In jalapic/apportR: Apportionment Tools and Data
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
Introduction
The apportR package contains several functions for performing apportionment procedures. The inspiration behind me writing this package can be found here. This package is in development and is very much a work in progress. This vignette details some sample problems that can be solved using the functions in this package, information regarding the example datasets contained in apportR as well as some background and worked examples of each function. At this stage of development I would welcome hearing from anyone who thinks that they could help improve the package.
Installation
The apportR package can be downloaded from github.
library(devtools)
devtools::install_github("jalapic/apportR")
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Loading the package
library(apportR)
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Examples
To illustrate the general problem at hand, here are three examples.
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Fair Division Example 1.
Mom has 50 identical pieces of candy that cannot be broken into bits. She tells her five children that she will divide the candy fairly at the end of the week according to the proportion of housework that each performs.
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The following tables shows the minutes spent in housework by each child and how many of the 50 candies each child should expect to get:
housework
candies <- (50*housework) / sum(housework)
candies
floor(candies)
sum(floor(candies))
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Anna expects 8.33%, Betty 4.33%, Carl 9.6%, Derek 11.33% and Ella 16.39% of the candies, but they cannot be broken up.
Assuming each gets the number of candies equal to their rounded down proportion at least, then Who should get the extra 2 candies that are not yet allocated?
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Fair Division Example 2.
There are 12 scholarships to be given out to each of 3 subjects at a university. They are to be awarded based upon the proportion of majors that each subject has.
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Here are the majors in each subject as well as the expected number of scholarships each would get out of the 12 available:
majors
scholarships <- (12*majors) / sum(majors)
scholarships
floor(scholarships)
sum(floor(scholarships))
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If each gets the rounded down value that they would expect, where do the remaining 2 scholarships go ? Presumably Psychology should get an extra one as its fractional is .9, but which of English and History should get the other one?
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Fair Division Example 3.
At the 1792 US House of Representatives elections, a total of 105 seats were to be awarded that would be shared between the 15 states of the union proportional to each state's population size at the 1790 US census.
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More reading can be found here
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Here are the states and their population sizes in 1790, followed by the expected number of seats for each state based on there being 105 available:
usa1790
seats <- (105*usa1790) / sum(usa1790)
seats
floor(seats)
sum(floor(seats))
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Who should get those 8 extra seats ??? This is a slightly more important question to get right than the fair division of candies (depending upon your perspective)!
I will describe some methods for working this out as well as the functions contained in apportR to represent these methods a bit later in this vignette.
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Datasets
The apportR package contains a number of small datasets useful for testing functions and/or illustrating methods. These are:
data(package='apportR')
#Data sets in package ‘apportR’:
#
#housework Housework example
#majors Majors example
#parador Parador example
#pop1 Population example 1
#usa1790 USA population by state in 1790.
#usa1820 USA population by state in 1820.
#usa1832 USA population by state in 1832.
#usa1880 USA population by state in 1880.
#usa1900 USA population by state in 1900.
#usa1907 USA population by state in 1907.
#usa1940 USA population by state in 1940.
#usa1990 USA population by state in 1990.
#usa2000 USA population by state in 2000.
#usa_census USA population by state in every census.
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All are named vectors except for the last usa_census which is a dataframe that contains the year that each state was admitted to the union as well as the population of every state at every official US census. It should be noted that these population figures do not include Native American or slave populations. (Note - for some reason, some of the population sizes differ between the named vectors and the corresponding columns in usa_census as a result of obtaining the data from two different sources for each. Some of this discrepancy may be due to e.g. whether to include overseas military and civilian personnel and their families towards the populations of their home states or not. I will try to update with wholly consistent data asap).
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Methods / Functions Overview
The methods for solving these sorts of questions essentially fall into two broad groups. There are the standard (or non-divisor) methods and the divisor methods. Many (if not all) of them have been discovered independently in various disciplines and so each may have several different names. I will mainly focus on the methods from the historical perspective of the US House of Representatives apportionment problem, as it is the most interesting.
Currently I include discussion of the standard (non-divisor) methods. I will add information about the divisor methods shortly.
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Standard Non-Divisor Methods
- Hamilton's Method, aka Largest Remainder Method, Vinton's Method, Hare-Niemeyer Method.
- Lowndes' Method
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Hamilton Method
The most simple method was first suggested by Alexander Hamilton in 1792.
Quite straightforwardly, the logic of this method is as follows:
- choose the number of seats available
- calculate the standard divisor (this is equal to the total population / number of seats)
- calculate the standard quota (this is equal to each state population / standard divisor)
- assign each state initally its lower quota (this is the rounded down value of each standard quota)
- calculate how many seats are still to be assigned
- assign extra seats to those states with the highest fractionals from their standad quota in descending order.
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Example
There were 120 seats to be allocated in the House in 1790. The population of each state and the total US population at the 1790 census was as follows:
usa1790
sum(usa1790)
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The standard divisor (which is equivalent to the average number of people that each representative is representing) and standard quotas of each state are therefore:
#standard divisor
std <- sum(usa1790) / 120
std
stq <- usa1790 / std
stq
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The first allocation of seats is the rounded down value of these standard quotas. When summing these values, it is clear that they only sum to 111 seats which is 9 short of the required total of 120.
lowq <- floor(stq)
lowq
sum(lowq)
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The states are then arranged in order of their fractional parts and the states with the 9 largest fractionals are awarded an extra seat. That would be NJ to NC.
rev(sort(stq%%1))
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This can be done using the hamilton function. The first argument is the named vector of population sizes and the second argument is the number of seats to allocate.
hamilton(usa1790, 120)
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Despite the 1792 bill proposing this method be used was passed but yet it was not brought into use at this time. Instead, this bill was the very first usage of a Presidential veto by George Washingon and a divsor method proposed by Thomas Jefferson was used. I will discuss this method in the divisor methods section. Washington vetoed the bill as Hamilton's method would have meant that some representatives would be representing fewer than 30,000 individuals which was against the constitution. It was first of only two times that Washington used his Presidential veto.
We can see this here - several states have too many representatives according to the Constitution:
usa1790 / hamilton(usa1790, 120)
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Despite this initial lack of support, Hamilton's method (also known as Vinton's method by the mid nineteenth century) was passed into law and adopted by 1850 and continued to be used until the turn of the 20th century. It stopped being used at this time because some strange anomalies or paradoxes were discovered. These are as follows:
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Alabama Paradox
In 1880 a goverment clerk named C. W. Seaton computed apportionments for all potential House sizes between 275 and 350 members. He then wrote to Congress describing that if the House of Representatives had 299 seats, Alabama would get 8 members but it would only receive 7 members if the House had 300 seats. This seemed to be a failure of common sense and became known as the Alabama Paradox.
Here are the state populations for 1880 and the number of seats that would be given under the Hamilton method if the House had 299 or 300 members:
usa1880
hamilton(usa1880, 299)
hamilton(usa1880, 300)
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Texas and Illinois would each gain a member whilst Alabama lost one !
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New States Paradox
A second paradox of this method also came to light. This is probably best illustrated using the example of Oklahoma which was admitted to the union in 1907. In 1900 the house had 386 seats. According to OK's population, they should have got 5 seats.
The US population in 1900 and seat allocation according to Hamilton's method was as follows:
usa1900
hamilton(usa1900, 386)
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but the following happens when adding in Oklahoma and allowing for an extra 5 seats:
usa1907
hamilton(usa1907, 391)
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As you can see, after OK is added in 1907, New York would lose a seat and gives it to Maine! Again, this seems to fail a common sense test.
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Population Paradox
A final paradox that has been found is that states that increase in population more relative to other states over a period of time can even lose seats.
Here is a fictious example. Say there are five states (A-E) and this is their population in hundreds of thousands at time 1 (states1) and time 2 (states2):
states1 <- c(150, 78, 173, 204, 295)
states2 <- c(150, 78, 181, 204, 296) #C increases by 8, E by 1
names(states1)<-names(states2)<-LETTERS[1:5]
states1
states2
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A, B and D have remained the same whilst E and C have increased in population. Now say there are 50 seats to allocate - let's see what happens:
hamilton(states1, 50)
hamilton(states2, 50)
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Even though E's population went up by 100,000 it actually lost a representative seat! whilst B whose population stayed the same increased its seat number and C whose population increased by 800,000 remained the same.
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Strengths & Weaknesses
The biggest advantage of this method is that it is very easy to calculate and to explain. This method also never violates the quota rule which states that the number of seats ought to be between the lower and upper quota of the standard quota (effectively the floor and ceiling value). The biggest weaknesses are that it throws up these strange paradoxes or common sense failures, and also that it does systematically favor larger states over smaller ones.
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Notes about the function
There are a few things to note about this function.
1. Named vector - the input vector must be named.
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2. Zero seats problem - for calculating seats in the House, there must always be at least one seat allocated to every state. The hamilton function does this automatically, so if for instance the standard quota of any state would be e.g. 0.333 and its fraction was not large enough to receive an extra surplus seat, then it will override this and allocate the seat. This happens several times in US census history for Delaware and Nevada.
Other users of this function may want to have the option of allocating 0 to a group. Therefore the sister function to hamilton is hamilton0 which allows for this.
For example, take the US census in 1880 and 300 available seats in the House:
usa1880
hamilton(usa1880, 300)
hamilton0(usa1880, 300)
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Clearly, Nevada would not get a seat unless allowances were made to ensure all states received at least one seat. Alabama would have lost out again in this instance !
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This zero issue can also be illustrated with a couple of fair division problems. Say we had five candies to divide to each of 3 children dependent upon how many hours of work they completed.
hours <- c(1,3,11)
names(hours) <- c("Xavier", "Yasmin", "Zach")
hours
#standard quotas
(5*hours)/ sum(hours)
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If we look at the standard quotas then we should initially give 0 to Xavier, 1 to Yasmin and 3 to Zach. The question then is what to do with the fifth candy. If we allow zeros then that extra one would be given to Zach. If we don't allow zeros, then that extra one will be given to Xavier.
hamilton(hours,5)
hamilton0(hours,5)
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This also leads us to an issue with the hamilton function when we automatically assign 1 to every group. If there are many individuals with small standard quotas, then it may become impossible to apportion.
For instance, if we had ten gold watches to give out proportionally to sales reps who sold the most cars in a car dealership and the data looked like this:
cars.sold <- c(1, 2, 15, 3, 1, 12, 4)
names(cars.sold) <- c(paste0("rep", 1:7))
cars.sold
#standard quotas
(cars.sold*10) / sum(cars.sold)
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Using Hamilton's method everyone would get at least 1 watch accounting for 7 of the watches, but rep3 and rep6 should each get 3 watches, making 11 to be allocated when there are only 10. Clearly this does not work. If it is possible to allocate zero watches to individuals, then we can apportion. Notice that the standard hamilton function reports the error.
hamilton(cars.sold, 10)
hamilton0(cars.sold, 10)
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3. Population and seat size - the procedure works by calculating the standard quota of each state i.e. each element of the initial inputted vector. Therefore it won't work in situations where 1) the values in the input vector are too small, 2) the number of seats to be allocated are either too small.
For example, let's say we had the following vector and wanted to allocate 3 seats:
z <- c(1,3,11)
names(z) <- LETTERS[1:3]
z
z1 <- sum(z) / 3 #standard divisor
z1
z2 <- z/z1 #standard quota
z2
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The above cannot work as the function will try to allocate 1 seat to each of A and B and then 2 to C. The minimum number of seats that can be allocated are 4.
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Future adaptations
There are several slight variations to the Hamilton Method, which change slightly how the standard quotas are calculated. I will add these in time. These include:
- Droop Quota method
- Hagenbach-Bischoff quota
- Imperiali quota
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Lowndes' Method
In 1822, Rep. Lowndes of South Carolina proposed an alternative method to Hamilton's. The main tenet of this approach was to give a greater emphasis to the smaller states. Consequently, this approach never received much support and did not become law.
The general logic is as follows:
- choose the number of seats available
- calculate the standard divisor (this is equal to the total population / number of seats)
- calculate the standard quota (this is equal to each state population / standard divisor)
- assign each state initally its lower quota (this is the rounded down value of each standard quota)
- list the number of persons per already assigned representative
- assign extra seats to those states with the highest number of persons per assigned representative untial all seats are filled
This function also accounts for the zero seats issue by automatically assigning at least one representative per state.
Here is an example of Lowndes' approach from the 1820 census with 213 seats to assign:
usa1820
lowndes(usa1820, 213)
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For comparison, here is how Lowndes' approach differs from Hamilton's for the 120 seats to be assigned in 1790.
hamilton(usa1790, 120)
lowndes(usa1790, 120)
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As is clearly evident, Lowndes' method disporportionately favors the smaller states.
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Divisor Methods
There are a large number of divisor methods. The general logic is as follows:
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choose the house size (number of seats)
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find a number (the divisor) such that when all the population sizes of every state are divided by this divisor and these numbers are rounded, that they sum to the number of seats.
The different divisor methods differ in how they round numbers. Further, there is rarely one unique value of the divisor that satisfies this equation, but all values that do for each method will lead to the same apportionment.
Although divisor methods are free from paradoxes that the Hamilton Method suffers, they can violate another rule of apportionment the quota rule - that the number of seats allocated should be no higher than rounded up value of the standard quota and no lower than the rounded down value of this number.
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Eventually the package will include the following divisor apportionment methods:
- Jefferson's Method, aka Greatest Divisor Method, de Hondt's Method - (used in House from 1791-1830)
- Adams' Method, aka Smallest Divisors Method, - (considered by Congress but not used)
- Webster's Method, aka Webster-Willcox Method, Major Fractions Method, Sainte-Lague Method, - (used in House in 1840, 1910 and 1930)
- Dean's Method, aka Harmonic Mean Method
- Huntington-Hill Method, aka Geometric Mean Method, Equal Proportions Method, - (used in House from 1940 to present)
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The package currently contains functions called jefferson, jefferson0 and adams that can perform those methods.
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Jefferson's Method
In 1792, ten days after George Washington vetoed the bill to use the Hamilton Method, Congress passed the method of apportionment sugested by Thomas Jefferson. They also decided to reduce the number of seats available from 120 to 105 to avoid the issue of having representatives representing too few people (i.e. fewer than 30,000).
Jefferson's Method works as follows:
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Find the initial divisor - this is equal to the sum of state populations divided by the number of seats available.
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Determine if the sum of the rounded down standard quotas produced by dividing state populations by the divisor are equal to the number of seats available.
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If they are not, then through trial and error, find a divisor such that the sum of the rounded down standard quotas produced by dividing state populations by the divisor are equal to the number of seats available.
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If a state's lower quota is equal to zero, assign them one representative seat regardless (the jefferson function automatically does this at present).
For example, here is the 1790 US state populations:
usa1790
divisor <- sum(usa1790) / 105
initial.divisor <- floor(divisor)
initial.divisor
first.sum <- sum(floor(usa1790/initial.divisor))
first.sum
first.sum == 105
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The best strategy here as we have too few seats is to decrease the value of the divisor to try and find a solution. This is possible and is what the jefferson function does:
jefferson(usa1790, 105)
rev(sort(jefferson(usa1790, 105)))
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... and to compare this with what Hamilton's Method would have produced:
hamilton(usa1790, 105)
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If Hamilton's had been used instead then 13 of the 15 states would have received the same number of Representatives. Unsurprisingly, Virginia were the beneficiaries under Jefferson's method with poor Delaware losing a seat.
This method was used by the House from 1792-1842 when it was considered that it was too favorable towards the larger States and a different method was needed.
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Here is how the Jefferson Method determined the allocation of seats for the 1832 elections to the House of Representatives with 240 seats being available. I have sorted the allocations in descending order.
usa1832
rev(sort(jefferson(usa1832, 240)))
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Quota Rule Violations
Although divisor methods do not have paradoxes associated with them, they can violate the quota rule which states that the number of seats given to each state should be at most the ceiling value of their standard quota and at least the floor value of their standard quota. The 1832 election above is interesting from this point of view because it actually breaks the quota rule. New York is allocated 40 seats, yet this is the standard quota for each state:
(240*usa1832)/sum(usa1832)
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NY should have received only 38 or 39 seats to satisfy the quota rule, but instead it received 40.
Interestingly, according to Balinski and Young’s Impossibility Theorem, it is not possible to come up with a perfect apporitonment system - there will always be quota rule violations or paradoxes but not both with any system.
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The quota rule violation can also be understood with this example of a fictitious country called Parador which has 5 states and needs to allocate 250 seats to its House. Below I will show the populations of each state, the standard, lower and upper quotas and the results of Jefferson's Method using the built-in dataset parador.
state = names(parador)
popn = parador
st.q = (250*parador)/sum(parador)
low.q = floor(st.q)
upp.q = ceiling(st.q)
seats = jefferson(parador, 250)
data.frame(state, popn, st.q, low.q, upp.q, seats)
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As can be seen, state B actually garners one extra seat that it should not do so according to the quota rule. One of the major criticisms of the Jefferson Method is that it overly favors large states.
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Notes about using the Jefferson Function
There are two important issues to consider as regards the Jefferson Method function.
1. All the divisor methods utilize a 'trial-and-error' method to find a divisor that will correctly apportion the seats to the states based on relative population size. For the jefferson function to work, there must therefore be a divisor that meets this criteria.
The way I have programmed the jefferson function to search for this divisor is to begin with the initial divisor and then try smaller and smaller divisors (initial divisors are too large). The user can adjust the increments by which the function searches for these divisors. The default is set to k=1 which means that the function will search for divisors by decreasing integer values one by one. This works fine for examples such as those above when using large population sizes. However, this might not be appropriate when using smaller initial values in groups. In this case, it may be better to set k to a smaller value such that the function will search for divisors between integers.
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Here is an example:
A teacher has 10 gold stars to divide between 6 students who are taking an exam. The teachers says they will award the stars based on how many questions each student gets correct.
answers <- c(1,4,5,14,16,3)
names(answers) <- paste0("Student ", LETTERS[1:6])
answers
(10*answers)/sum(answers) #standard quotas
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Let's try and apply the jefferson function with the default parameter setting for k=1:
jefferson(answers,10)
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This tells us that it was not able to find an appropriate divisor and that we should try reducing our value of k.
jefferson(answers, 10, k=0.1)
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Now, we are able to find a solution. My suggestion for adjusting k is to gradually decrease it by one decimal place at a time.
Here is another example.
In the country of Tecala, there are four states that wish to elect 160 seats to their house. The population of each state was 3.31, 2.67, 1.33 & 0.69 (in millions).
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popn <- c(3.31, 2.67, 1.33, 0.69)
names(popn)<- c("Apure", "Barinas", "Carabobo", "Dolores")
popn
jefferson(popn, n=160) # 160 seats
jefferson(popn, n=160, k=0.1) # 160 seats
jefferson(popn, n=160, k=0.01) # 160 seats
jefferson(popn, n=160, k=0.001) # 160 seats
jefferson(popn, n=160, k=0.0001) # 160 seats
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2. Allowing zero apportions. It is possible to allow for zero apportions using jefferson's sister function jefferson0.
For instance, if the teacher in the example above did not feel that all students should get gold stars but only those who were deserving should get them, then we could allocate using jefferson0:
jefferson0(answers, 10, k=0.1)
jefferson(answers, 10, k=0.1)
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Adams' Method
In 1832, former president John Quincy Adams essentially suggested an inverse method to Jefferson's Method. This method was considered but never passed into law. Essentially, this method operates identically to Jefferson's except that the rounding method used to determine representatives from the standard quota is a ceiling rounding (i.e. round the standard quota up). This has the result of overly benefitting the smaller states and hence why it didn't receive much support.
Here is an example for the adams function for the 1832 House with a size of 240 seats. The function is compared to the Hamilton and Jefferson Methods.
usa1832
adams(usa1832, 240)
rev(sort(jefferson(usa1832, 240))) #function doesn't automatically sort currently
hamilton(usa1832, 240)
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Notes about using the Adams Function
- As with the
jefferson and jefferson0 functions, the adams function will automatically stop and return an error message if it cannot find an appropriate divisor. It will encourage the user to use a smaller incremental for the 'trial-and-error' process of finding a divisor.
For example, say we had collected data on how many workers attended an office on each day of the week and we wanted to create a chart such as a waffle chart to visually represent this. However, we wanted a nice rectangular chart that requires 100 percentage points to be allocated. We could use the adams function.
set.seed(111)
workers <- round(runif(7, 1, 1000))
names(workers) <- c("Mon", "Tues", "Weds", "Thurs", "Fri", "Sat", "Sun")
workers
adams(workers) #defaults are n=100, k=1
adams(workers, k=.1)
waffle::waffle(adams(workers, k=.1), rows=5)
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- Like the Lowndes Method above, the Adams Method uses a rounding up method, so it is not possible to apportion zeros to groups.
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Webster's Method
In the 1830s another divisor method was suggested by Senator Daniel Webster. Webster's method can be summarized as follows:
- choose the number of seats available
- calculate the standard divisor (this is equal to the total population / number of seats)
- calculate the standard quota (this is equal to each state population / standard divisor)
- calculate the arithmetic mean of the lower bound (LQ) and upper bound (UQ) of the standard quota of each state
- assign a state its upper quota number of states if the standard quota is greater than the arithmetic mean of the LQ and UQ
- assign a state its lower quota number of states if the standard quota is less than or equal to the arithmetic mean of the LQ and UQ
- check if the total number of seats assigned are equal to the number of seats desired.
- if the total number of seats are not equal to the number of seats desired, find a divisor that solves this.
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This method was used in the 1840, 1910 and 1930 apportionments.
I will illustrate Webster's method using population data from the 1832 apportionment to find 240 seats:
usa1832
divisor1832 <- sum(usa1832)/240
stq1832 <- usa1832/divisor1832
webster.values <- ifelse(stq1832%%1 > 0.5, ceiling(stq1832), floor(stq1832))
cbind(stq1832, webster.values)
sum(webster.values)
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Here, the number of seats produced by this method are 241, which is one too many for the desired 240 seats. The Webster method proceeds by trying different divisors until a solution is found. This is done by the webster function in apportR.
webster(usa1832,240)
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This shows that the unlucky state was Kentucky who had to give up a seat. Unsurprisingly, the standard quota of Kentucky had the fractional closest to 0.5.
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Dean's Method
Dean's method was devised by Prof James Dean, a professor of astronomy and mathematics at Dartmouth and University of Vermont. It operates in a very similar fashion to Webster's method, but instead of rounding based on the arithmetic mean, the harmonic mean is used.
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harmonic1832 <- 2 / ( (1/floor(stq1832)) + (1/ceiling(stq1832)) )
dean.values <- ifelse(stq1832 > harmonic1832, ceiling(stq1832), floor(stq1832))
cbind(stq1832, harmonic1832, dean.values)
sum(dean.values)
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This time the initial value is 2 seats too many. The dean function will find a solution if possible.
dean(usa1832, 240)
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Dean’s method was never used by Congress. One issue with Dean's is that it does have bias towards smaller states. For instance, in the example above, NY's standard quota is 38.59 yet it only gets 38 seats. Alternatively, Louisiana's standard quota is only 3.46 yet it gets 4 seats. This is because the harmonic mean approaches the arithmetic mean at larger numbers, meaning for smaller numbers, there is a greater possibility of rounding upwards.
Because finding a divisor that works can be computationally time consuming, we can speed up the process by skipping certain divisors. Setting the parameter k to be a higher value, e.g. 20, will speed up the function considerably. Obviously, we may miss a divisor, in which case it would return an error message and advise adjusting k. For situations where the named vector has much smaller values initially, the user may actually want to reduce k.
library(microbenchmark)
microbenchmark(dean(usa1832, 240))
microbenchmark(dean(usa1832, 240,20))
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This increase in speed achieved by increasing the k constant can also be utilized with the other divisor methods.
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Huntington-Hill's Method
Huntington-Hill's method was initially proposed in 1911 by Joseph A. Hill, Chief Statistician of the Bureau of the Census and later added to by Prof Edward Huntington, professor of mechanics & mathematics at Harvard. This operates in a very similar fashion to Webster's and Dean's method, but use the geometric mean to assign rounding.
This method has been used by the House since 1940 and is still the currently used method. Below is an example of applying this method to the 1832 data.
gmean1832 <- (sqrt(floor(stq1832) * ceiling(stq1832)))
hhill.values <- ifelse(stq1832 > gmean1832, ceiling(stq1832), floor(stq1832))
cbind(stq1832, gmean1832, hhill.values)
sum(dean.values)
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This method is still somewhat biased to smaller states as the geometric mean is smaller than the arithmetic mean, but this difference is not as pronounced as the harmonic mean used in Dean's method.
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Important notes about Webster, Dean and Huntington-Hill methods
In the case of assigning seats to a parliament, Webster's method could theoretically apportion 0 seats but Dean's method and Huntington-Hill's methods will not - they will always assign at least one seat. This can be understood by demonstrating what happens with Nevada in the 1940 census when the House had 435 seats.
usa1940
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Let's look at the standard quotas of Delaware, Wyoming and Nevada:
tail(usa1940 / (sum(usa1940)/435), 3)
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Nevada has the lowest standard quota and the only one under 0.5. Now let's calculate the lower quota and upper quota and the arithmetic mean, harmonic mean and geometric means of these two numbers:
x1940 <- data.frame(stq = tail(usa1940 / (sum(usa1940)/435), 3))
x1940$lq <- floor(x1940$stq)
x1940$uq <- ceiling(x1940$stq)
x1940$arith <- mean(c(floor(x1940$stq), ceiling(x1940$stq)))
x1940$harmon <- 2 / ( (1/floor(x1940$stq)) + (1/ceiling(x1940$stq)) )
x1940$geom <- (sqrt(floor(x1940$stq) * ceiling(x1940$stq)))
x1940
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The harmonic and geometric means of 0 and 1 are 0 and therefore the standard quota of all states is always going to be higher than this meaning that these states are going to be assigned at least 1 seat automatically. With Webster's method that uses the arithmetic mean for rounding, it is possible for the standard quota of any state to be lower than the arithmetic mean (0.5), resulting in a state being allocated 0 seats.
Obviously, this fails the common sense test and Webster's method has to be altered accordingly to be able to assign a seat to a state in this situation. All three functions, webster, dean and hhill will automatically always assign at least 1 seat.
For example, here are the results of the three methods compared to one another:
webster1940<-webster(usa1940, 435)
hhill1940<-hhill(usa1940, 435)
dean1940<-dean(usa1940, 435)
webster1940 <- webster1940[names(usa1940)]
hhill1940 <- hhill1940[names(usa1940)]
dean1940 <- dean1940[names(usa1940)]
data.frame(webster=webster1940, hhill=hhill1940, dean=dean1940)
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As can be seen from the table above, the Webster Method allocated 18 seats to Michigan and 6 to Arkansas, whereas Huntington-Hill's method (and Dean's Method) gave 7 to Arkansas and 17 to Michigan.
Unsurprisingly, a Democratic representative from Arkansas sponsored a bill to use Huntington-Hill’s method to apportion the House. This was opposed by all the Republican representatives and the Democrats from Michigan. Nevertheless, this bill passed and Huntington-Hill's method has been used ever since, and the number of seats has remained at 435.
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Assigning Zero Apportions
In other situations, it may be necessary to apportion zero to one subject in the final apportions. This could happen when using the Webster method and a subject's standard quota is beneath 0.5, but it could also occur when using Dean's method or Huntington-Hill's method if the standard quota is also 0 (i.e. if the individual subject in the named vector was initially zero).
For example, if special allowances had not been made in the 1940 apportionment, then Nevada would have received 0 seats under Webster's method as their standard quota was actually below 0.5. This is shown here with the function webster0 which allows for 0 seats to be allocated:
webster0(usa1940, 435)
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Nevada's seat would be given to Arkansas in this situation.
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To illustrate the possibility of apportioning zeros using the Dean or Huntington-Hill methods, imagine the following scenario.
If 10 scholarships were to be given out to subject based on the proportion of students majoring in that subject that year but one of the subjects actually contained zero students.
Comparing each method allowing and not-allowing for zero apportionment:
majs <- c(100, 154, 0, 22, 5)
names(majs) <- c("English", "History", "Swahili", "French", "Aeronautics")
majs
webster(majs, 10)
dean(majs, 10)
hhill(majs, 10)
webster0(majs, 10)
dean0(majs, 10)
hhill0(majs, 10)
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Clearly, when allowing for zero apportionment, Dean's Method and Huntington-Hill's Method only give 0's when a subject initially contained zeros, whereas Webster's Method will also apportion zeros when the standard quota is less than 0.5 as is the case here for Aeronautics.
Going on from this, it's also clear that the results for Dean's method and for Huntington-Hill's method are the same as if the 'Swahili' group never existed, i.e.
majs1 <- c(100, 154, 22, 5)
names(majs1) <- c("English", "History", "French", "Aeronautics")
majs1
dean0(majs1, 10)
hhill0(majs1, 10)
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1876 Presidential Election
Perhaps the biggest snafu in US Presidential electoral history occurred as a result of mis-apportionment.
The House of Representatives in 1872 was apportioned according to the US census of 1870 and the House size was 292. Even though the legal method in place at the time was Hamilton's method, the apportionment that occurred had the same results as Dean's Method (even if it's not clear that Dean's Method was actually used).
usa1870
hamilton(usa1870, 292)
dean(usa1870, 292)
you will notice that four states ended up with different number of representatives than they should have received under the legally constituted Hamilton's Method. New York and Illinois recieved one seat too few whilst New Hampshire and Florida received one extra seat each.
This ended up being historically very significant. In the Presidential election of 1876, Samuel Tilden won the state of New York and received 35 electoral votes. He would have received 36 if the Hamilton apportionment had been used. His opponent, Rutherford B. Hayes, won the other three states and had the Hamilton Method been applied would have received one fewer electoral vote in both Florida and New Hampshire (he would also have actually gained one electoral college vote in Illinois). In the election, Hayes beat Tilden in the Electoral College by a vote of 185 to 184. So, if the correct apportionment had actually been followed, the vote would have been 185 to 184 in favor of Tilden!
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This continues to have relevance for today. Had Jefferson's Method been used to apportion the House prior to the 2000 US Presidential election, then Gore would have defeated Bush in the electoral college count! Had Hamilton's method been used then it would have been a tie!
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Note of Caution
This package is very much a work in progress. I am looking to improve functions to make them more generalizable as well as adding new functions. Please reach out if you have suggestions or are interested in this project.
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Ideas for future
I'm keen to hear from anyone who has ideas for this project. One vision I have is to generate map plots that visualize how different states have gained and lost representative seats over time. I'd also like to map forecasts of how seats in the House may change given current population projections.
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References and Further Reading
There are many online resources that describe the apportionment problem, here are some that are worthwhile:
- Mathematical Association of America
- American Mathematical Association
- The Mathematics of Apportionment
jalapic/apportR documentation built on May 18, 2019, 11:17 a.m.
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Introduction
The apportR package contains several functions for performing apportionment procedures. The inspiration behind me writing this package can be found here. This package is in development and is very much a work in progress. This vignette details some sample problems that can be solved using the functions in this package, information regarding the example datasets contained in apportR as well as some background and worked examples of each function. At this stage of development I would welcome hearing from anyone who thinks that they could help improve the package.
Installation
The apportR package can be downloaded from github.
library(devtools) devtools::install_github("jalapic/apportR")
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Loading the package
library(apportR)
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Examples
To illustrate the general problem at hand, here are three examples.
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Fair Division Example 1.
Mom has 50 identical pieces of candy that cannot be broken into bits. She tells her five children that she will divide the candy fairly at the end of the week according to the proportion of housework that each performs.
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The following tables shows the minutes spent in housework by each child and how many of the 50 candies each child should expect to get:
housework candies <- (50*housework) / sum(housework) candies floor(candies) sum(floor(candies))
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Anna expects 8.33%, Betty 4.33%, Carl 9.6%, Derek 11.33% and Ella 16.39% of the candies, but they cannot be broken up. Assuming each gets the number of candies equal to their rounded down proportion at least, then Who should get the extra 2 candies that are not yet allocated?
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Fair Division Example 2.
There are 12 scholarships to be given out to each of 3 subjects at a university. They are to be awarded based upon the proportion of majors that each subject has.
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Here are the majors in each subject as well as the expected number of scholarships each would get out of the 12 available:
majors scholarships <- (12*majors) / sum(majors) scholarships floor(scholarships) sum(floor(scholarships))
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If each gets the rounded down value that they would expect, where do the remaining 2 scholarships go ? Presumably Psychology should get an extra one as its fractional is .9, but which of English and History should get the other one?
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Fair Division Example 3.
At the 1792 US House of Representatives elections, a total of 105 seats were to be awarded that would be shared between the 15 states of the union proportional to each state's population size at the 1790 US census.
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More reading can be found here
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Here are the states and their population sizes in 1790, followed by the expected number of seats for each state based on there being 105 available:
usa1790 seats <- (105*usa1790) / sum(usa1790) seats floor(seats) sum(floor(seats))
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Who should get those 8 extra seats ??? This is a slightly more important question to get right than the fair division of candies (depending upon your perspective)!
I will describe some methods for working this out as well as the functions contained in apportR to represent these methods a bit later in this vignette.
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Datasets
The apportR package contains a number of small datasets useful for testing functions and/or illustrating methods. These are:
data(package='apportR') #Data sets in package ‘apportR’: # #housework Housework example #majors Majors example #parador Parador example #pop1 Population example 1 #usa1790 USA population by state in 1790. #usa1820 USA population by state in 1820. #usa1832 USA population by state in 1832. #usa1880 USA population by state in 1880. #usa1900 USA population by state in 1900. #usa1907 USA population by state in 1907. #usa1940 USA population by state in 1940. #usa1990 USA population by state in 1990. #usa2000 USA population by state in 2000. #usa_census USA population by state in every census.
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All are named vectors except for the last usa_census which is a dataframe that contains the year that each state was admitted to the union as well as the population of every state at every official US census. It should be noted that these population figures do not include Native American or slave populations. (Note - for some reason, some of the population sizes differ between the named vectors and the corresponding columns in usa_census as a result of obtaining the data from two different sources for each. Some of this discrepancy may be due to e.g. whether to include overseas military and civilian personnel and their families towards the populations of their home states or not. I will try to update with wholly consistent data asap).
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Methods / Functions Overview
The methods for solving these sorts of questions essentially fall into two broad groups. There are the standard (or non-divisor) methods and the divisor methods. Many (if not all) of them have been discovered independently in various disciplines and so each may have several different names. I will mainly focus on the methods from the historical perspective of the US House of Representatives apportionment problem, as it is the most interesting.
Currently I include discussion of the standard (non-divisor) methods. I will add information about the divisor methods shortly.
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Standard Non-Divisor Methods
- Hamilton's Method, aka Largest Remainder Method, Vinton's Method, Hare-Niemeyer Method.
- Lowndes' Method
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Hamilton Method
The most simple method was first suggested by Alexander Hamilton in 1792.
Quite straightforwardly, the logic of this method is as follows:
- choose the number of seats available
- calculate the standard divisor (this is equal to the total population / number of seats)
- calculate the standard quota (this is equal to each state population / standard divisor)
- assign each state initally its lower quota (this is the rounded down value of each standard quota)
- calculate how many seats are still to be assigned
- assign extra seats to those states with the highest fractionals from their standad quota in descending order.
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Example
There were 120 seats to be allocated in the House in 1790. The population of each state and the total US population at the 1790 census was as follows:
usa1790
sum(usa1790)
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The standard divisor (which is equivalent to the average number of people that each representative is representing) and standard quotas of each state are therefore:
#standard divisor std <- sum(usa1790) / 120 std stq <- usa1790 / std stq
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The first allocation of seats is the rounded down value of these standard quotas. When summing these values, it is clear that they only sum to 111 seats which is 9 short of the required total of 120.
lowq <- floor(stq) lowq sum(lowq)
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The states are then arranged in order of their fractional parts and the states with the 9 largest fractionals are awarded an extra seat. That would be NJ to NC.
rev(sort(stq%%1))
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This can be done using the hamilton function. The first argument is the named vector of population sizes and the second argument is the number of seats to allocate.
hamilton(usa1790, 120)
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Despite the 1792 bill proposing this method be used was passed but yet it was not brought into use at this time. Instead, this bill was the very first usage of a Presidential veto by George Washingon and a divsor method proposed by Thomas Jefferson was used. I will discuss this method in the divisor methods section. Washington vetoed the bill as Hamilton's method would have meant that some representatives would be representing fewer than 30,000 individuals which was against the constitution. It was first of only two times that Washington used his Presidential veto.
We can see this here - several states have too many representatives according to the Constitution:
usa1790 / hamilton(usa1790, 120)
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Despite this initial lack of support, Hamilton's method (also known as Vinton's method by the mid nineteenth century) was passed into law and adopted by 1850 and continued to be used until the turn of the 20th century. It stopped being used at this time because some strange anomalies or paradoxes were discovered. These are as follows:
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Alabama Paradox
In 1880 a goverment clerk named C. W. Seaton computed apportionments for all potential House sizes between 275 and 350 members. He then wrote to Congress describing that if the House of Representatives had 299 seats, Alabama would get 8 members but it would only receive 7 members if the House had 300 seats. This seemed to be a failure of common sense and became known as the Alabama Paradox.
Here are the state populations for 1880 and the number of seats that would be given under the Hamilton method if the House had 299 or 300 members:
usa1880 hamilton(usa1880, 299) hamilton(usa1880, 300)
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Texas and Illinois would each gain a member whilst Alabama lost one !
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New States Paradox
A second paradox of this method also came to light. This is probably best illustrated using the example of Oklahoma which was admitted to the union in 1907. In 1900 the house had 386 seats. According to OK's population, they should have got 5 seats.
The US population in 1900 and seat allocation according to Hamilton's method was as follows:
usa1900 hamilton(usa1900, 386)
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but the following happens when adding in Oklahoma and allowing for an extra 5 seats:
usa1907 hamilton(usa1907, 391)
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As you can see, after OK is added in 1907, New York would lose a seat and gives it to Maine! Again, this seems to fail a common sense test.
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Population Paradox
A final paradox that has been found is that states that increase in population more relative to other states over a period of time can even lose seats.
Here is a fictious example. Say there are five states (A-E) and this is their population in hundreds of thousands at time 1 (states1) and time 2 (states2):
states1 <- c(150, 78, 173, 204, 295) states2 <- c(150, 78, 181, 204, 296) #C increases by 8, E by 1 names(states1)<-names(states2)<-LETTERS[1:5] states1 states2
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A, B and D have remained the same whilst E and C have increased in population. Now say there are 50 seats to allocate - let's see what happens:
hamilton(states1, 50) hamilton(states2, 50)
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Even though E's population went up by 100,000 it actually lost a representative seat! whilst B whose population stayed the same increased its seat number and C whose population increased by 800,000 remained the same.
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Strengths & Weaknesses
The biggest advantage of this method is that it is very easy to calculate and to explain. This method also never violates the quota rule which states that the number of seats ought to be between the lower and upper quota of the standard quota (effectively the floor and ceiling value). The biggest weaknesses are that it throws up these strange paradoxes or common sense failures, and also that it does systematically favor larger states over smaller ones.
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Notes about the function
There are a few things to note about this function.
1. Named vector - the input vector must be named.
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2. Zero seats problem - for calculating seats in the House, there must always be at least one seat allocated to every state. The hamilton function does this automatically, so if for instance the standard quota of any state would be e.g. 0.333 and its fraction was not large enough to receive an extra surplus seat, then it will override this and allocate the seat. This happens several times in US census history for Delaware and Nevada.
Other users of this function may want to have the option of allocating 0 to a group. Therefore the sister function to hamilton is hamilton0 which allows for this.
For example, take the US census in 1880 and 300 available seats in the House:
usa1880 hamilton(usa1880, 300) hamilton0(usa1880, 300)
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Clearly, Nevada would not get a seat unless allowances were made to ensure all states received at least one seat. Alabama would have lost out again in this instance !
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This zero issue can also be illustrated with a couple of fair division problems. Say we had five candies to divide to each of 3 children dependent upon how many hours of work they completed.
hours <- c(1,3,11) names(hours) <- c("Xavier", "Yasmin", "Zach") hours #standard quotas (5*hours)/ sum(hours)
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If we look at the standard quotas then we should initially give 0 to Xavier, 1 to Yasmin and 3 to Zach. The question then is what to do with the fifth candy. If we allow zeros then that extra one would be given to Zach. If we don't allow zeros, then that extra one will be given to Xavier.
hamilton(hours,5) hamilton0(hours,5)
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This also leads us to an issue with the hamilton function when we automatically assign 1 to every group. If there are many individuals with small standard quotas, then it may become impossible to apportion.
For instance, if we had ten gold watches to give out proportionally to sales reps who sold the most cars in a car dealership and the data looked like this:
cars.sold <- c(1, 2, 15, 3, 1, 12, 4) names(cars.sold) <- c(paste0("rep", 1:7)) cars.sold #standard quotas (cars.sold*10) / sum(cars.sold)
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Using Hamilton's method everyone would get at least 1 watch accounting for 7 of the watches, but rep3 and rep6 should each get 3 watches, making 11 to be allocated when there are only 10. Clearly this does not work. If it is possible to allocate zero watches to individuals, then we can apportion. Notice that the standard hamilton function reports the error.
hamilton(cars.sold, 10) hamilton0(cars.sold, 10)
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3. Population and seat size - the procedure works by calculating the standard quota of each state i.e. each element of the initial inputted vector. Therefore it won't work in situations where 1) the values in the input vector are too small, 2) the number of seats to be allocated are either too small.
For example, let's say we had the following vector and wanted to allocate 3 seats:
z <- c(1,3,11) names(z) <- LETTERS[1:3] z z1 <- sum(z) / 3 #standard divisor z1 z2 <- z/z1 #standard quota z2
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The above cannot work as the function will try to allocate 1 seat to each of A and B and then 2 to C. The minimum number of seats that can be allocated are 4.
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Future adaptations
There are several slight variations to the Hamilton Method, which change slightly how the standard quotas are calculated. I will add these in time. These include:
- Droop Quota method
- Hagenbach-Bischoff quota
- Imperiali quota
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Lowndes' Method
In 1822, Rep. Lowndes of South Carolina proposed an alternative method to Hamilton's. The main tenet of this approach was to give a greater emphasis to the smaller states. Consequently, this approach never received much support and did not become law.
The general logic is as follows:
- choose the number of seats available
- calculate the standard divisor (this is equal to the total population / number of seats)
- calculate the standard quota (this is equal to each state population / standard divisor)
- assign each state initally its lower quota (this is the rounded down value of each standard quota)
- list the number of persons per already assigned representative
- assign extra seats to those states with the highest number of persons per assigned representative untial all seats are filled
This function also accounts for the zero seats issue by automatically assigning at least one representative per state.
Here is an example of Lowndes' approach from the 1820 census with 213 seats to assign:
usa1820 lowndes(usa1820, 213)
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For comparison, here is how Lowndes' approach differs from Hamilton's for the 120 seats to be assigned in 1790.
hamilton(usa1790, 120) lowndes(usa1790, 120)
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As is clearly evident, Lowndes' method disporportionately favors the smaller states.
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Divisor Methods
There are a large number of divisor methods. The general logic is as follows:
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choose the house size (number of seats)
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find a number (the divisor) such that when all the population sizes of every state are divided by this divisor and these numbers are rounded, that they sum to the number of seats.
The different divisor methods differ in how they round numbers. Further, there is rarely one unique value of the divisor that satisfies this equation, but all values that do for each method will lead to the same apportionment.
Although divisor methods are free from paradoxes that the Hamilton Method suffers, they can violate another rule of apportionment the quota rule - that the number of seats allocated should be no higher than rounded up value of the standard quota and no lower than the rounded down value of this number.
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Eventually the package will include the following divisor apportionment methods:
- Jefferson's Method, aka Greatest Divisor Method, de Hondt's Method - (used in House from 1791-1830)
- Adams' Method, aka Smallest Divisors Method, - (considered by Congress but not used)
- Webster's Method, aka Webster-Willcox Method, Major Fractions Method, Sainte-Lague Method, - (used in House in 1840, 1910 and 1930)
- Dean's Method, aka Harmonic Mean Method
- Huntington-Hill Method, aka Geometric Mean Method, Equal Proportions Method, - (used in House from 1940 to present)
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The package currently contains functions called jefferson, jefferson0 and adams that can perform those methods.
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Jefferson's Method
In 1792, ten days after George Washington vetoed the bill to use the Hamilton Method, Congress passed the method of apportionment sugested by Thomas Jefferson. They also decided to reduce the number of seats available from 120 to 105 to avoid the issue of having representatives representing too few people (i.e. fewer than 30,000).
Jefferson's Method works as follows:
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Find the initial divisor - this is equal to the sum of state populations divided by the number of seats available.
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Determine if the sum of the rounded down standard quotas produced by dividing state populations by the divisor are equal to the number of seats available.
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If they are not, then through trial and error, find a divisor such that the sum of the rounded down standard quotas produced by dividing state populations by the divisor are equal to the number of seats available.
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If a state's lower quota is equal to zero, assign them one representative seat regardless (the
jeffersonfunction automatically does this at present).
For example, here is the 1790 US state populations:
usa1790 divisor <- sum(usa1790) / 105 initial.divisor <- floor(divisor) initial.divisor first.sum <- sum(floor(usa1790/initial.divisor)) first.sum first.sum == 105
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The best strategy here as we have too few seats is to decrease the value of the divisor to try and find a solution. This is possible and is what the jefferson function does:
jefferson(usa1790, 105) rev(sort(jefferson(usa1790, 105)))
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... and to compare this with what Hamilton's Method would have produced:
hamilton(usa1790, 105)
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If Hamilton's had been used instead then 13 of the 15 states would have received the same number of Representatives. Unsurprisingly, Virginia were the beneficiaries under Jefferson's method with poor Delaware losing a seat.
This method was used by the House from 1792-1842 when it was considered that it was too favorable towards the larger States and a different method was needed.
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Here is how the Jefferson Method determined the allocation of seats for the 1832 elections to the House of Representatives with 240 seats being available. I have sorted the allocations in descending order.
usa1832 rev(sort(jefferson(usa1832, 240)))
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Quota Rule Violations
Although divisor methods do not have paradoxes associated with them, they can violate the quota rule which states that the number of seats given to each state should be at most the ceiling value of their standard quota and at least the floor value of their standard quota. The 1832 election above is interesting from this point of view because it actually breaks the quota rule. New York is allocated 40 seats, yet this is the standard quota for each state:
(240*usa1832)/sum(usa1832)
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NY should have received only 38 or 39 seats to satisfy the quota rule, but instead it received 40.
Interestingly, according to Balinski and Young’s Impossibility Theorem, it is not possible to come up with a perfect apporitonment system - there will always be quota rule violations or paradoxes but not both with any system.
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The quota rule violation can also be understood with this example of a fictitious country called Parador which has 5 states and needs to allocate 250 seats to its House. Below I will show the populations of each state, the standard, lower and upper quotas and the results of Jefferson's Method using the built-in dataset parador.
state = names(parador) popn = parador st.q = (250*parador)/sum(parador) low.q = floor(st.q) upp.q = ceiling(st.q) seats = jefferson(parador, 250) data.frame(state, popn, st.q, low.q, upp.q, seats)
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As can be seen, state B actually garners one extra seat that it should not do so according to the quota rule. One of the major criticisms of the Jefferson Method is that it overly favors large states.
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Notes about using the Jefferson Function
There are two important issues to consider as regards the Jefferson Method function.
1. All the divisor methods utilize a 'trial-and-error' method to find a divisor that will correctly apportion the seats to the states based on relative population size. For the jefferson function to work, there must therefore be a divisor that meets this criteria.
The way I have programmed the jefferson function to search for this divisor is to begin with the initial divisor and then try smaller and smaller divisors (initial divisors are too large). The user can adjust the increments by which the function searches for these divisors. The default is set to k=1 which means that the function will search for divisors by decreasing integer values one by one. This works fine for examples such as those above when using large population sizes. However, this might not be appropriate when using smaller initial values in groups. In this case, it may be better to set k to a smaller value such that the function will search for divisors between integers.
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Here is an example:
A teacher has 10 gold stars to divide between 6 students who are taking an exam. The teachers says they will award the stars based on how many questions each student gets correct.
answers <- c(1,4,5,14,16,3) names(answers) <- paste0("Student ", LETTERS[1:6]) answers (10*answers)/sum(answers) #standard quotas
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Let's try and apply the jefferson function with the default parameter setting for k=1:
jefferson(answers,10)
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This tells us that it was not able to find an appropriate divisor and that we should try reducing our value of k.
jefferson(answers, 10, k=0.1)
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Now, we are able to find a solution. My suggestion for adjusting k is to gradually decrease it by one decimal place at a time.
Here is another example.
In the country of Tecala, there are four states that wish to elect 160 seats to their house. The population of each state was 3.31, 2.67, 1.33 & 0.69 (in millions).
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popn <- c(3.31, 2.67, 1.33, 0.69) names(popn)<- c("Apure", "Barinas", "Carabobo", "Dolores") popn jefferson(popn, n=160) # 160 seats jefferson(popn, n=160, k=0.1) # 160 seats jefferson(popn, n=160, k=0.01) # 160 seats jefferson(popn, n=160, k=0.001) # 160 seats jefferson(popn, n=160, k=0.0001) # 160 seats
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2. Allowing zero apportions. It is possible to allow for zero apportions using jefferson's sister function jefferson0.
For instance, if the teacher in the example above did not feel that all students should get gold stars but only those who were deserving should get them, then we could allocate using jefferson0:
jefferson0(answers, 10, k=0.1) jefferson(answers, 10, k=0.1)
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Adams' Method
In 1832, former president John Quincy Adams essentially suggested an inverse method to Jefferson's Method. This method was considered but never passed into law. Essentially, this method operates identically to Jefferson's except that the rounding method used to determine representatives from the standard quota is a ceiling rounding (i.e. round the standard quota up). This has the result of overly benefitting the smaller states and hence why it didn't receive much support.
Here is an example for the adams function for the 1832 House with a size of 240 seats. The function is compared to the Hamilton and Jefferson Methods.
usa1832 adams(usa1832, 240) rev(sort(jefferson(usa1832, 240))) #function doesn't automatically sort currently hamilton(usa1832, 240)
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Notes about using the Adams Function
- As with the
jeffersonandjefferson0functions, theadamsfunction will automatically stop and return an error message if it cannot find an appropriate divisor. It will encourage the user to use a smaller incremental for the 'trial-and-error' process of finding a divisor.
For example, say we had collected data on how many workers attended an office on each day of the week and we wanted to create a chart such as a waffle chart to visually represent this. However, we wanted a nice rectangular chart that requires 100 percentage points to be allocated. We could use the adams function.
set.seed(111) workers <- round(runif(7, 1, 1000)) names(workers) <- c("Mon", "Tues", "Weds", "Thurs", "Fri", "Sat", "Sun") workers adams(workers) #defaults are n=100, k=1 adams(workers, k=.1) waffle::waffle(adams(workers, k=.1), rows=5)
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- Like the Lowndes Method above, the Adams Method uses a rounding up method, so it is not possible to apportion zeros to groups.
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Webster's Method
In the 1830s another divisor method was suggested by Senator Daniel Webster. Webster's method can be summarized as follows:
- choose the number of seats available
- calculate the standard divisor (this is equal to the total population / number of seats)
- calculate the standard quota (this is equal to each state population / standard divisor)
- calculate the arithmetic mean of the lower bound (LQ) and upper bound (UQ) of the standard quota of each state
- assign a state its upper quota number of states if the standard quota is greater than the arithmetic mean of the LQ and UQ
- assign a state its lower quota number of states if the standard quota is less than or equal to the arithmetic mean of the LQ and UQ
- check if the total number of seats assigned are equal to the number of seats desired.
- if the total number of seats are not equal to the number of seats desired, find a divisor that solves this.
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This method was used in the 1840, 1910 and 1930 apportionments.
I will illustrate Webster's method using population data from the 1832 apportionment to find 240 seats:
usa1832 divisor1832 <- sum(usa1832)/240 stq1832 <- usa1832/divisor1832 webster.values <- ifelse(stq1832%%1 > 0.5, ceiling(stq1832), floor(stq1832)) cbind(stq1832, webster.values) sum(webster.values)
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Here, the number of seats produced by this method are 241, which is one too many for the desired 240 seats. The Webster method proceeds by trying different divisors until a solution is found. This is done by the webster function in apportR.
webster(usa1832,240)
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This shows that the unlucky state was Kentucky who had to give up a seat. Unsurprisingly, the standard quota of Kentucky had the fractional closest to 0.5.
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Dean's Method
Dean's method was devised by Prof James Dean, a professor of astronomy and mathematics at Dartmouth and University of Vermont. It operates in a very similar fashion to Webster's method, but instead of rounding based on the arithmetic mean, the harmonic mean is used.
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harmonic1832 <- 2 / ( (1/floor(stq1832)) + (1/ceiling(stq1832)) ) dean.values <- ifelse(stq1832 > harmonic1832, ceiling(stq1832), floor(stq1832)) cbind(stq1832, harmonic1832, dean.values) sum(dean.values)
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This time the initial value is 2 seats too many. The dean function will find a solution if possible.
dean(usa1832, 240)
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Dean’s method was never used by Congress. One issue with Dean's is that it does have bias towards smaller states. For instance, in the example above, NY's standard quota is 38.59 yet it only gets 38 seats. Alternatively, Louisiana's standard quota is only 3.46 yet it gets 4 seats. This is because the harmonic mean approaches the arithmetic mean at larger numbers, meaning for smaller numbers, there is a greater possibility of rounding upwards.
Because finding a divisor that works can be computationally time consuming, we can speed up the process by skipping certain divisors. Setting the parameter k to be a higher value, e.g. 20, will speed up the function considerably. Obviously, we may miss a divisor, in which case it would return an error message and advise adjusting k. For situations where the named vector has much smaller values initially, the user may actually want to reduce k.
library(microbenchmark) microbenchmark(dean(usa1832, 240)) microbenchmark(dean(usa1832, 240,20))
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This increase in speed achieved by increasing the k constant can also be utilized with the other divisor methods.
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Huntington-Hill's Method
Huntington-Hill's method was initially proposed in 1911 by Joseph A. Hill, Chief Statistician of the Bureau of the Census and later added to by Prof Edward Huntington, professor of mechanics & mathematics at Harvard. This operates in a very similar fashion to Webster's and Dean's method, but use the geometric mean to assign rounding.
This method has been used by the House since 1940 and is still the currently used method. Below is an example of applying this method to the 1832 data.
gmean1832 <- (sqrt(floor(stq1832) * ceiling(stq1832))) hhill.values <- ifelse(stq1832 > gmean1832, ceiling(stq1832), floor(stq1832)) cbind(stq1832, gmean1832, hhill.values) sum(dean.values)
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This method is still somewhat biased to smaller states as the geometric mean is smaller than the arithmetic mean, but this difference is not as pronounced as the harmonic mean used in Dean's method.
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Important notes about Webster, Dean and Huntington-Hill methods
In the case of assigning seats to a parliament, Webster's method could theoretically apportion 0 seats but Dean's method and Huntington-Hill's methods will not - they will always assign at least one seat. This can be understood by demonstrating what happens with Nevada in the 1940 census when the House had 435 seats.
usa1940
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Let's look at the standard quotas of Delaware, Wyoming and Nevada:
tail(usa1940 / (sum(usa1940)/435), 3)
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Nevada has the lowest standard quota and the only one under 0.5. Now let's calculate the lower quota and upper quota and the arithmetic mean, harmonic mean and geometric means of these two numbers:
x1940 <- data.frame(stq = tail(usa1940 / (sum(usa1940)/435), 3)) x1940$lq <- floor(x1940$stq) x1940$uq <- ceiling(x1940$stq) x1940$arith <- mean(c(floor(x1940$stq), ceiling(x1940$stq))) x1940$harmon <- 2 / ( (1/floor(x1940$stq)) + (1/ceiling(x1940$stq)) ) x1940$geom <- (sqrt(floor(x1940$stq) * ceiling(x1940$stq))) x1940
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The harmonic and geometric means of 0 and 1 are 0 and therefore the standard quota of all states is always going to be higher than this meaning that these states are going to be assigned at least 1 seat automatically. With Webster's method that uses the arithmetic mean for rounding, it is possible for the standard quota of any state to be lower than the arithmetic mean (0.5), resulting in a state being allocated 0 seats.
Obviously, this fails the common sense test and Webster's method has to be altered accordingly to be able to assign a seat to a state in this situation. All three functions, webster, dean and hhill will automatically always assign at least 1 seat.
For example, here are the results of the three methods compared to one another:
webster1940<-webster(usa1940, 435) hhill1940<-hhill(usa1940, 435) dean1940<-dean(usa1940, 435) webster1940 <- webster1940[names(usa1940)] hhill1940 <- hhill1940[names(usa1940)] dean1940 <- dean1940[names(usa1940)] data.frame(webster=webster1940, hhill=hhill1940, dean=dean1940)
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As can be seen from the table above, the Webster Method allocated 18 seats to Michigan and 6 to Arkansas, whereas Huntington-Hill's method (and Dean's Method) gave 7 to Arkansas and 17 to Michigan.
Unsurprisingly, a Democratic representative from Arkansas sponsored a bill to use Huntington-Hill’s method to apportion the House. This was opposed by all the Republican representatives and the Democrats from Michigan. Nevertheless, this bill passed and Huntington-Hill's method has been used ever since, and the number of seats has remained at 435.
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Assigning Zero Apportions
In other situations, it may be necessary to apportion zero to one subject in the final apportions. This could happen when using the Webster method and a subject's standard quota is beneath 0.5, but it could also occur when using Dean's method or Huntington-Hill's method if the standard quota is also 0 (i.e. if the individual subject in the named vector was initially zero).
For example, if special allowances had not been made in the 1940 apportionment, then Nevada would have received 0 seats under Webster's method as their standard quota was actually below 0.5. This is shown here with the function webster0 which allows for 0 seats to be allocated:
webster0(usa1940, 435)
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Nevada's seat would be given to Arkansas in this situation.
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To illustrate the possibility of apportioning zeros using the Dean or Huntington-Hill methods, imagine the following scenario.
If 10 scholarships were to be given out to subject based on the proportion of students majoring in that subject that year but one of the subjects actually contained zero students.
Comparing each method allowing and not-allowing for zero apportionment:
majs <- c(100, 154, 0, 22, 5) names(majs) <- c("English", "History", "Swahili", "French", "Aeronautics") majs webster(majs, 10) dean(majs, 10) hhill(majs, 10) webster0(majs, 10) dean0(majs, 10) hhill0(majs, 10)
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Clearly, when allowing for zero apportionment, Dean's Method and Huntington-Hill's Method only give 0's when a subject initially contained zeros, whereas Webster's Method will also apportion zeros when the standard quota is less than 0.5 as is the case here for Aeronautics.
Going on from this, it's also clear that the results for Dean's method and for Huntington-Hill's method are the same as if the 'Swahili' group never existed, i.e.
majs1 <- c(100, 154, 22, 5) names(majs1) <- c("English", "History", "French", "Aeronautics") majs1 dean0(majs1, 10) hhill0(majs1, 10)
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1876 Presidential Election
Perhaps the biggest snafu in US Presidential electoral history occurred as a result of mis-apportionment.
The House of Representatives in 1872 was apportioned according to the US census of 1870 and the House size was 292. Even though the legal method in place at the time was Hamilton's method, the apportionment that occurred had the same results as Dean's Method (even if it's not clear that Dean's Method was actually used).
usa1870 hamilton(usa1870, 292) dean(usa1870, 292)
you will notice that four states ended up with different number of representatives than they should have received under the legally constituted Hamilton's Method. New York and Illinois recieved one seat too few whilst New Hampshire and Florida received one extra seat each.
This ended up being historically very significant. In the Presidential election of 1876, Samuel Tilden won the state of New York and received 35 electoral votes. He would have received 36 if the Hamilton apportionment had been used. His opponent, Rutherford B. Hayes, won the other three states and had the Hamilton Method been applied would have received one fewer electoral vote in both Florida and New Hampshire (he would also have actually gained one electoral college vote in Illinois). In the election, Hayes beat Tilden in the Electoral College by a vote of 185 to 184. So, if the correct apportionment had actually been followed, the vote would have been 185 to 184 in favor of Tilden!
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This continues to have relevance for today. Had Jefferson's Method been used to apportion the House prior to the 2000 US Presidential election, then Gore would have defeated Bush in the electoral college count! Had Hamilton's method been used then it would have been a tie!
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Note of Caution
This package is very much a work in progress. I am looking to improve functions to make them more generalizable as well as adding new functions. Please reach out if you have suggestions or are interested in this project.
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Ideas for future
I'm keen to hear from anyone who has ideas for this project. One vision I have is to generate map plots that visualize how different states have gained and lost representative seats over time. I'd also like to map forecasts of how seats in the House may change given current population projections.
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References and Further Reading
There are many online resources that describe the apportionment problem, here are some that are worthwhile:
- Mathematical Association of America
- American Mathematical Association
- The Mathematics of Apportionment
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