Divisor Methods Research Articles

Apportioning the House of Representatives in 2000: The Effects of Alternative Policy Scenarios*

Objective. The goal of this article is to develop and examine eight different scenarios in which the U.S. House of Representatives is reapportioned for the year 2000 on the basis of eight scenarios in which the apportionment population is differentially defined. In each scenario we adjust the 2000 apportionment population of each of the 50 states on the basis of various aspects of immigration, eligibility to vote, the overseas population, and statistical sampling.Method. The method we use to apportion the House under each of the eight scenarios is the method of Equal Proportions, a divisor method that has been used to apportion the House since 1940. It endeavors to assign equitably to the 50 states, on the basis of population size, the 385 House seats that are available to be apportioned. Not all 435 House seats are assigned to the states using this method because each state automatically gets one seat.Results. We show that when the House of Representatives is reapportioned on the basis of any of the eight scenarios, there is a varying amount of redistribution of seats among the states. Many states end up gaining or losing seats, but not all are affected.Conclusion. There have been several legal challenges following the 2000 and earlier apportionments. The results of our analyses suggest that there will continue to be legal challenges to the 2000 and later apportionments. Since it is not likely that the Congress will permit a larger number of representatives than 435, apportionment challenges are really a zero‐sum game. Some states will gain political power, and others will lose it. This redistribution of power and its politically relevant consequences lead us to expect that apportionment challenges regarding the kinds of political and demographic issues raised in this article are not at all unlikely in the future.

Dividing the Indivisible

Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each other’s preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is non-monotonicity – choosing earlier may hurt rather than help a party. We introduce a mechanism, combining sequential choices with a structured form of trading, that results in sincere choices for two parties that avoids these problems. Although there are dif.culties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to make consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.

On the [formula omitted]-divisor method and its application

For a smooth projective 3-fold of general type, we prove that the relative canonical stability μ s (3)⩽8. This is induced by our improved result of Kollár: the m-canonical map of X is birational onto its image whenever m⩾5 k+6, provided P k ( X)⩾2. The Q -divisor method is intensively developed to prove our results.

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-Lague), 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.

Surface volumes of rounding polytopes

The paper determines the vertices and surface volumes of all rounding polytopes for commonly used rounding methods: the quota method of greatest remainders, and the divisor methods. These methods are used to round continuous non-negative weights summing to one to non-negative integers summing to a predetermined accuracy, e.g. to 100 when rounding to percentages. Our results are of interest when average properties of rounding methods are investigated, and an example from political science is included.

Seat biases of apportionment methods for proportional representation

In proportional representation systems, an important issue is whether a given apportionment method favors larger parties at the expense of smaller parties. For an arbitrary number of parties, ordered from largest to smallest by their vote counts, we calculate (apparently for the first time) the expected differences between the seat allocation and the ideal share of seats, separately for each party, as a function of district magnitude, with a particular emphasis on three traditional apportionment methods. These are (i) the quota method with residual fit by greatest remainders, associated with the names of Hamilton and Hare, (ii) the divisor method with standard rounding (Webster, Sainte-Laguë), and (iii) the divisor method with rounding down (Jefferson, Hondt). For the first two methods the seat bias of each party turns out to be practically zero, whence on average no party is advantaged or disadvantaged. On the contrary, the third method exhibits noticeable seat biases in favor of larger parties. The theoretical findings are confirmed via empirical data from the German State of Bavaria, the Swiss Canton Solothurn, and the US House of Representatives.

A majorization comparison of apportionment methods in proportional representation

From the inception of the proportional representation movement it has been an issue whether larger parties are favored at the expense of smaller parties in one apportionment of seats as compared to another apportionment. A number of methods have been proposed and are used in countries with a proportional representation system. These apportionment methods exhibit a regularity of order, as discussed in the present paper, that captures the preferential treatment of larger versus smaller parties. This order, namely majorization, permits the comparison of seat allocations in two apportionments. For divisor methods, we show that one method is majorized by another method if and only if their signpost ratios are increasing. This criterion is satisfied for the divisor methods with power-mean rounding, and for the divisor methods with stationary rounding. Majorization places the five traditional apportionment methods in the order as they are known to favor larger parties over smaller parties: Adams, Dean, Hill, Webster, and Jefferson.

Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System

Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each others' preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is nonmonotonicty - choosing earlier may hurt rather than help a party. We introduce a mechanism that combines sequential choices with a structured form of trading that results in sincere choices for two parties. Although there are difficulties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to making consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.

Het Belgische kiessysteem : de klassieke en alternatieve methode D'Hondt

In comparative electora/ studies, the Belgian electoral formula is classified in different, sometimes rather conflicting, classes. This paper is primarily concerned with the question if the Belgian electoral system truely is a D'Hondt system ? In the first part I argued that the D'Hondt method, as a divisor sequence, can be applied alternatively. This modified method departs from a Hare-quota in the first phase. To allocate the rest seats in the second phase, the parties' aggregated vote totals (and not rest totals) are divided by the number of seats obtained in phase one, added by 1, then 2, then 3 etc. Like all divisor methods, the highest averages are used to assign the rest seats. This alternative D'Hondt method produces exactly the same resu!ts as the classical method D'Hondt. Thus, it is also used in the Belgian two-tier-districting system. Furthermore, the Belgian electoral system can be classified as a proper D'Hondt system (in its form and results) while ignoring the primary districts.

ON THE UNBIASEDNESS OF THE PARAMETRIC DIVISOR METHOD FOR THE APPORTIONMENT PROBLEM

Apportionment problem has a very long history of more than 200 years and has been challenged actively by many applied mathematicians and operations researchers. In the last 200 years many appor­ tionment methods have been proposed and various types of properties, which they should desirably satisfy, have been proposed and investigated although the apportionment problem itself has not been completely solved yet. Unbiasedness is an important property for the apportJionment method, and has been studied intensively by Balinski and Young and others. In this paper we investigate the unbiasedness of parametric divisor method, comparing with that of most traditional apportionment methods. First we characterize parametric divisor method by explicitly showing the corresponding loeal measure of bias and deriving several useful facts concerning the unbiasedness of apportionment methods. Then we show the numerical results using Japan's House of Representatives data and exhibit parametric divisor method is preferable to other apportionment methods from the viewpoint of the unbiasedness by adJusting the parameter value appropriately in the certain range. Finally we discuss the possibility for the parametric divisor method to be accepted by our society in the future.

THE PROBLEM WITH APPORTIONMENT

The problem of apportionment is to allocate the seats of a House of Representatives or Diet to states or constituencies proportionally to their respective populations. This paper reviews the problem, its background and pitfalls, and the various approaches that have been applied to its analysis.

Use of chemometric factor analysis for chromatographic integration: application to diode-array high-performance liquid chromatography of mixtures of chlorophyll a degradation products

Quantification errors of using the vertical divisor method for single wavelength chromatographic integration of closely eluting peaks are discussed. A new approach of using the chemometric factor analysis method, heuristic evolving latent projections, prior to integration is discussed, and typical single-wavelength errors in estimated integrals are less than 1% on simulated data. The method was applied to three mixtures of six closely eluting chlorophyll degradation products. Single-wavelength quantification is good, provided that the wavelengths chosen are close to the maxima of the peaks. Further improvements can be made by summing over all wavelengths. Finally, an approach to normalization of chromatographic integrals is discussed, which is particularly valuable when pure standards are not available.

Application of the divisor method to multiple peak DLTS spectra

The DLTS divisor method, derived in a previous paper, is applied to multiple and overlapping DLTS peaks. An initial analysis is made of two moderately separated peaks. Then, the same peaks are analyzed with 10 percent noise present, using four thermal scans and no signal averaging. The results are compared with those obtained from other methods. Finally, to test the resolution, the peaks are moved together until the method no longer applies. It is found that the divisor method successfully resolves peaks with separation greater than 0.07 eV.

Algebraic Computations of Scaled Padé Fractions

Two companion algorithms are developed for constructing Pade fractions along an off-diagonal path of the Padé table for a function ${{ - A(z)} / {B(z)}}$, where $A(z)$ and $B(z)$ are formal power series over a field. One of the algorithms computes the first n Padé fractions along the off-diagonal in time $O(n^2 )$. When $A(z)$ and $B(z)$ are finite power series (i.e., polynomials), it is shown that the algorithm is equivalent to Euclid’s extended algorithm for computing greatest common divisors. The other algorithm, a generalization of the first, proceeds along the off-diagonal in quadratic steps, and is of complexity $O(n\log ^2 n)$. When $A(z)$ and $B(z)$ are polynomials, the second algorithm becomes a fast Euclid’s extended algorithm for computing greatest common divisors. The algorithm is of the same complexity as other fast greatest common divisor methods, but its iterative nature provides a practical advantage during implementation. The algorithms may also be used for computing Padé fractions along an anti-diagonal path of the Padé table. The fast algorithm is of the same complexity as other fast algorithms for anti-diagonal computations. However, it has the advantage of being able to determine easily any specific Padé fraction along the anti-diagonal.

Some Properties of Divisor Methods for Legislative Apportionment and Proportional Representation

Some rather elegant properties of linear divisor methods are derived and used to establish upper and lower bounds on the possible variation between apportionment and exact quota entitlement. A probability distribution is derived for this variation, and it is shown that the probability of the variation exceeding one seat is very small with the major fractions linear divisor method.A less rigorous analysis of the nonlinear equal proportions method shows that in practice it is very similar to the major fractions method, but with a very slight bias in favor of small parties (or states). It is concluded that there is no “best” apportionment method, but a knowledge of the properties of the various methods enables a political choice of the most appropriate method.

Some Properties of Divisor Methods for Legislative Apportionment and Proportional Representation

Some rather elegant properties of linear divisor methods are derived and used to establish upper and lower bounds on the possible variation between apportionment and exact quota entitlement. A probability distribution is derived for this variation, and it is shown that the probability of the variation exceeding one seat is very small with the major fractions linear divisor method.A less rigorous analysis of the nonlinear equal proportions method shows that in practice it is very similar to the major fractions method, but with a very slight bias in favor of small parties (or states). It is concluded that there is no “best” apportionment method, but a knowledge of the properties of the various methods enables a political choice of the most appropriate method.

Apportionment Approach for Just-In-Time Sequencing Problem

The Just-in-Time sequencing of different products is a well-known socio-industrial problem which is supposed to minimize the maximum and total deviations between actual and ideal productions; and the apportionment problem is a socio-political problem which aims to allocate representatives to a state as close as its exact quota. Significant amounts of research have been done in these two problems independently. The relation between them has been studied from last decade. In this article, both problems are addressed with some just-in-time sequencing algorithms and their characterizations via apportionment.Key words: apportionment, divisor methods, algorithms, Just-in-Time sequencing.Journal of the Institute of Engineering, Vol. 7, No. 1, July, 2009 pp.76-89doi: 10.3126/jie.v7i1.2065