Proportional Representation
Abstract
The book offers a rigorous description of the procedures that proportional representation systems use to translate vote counts into seat numbers. Since the methodological analysis is guided by practical needs, plenty of empirical instances are provided and reviewed to motivate the development, and to illustrate the results. Concrete examples, like the 2009 elections to the European Parliament in each of the 27 Member States and the 2013 election to the German Bundestag, are analyzed in full detail. The level of mathematical exposition, as well as the relation to political sciences and constitutional jurisprudence makes this book suitable for special graduate courses and seminars.
... The main motivation is the growing need to incorporate dimensions beyond the classic political and geographical aspects in elections, as modern societies become more complex. The increasing need for such methods has driven increasing attention to the subject in the fields of social choice, mathematics, and computer science [Flanigan, Gölz, Gupta, Hennig, and Procaccia, 2021, Lang and Skowron, 2018, Pukelsheim, 2017. In practice, New Zealand's parliament has included ethnic representation for more than 50 years, while Bosnia and Herzegovina's Parliament was proposed to include a division of three types of "Constituent People": Bosniacs, Croats, and others [Demange, 2013]. ...
... The apportionment problem, and in particular divisor methods, have been extensively studied from disciplines such as operations research, computer science, and political science; the books by Balinski and Young [2010] and Pukelsheim [2017] provide historical and theoretical surveys of this topic. In their seminal work, Balinski and Demange [1989a,b] extended the notion of proportionality and divisor methods to the case in which the apportionment is ruled by two dimensions, e.g., political parties and geographic districts, studying this extension from an axiomatic and algorithmic point of view. ...
... The presented methods can be valuable and feasible proposals for the election of representative bodies. Proportional or mixed-member proportional representation systems are two families of electoral systems widely used worldwide, and some countries have also started to use biproportional apportionment methods [Maier et al., 2010, Pukelsheim, 2017. Our theoretical and applied results lay the groundwork for incorporating features such as representation threshold-to favor the conformation of broader political projects-or the election of the most-voted candidate in every district-to enhance local representation-into methods based on multidimensional proportionality. ...
How to elect the representatives in legislative bodies is a question that every modern democracy has to answer. This design task has to consider various elements so as to fulfill the citizens' expectations and contribute to the maintenance of a healthy democracy. The notion of proportionality, in that the support of a given idea in the house should be nearly proportional to its support in the general public, lies at the core of this design task. In the last decades, demographic aspects beyond political support have been incorporated by requiring that they are also fairly represented in the body, giving rise to a multidimensional version of the apportionment problem. In this work, we provide an axiomatic justification for a recently proposed notion of multidimensional proportionality and extend it to encompass two relevant constraints often used in electoral systems: a threshold on the number of votes that a list needs in order to be eligible and the election of the most-voted candidate in each district. We then build upon these results to design methods based on multidimensional proportionality. We use the Chilean Constitutional Convention election (May 15-16, 2021) results as a testing ground -- where the dimensions are given by political lists, districts, and genders -- and compare the apportionment obtained under each method according to three criteria: proportionality, representativeness, and voting power. While local and global methods exhibit a natural trade-off between local and global proportionality, including the election of most-voted candidates on top of methods based on 3-dimensional proportionality allows us to incorporate both notions while ensuring higher levels of representativeness and a balanced voting power.
... Unlike prior work, we focus on multi-group representation in retrieval rather than multi-group fairness in classification and regression. Lastly, proportional representation strategies arose in the political sciences literature [56][57][58][59][60] -we address this body of work in Appendix C. ...
... Proportional Representation (PR) in political sciences. As described in the related works in Section 1, proportional representation (broadly construed) has a long history in social choice theory and political sciences [56][57][58][59][60] and has been a guiding principle in the design of political systems, aiming to ensure representation reflects underlying population preferences and demographics. It can be traced back to the late 18th century, with early proponents such as Honoré Gabriel Riqueti, Comte de Mirabeau, who discussed the idea in 1780. ...
... In the late 19th and early 20th centuries, the works of Carl Andrae, Victor D'Hondt, and John Stuart Mill further explored and contributed to the discussion of proportional representation in the political science literature. See [56][57][58][59][60]. ...
Image search and retrieval tasks can perpetuate harmful stereotypes, erase cultural identities, and amplify social disparities. Current approaches to mitigate these representational harms balance the number of retrieved items across population groups defined by a small number of (often binary) attributes. However, most existing methods overlook intersectional groups determined by combinations of group attributes, such as gender, race, and ethnicity. We introduce Multi-Group Proportional Representation (MPR), a novel metric that measures representation across intersectional groups. We develop practical methods for estimating MPR, provide theoretical guarantees, and propose optimization algorithms to ensure MPR in retrieval. We demonstrate that existing methods optimizing for equal and proportional representation metrics may fail to promote MPR. Crucially, our work shows that optimizing MPR yields more proportional representation across multiple intersectional groups specified by a rich function class, often with minimal compromise in retrieval accuracy.
... The optimal allocation of legislative seats to groups according to their population size or vote count is a persistently critical and often considered the most "susceptible" issue in legal and political studies [1]. Despite over two centuries of comprehensive exploration through debates (e.g., [1][2][3], reports (e.g., [4][5][6][7][8][9]), algorithmic solutions (e.g., [10][11][12][13][14][15]), mathematical theories (e.g., [16][17][18][19][20][21][22]), empirical studies (e.g., [23,24]), among other studies, a significant disparity still exists between theoretical constructs and real-world scenarios [24,25]. ...
... Apportionment methods fall into two categories. Divisor methods employ fixed rounding rules and dynamic scaling rules, while quota methods feature fixed scaling rules and dynamic rounding rules [18]. Well-known divisor algorithms include Adam's, Jefferson's, Webster's and Hill's methods, each differing by its rounding rule (we will review them in Section 3). ...
... Conversely, Hamilton's method is a notable quota algorithm (for more, see [39]). Further discussions can be found in [15,18]. ...
The allocation of seats in a legislative body to groups based on their size is a crucial issue in legal and political studies. However, recent findings suggest that an optimal allocation of seats may not be proportional to the size of the groups. For instance, the European Parliament (EP) utilizes a subproportional system known as degressive proportionality. Unfortunately, current apportionment methods for the EP lack a rigorous axiomatic analysis and fail to adequately address equality. Building upon recent research on equality in subproportional settings, this paper proposed a novel generalization of existing axioms and divisor methods for proportionality to encompass subproportionality with relative equality. Specifically, we consider a function f(p)=a+bpγ on the standard number of seats for a group of size p, where a, b and γ are given non-negative constants, and a is an integer. This theory is exemplified through an empirical study focused on the EP.
... There are many conceivable such methods, but [3] show that the divisor methods (introduced below) are the only methods that guarantee pairwise vote monotonicity (population monotonicity in [3]), which requires that a party i cannot lose seats to a party j when i gains votes while j loses votes (and all other parties remain unchanged). For a comprehensive introduction into the topic with its historical, political, and mathematical dimensions, including desirable and undesirable properties of various apportionment methods and corresponding impossibility results, we refer the reader to the books of [3,9]. ...
... A simple refinement, the jump-and-step algorithm described by [9] (see also Sect. 4.1), avoids any dependency on k. ...
... These bounds seem to be folklore; they are mentioned explicitly for example by [6,14]. This running time is not optimal, but the algorithm is simple and performs provably well in certain average-case scenarios [9,Sect. 6.7]. ...
Proportional apportionment is the problem of assigning seats to states (resp. parties) according to their relative share of the population (resp. votes), a field heavily influenced by the early work of Michel Balinski, not least his influential 1982 book with Peyton Young (Fair representation, 2nd edn. Brookings Institution Press, Washington, D.C., 2001). In this article, we consider the computational cost of divisor methods (also known as highest averages methods), the de-facto standard solution that is used in many countries. We show that a simple linear-time algorithm can exactly simulate all instances of the family of divisor methods of apportionment by reducing the problem to a single call to a selection algorithm. All previously published solutions were iterative methods that either offer no linear-time guarantee in the worst case or require a complex update step that suffers from numerical instability.
... The formal description of the mechanism and its analysis can be found in Section 4. This mechanism can be seen as a two stage optimization problem in which the first level ensures party proportionality, and then the second level guarantees biproportionality. We remark that the number of seats for each party are computed from the instance and they are not part of the input, which contrasts with the typical biproportional method where the voting matrix and the seats distributions are fixed and part of the input [21,3]. This interaction of two levels of apportionment requires a careful treatment, and we make progress on one of the research directions mentioned by Demange [10] on the theoretical understanding of the interaction between these two levels of proportionality. ...
... The divisor methods are widely used at national and regional level and we refer to the book by Balinski and Young [5] and the recent book by Pukelsheim [21] for a deep treatment of the theory and use of these methods. Rote and Zachariasen [23] and later Gaffke and Pukelsheim [14,13] provided a network flow approach for computing a biproportional solution. ...
... Observe that when x is fractional the rounding function coincides with the floor of x. This method belongs to a broader family known as divisor methods, and they are used in various countries at national and regional level [5,21]. Their main feature is the fact that they capture the notion of proportionality in the case where allocations have to be integral. ...
In the classic apportionment problem the goal is to decide how many seats of a parliament should be allocated to each party as a result of an election. The divisor methods provide a way of solving this problem by defining a notion of proportionality guided by some rounding rule. Motivated by recent challenges in the context of electoral apportionment, we consider the question of how to allocate the seats of a parliament under parity constraints between candidate types (e.g. equal number of men and women elected) while at the same time satisfying party proportionality. We consider two different approaches for this problem. The first mechanism, that follows a greedy approach, corresponds to a recent mechanism used in the Chilean Constitutional Convention 2021 election. We analyze this mechanism from a theoretical point of view. The second mechanism follows the idea of biproportionality introduced by Balinski and Demange [Math. Program. 1989, Math. Oper. Res. 1989]. In contrast with the classic biproportional method by Balinski and Demange, this mechanism is ruled by two levels of proportionality: Proportionality is satisfied at the level of parties by means of a divisor method, and then biproportionality is used to decide the number of candidates allocated to each type and party. We provide a theoretical analysis of this mechanism, making progress on the theoretical understanding of methods with two levels of proportionality. A typical benchmark used in the context of two-dimensional apportionment is the fair share (a.k.a matrix scaling), which corresponds to an ideal fractional biproportional solution. We provide lower bounds on the distance between these two types of solutions, and we explore their consequences in the context of two-dimensional apportionment.
... There is a long and rich body of literature for the apportionment problem and the divisor methods, intersecting different areas such as operations research, computer science and political science. For a formal treatment of the theory and a historical survey, we refer to the book of Balinski and Young [9] and to the recent book by Pukelsheim [27]. For a deeper treatment of social choice and new methods, we also refer to the book and article by Baliski and Laraki [6,7]. ...
... In the context of voting, the classic Jefferson/D'Hondt method corresponds to the divisor method associated to the stationary signpost sequence with ∆ = 0. Other classic methods are the one by Adams, corresponding to the divisor method associated to the stationary signpost sequence with ∆ = 1, and the method by Webster/Sainte-Laguë, corresponding to the divisor method associated to the stationary signpost sequence with ∆ = 1/2. For an extensive treatment of the theory of divisor methods and their historical aspects, we refer to the book by Balinski and Young [9] and the one by Pukelsheim [27]. ...
... When d = 2, the above problem is as hard as a transportation problem in a bipartite network, and in consequence, one can recover an optimal solution of this problem by relaxing integrality and solving the linear relaxation. Furthermore, it can be shown when d = 2 that any extreme point defines a proportional apportionment, where multipliers are obtained by computing the exponential of the corresponding dual solution [29, 19,27]. Therefore, in the general d-dimensional setting, the first natural question that we address is the following: Can we characterize the set of proportional apportionments in terms of the set of optimal solutions of the linear relaxation of (6)-(12)? ...
Deciding how to allocate the seats of a house of representatives is one of the most fundamental problems in the political organization of societies, and has been widely studied over already two centuries. The idea of proportionality is at the core of most approaches to tackle this problem, and this notion is captured by the divisor methods, such as the Jefferson/D'Hondt method. In a seminal work, Balinski and Demange extended the single-dimensional idea of divisor methods to the setting in which the seat allocation is simultaneously determined by two dimensions, and proposed the so-called biproportional apportionment method. The method, currently used in several electoral systems, is however limited to two dimensions and the question of extending it is considered to be an important problem both theoretically and in practice. In this work we initiate the study of multidimensional proportional apportionment. We first formalize a notion of multidimensional proportionality that naturally extends that of Balinski and Demange. By means of analyzing an appropriate integer linear program we are able to prove that, in contrast to the two-dimensional case, the existence of multidimensional proportional apportionments is not guaranteed and deciding its existence is NP-complete. Interestingly, our main result asserts that it is possible to find approximate multidimensional proportional apportionments that deviate from the marginals by a small amount. The proof arises through the lens of discrepancy theory, mainly inspired by the celebrated Beck-Fiala Theorem. We finally evaluate our approach by using the data from the recent 2021 Chilean Constitutional Convention election.
... Also most of these models are two-state models while in some cases there are more than two viable options to choose from, e.g., usually more than 15 parties participate in Lithuanian parliamentary election (with more than 4 of them winning seats in the parliament by the popular vote). In the political science and mathematical literature one would find a more varied approaches [43,44], but usually their primary goal is to provide election procedures, which would represent the opionion of the electorate the best. In Section 2 of this paper we will discuss an alternative possibility, based on Kirman's model [45], to formulate an agent-based model for the voting behavior. ...
Competition between varying ideas, people and institutions fuels the dynamics of socio-economic systems. Numerous analyses of the empirical data extracted from different financial markets have established a consistent set of stylized facts describing statistical signatures of the competition in the financial markets. Having an established and consistent set of stylized facts helps to set clear goals for theoretical models to achieve. Despite similar abundance of empirical analyses in sociophysics, there is no consistent set of stylized facts describing the opinion dynamics. In this contribution we consider the parties' vote share distributions observed during the Lithuanian parliamentary elections. We show that most of the time empirical vote share distributions could be well fitted by numerous different distributions. While discussing this peculiarity we provide arguments, including a simple agent-based model, on why the beta distribution could be the best choice to fit the parties' vote share distributions.
... Preamble. Mathematics is fundamental to the design and analysis of voting systems (see, for example, the books [3,10,16,36,43]). Mathematical models for human behaviour frequently involve probability, and they invariably rely upon assumptions whose validity is ripe for debate. ...
We examine two aspects of the mathematical basis for two-tier voting systems, such as that of the Council of the European Union. These aspects concern the use of square-root weights and the choice of quota. Square-root weights originate in the Penrose square-root system, which assumes that votes are cast independently and uniformly at random, and is based around the concept of equality of influence of the voters across the Union. There are (at least) two distinct definitions of influence in current use in probability theory, namely, absolute and conditional influence. These are in agreement when the underlying random variables are independent, but not generally. We review their possible implications for two-tier voting systems, especially in the context of the so-called collective bias model. We show that the two square-root laws invoked by Penrose are unified through the use of conditional influence. In an elaboration of the square-root system, Slomczynski and Zyczkowski have proposed an exact value for the quota to be achieved in a successful vote of a two-tier system, and they have presented numerical and theoretical evidence in its support. We indicate some numerical and mathematical issues arising in the use of a Gaussian (or normal) approximation in this context, and we propose that other values of q may be as good if not better than . We discuss certain aspects of the relationship between theoreticians and politicians in the design of a two-tier voting system, and we reach the conclusion that the choice of quota in the square-root system is an issue for politicians informed by theory.
... There is a rich body of literature on the theory and applications of apportionment methods; for a comprehensive treatment of this topic, we refer the reader to the book of Balinski and Young [2010] and the book of Pukelsheim [2017]. Closely related to our work is the stream of literature dealing with the design of house-monotone and quota-compliant methods. ...
The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states' populations? This--seemingly simple--task has led to a rich literature and has become well known in the context of the US House of Representatives. In this paper, we connect the design of monotone apportionment methods to classic problems from discrete geometry and combinatorial optimization and explore the extent to which randomization can enhance proportionality. We first focus on the well-studied family of stationary divisor methods, which satisfy the strong population monotonicity property, and show that this family produces only a slightly superlinear number of different outputs as a function of the number of states. While our upper and lower bounds leave a small gap, we show that--surprisingly--closing this gap would solve a long-standing open problem from discrete geometry, known as the complexity of k-levels in line arrangements. The main downside of divisor methods is their violation of the quota axiom, i.e., every state should receive or seats, where is the proportional share of the state. As we show that randomizing over divisor methods can only partially overcome this issue, we propose a relaxed version of divisor methods in which the total number of seats may slightly deviate from the house size. By randomizing over them, we can simultaneously satisfy population monotonicity, quota, and ex-ante proportionality. Finally, we turn our attention to quota-compliant methods that are house-monotone, i.e., no state may lose a seat when the house size is increased. We provide a polyhedral characterization based on network flows, which implies a simple description of all ex-ante proportional randomized methods that are house-monotone and quota-compliant.
... Arguably the closest in flavor is randomized rounding for flows [21], which has constraints similar to our inheritance ones. Furthermore, rounding the number of seats given to each child assembly (a step in several of our algorithms) is reminiscent of randomized apportionment [2,20]. However, our inheritance and ex post child constraints do not fit neatly within existing frameworks and make it challenging to use off-the-shelf techniques directly. ...
A citizens' assembly is a group of people who are randomly selected to represent a larger population in a deliberation. While this approach has successfully strengthened democracy, it has certain limitations that suggest the need for assemblies to form and associate more organically. In response, we propose federated assemblies, where assemblies are interconnected, and each parent assembly is selected from members of its child assemblies. The main technical challenge is to develop random selection algorithms that meet new representation constraints inherent in this hierarchical structure. We design and analyze several algorithms that provide different representation guarantees under various assumptions on the structure of the underlying graph.
... PR is considered the ideal system for ensuring the equality of individuals since it considers the contribution of all people and votes (see, e.g., Lijphart, 1998). It has been adopted worldwide in modern comparative politics, apportionment, and elections (see, e.g., Allen and Taagepera, 2017;Benoit, 2000;Lijphart, 1998Lijphart, , 2012Pukelsheim, 2017;Puyenbroeck, 2008;Samuels and Snyder, 2001;Taagepera and Grofman, 2003;US House of Representatives, 2022). ...
The proportion of seats to population (PSP) has long been used to quantify the contributions (weights) of individuals or votes in representation equality studies. We show that the PSP has a bias in estimating the true contribution; thus, proportionality schemes (proportional representation) are insufficient for ensuring equality. To address this issue, we introduce a standard function f(p) for the number of seats for a population p and a population seat index (PSI) to replace the PSP for assigning s seats to population p, where is the inverse of f. In contrast to the PSP, the PSI has no bias. We use it to derive an apportioning scheme with absolute or relative individual equality. If for a constant , this scheme distributes seats proportionally to the -th powers of the populations. Real-world observations indicate that with a constant , showing that proportionality schemes represent individuals in less populous groups less than individuals in more populous groups, while the proposed subproportionality scheme guarantees equality.
... In order to structurally rationalise the legitimacy of authority of the judiciary in the democratic state, people must be involved in the decision-making process of who sits as a judge, like they used to do in case of the nyay panchayats (Meschievitz &Galanter, 1982) (Muhlberger andPaine, 1993). Such elections could be conducted through the method of proportional representation in order to ensure maximum representation (Pukelsheim, 2017) (Volcansek, 1981), however, like panchayats the office should be a five-member commission instead of a single person with options for recall and no one should be eligible for re-election in the same constituency (Sharma, 2001). ...
The structure of Indian Judiciary is very similar to the common law British structure and it was designed to be exploitative in nature. It being a power institution that constitutes a part of the modern state, Indian Judiciary not only exercises authority over all of India it is also at the same time a democratic institution. The structure of the colonial euro-centric institutions is such that the locus and focus of responsibility can never be realised at the same time and at the same place. This naturally creates the paradox in institutional responsibility which is a natural consequence of irrational bureaucratic structure within the institution of judiciary. The first half of this paper starts critically reviewing the problems facing the Judiciary in India from the point of view of its euro-centric structure and the various malaise that this irrational structure results into. In the second half the paper reviews and recommends the application of New maanagerial philosophies to the structural aspects of Indian Judiciary with the aim of structurally rationalising it.
... Let r JDH k be an allocation rule defined as follows: let w i , i ∈ [p], be the share of votes such that the i-th party's candidate is ranked first, and let M be such interval that for each M ∈ M, i∈[p] M w i = k. Then the i-th coordinate of r JDH k is given by M w i /k, where M ∈ M (the rule is independent of the choice of M ) [31,16,17,4,45]. Remark 1. ...
Electoral spoilers are such agents that there exists a coalition of agents whose total gain when a putative spoiler is eliminated exceeds that spoiler's share in the election outcome. So far spoiler effects have been analyzed primarily in the context of single-winner electoral systems. We consider this problem in the context of multi-district party elections. We introduce a formal measure of a party's excess electoral impact, treating "spoilership" as a manner of degree. This approach allows us to compare multi-winner social choice rules according to their degree of spoiler susceptibility. We present experimental results, as well as analytical results for toy models, for seven classical rules (k-Borda, Chamberlin--Courant, Harmonic-Borda, Jefferson--D'Hondt, PAV, SNTV, and STV). Since the probabilistic models commonly used in computational social choice have been developed for non-party elections, we extend them to be able to generate multi-district party elections.
... In particular, Budish et al. (2013) and Akbarpour and Nikzad (2020) build implementation methods for random allocation mechanisms based on techniques from deterministic and randomized rounding developed in Edmonds (2003) and Gandhi et al. (2006). Our constraints, in addition to following a "bihierarchical" structure, also extend in the time dimension in order to accommodate 3 See Pukelsheim (2017) for detailed results and insights on biproportional apportionment problems. the multi-period considerations. ...
In many settings affirmative action policies apply at two levels simultaneously, for instance, at university as well as at its departments. We show that commonly used methods in reserving positions for beneficiaries of affirmative action are often inadequate in such settings. We present a comprehensive evaluation of existing procedures to formally document their shortcomings. We propose a new solution with appealing theoretical properties and quantify the benefits of adopting it using recruitment advertisement data from India.
... First, we are going to introduce some important apportionment methods (cf. Balinski and Young 1982;Pukelsheim 2017). ...
In 1998 a long-lost proposal for an election law by Gottlob Frege (1848–1925) was rediscovered in the Thüringer Universitäts- und Landesbibliothek in Jena, Germany. The method that Frege proposed for the election of representatives of a constituency features a remarkable concern for the representation of minorities. Its core idea is that votes cast for unelected candidates are carried over to the next election, while elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a proportional representation of political opinions in the long run. We find that through a slight modification of Frege’s original method even stronger proportionality guarantees can be achieved. This modified version of Frege’s method also provides a novel solution to the apportionment problem, which is distinct from all of the best-known apportionment methods, while still possessing noteworthy proportionality properties.
... This property is called the lower Hare-quota. It is a known property of the D'Hondt's method [53,Section 11.4]. But D'Hondt's method satisfies all the other instantiations of (P1) and (P2), which-as far as I know-was not noted before. ...
Competitive equilibrium (CE) is a fundamental concept in market economics. Its efficiency and fairness properties make it particularly appealing as a rule for fair allocation of resources among agents with possibly different entitlements. However, when the resources are indivisible, a CE might not exist even when there is one resource and two agents with equal incomes. Recently, Babaioff and Nisan and Talgam-Cohen (2017–2019) have suggested to consider the entire space of possible incomes, and check whether there exists a CE for almost all income-vectors—all income-space except a subset of measure zero. They proved various existence and non-existence results, but left open the cases of four goods and three or four agents with monotonically-increasing preferences. This paper proves non-existence in both these cases, thus completing the characterization of CE existence for almost all incomes in the domain of monotonically increasing preferences. Additionally, the paper provides a complete characterization of CE existence in the domain of monotonically decreasing preferences, corresponding to allocation of chores. On the positive side, the paper proves that CE exists for almost all incomes when there are four goods and three agents with additive preferences. The proof uses a new tool for describing a CE, as a subgame-perfect equilibrium of a specific sequential game. The same tool also enables substantially simpler proofs to the cases already proved by Babaioff et al. Additionally, this paper proves several strong fairness properties that are satisfied by any CE allocation, illustrating its usefulness for fair allocation among agents with different entitlements.
... In real applications, where no ambiguity is allowed, the problem of choice of proportional division is solved by arbitrary algorithm to determine such a division. The description of algorithms based on divisor method along with the cases of their applications can be found in Pukelsheim (2017) and Colomer (2016). https://dx.doi.org/10.15405/epsbs.2019.02.02 ...
... Also most of these models are two-state models while in some cases there are more than two viable options to choose from, e.g., usually more than 15 parties participate in Lithuanian parliamentary election (with more than 4 of them winning seats in the parliament by the popular vote). In the political science and mathematical literature one would find a more varied approaches [43,44], but usually their primary goal is to provide election procedures, which would represent the opionion of the electorate the best. In Section 2 of this paper we will discuss an alternative possibility, based on Kirman's model [45], to formulate an agent-based model for the voting behavior. ...
Competition between varying ideas, people and institutions fuels the dynamics of socio-economic systems. Numerous analyses of the empirical data extracted from different financial markets have established a consistent set of stylized facts describing statistical signatures of the competition in the financial markets. Having an established and consistent set of stylized facts helps to set clear goals for theoretical models to achieve. Despite similar abundance of empirical analyses in sociophysics, there is no consistent set of stylized facts describing the opinion dynamics. In this contribution we consider the parties' vote share distributions observed during the Lithuanian parliamentary elections. We show that most of the time empirical vote share distributions could be well fitted by numerous different distributions. While discussing this peculiarity we provide arguments, including a simple agent-based model, on why the beta distribution could be the best choice to fit the parties' vote share distributions.
... This perfectly solves the problem of accountability but the representativity of such a system is known to be poor because it tends to be detrimental for minorities, especially for a minority that is spread in all districts. On the other hand, party-list proportional-representation systems [19] can be quite good on representativity, provided that the threshold of representation is small, but very poor on accountability. ...
The goal of this paper is twofold. First and foremost, we aim to experimentally and quantitatively show that the choice of a multiwinner voting rule can play a crucial role on the way minorities are represented. We also test the possibility for some of these rules to achieve proportional representation.
... This system allows a representation in a parliament which is both proportional with respect to parties and with respect to regions. For the biproportional apportionment method in theory and practice see Pukelsheim [26] and references therein. ...
In this article we describe some concepts, ideas and results from the mathematical theory of voting. We give a mathematical description of voting systems and introduce concepts to measure the power of a voter. We also describe and investigate two-tier voting systems, for example the Council of the European Union. In particular, we prove criteria which give the optimal voting weights in such systems. 2010 Mathematics Subject Classification. Primary 91B12; Secondary 91B80, 82B05.
Affirmative action in India reserves explicit proportions of seats and jobs in publicly funded institutions for various beneficiary groups. Because seats are indivisible and arise in small numbers over time, implementation of this policy requires that beneficiary groups take turns claiming seats, for which purpose India relies on a device called a roster. We study the problem of constructing a roster, which involves addressing a series of connected apportionment problems. To identify suitable apportionment methods, six essential requirements direct our search to a large class of divisor methods. We show that the Webster–Sainte-Laguë method is the unique divisor method that satisfies several practical properties and fairness criteria. Comparative analysis between an existing Indian roster and the application of the Webster–Sainte-Laguë method highlights that method’s benefits.
For the 2016 U.S. presidential primary, the Georgia Republican Party created a new method, which we call the Iterated Lower Quota (ILQ) method, to apportion delegates using a series of rounds that is reminiscent of a fair division procedure. We analyze the ILQ method and compare it to Hamilton’s method (the most common method of delegate apportionment) using simplicial geometry. We use the geometry to estimate the likelihood that the two methods agree. To understand the bias exhibited by the ILQ method towards the strongest candidates, we calculate the threshold values for ILQ and estimate the expected deviation of the allocations from quota. We determine the maximum number of rounds required to execute the ILQ method, and investigate the apportionment paradoxes to which ILQ is susceptible. In practice, before using the ILQ method, the Georgia Republican Party uses cutoffs to eliminate candidates with low vote totals. We explain how these cutoffs, also used in the Democratic Party’s primaries and in other Republican state primaries, affect the theoretical results on ILQ. We conclude by considering the advantages and disadvantages of ILQ in the context of delegate allocation.
In apportionment elections, a fixed number of seats in a parliament are distributed to parties according to their vote counts. Common procedures are divisor sequence methods like D’Hondt or Sainte-Laguë. In many countries, an electoral threshold is used to prevent very small parties from entering the parliament. Parties with fewer than a given number of votes are simply removed. We (experimentally) show that by exploiting this threshold, the effectiveness of strategic campaigns (where an external agent seeks to change the outcome by bribing voters) can be increased significantly, and prove that it is computationally easy to determine the required actions. To resolve this, we propose an alternative second-chance mode where voters of parties below the threshold receive a second chance to vote for another party. We establish complexity results showing that this makes elections more resistant to strategic campaigns.
This chapter proceeds to define the baseline analytical framework for assessing the political consequences of electoral systems. Since the research is concerned with the objectives of reformers in changing electoral systems, the chapter focuses primarily on the systemic consequences of electoral reforms, both on the inter-party (the allocation of seats to parties) and intra-party (the allocation the seats to candidates) dimensions. Drawing upon a discussion of existing theoretical and empirical studies, the chapter provides both the preliminary expectations and conceptual and theoretical baseline for the empirical analysis provided in the next chapters, where the overall seats–votes proportionality (inter-party dimension) and personalisation of electoral systems (intra-party dimension) are analysed as specific political consequences of electoral reforms in Central Europe.
Chapter 9 looks at the most recent electoral reform in Czechia, which was to change its parliamentary electoral system in 2021 after the Constitutional Court annulled several provisions of the Parliamentary Electoral Act as unconstitutional. The shortcomings of the 2002 electoral system have been identified; discussed are changes in electoral system and the political consequences thereof. Despite the many flaws in the Court’s reasoning and the inconvenient timing of the announcement of the judgment just 8 months before the upcoming parliamentary elections, there was some space for the political elites to remedy the deficiencies of the 2002 electoral system. The case study concludes that the political representation did not take advantage of but rather missed the opportunity to remedy the Czech parliamentary electoral system.
The tandem system proposes a double proportional electoral system for the European Parliament. It offers a forum for europarties to contest an election with power, visibility and influence. The tandem system proceeds in three steps. The first step apportions all parliamentary seats among europarties by aggregating the electorate’s votes at Union level. Thus, with regard to the division of the Union’s citizens by political persuasion, the tandem system obeys the One Person–One Vote principle. The second step, disaggregation of the unionwide apportionment, allots the seats by Member State and Europarty in a way safeguarding the seat contingents of the Member States. Thus, with regard to the Union’s layout by Member State, the tandem system respects the principle of degressive representation. The third step assigns the seats of a party in a Member State to domestic candidates by means of the same provisions that Member States have been employing in the past, thus complying with the Union’s principle of subsidiarity.
Ideally, a representative democracy awards a genuine vote to each adult. We study this issue in competitive democracies with an election model combining district apportionment and proportional representation. Four classic seat allocation rules, including d’Hondt, are reframed as Dutch auctions, allowing important properties to be easily derived. The pros and cons of these methods are stated in terms of economic inequality; Sainte Laguë’s is shown to best carry the genuine vote ideal, both for elections and for apportionment. We next expound the interplay between these two components in generating an inequitable treatment of voters and develop the scale-free index of inequity best fitted to their concern. We apply it to 40 countries for the apportionment of electoral districts. Lastly, we compute the same inequity index for recent parliamentary elections in 80 countries, finding that the majority system mistreats electors, thus putting a ‘price’ on government stability.
Although the change in the electoral system used in elections to the Chamber of Deputies has attracted the attention of lawyers, political scientists, and sociologists, we still lack a comprehensive comparative analysis of the new system with the original one or other alternatives. The main reason for this is the lack of empirical data. This article overcomes this problem using a simulation of electoral results that correspond to the real Czech election environment. On the basis of this simulated dataset it is possible to generate generalisable conclusions about the proportionality, integration effect, and legitimacy of three electoral formulas: the original D’Hondt divisor, the newly adopted Imperiali quota, and the Hare quota. Our results show that the Hare quota is clearly the best choice. Its advantages in proportionality significantly outweigh its disadvantages in integrative effect. Moreover, this formula sees the least disruption to the logical sequence of results, i.e. where a party with fewer votes gets more seats, a phenomenon that is undesirable and undermines the legitimacy of elections. We are convinced that among the three formulas compared the Hare quota is the one that best fits the constitutional requirements of the electoral system as interpreted by the Constitutional Court, and that – unless the legislature is planning to change other parameters of the electoral system – it is the one that should be implemented.
This article gives an algorithm that will build an apportionment diagram for any apportionment method when there are three states. We then provide several images that are produced by the algorithm and discuss what these images can reveal about the given apportionment method.
In the classic apportionment problem, the goal is to decide how many seats of a parliament should be allocated to each party as a result of an election. The divisor methods solve this problem by defining a notion of proportionality guided by some rounding rule. Motivated by recent challenges in the context of electoral apportionment, we consider the question of how to allocate the seats of a parliament under parity constraints between candidate types (e.g., an equal number of men and women elected) while at the same time satisfying party proportionality. We study two different approaches to solve this question. We first provide a theoretical analysis of a recently devised mechanism based on a greedy approach. We then propose and analyze a mechanism that follows the idea of biproportionality introduced by Balinski and Demange. In contrast with the classic biproportional method by Balinski and Demange, this mechanism is ruled by two levels of proportionality: Proportionality is satisfied at the level of parties by means of a divisor method, and then biproportionality is used to decide the number of candidates allocated to each type and party. A typical benchmark used in the context of two-dimensional apportionment is the fair share (a.k.a matrix scaling), which corresponds to an ideal fractional biproportional solution. We provide lower bounds on the distance between these two types of solutions, and we explore their consequences in the context of two-dimensional apportionment.
Significance
A cornerstone in the modern political organization of societies is the existence of a deliberative assembly, reflecting the needs of different population segments. As modern societies become more complex, representation according to dimensions beyond political affiliation and geography is demanded; examples include gender balance and ethnicity. As this dimensionality increases, the task becomes more challenging and requires more sophisticated mathematical tools. In this paper, we initiate the study of multidimensional apportionments and show that, in three and more dimensions, their existence is not guaranteed. However, our main result states that it is possible to elect a house nearly respecting proportionality of representation along several dimensions simultaneously. We finally illustrate the potential of our approach with recent election data.
Several different arguments support the use of Webster’s method when seats in a parliament are to be apportioned proportionally according to populations. This note—instigated by a new property—summarizes the reasons.
Designing Constituencies in the French Political Tradition
The article analyzes the French redistricting process (redécoupage) since 1789, tracing both the evolution of relevant legal norms through the transformations in the political regime and the electoral system, and the actual political practice – what were the driving forces behind consecutive redistrictings, how the districting rules and district boundaries themselves were manipulated for partisan and individual political gain, and how did the resultant district maps measure against standards such as electoral equality and political neutrality. The article concludes that while the French redistricting process remains highly politicized when compared with other European countries, the 2008 constitutional reforms and the changes resulting from them were substantial steps towards greater transparency. Moreover, there is no persuasive evidence that recent redécoupages were systematically manipulated for political purposes by the governing parties.
Given a real number , we define the Webster sequence of density to be , where is the ceiling function. It is known that if and are irrational with , then and partition . On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of into sequences with irrational. In this paper, we consider the question of how close one can come to a three-part partition of into Webster sequences with prescribed densities . We show that if are irrational with , there exists a partition of into sequences of densities , in which one of the sequences is a Webster sequence and the other two are "almost" Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the nth element of each sequence in constant time and the second algorithm gives the assignment of the nth positive integer to a sequence in constant time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems.
The German mixed-member proportional (MMP) system is considered a role model worldwide. Nevertheless, it has a neglected side-effect: it may produce greatly enlarged parliaments, like the 2017 Bundestag with 111 additional seats. Thus, it is highly relevant to know under which conditions MMP systems lead to such seat enlargements. The article explores this question for the German Länder that have used various MMP versions and seen occasional parliamentary inflations. The analysis demonstrates that a two-stage model of party-system features and institutional factors explain enlargements under MMP systems in 156 Länder elections from 1947 to 2019. Concerning the party system features, enlargements are driven by high seat concentrations in single-member districts and low list-vote shares of the largest party. Institutionally, high ratios of SMD seats and full levelling of surplus seats affect parliamentary oversize. These results have important implications for MMP systems in Germany and other countries.
We propose a simple yet new formula for estimating national seat shares and quantifying seat biases in elections employing the Jefferson–D’Hondt (JDH) method for seat allocation. It is based solely on the national vote shares and fixed parameters of the given electoral system. The proposed formula clarifies the relationship between seat bias on the one hand, and the number of parties and the number of districts on the other. We demonstrate that the formula provides a good estimate of seat allocations in real-life elections even in the case of minor violations of the underlying assumptions. With that aim in mind, we have tested it for all nine EU countries that employ the JDH method in parliamentary elections. Moreover, we discuss the applications of the formula for modeling the effects of vote swings, coalition formation and breakup, spoiler effects, electoral engineering, artificial thresholds and political gerrymandering. By not requiring district-level vote shares, our formula simplifies electoral simulations using the JDH method.
Moving beyond the analytical characteristics of apportionment methods or election systems, this article focuses on their outcomes in practice. We illustrate how apportionment and partisan biases working with a high threshold created an electoral environment conducive to the establishment of a predominant party system. We use the historical example of the Turkish experience. We trace the historical development of disproportionality for the entire multi-party elections for the 1950–2015 period. Focusing on the five most recent elections of this period since 2002, we demonstrate how the biases introduced by the apportionment method in use and the 10% threshold have advantaged the leading Justice and Development Party (Adalet ve Kalkınma Partisi, AKP). Our study suggests that a partisan bias favoring AKP still continues to exist at a lower level even after correcting the apportionment and the threshold biases. We underline how these biases form the foundation for a conservative over-representation and emphasize the path-dependent dynamics that keep challengers to the AKP away from the electoral scene, effectively helping to continue its hegemonic position in the system.
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