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Apportionment

Show #2 of 7

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Hamilton’s Method of Apportionment --A bundle of contradictions??

Hamilton’s Method Apportionment Vocabulary &

Example

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Hamilton’s Method: Vocabulary Because apportionment

applications so often deal with representative governments, the vocabulary for the “generic” apportionment application will use words that relate to that type of problem.

When we do specific applications, we will make sure to use the applicable terms.

For example, in generic terms, we will be apportioning “seats” to particular “states” based on their “populations”

Whereas in a specific application the “seats” may be canned goods

the “states” may be charity organizations

and the “populations” may be the number of people served per day.

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Hamilton’s Method: Vocabulary & Example We will begin these applications by creating a chart

with 6 columns Label the columns as seen here...

The number of rows will depend on the number of “states”.

State population SQ LQ Rank Apmt

The ORANGE type indicates that these columns will be labeled appropriately for the given application.

apportionment

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Hamilton’s Method: Vocabulary & ExampleAPPLICATION: Mathland is a small country

that consists of 3 states; Algebra, Geometry, and Trigonometry. The populations of the 3 states will be given in the chart.

Suppose that one year, their government’s representative body allows 176 seats to be filled.

The number of seats awarded to each state will be determined using Hamilton’s Method of apportionment.

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Hamilton’s Method: Vocabulary & Example*handout

SD:

TOTAL:

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176

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Hamilton’s Method: Vocabulary & Example*handout

SD:

TOTAL:

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176

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Hamilton’s Method: Vocabulary & ExampleUse the given information to calculate the TOTAL population

and the “standard divisor”

The Standard Divisor is the “number of people per representative” [calculated by dividing TOTAL by #seats]

Here, the SD means that for every 100 people a state will receive 1 representative

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176

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Hamilton’s Method: Vocabulary & ExampleThe STANDARD QUOTA

is the exact quotient upon dividing a state’s population by the SD

The SQ is the exact # of seats that a state would be allowed if fractional seats could be awarded.

Notice that the “# of seats awarded” can also be thought of as the “# of representatives” for a state.

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176

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Hamilton’s Method: Vocabulary & Example

The STANDARD QUOTA

is the exact quotient upon dividing a state’s population by the SD

For example… The SQ for Algebra is: 9230/100 = 92.3 The SQ for Geometry is: 8231/100 = 82.31 etc…

You will note that, very often, the values upon division must be rounded. There is a danger in rounding to too few decimal places, as the decimal

values will be used to determine which states are awarded the remaining seats.

92.3

82.31

1.39

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176

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Hamilton’s Method: Vocabulary & Example

The LOWER QUOTA

is the integer part of the SQ

For example… The LQ for Algebra is: 92 etc…

It may be thought of as “rounding down” to the lower of the two integers the SQ lies between.

92

82

1

92.3

82.31

1.39

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176

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Hamilton’s Method: Vocabulary & Example

If each state was awarded its LQ,

What would the total # of seats apportioned be?

That leaves 1 seat empty!

And we certainly can’t have that!

92

82

1

92.3

82.31

1.39

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176 175

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Hamilton’s Method: Vocabulary & Example

The 1 empty seat will be awarded by ranking the decimal portions of the SQ.

Since we only have 1 seat to fill, we need only rank the highest decimal portion.

1st

92

82

1

92.3

82.31

1.39

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176 175

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Hamilton’s Method: Vocabulary & Example

So, in this application, the state with the highest decimal portion is awarded its UPPER QUOTA

The Upper Quota is the first integer that is higher that the SQ.

It may be thought of as “rounding up” to the higher of the two integers the SQ lies between.

And now the TOTAL # reps is equal to the allowable #seats!

92

82

1st 2

92

82

1

92.3

82.31

1.39

SD: 100

TOTAL: 17600

State population SQ LQ Rank #reps

Algebra 9230

Geometry 8231

Trig 139

#seats 176 175 176

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Hamilton’s Method seems simple enough...

In practice, Hamilton’s Method of apportionment is very straightforward.

In fact, for one-time applications, it’s probably the easiest method to use.

The only confusion might occur if two decimal values are the same and both can not be ranked.

In that case, usually the state with the higher integer value would be awarded the extra seat.

So, what’s the problem?

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Hamilton’s Method seems simple enough...

Problems may occur when Hamilton’s Method is used repeatedly in an application which has certain changes (in # of seats, populations, and/or # of states) over a period of time.

In practice, these problems rarely occur.

But, when they do, they create controversies about the “fairness” of the re-apportionment.

In fact, an apportionment method equivalent to Hamilton’s was actually used for awhile in the USHR…

But the discovery of these problems caused it to be abandoned.

So, what’s the problem?

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End of show #2Going on?...

Apportionment:

Show #3: Hamilton’s Method -- Alabama Paradox

Prepared by Kimberly Conti, SUNY College @ Fredonia

Suggestions and comments to: Kimberly.Conti@fredonia.edu