Dynamic adjustment: an electoral method for relaxed double proportionality

We describe an electoral system for distributing seats in a parliament. It gives proportionality for the political parties and close to proportionality for constituencies. The system suggested here is a version of the system used in Sweden and other Nordic countries with permanent seats in each constituency and adjustment seats to give proportionality on the national level. In the national election of 2010 the current Swedish system failed to give proportionality between parties. We examine here one possible cure for this unwanted behavior. The main difference compared to the current Swedish system is that the number of adjustment seats is not fixed, but rather dynamically determined to be as low as possible and still insure proportionality between parties.     

The give-up problem for blocked regional lists with multi-winners

The current electoral law for the Italian Parliament prescribes blocked, linearly ordered lists of candidates for each party within each constituency. The peculiarity of the Italian electoral system is that a party can present the same candidate in different constituencies. There are several seats at stake in each constituency; these seats are allocated to the parties proportionally to the total number of votes they get. If the blocked list mechanism-which assigns the seats obtained by a party in a constituency to the first candidates of the corresponding ordered list-causes some candidates to win in more than one constituency, they may retain only one of the seats, giving up all the remaining ones. Thus, the problem arises for a party to find a suitable “schedule of give-ups” that produces the final set of winners for that party. In order to do this, we assume that such decision is centralized and based on some models of global (inter-regional) preferences over the set of candidates. In this paper, we introduce two classes of models to formulate the “give-up problem”, i.e., utility and ordinal models, and we show that for both of them some natural formulations of the problem can be efficiently solved by network flows techniques.     

Ghost seats in parliaments

In several European Countries citizens’ movements have formed parties whose programme is to reform electoral legislation so that blank votes are transformed into vacant seats in parliament. The first objective of this paper is to offer some tools in order to quantify the potential effect of a reform that would allow citizens to cast a dissatisfaction vote. We show that such a reform would reduce the ease to pass a proposal and modify the distribution of power among parties. The second objective of this paper is to show that this reform could have an impact in parliaments. This is illustrated by the case of the Basque Parliament from 1980 to 2016. At the theoretical level this paper provides some relations between measures of success.     

Some Consequences of a Three-Party Split in a UK General Election

The British electoral system is unique in Europe in being of the first-past-the-post variety. The apparent emergence of a strong third party renders any prediction exercise a good deal more difficult, although some political commentators appear oblivious to that fact. It would appear that a transition matrix approach is capable of providing the deeper insights needed to explore the consequences of alternative voting-behaviour patterns. Unfortunately data of this form are not currently collected, but it is possible to postulate that transition matrices of a particular form could be of interest to the three parties. By associating such transition probabilities with the 1983 results for each of the 633 mainland constituencies, one can derive interesting relationships between the number of seats secured by each party.A range of computer analyses was performed, and this article sets out some of the more interesting results, some of which came as something of a surprise.     

Swing Analysis and Electoral Forecasting

This paper examines the procedure by which votes are converted into seats at U.K. General Elections. In particular it seeks to answer these questions.

1. i)
   What is the meaning of swing when more than two parties fight an election?
    
2. ii)
   How can the distribution of seats at an election be determined from a prediction of national swing?
    
3. iii)
   How can swing analysis be extended to help determine an electoral strategy for political parties?
    
4. iv)
   What explanation can be provided for the swing in Scotland?
    
5. v)
   What explanation can be provided for variations in the English swing?
    

     

Error minimization methods in biproportional apportionment

One of the most active research lines in the area of electoral systems to date deals with the Biproportional Apportionment Problem, which arises in those proportional systems where seats must be allocated to parties within territorial constituencies. A matrix of the vote counts of the parties within the constituencies is given, and one has to convert the vote matrix into an integer matrix of seats ??as proportional as possible?? to it, subject to the constraints that each constituency be granted its pre-specified number of seats, each party be allotted the total number of seats it is entitled to on the basis of its national vote count, and a zero-vote zero-seat condition be satisfied. The matrix of seats must simultaneously meet the integrality and the proportionality requirement, and this not infrequently gives rise to self-contradictory procedures in the electoral laws of some countries. Here we discuss a class of methods for Biproportional Apportionment characterized by an ??error minimization?? approach. If the integrality requirement is relaxed, fractional seat allocations (target shares) can be obtained so as to achieve proportionality at least in theory. In order to restore integrality, one then looks for integral apportionments that are as close as possible to the ideal ones in a suitable metric. This leads to the formulation of constrained optimization problems called ??best approximation problems?? which are solvable in polynomial time through the use of network flow techniques. These error minimization methods can be viewed as an alternative to the classical axiomatic approach introduced by Balinski and Demange (in Math Oper Res 14:700?C719, 1989a; Math Program 45:193?C210, 1989b). We provide an empirical comparison between these two approaches with a real example from the Italian Elections and a theoretical discussion about the axioms that are not necessarily satisfied by the error minimization methods.     

A mathematical model for estimating the potential margin of state undecided voters for a candidate in a US Federal election

A US Federal election in which candidates from two major political parties compete for the votes of those undecided voters in a state who usually do not vote in US elections is considered. A mathematical model for evaluating the expectation of the margin of votes to be received from such voters by either candidate as a result of the election campaigns of all the competing candidates is proposed. On the basis of this model, finding the estimation under consideration is reducible to finding the minimum of the maximin function of the difference of two bilinear functions with one and the same first vector argument whose second vector arguments belong to a polyhedron of connected variables (strategies of the candidates), and this minimum is sought on another polyhedron.     

Bounds on Malapportionment

Uniformly sized constituencies give voters similar influence on election outcomes. When constituencies are set up, seats are allocated to the administrative units, such as states or counties, using apportionment methods. According to the impossibility result of Balinski and Young, none of the methods satisfying basic monotonicity properties assign a rounded proportional number of seats (the Hare-quota). We study the malapportionment of constituencies and provide a simple bound as a function of the house size for an important class of divisor methods, a popular, monotonic family of techniques.     

Bidimensional allocation of seats via zero-one matrices with given line sums

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously. In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.     

Asymptotic bias of some election methods

Consider an election where N seats are distributed among parties with proportions p ₁,…,p _m of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a party, i.e. the difference between the number of seats given to the party and the (real) number Np _i that yields exact proportionality. Our approach is to keep p ₁,…,p _m fixed and let N→∞, with N random in a suitable way. In particular, we give formulas showing the bias favouring large or small parties for the different election methods.     

Parametric vs. divisor methods of apportionment

This paper formulates a new criterion that distinguishes the set of parametric methods within the set of all the divisor methods of apportionment. The criterion—that a method transfer seats as it “should”—asks that as population (or the votes of parties in a PR system) is shifted more and more from one state to another state (from one party to another party) at some point the first state (or party) is apportioned one less seat, the second state (or party) one more seat, and the remaining apportionments are as they were. It goes on to examine several properties of parametric methods.     

Hybrid Hamilton-Webster and the Greek apportionment

The method of largest remainders (Hamilton) is used for allotment of 288 of the seats among the 56 Greek constituencies. This method possesses various paradoxes as was observed through two centuries of application. So we propose a modification where the seats are allocated at a first stage by the lower Hare-Quota and the additional ones using the method of major fractions (Webster) restricted simultaneously by the upper quota. This method may produce paradoxes but they are observed extremely rare. Extended simulations over the Greek electoral data indicate that the frequency that the new method violates monotonicity is by far less than the frequency that Webster method violates quota.     

The electoral system for the Italian Senate: an analogy with deterministic chaos?

The electoral system adopted for the allocation of seats in the Italian Senate utilizes a complex mechanism of awards at a regional level with the aim of strengthening, when necessary, the winning coalition and so improve overall government stability. The results presented here demonstrate that in a significant number of cases, the effect of the mechanism is opposite to that desired, to wit, weakening the resultant government by awarding more seats to the minority coalition. Indeed the award to the minority can even be such that the minority coalition becomes the majority and wins the election. The application of the award mechanism is strongly unpredictable as it depends crucially on the precise number of seats independently obtained in each region, and that each adjustment thereof can be positive, zero or negative; a characteristic that closely resembles the behaviour of a chaotic dynamical system whose trajectory, although purely deterministic, depends on infinitely precise details and is therefore unpredictable. To perform the systematic numerical analysis of the award effectiveness, we introduce characteristic polynomials, one for each electoral district, which carry information about all possible outcomes and award applications. Their product yields a polynomial containing the dependence of the result at national level on each of the regional awards.     

Bicolored graph partitioning, or: gerrymandering at its worst

This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a “balanced” red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist “partisan” district maps such that the discrepancy between their number of “red” (or “blue”) districts for that bicoloring is extremely large, in fact as large as allowed by color balance.     

一个公平分配席位的新方案

提出了公平累加分配席位的方法,并证明了该方法在一定条件下满足公平分摊原则和席位单调性原则.并进一步指出该方法具备更多的公平性质.     

Game theory and the french apparentements of 1951

The French apparentement electoral system is modeled as an-person cooperative game without side payments. The players are seat-maximizing political parties. Each electoral district constitutes an independent game. The Von-Neumann and Morgenstern solution is applied to generate theoretical postdictions for a cross section of observations from the 1951 French parliamentary elections. In statistical terms, this solution has a modest, but significant, degree of explanatory power.     

Political engineering: optimizing a U.S. Presidential candidate’s platform

We explore the application of operations research to the problem of defining/refining the political strategy for a candidate in a U.S. Presidential election. We use Hierarchical Bayesian techniques to model criteria used by a stratified random sample of registered voters to evaluate a candidate/platform. We then use the estimated utility parameters as inputs to a model that finds the positions a candidate can take on the salient issues of the election that will optimize expected Electoral College votes conditional on the positions respondents perceive to have been taken by the opposing party’s nominee. This approach is unique in that it (i) considers the value that individual voters associate with various positions the candidates can take on various issues, (ii) considers the chronicity of the electorate’s perceptions of a candidate’s positions on the salient issues, and (iii) yields a solution that will optimize expected Electoral College votes. We demonstrate this model on data collected immediately prior to the 2004 U.S. Presidential election (the most recent U.S. Presidential election not involving any potential candidate for the upcoming 2012 U.S. Presidential election), and we show how these data and the model can also be used to assess the perceived clarity of a candidate’s positions, the sensitivity of a candidate’s support to her/his perceived positions, and the viability of a third party candidate.     

Thresholds of the Divisor Methods

We give an algorithm which permits calculating the maximum and minimum vote shares that allow a party to obtain h seats, that is, the threshold of exclusion and the threshold of representation. These have already been studied for some methods (such as d'Hondt or Sainte-Laguë), and are here generalized to any divisor method, and to any number of seats. The thresholds depend on the size of the constituency, the number of parties running in the constituency, and the divisor method used. Finally, we give some consequences, including a characterization of the d'Hondt method.     

Apportionment methods

Most democratic countries use apportionment methods to transform election results into whole numbers, which usually give the number of seats in a legislative body that the parties obtained. Which apportionment method does this best can be specified by measuring the error between the allocated result and the ideal proportion. We show how to find an apportionment method which is best suited to a given error function. We also discuss several properties of apportionment methods that have been labelled paradoxa. In particular we explain the highly publicised “Alabama” Paradox for the Hare/Hamilton method and show that other popular apportionment methods come with their very own paradoxa.