over the period 1976-1984. The model permitted an examination of the concepts of electoral bias and representation (majoritarianism). The analysis indicated that some degree of bias exists in the states, and they tend more toward a proportional votes-seats relationship than has generally been found for Congress and comparative legislative bodies.
Article
Caretaker periods mark the transitions between the termination of one government and the formation of another. Caretaker conventions exist to ensure that the country is never left without a fully functioning executive, and to prevent a government whose democratic mandate has expired from making decisions that will inappropriately bind the incoming government. This article examines the UK's current caretaker conventions and argues that the UK is particularly vulnerable to problematic caretaker periods. Historically, its constitutional traditions and party system have made these periods short and rare. As a result, the conventions governing these transitional periods have remained underspecified. This article discusses why caretaker periods are likely to become more frequent and prolonged in the UK, analyses the UK's conventions and suggests how they can be reformed to pre-empt some of the challenges inherent to these constitutionally exceptional periods.
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In the U.S. redistricting is deeply politicized and often synonymous with gerrymandering -- the manipulation of boundaries to promote the goals of parties, incumbents, and racial groups. In contrast, Mexico’s federal redistricting has been implemented nationwide since 1996 through automated algorithms devised by the electoral management body (EMB) in consultation with political parties. In this setting, parties interact strategically and generate counterproposals to the algorithmically generated plans in a closed-door process that is not revealed outside the bureaucracy. Applying geospatial statistics and large-scale optimization to a novel dataset that has never been available outside of the EMB, we analyze the effects of automated redistricting and partisan strategic interaction on representation. Our dataset comprises the entire set of plans generated by the automated algorithm, as well as all the counterproposals made by each political party during the 2013 redistricting process. Additionally, we inspect the 2006 map with new data and two proposals to replace it towards 2015 in search for partisan effects and political distortions. Our analysis offers a unique insight into the internal workings of a purportedly autonomous EMB and the partisan effects of automated redistricting on representation.
Article
A large body scholarship demonstrates that the population size of an electoral district affects elections in important ways, yet little is known about the implications of population size for campaigning and fundraising. I posit that the challenges of running a campaign in a populous electorate require candidates to focus their fundraising efforts on the wealthy. I analyze campaign finance records published by the Federal Election Commission during the 2006–2014 Senate elections and find that Senate candidates running in large states receive fewer donations per capita from in-state donors, but they tend to receive larger donations on average and more money from contributions of $1,500 and above. In sum, candidates running in populous states appear to rely upon comparably smaller pools of wealthy constituents writing larger checks to finance their campaigns. In the context of rising campaign costs, these findings suggest that constituency population growth may exacerbate representational inequalities between citizens and contribute to the growing influence of the wealthy in U.S. politics.
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In the 1990s, the shortest path counting problem and its application in measuring the “importance” of a traffic road network were proposed; thereafter, it was generalized to the path counting problem. To measure the robustness of a network-structured system, the edge deletion connectivity function (EDCF) was defined, and the expected edge deletion connectivity function (EEDCF) for the network was obtained. The survivability function (SF) was proposed to approximate the EEDCF using an appropriate nonlinear function with two parameters. It was demonstrated that the SF can be used for solving various complex problems in several types of social systems. The EDCF indicates the relationship between the ratio of edges deleted from a network and the ratio of the number of edges in the network after deleting an arbitrary number of edges, to the total number of edges in the original network. The expected ratio of the number of paths connecting two different nodes in the network constitutes an EEDCF. The EEDCF was estimated using the Monte Carlo method. Subsequently, the SF was defined to approximate the EEDCF. The SF can be used to solve various social problems. This paper presents two examples of these applications—quantitatively measuring the robustness of a traffic road network in an urban area, and approximating the relationship between the vote share and seat share in the national election for the single-member voting system in Japan.
Article
The so-called cube ‘law’ has become ‘part of the political folklore of Great Britain’. Indeed it seems also to have passed into the general folklore of political science, having been applied to electoral systems having single-member constituencies contested by two major parties in the United States, New Zealand, Canada, Australia and South Africa.
Article
The British electoral system has traditionally been defended on the grounds that its systematic exaggeration of the lead of the Conservatives over Labour or vice versa secured for the largest party an overall majority of seats on a minority of the votes and thereby enabled the electorate directly to determine which party holds office. Recent changes in the electoral geography of Britain have eroded that systematic exaggeration at Westminster elections and may soon remove it entirely. Meanwhile the electoral system is discouraging the pursuit of geographically aggregative policies. Consequently, the single member plurality system is no longer a suitable instrument to achieve the ends of its defenders but, despite their availability, alternative systems which could meet those aims have not been seriously considered.
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