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 Ling Yeong Tyng Faculty of Computer Science & Information T echnology University Malaysia Sarawak Source: Peter Tanenbaum Chapter 4 The Mathematics of  Apportio nment

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Ling Yeong Tyng

Faculty of Computer Science & Information Technology

University Malaysia Sarawak

Source: Peter Tanenbaum

Chapter 4The Mathematics of

 Apportionment

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Excursions in Modern Mathematics, 7e: 4.1 - 2 Copyright © 2010 Pearson Education, Inc.

4 The Mathematics of Apportionment 

4.1 Apportionment Problems

4.2 Hamilton’s Method and the Quota Rule 

4.3 The Alabama and Other Paradoxes 

4.4 Jefferson’s Method 

4.5 Adam’s Method 

4.6 Webster’s Method 

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Parador is a small republic located in Central

 America and consists of six states:  Azucar,

Bahia, Cafe, Diamante, Esmeralda, and

Felicidad ( A, B, C , D, E , and F  for short).

There are 250 seats in the Congress, which,according to the laws of  Parador, are to be

apportioned among the six states in

proportion to their respective populations.What is the “correct” apportionment? Table

4-3 shows the population figures for the six

states according to the most recent census. 

Example 4.3 The Congress of Parador

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The first step we will take to tackle this

apportionment problem is to compute the

population to seats ratio, called the standard

divisor (SD). In the case of  Parador, the

standard divisor is 12,500,000 / 250 = 50,000. The standard divisor tells us that in Parador,

each seat in the Congress corresponds to

50,000 people. 

Example 4.3 The Congress of Parador

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We can now use this yardstick to find thenumber of seats that each state should get by

the proportionality criterion – all we have to do

is divide the state’s population by 50,000. For

example, take state A. If we divide the

population of A by the standard divisor, we

get 1,646,000 / 50,000 = 32.92. This number is

called the standard quota (sometimes alsoknown as the exactor fair quota) of state A. 

Example 4.3 The Congress of Parador

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Using the standard divisor SD = 50,000, wecan quickly find the standard quotas of each

of the other states.These are shown in Table

4-4 (rounded to two decimal places). Notice

that the sum of the standard quotas equals

250, the number of seats being apportioned. 

Example 4.3 The Congress of Parador

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Once we have computed the standardquotas, we get to the heart of theapportionment problem: How should weround these quotas into whole numbers?  At

first glance, this seems like a dumb question. After all, we all learned in school how toround decimals to whole numbers –rounddown if the fractional part is less than 0.5,

round up otherwise. This kind of rounding iscalled rounding to the nearest integer , orsimply conventional rounding .

Example 4.3 The Congress of Parador

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Unfortunately, conventional rounding will notwork in this example, and Table 4-5 showswhy not –we would end up giving out 251seats in the Congress, and there are only 250

seats to give out!

Example 4.3 The Congress of Parador

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The gold standard for  apportionmentapplications is the allocation of seats in alegislature, and thus it is standard practice to

borrow the terminology of legislativeapportionment and apply it to apportionmentproblems in general. 

Apportionment: Basic Concepts and

Terminology

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The basic elements of every apportionmentproblem are as follows:

The “states” This is the term we will use to describe theparties having a stake in the apportionment.

Unless they have specific names (Azucar,Bahia, etc.), we will use A1, A2,…, AN , todenote the states.

Apportionment: Basic Concepts and

Terminology

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The “seats” 

This term describes the set of M  identical,

indivisible objects that are being divided

among the N  states. For convenience, we willassume that there are more seats than there

are states, thus ensuring that every state can 

potentially get a seat. (This assumption doesnot imply that every state must get a seat!)

Apportionment: Basic Concepts and

Terminology

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The “populations” 

This is a set of N  positive numbers (for

simplicity we will assume that they are whole

numbers) that are used as the basis for theapportionment of the seats to the states. We

will use p1, p2,…, pN , to denote the state’s

respective populations and P  to denote thetotal population P  = p1 + p2 +…+ pN ).

Apportionment: Basic Concepts and

Terminology

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Two of the most important concepts of the

chapter are the standard divisor  and the

standard quotas. 

We can now formally define these concepts

using our  new terminology and notation. 

Apportionment: Basic Concepts and

Terminology

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The standard divisor(SD) 

This is the ratio of population to seats. It

gives us a unit of measurement (SD people =

1 seat) for our apportionment calculations. 

Apportionment: Basic Concepts and

Terminology

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The standard quotas The standard quota of a state is the exactfractional number of seats that the state

would get if fractional seats were allowed.We will use the notation q1, q2,…, qN  todenote the standard quotas of the respectivestates. To find a state’s standard quota, we

divide the state’s population by the standarddivisor.

Apportionment: Basic Concepts and

Terminology

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Upper and lower quotas The lower quota is the standard quotarounded down, L. 

the upper quota is the standard quotarounded up, U .

For example, the standard quota q1 = 32.92 has lower quota L1 = 32 and upper quota U 1 = 33. 

Apportionment: Basic Concepts and

Terminology

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Our main goal in this chapter is to discover a

“good” apportionment method –a reliable

procedure that (1) will always produce a validapportionment (exactly M  seats are

apportioned) and (2) will always produce a

“fair” apportionment. In this quest we willdiscuss several different methods and find

out what is good and bad about each one. 

Apportionment Methods

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4 The Mathematics of Apportionment 

4.1 Apportionment Problems4.2 Hamilton’s Method and the Quota

Rule

4.3 The Alabama and Other Paradoxes 

4.4 Jefferson’s Method 

4.5 Adam’s Method 

4.6 Webster’s Method 

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Step 1  Calculate each state’s standardquota.

Step 2  Give to each state its lower

quota.Step 3  Give the surplus seats (one at a

time) to the states with the

largest residues (fractionalparts) until there are no more

surplus seats. 

HAMILTON’S METHOD 

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Hamilton’s method (also known as Vinton’smethod or the method of largest remainders)was used in the United States only between

1850 and 1900.

Example 4.4 Parador’s Congress 

(Hamilton’s Method) 

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Hamilton’s method is still used today to

apportion the legislatures of Costa Rica,Namibia, and Sweden.

 At first glance, Hamilton’s method appears to

be quite fair. It could be reasonably arguedthat Hamilton’s method has a major flaw in the way it relies entirely on the size of theresidues without consideration of what those

residues represent as a percent of the state’spopulation. In so doing, Hamilton’s methodcreates a systematic bias in favor of largerstates over smaller  ones.

Hamilton’s Method 

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This is bad –a good apportionment method

should be population neutral, meaning that itshould not be biased in favor of large states

over small ones or  vice versa.

To be totally fair, Hamilton’s method has twoimportant things going for it: (1) It is very easy

to understand, and (2) it satisfies an

extremely important requirement for fairnesscalled the quota rule.

Hamilton’s Method 

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No state should be apportioned anumber of seats smaller than its lower  quota or larger than its upper quota.

(When a state is apportioned a number  

smaller than its lower quota, we call it

a lower-quota violation; when a state

is apportioned a number larger than its

upper quota, we call it an upper-quota violation.)

QUOTA RULE

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4 The Mathematics of Apportionment 

4.1 Apportionment Problems4.2 Hamilton’s Method and the Quota Rule 

4.3 The Alabama and Other Paradoxes

4.4 Jefferson’s Method 

4.5 Adam’s Method 

4.6 Webster’s Method 

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Jefferson’s method is based on an approach

very different from Hamilton’s method. It is in

the handling of the surplus seats (Step 3) that

Hamilton’s method runs into trouble. So here

is an interesting idea: Let’s tweak things sothat when the quotas are rounded down,

there are no surplus seats! The answer is by

changing the divisor, which then changes thequotas. The idea is that by using a smaller  divisor, we make the quotas bigger.

Let’s work an example. 

Jefferson’s Method

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It’s nothing short of magical! And, of course, you are wondering, where did that 49,500come from? Let’s call it a lucky guess for now.

Example 4.8 Parador’s Congress 

(Jefferson’s Method) 

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Example 4.8 illustrates a key point:  Apportionments don’t have to be based exclusively on the standard divisor.

Jefferson’s method is but one of a group of

apportionment methods based on the

principle that the standard yardstick 1 seat =SD of people is not set in concrete and that, if

necessary, we can change to a differentyardstick: 1 seat = D people, where D is a

suitably chosen number.

Apportionment and Standard Divisor

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The number D is called a divisor  (sometimes we use the term modified divisor), andapportionment methods that use modifieddivisors are called divisor  methods. Different

divisor methods are based on differentphilosophies of how the modified quotas should be rounded to whole numbers, butthey all follow one script: When you are done

rounding the modified quotas, all M  seatshave been apportioned (no more, and noless). To be able to do this, you just need tofind a suitable divisor D.

Modified Divisor and Divisor Methods

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Step 1  Find a “suitable” divisor D.

Step 2  Using D as the divisor, compute

each state’s modified quota (modified quota = state

population/D). 

Step 3  Each state is apportioned its

modified lower quota. 

JEFFERSON’S METHOD 

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How does one find a suitable divisor D?

Finding D  

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The first guess should be a divisor somewhatsmaller than SD = 50,000. Start with D =49,000. Using this divisor, we calculate the

quotas, round them down, and add. We get atotal of T  = 252 seats. We overshot our targetby two seats! Refine our guess by choosing alarger  divisor D (the point is to make the

quotas smaller). A reasonable next guess(halfway between 50,000 and 49,000) is49,500. We go through the computation, andit works! 

Example 4.8 Parador’s Congress 

(Jefferson’s Method) 

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4 The Mathematics of Apportionment 

4.1 Apportionment Problems4.2 Hamilton’s Method and the Quota Rule 

4.3 The Alabama and Other Paradoxes

4.4 Jefferson’s Method 

4.5 Adams’s Method 

4.6 Webster’s Method 

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Like Jefferson’s method, Adams’s method isa divisor method, but instead of  rounding the

quotas down, it rounds them up. For this to

work the modified quotas have to be madesmaller , and this requires the use of a divisor

D larger than the standard divisor SD.

Adams’s Method

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Step 1  Find a “suitable” divisor D.

Step 2  Using D as the divisor, compute

each state’s modified quota (modified quota = state

population/D). 

Step 3  Each state is apportioned its

modified upper quota. 

 ADAMS’S METHOD 

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By now you know the background story byheart. The populations are shown once againin the second row of Table 4-15. The

challenge, as is the case with any divisormethod, is to find a suitable divisor D. Weknow that in Adams’s method, the modifieddivisor D will have to be bigger than the

standard divisor of 50,000. We start with theguess D = 50,500. The total is T  = 251, oneseat above our target of 250, so we need to

make the quotas just a tad smaller. 

Example 4.9 Parador’s Congress 

(Adams’s Method) 

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Increase the divisor a little bit, try D = 50,700.This divisor works! The apportionment under

 Adams’s method is shown in the last row. 

Example 4.9 Parador’s Congress 

(Adams’s Method) 

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Example 4.9 highlights a serious weakness ofthis method –it can produce lower-quota 

violations! This is a different kind of violation,

but just as serious as the one in Example 4.8 –state B got 1.72 fewer seats than what it

rightfully deserves! We can reasonably

conclude that Adams’s method is no better(or worse) than Jefferson’s method – just

different. 

Adams’s Method and the Quota Rule 

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4 The Mathematics of Apportionment 

4.1 Apportionment Problems4.2 Hamilton’s Method and the Quota Rule 

4.3 The Alabama and Other Paradoxes

4.4 Jefferson’s Method 

4.5 Adam’s Method 

4.6 Webster’s Method 

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What is the obvious compromise between

rounding all the quotas down (Jefferson’smethod) and rounding all the quotas up(Adams’ method)? What about conventional

rounding (Round the quotas down when the fractional part is less than 0.5 and upotherwise.)? 

Now that we know that we can use modified

divisors to manipulate the quotas, it is alwayspossible to find a suitable divisor  that willmake conventional rounding work. This is theidea behind Webster’s method. 

Webster’s Method

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Step 1  Find a “suitable” divisor D.

Step 2  Using D as the divisor, compute

each state’s modified quota (modified quota = state

population/D). 

Step 3  Find the apportionment by

rounding each modified quotathe conventional way. 

WEBSTER’S METHOD 

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Our first decision is to make a guess at thedivisor D: Should it be more than the standard 

divisor (50,000), or should it be less? Use the

standard quotas as a starting point. When weround off the standard quotas to the nearest

integer, we get a total of 251 (row 4 of Table 4-

16). This number is too high (just by one seat), which tells us that we should try a divisor D a

tad larger than the standard divisor. We try

D = 50,100. 

Example 4.10 Parador’s Congress 

(Webster’s Method) 

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Row 5 shows the respective modified quotas, 

and the last row shows these quotas roundedto the nearest integer. Now we have a valid

apportionment! The last row shows the final

apportionment under Webster’s method. 

Example 4.9 Parador’s Congress 

(Adam’s Method) 

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Webster’s Method 

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When the standard divisor works as a

suitable divisor for Webster’s method, every

state gets an apportionment that is within 0.5

of its standard quota. This is as good an

apportionment as one can hope for. If the

standard divisor doesn’t quite do the job,

there will be at least one state with an

apportionment that differs by more than 0.5

from its standard quota.

Webster’s Method 

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In general, Webster’s method tends to

produce apportionments that don’t stray too

far from the standard quotas, although

occasional violations of the quota rule (both

lower- and upper-quota violations), but such

violations are rare in real-life apportionments.

are possible. Webster’s method has a lot

going for it –it does not suffer from any

paradoxes, and it shows no bias between

small and large states.

Webster’s Method 

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Surprisingly, Webster’s method had a rather

short tenure in the U.S.House of

Representatives. It was used for the

apportionment of 1842, then replaced by

Hamilton’s method, then reintroduced for the

apportionments of 1901, 1911, and 1931, and

then replaced again by the Huntington-Hill

method, the apportionment method we

currently use. 

Webster’s Method