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Apportionment

Show #6 of 7

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More Apportionment Methods...

• No matter how simple Hamilton’s Method of apportionment is, the paradoxes we noted are often troubling enough to cause the method to be abandoned.

• Jefferson’s Method of apportionment, although a bit more complicated, does not have any paradoxes associated with it.

• That’s not to say it’s perfect…• We will find that it, also, has a flaw!

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JEFFERSON’S METHOD of apportionment

• Introduction

• Vocabulary & Example

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What’s the difference???When using Hamilton’s Method, recall that every state

received either its• LOWER QUOTA (standard quota rounded down)

Or its• UPPER QUOTA (standard quota rounded up)

for its final apportionment.

And recall that we decided which states would receive their UPPER QUOTA by ranking the decimal portion of the standard quota

Hence, Hamilton’s Method is referred to as a “method of largest fractions”

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What’s the difference???

• In the examples we did, we never saw a case where the LOWER QUOTAS added up to the correct apportionment right away.

• There are examples when that does happen…• but it doesn’t happen often.

• Perhaps one of the things people like about Hamilton’s Method is that the STANDARD DIVISOR (total population/ #seats) is so logical to calculate and understand.

• And this value is used to find the STANDARD QUOTA, which is then used to calculate the correct apportionment.

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A new DIVISOR???• Wouldn’t it be great if, after using that divisor and

finding the LOWER QUOTAS• the sum of the LOWER QUOTAS was already the

correct apportionment?

• There is no law that says we have to use that STANDARD DIVISOR.

• Can’t we find some other divisor that would work so that the sum of the lower quotas for each state adds to the desired value?

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A new DIVISOR???• Of course, we don’t want this divisor to vary

greatly from the Standard Divisor;• After all, the sum of the Lower Quotas was never

too far off.

• Let’s just see what happens if we take a look back at a previous example, but use different divisors.

• (We’ll call them MODIFIED DIVISORS: MD)

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MATHLAND example

Apportionment of the representative body of Mathland

STATE population SQ LQ Rank #reps

Algebra 9,230 92.300 92 92Geometry 8,231 82.310 82 82

Trig 139 1.390 1 1st 2

TOTAL: 17,600 175 176# SEATS: 176

SD 100.00

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A new DIVISOR???

So, with the SD = 100 people/rep.

The Lower Quotas add to 175

And we have 176 seats to apportion.

• What if we try a divisor that is a bit HIGHER than 100?

• Let’s try 100.5 people/rep.

• How would that change the quota for each state?

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Some “modified” vocabulary...

• Let’s call the new divisors we try:

• MODIFIED DIVISORS (MD)

• And the new quotas that occur:

• MODIFIED QUOTAS (MQ)

• As before MQ = state population/ MD

Create the chart & fill it in...

STATE population MQ LQ

Algebra 9,230Geometry 8,231

Trig 139

total: 17,600#seats: 176

SD 100.00MD 100.5

handout

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Some “modified” vocabulary...

MODIFIED QUOTAS (MQ):

As before MQ = state population/ MD

• The sum here is too low


Algebra 9,230 91.84 91Geometry 8,231 81.90 81

Trig 139 1.38 1

total: 17,600#seats: 176 173

SD 100.00MD 100.5

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Another “modification”...

• Let’s try a modified divisor LOWER than 100

• This time, let’s try 99

• Create the chart and do calculations before continuing!

• HEY! That’s too high!



Trig 139 1.40 1

total: 17,600#seats: 176 177

SD 100.00MD 99

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Try again???

• Let’s try another modified divisor

• This time, let’s try 99.2


• WOW! That works perfectly!



Trig 139 1.40 1

total: 17,600#seats: 176 176

SD 100.00MD 99.2

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Is “perfection” unique???

• Let’s try another modified divisor

• This time, let’s try 99.18


• BINGO! That works, too!



Trig 139 1.40 1

total: 17,600#seats: 176 176

SD 100.00MD 99.18

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What’s the big idea?• This idea of finding

a “perfect divisor” is the basis of Jefferson’s Method of apportionment.

• That’s why his method is called a

• DIVISOR METHOD

• So we really only need to find ANY MODIFIED DIVISOR that will work the way we need it to.

• But, hey …• There are infinitely

many choices...

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Infinitely many??? That’s a lot!

• The next few slides will introduce a way to calculate an apportionment using Jefferson’s Method

• WITHOUT having to GUESS & check your own ideas for MODIFIED DIVISORS.

• It will always work… so I hope you like it!

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Jefferson’s Method… handout

• For this method, you will need to know both the LQ and the UQ

• The MD#1 will be calculated by dividing the state’s population

by the Upper Quota (LQ+1) • The MD#2 will be calculated by dividing the state’s population

by the LQ+2• Most of the time, MD#2 will not be important

STATE pop. SQ LQ UQ=LQ+1 LQ+2 MD#1 MD#2 rank #reps(by MD)


Trig 139 1.39 1

TOTAL: 17,600 175# SEATS: 176

SD 100

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Jefferson’s Method… you do the math!

• The reason for including MD#2 will be evident in the next slide show!



Trig 139 1.39 1

TOTAL: 17,600 175# SEATS: 176

SD 100

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Jefferson’s Method… you do the math!

• Since Jefferson’s Method is a DIVISOR METHOD, you will RANK the states’ entitlement to the “empty” seat(s) using the MODIFIED DIVISORS

• NOT just the decimal portions this time!

• The ENTIRE MODIFIED DIVISOR!!!!

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FINALLY...Jefferson’s Method…

Since Algebra has the highest modified divisor, it is awarded the unfilled seat.

Just as in Hamilton’s Method, only the “ranked” states receive their UQ


Algebra 9,230 92.30 92 93 94 99.247 98.191 1st 93Geometry 8,231 82.31 82 83 84 99.169 97.988 82

Trig 139 1.39 1 2 3 69.500 46.333 1

TOTAL: 17,600 175 176# SEATS: 176

SD 100One more

seat to fill...

This is the

highest MD

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And so it goes...• Your text does NOT show this method of

calculating an apportionment using Jefferson’s Method.

• It does, however, allude to this method in the reading…

• Since this is consistent with the chart, calculations and vocabulary used for Hamilton’s Method, I suggest you seriously consider the benefits of using this chart format!

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Have we found perfection?

• As mentioned in the beginning, Jefferson’s Method does have a flaw.

• The problem comes with the educated choice of modified divisors.

• If a state receives either its UPPER QUOTA or LOWER QUOTA as its final apportionment, it is said to adhere to the QUOTA RULE

• It seems fair, doesn’t it?• There are times, however, when using Jefferson’s

Method, that this rule is violated.

• Another cause for controversy??????

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End of show #6

Going on?...

Apportionment:

Show #7: Jefferson’s Method: Quota Rule violation

Prepared by Kimberly Conti, SUNY College @ Fredonia

Suggestions and comments to: [email protected]

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