The Independence of Irrelevant Alternatives Criterion

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The Independence of Irrelevant Alternatives
Criterion

• If an election is held and a winner is declared,
  this winning candidate should remain the winner
  in any recalculation of votes as a result of one
  or more of the losing candidates dropping out or
  being disqualified.
• Lets look at how this can be violated.

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As a publicity stunt for its soon-to-be-published
cookbook, the Culinary Club of Smallville decided
to have a 'Best Pie' contest. The entries were
narrowed down to three for the final round of the
contest, Al's Apple Pie, Chris' Cream Pie, and
Pat's Peach Pie. Here is a summary of the results
Using the Borda Count Method A 3(27) 2(0)
1(26) 107 C 3(2) 2(51) 1(0) 108 P
3(24) 2(2) 1(27) 103
Our new results A 2(27) 1(26) 80 C 2(26)
1(27) 79

• Was the criterion violated?
• Were the conditions met?
• Was the expected result reached?

Before the results can be publicized, Pat (who
is upset about a third place finish) demands that
her Peach Pie entry be retroactively withdrawn
from the contest. Bowing to her wishes, the club
removes the Peach Pie and recalculates.
The Independence of Irrelevant Alternatives
Criterion was violated.
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APPORTIONMENT

• An overview
• and
• introductory example

The vocabulary and historical references are
taken from Excursions in Modern Mathematics, 4th
edition. By Tannenbaum Arnold. Prentice Hall,
Inc. 2000
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APPORTIONMENT

• An APPORTIONMENT PROBLEM
• is a problem that involves dividing up items so
  that a sum is maintained.

• The items being divided in these types of
  problems are indivisible objects.

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APPORTIONMENT

• Awarding sections of certain courses based on
  projected enrollment and available time slots.
• Dividing up goods/funds for a certain number of
  agencies based on the number of people they
  serve.

• EXAMPLES
• Awarding seats in a representative government,
  where the total number of delegates is fixed.

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APPORTIONMENT U.S. History

• The problem of fair representation was a major
  cause of the Revolutionary War.
• During the 1787 Constitutional Convention, our
  current method of representation was proposed
• each state would have 2 Senators
• and a number of Representatives based on the
  population of the state.
• In 1790, the first U.S. census was taken.
• This census was the basis for the first
  apportionment of the House of Representatives

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APPORTIONMENT U.S. History

• Two methods of apportionment were considered by
  Congress.
• One was proposed by Alexander Hamilton
• The other by Thomas Jefferson.
• HAMILTONS METHOD was very straightforward, and
  was adopted as the method of choice by the
  Congress.
• President Washington had some reservations about
  this method, and vetoed the bill.
• This was the FIRST veto in American history!

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APPORTIONMENT U.S. History

• After much debate, Jeffersons method was passed,
  and was the first method that was actually used
  to apportion the U.S. House of Representatives.
• It was used until 1842.

• Throughout our countrys history, FOUR different
  apportionment methods have been used.
• The method used today is the HUNTINGTON-HILL
  METHOD.
• The current method was adopted in 1941.
• Since that time, the size of the H.R. has been
  fixed at 435 representatives.

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A brief example...

• A fifth grade class is having a canned food drive
  to help support 3 local agencies.
• At the end of each week, they must deliver all of
  the cans they have collected.
• Suppose that one week, they collect 100 cans.
• How many cans would they give to each agency?

• Certainly, you could divide
• 100/3 33.3
• And decide to give two of the agencies 33 cans,
  and the other 34 cans.

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A brief example...

• Now suppose that you know
• AGENCY 1 serves 1000 people each week
• AGENCY 2 serves 200 people each week
• AGENCY 3 serves 100 people each week.

• Now you might decide to do things a bit
  differently!
• Maybe the number of cans donated should reflect
  the number served!?

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A brief example...

• This is a simple example of an apportionment
  problem.
• The things being APPORTIONED (divided up) are the
  cans.
• We can not give fractions of cans
• So we must decide on a fair method of dividing
  these cans.

• Now suppose that you know
• AGENCY 1 serves 1000 people each week
• AGENCY 2 serves 200 people each week
• AGENCY 3 serves 100 people each week.

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Example continued...
To figure out exactly how many cans should be
given, We will divide the served by each
agency by 13...
We cant give fractions of cans so lets just
chop off the decimals
DIVISOR 1300/100 13 GIVE ONE CAN FOR EVERY 13
PEOPLE SERVED!
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Example continued...
The extra cans will go to...

• A valid apportionment method must have a rule for
  distributing extras.
• Our rule will be
• The agency with the highest DECIMAL PART of its
  quotient will get the first extra
• the 2nd highest decimal will get the 2nd extra
• and so on until all cans are distributed.

DIVISOR 1300/100 13 GIVE ONE CAN FOR EVERY 13
PEOPLE SERVED!
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Example conclusion...
Now all of the cans have been distributed
DIVISOR 1300/100 13 GIVE ONE CAN FOR EVERY 13
PEOPLE SERVED!
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About the example...

• This was the method Hamilton proposed for
  apportionment applications.
• In practice, some problems may occur when using
  this method over a period of time.
• But the simplicity of the method makes it useful
  for many applications such as this.

• This example uses a simple method of
  apportionment called
• the method of largest fractions
• Here it seems like conventional rounding
• But it may not ALWAYS be the same as conventional
  rounding, as it does not require the decimal
  value to be greater than or equal to .5 in order
  to award the higher integer value.

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Hamiltons Method of Apportionment --A bundle of
contradictions??

• Hamiltons Method
• Apportionment Vocabulary Example

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Hamiltons 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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Hamiltons Method Vocabulary Example

• We will begin these applications by creating a
  chart with 6 columns
• Label the columns as seen here...

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

• The number of rows will depend on the number of
  states.