Quantitative Literacy

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8.1 Measuring voting power: Does my vote count? (15 of 31)

The following table lists the 24 – 1 = 15 possible coalitions.

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8.1 Measuring voting power: Does my vote count? (16 of 31)

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8.1 Measuring voting power: Does my vote count? (17 of 31)

The accompanying table lists the winning coalitions only, along with the critical voting blocs in each case.

Winning Coalitions Only

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8.1 Measuring voting power: Does my vote count? (18 of 31)

There are 10 instances in which a voting bloc is critical:

Roosevelt is critical in 6 of these 10 instances. Therefore, this Banzhaf index is 6/10 = 60%.

Smith is critical in 2 of these 10 instances. Therefore, this Banzhaf index is 2/10 = 20%.

"Other" is critical in 2 of these 10 instances. Therefore, this Banzhaf index is also 2/10 = 20%.

Garner is not critical in any of these 10 instances. Therefore, this Banzhaf index is 0%.

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8.1 Measuring voting power: Does my vote count? (19 of 31)

Banzhaf power index: A voter’s Banzhaf index is the number of times that voter is critical in some winning coalition divided by the total number of instances in which any voter is critical. The index is expressed as a fraction or as a percentage.

Swing voter: Suppose the voters vote in order and their votes are added as they vote. The swing voter is the voter whose vote makes the total meet the quota and thus decides the outcome. Which is the swing voter depends on the order in which the votes are cast.

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8.1 Measuring voting power: Does my vote count? (20 of 31)

Example: Members of the European Union have votes on the Council determined roughly by a country’s population but progressively weighted in favor of smaller countries. Ireland has 7 votes, Cyprus has 4 votes, and Malta has 3 votes. Suppose that these three countries serve as a committee where a simple majority wins, so the quota is 8 votes. Make a table with all the permutations of the voters and the swing voter in each case.

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8.1 Measuring voting power: Does my vote count? (21 of 31)

Solution: There are n! different permutations of n objects, where

𝑛!=𝑛×(𝑛−1)×…×1

So the 3 objects have 3!=3×2×1=6 permutations. The following table lists the permutations and swing voter in each case.

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8.1 Measuring voting power: Does my vote count? (22 of 31)

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8.1 Measuring voting power: Does my vote count? (23 of 31)

The Shapley-Shubik power index: is calculated as the fraction (or percentage) of all permutations of the voters in which that voter is the swing.

Example: Compute the Shapley-Shubik power index for the committee of Ireland, Cyprus, and Malta from the previous example.

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8.1 Measuring voting power: Does my vote count? (24 of 31)

Solution: There are six permutations of the voters. Ireland is the swing in 4 of the 6 cases so the index for Ireland is 4/6 = 2/3 or about 66.67%.

Cyprus is the swing in 1 of the 6 cases so its index is 1/6 or 16.67%.

Malta also is the swing in 1 of the 6 cases so its index is 1/6 or 16.67%.

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8.1 Measuring voting power: Does my vote count? (25 of 31)

Example: A few states in the 2016 U.S. presidential election were especially important in determining the outcome. They are referred to as “swing states.” Four of these states were Virginia with 13 electoral votes, Wisconsin with 10, Colorado with 9, and Iowa with 6. Assume that a simple majority vote from only these four states would determine the election.

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8.1 Measuring voting power: Does my vote count? (26 of 31)

How many permutations of these four states are there?

What is the quota?

Make a table listing all of the permutations of the states and the swing voter in each case.

Find the Shapley-Shubik index for each state.

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8.1 Measuring voting power: Does my vote count? (27 of 31)

Solution:

The number of permutations of four items is

4! = 4 × 3 × 2 × 1 = 24

There are 38 votes, and it requires a majority to win, so the quota is 20.

The following table shows the different permutations and the swing vote state in each case.

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8.1 Measuring voting power: Does my vote count? (28 of 31)

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8.1 Measuring voting power: Does my vote count? (29 of 31)

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8.1 Measuring voting power: Does my vote count? (30 of 31)

Looking at the first row, Virginia’s 13 electoral votes are not enough to make 20, but Wisconsin’s 10 added to it exceeds 20. Therefore, Wisconsin is the swing voter in this order. Looking at the fifth row, we see that Virginia’s 13 electoral votes plus Iowa’s 6 are not enough to make 20, but Wisconsin’s 10 added to that exceeds 20. Therefore, Wisconsin is the swing voter in this order too.

The Shapley-Shubik Index of a given voter is calculated as the fraction of all permutations of the voters in which that voter is the swing.

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8.1 Measuring voting power: Does my vote count? (31 of 31)

Virginia: 10/24 or about 41.67%; Wisconsin: 6/24 or 25%; Colorado: 6/24 or 25%; and Iowa: 2/24 or about 8.33%.

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8.2 Voting systems: How do we choose a winner? (1 of 28)

Voting System: A set of rules under which a winner in an election is determined.

Plurality Voting: The system of voting in which the candidate that receives more votes than any other candidate is the winner.

Example: Of four candidates and 100 votes, what is the smallest number of votes needed to win?

Solution: If the four candidates have an equal number of votes, they would have 100/4 =25 each. So a candidate could have a plurality with as few as 26 votes.

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8.2 Voting systems: How do we choose a winner? (2 of 28)

Spoiler: A candidate who has no realistic chance of winning but whose presence in the election affects the outcome.

Example: In the 1996 presidential race, the three major candidates were Bill Clinton, Robert Dole, and H. Ross Perot. Here is the outcome for the state of Florida.

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8.2 Voting systems: How do we choose a winner? (3 of 28)

What percentage of all votes cast in Florida were for Clinton? Did any candidate achieve a majority of votes cast?

Florida determines the winner of a presidential election by plurality, so Clinton was declared the winner of all 25 of Florida’s electoral votes. Let’s suppose that the vote was conducted with only the top three candidates and that all their supporters continued to vote for them. Would it be possible for any of the three to achieve a majority if all the votes cast for “Others” were awarded to Clinton?

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8.2 Voting systems: How do we choose a winner? (4 of 28)

It was speculated that by far most Perot voters would have voted for Dole if Perot were not in the race. Let’s suppose that 83% of Perot voters would have voted for Dole and 17% for Clinton. How many votes would Clinton have had and how many would Dole have had? Who would have won the election? Would you call Perot a spoiler in this case?

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8.2 Voting systems: How do we choose a winner? (5 of 28)

Solution:

Of the 5,303,794 votes cast, Clinton received 2,546,870 votes, which is 2,546,870/5,303,794 or about 48% of the votes. This is not a majority. Clinton won more votes than any other candidate, so no candidate received a majority of votes cast.

Suppose that all of the 28,518 “Other” votes went to Clinton. Then he would have received 2,575,388 votes. But half of 5,303,794 votes is 2,651,897, so he still would not have had a majority. Certainly the same is true of the other two candidates.

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8.2 Voting systems: How do we choose a winner? (6 of 28)

If 83% of Perot’s votes went to Dole, then Dole would have received 83% of 483,870 or an extra 401,612 votes, for a total of 2,646,148. Clinton would have received 17% of 483,870 or an extra 82,258 votes, for a total of 2,629,128. Therefore Dole would have won with 17,020 more votes than Clinton. If this were indeed true, then Perot would have been a spoiler.

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8.2 Voting systems: How do we choose a winner? (7 of 28)

Preferential Voting System: Systems in which voters express their ranked preferences between various candidates, usually with a ranked ballot that is used to avoid several rounds of voting and the voter lists his or her candidate preferences. Two examples follow.

Top-Two Runoff System: If no candidate receives majority, there is a new election with only the two highest vote-getters.

Elimination Runoff System: If no candidate receives majority, the lowest vote-getter is eliminated and a vote is taken again among those who are left. This repeats until a majority is reached.

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8.2 Voting systems: How do we choose a winner? (8 of 28)

Example: Consider the following ranked ballot outcome for 10 voters choosing among three candidates:

Determine the winner under the elimination runoff system.

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8.2 Voting systems: How do we choose a winner? (9 of 28)

Solution:

No candidate has first-choice majority. Betty has the least so she is eliminated from the first round.

With Betty eliminated, the table is now as follows:

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8.2 Voting systems: How do we choose a winner? (10 of 28)

Now the first-choice votes are for Alfred and 4 + 2 = 6 for Gabby. In this runoff, Gabby has majority and is the winner.

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8.2 Voting systems: How do we choose a winner? (11 of 28)

Borda count: A method of ranked balloting that assigns for each ballot: 0 points to the choice ranked last, 1 point to next higher choice, and so on. The Borda winner is the candidate with the highest Borda count.

Example: To decide on food, five friends mark ranked ballots by preference, using the table:

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8.2 Voting systems: How do we choose a winner? (12 of 28)

Did one of the foods receive a majority for first-choice?

Use the Borda count to determine which food should be ordered.

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8.2 Voting systems: How do we choose a winner? (13 of 28)

Solution:

First-place votes are indicated by the number 2. There were three first-place votes for pizza, which is a majority of the five first-place votes.

The Borda count is:

for pizza: 2+2+2+0+0 = 6

for tacos: 1+1+1+2+2 = 7

for burgers: 1+1=2

According to the Borda count, the group should order tacos. This is true in spite of pizza receiving majority.

This lends to our understanding that no voting system is perfect with three or more candidates.

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8.2 Voting systems: How do we choose a winner? (14 of 28)

Example: The three finalists for the 2016 Heisman Trophy follow:

Determine the Borda counts and the Borda winner.

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8.2 Voting systems: How do we choose a winner? (15 of 28)

Solution:

Borda count for L. Jackson=(526×2)+(251×1)+(64×0)=1303

Borda count for D. Watson =(269×2)+(302×1)+(113×0)=840

Borda count for B. Mayfield =(26×2)+(72×1)+(139×0)=124

Lamar Jackson has the highest Borda count and is the Borda winner.

He was the winner of the Heisman Trophy in 2016.

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8.2 Voting systems: How do we choose a winner? (16 of 28)

Example: Consider the following ranked ballot outcome for 100 voters choosing from options A, B, C, D:

Who wins under plurality voting?

Who wins under the top-two runoff system?

Who wins under the elimination runoff system?

Who wins under the Borda count system?

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8.2 Voting systems: How do we choose a winner? (17 of 28)

Solution:

Under plurality voting only first-choice picks are considered. In that case, candidate A has the most first-choice votes with 28 out of 100.

For a runoff with only the top two candidates, C and D are eliminated from the table, as below:

In this runoff, B has 72 first-choice votes; this is clearly a majority, so B is the winner.

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8.2 Voting systems: How do we choose a winner? (18 of 28)

In the first round of elimination runoff, D has the fewest votes and is eliminated. The table follows:

Now B has the fewest votes and is eliminated:

C wins in this runoff by a majority of 72 votes.

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8.2 Voting systems: How do we choose a winner? (19 of 28)

The Borda count incorporated into the original table:

So the Borda count for each candidate is:

A=28×3+25×0+24×0+23×0=84

B=28×1+25×3+24×1+23×1=150

C=28×0+25×2+24×3+23×2=168

D=28×2+25×1+24×2+23×3=198

Hence D is the winner based on Borda count.

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8.2 Voting systems: How do we choose a winner? (20 of 28)

Common Preferential Systems

Plurality: The candidate with the most votes wins.

Top-two runoff: If no one garners a majority of the votes, a second election is held with the top two vote-getters as the only candidates.

Elimination runoff: Successive elections are held where the candidate with the smallest number of votes is eliminated. This continues until there is a majority winner.

Borda count: Voters rank the candidates first to last. The last-place candidate gets 0 points, the next 1 point, and so on. The candidate with the most points wins.

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8.2 Voting systems: How do we choose a winner? (21 of 28)

A Condorcet winner is a candidate who beats each of the other candidates in a head-to-head election.

Example: Suppose in an election there are seven voters and three candidates, A, B, and C; the voters’ preferences follow:

Is there a Condorcet winner? If so, which candidate?

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8.2 Voting systems: How do we choose a winner? (22 of 28)

Solution:

Find the results of each head-to-head contest:

Consider A and B: there are 3+2=5 voters who rank A over B, and only 2 who rank B over A. So A wins versus B.

The results of the other head-to-head contests are:

A and C: C wins by 1 (4 to 3)

B and C: C wins by 1 (4 to 3)

There is a Condorcet winner―it is candidate C.

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8.2 Voting systems: How do we choose a winner? (23 of 28)

Example: In an election there are seven voters and candidates A, B, C, and D. The tally of ranked ballots follows:

Who wins the plurality system?

Who wins the top-two runoff system?

Who wins in the elimination runoff system?

Who wins the Borda count?

Is there a Condorcet winner? If so, which candidate is it?

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8.2 Voting systems: How do we choose a winner? (24 of 28)

Solution:

In a plurality voting, only first choices are considered. Candidate A has 3 votes, B has 1, and C has 2 votes. A has the most, so A wins.

The first-place winner is A. Candidate C is second with 2 votes. In a runoff with A and C, A wins with 5 votes to 2.

Because B and D get only one vote each, they are eliminated and A wins.

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8.2 Voting systems: How do we choose a winner? (25 of 28)

The Borda count is as follows:

A=3×3+1×2+3×1=14

B=1×3+4×2+2×1=13

C=2×3+2×1=8

D=1×3+2×2=7

So, A wins the Borda count.

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8.2 Voting systems: How do we choose a winner? (26 of 28)

There is a Condorcet winner. The head-to-head outcomes are as follows:

B beats A (4 to 3)

B beats C (5 to 2)

B beats D (4 to 3)

So, B is the Condorcet winner.

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8.2 Voting systems: How do we choose a winner? (27 of 28)

The Condorcet winner criterion says that if there’s a Condorcet winner, then he or she should be the winner of the whole election.

The condition of Independence of irrelevant alternatives states: Supposing candidate A wins an election and B loses, and another election follows in which no voter changes their preference concerning A and B, B should still lose to A no matter what happens concerning the other candidates.

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8.2 Voting systems: How do we choose a winner? (28 of 28)

Arrow’s impossibility theorem: If there are three or more candidates, there is no voting system (other than a dictatorship) for which the Condorcet winner criterion and the Independence of irrelevant alternatives hold.

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8.3 Fair Division: What is a fair share? (1 of 32)

Divide-and-Choose Procedure: One person divides the items into two parts and the other person chooses which part he or she wants.

The Lone-Divider Method applies this procedure to parties of 3:

Person 1 divides the assets into three parts, persons 2 and 3 then choose between the piles.

If they choose differently, person 1 gets the remaining pile.

If they choose the same, person 1 chooses which pile they want and persons 2 and 3 mix the remaining piles and perform the divide-and-choose procedure.

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8.3 Fair Division: What is a fair share? (2 of 32)

Adjusted winner procedure: Two people assign points to bid on each item, assigning a total of 100 points.

Initial division of the items gives each item to the person offering the highest bid.

The division is then adjusted based on the ratio of the bids for each item so that ultimately each person receives a group of items whose bid totals are the same for each person.

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8.3 Fair Division: What is a fair share? (3 of 32)

Example: Suppose my sister and I want to divide an inheritance. The assets consist of a guitar, a jewelry collection, a car, a small library, and a certain amount of cash. To start the procedure, each of us takes 100 points and divides those points among the assets. In a division of assets, point values of two siblings follow:

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8.3 Fair Division: What is a fair share? (4 of 32)

Initial round: I get the guitar and the library for a total of 50 points; the sister gets the car and the cash for a total of 70 points.

The tied item, the jewelry, goes to the point leader, my sister.

So she is 80% satisfied while I am only 50% satisfied. The points must be adjusted.

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8.3 Fair Division: What is a fair share? (5 of 32)

To make the values even, my sister must share some of her property. Each item my sister won can be calculated as:

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8.3 Fair Division: What is a fair share? (6 of 32)

Note my sister’s bid goes on top because she is the point leader.

Arrange the ratios in increasing order:

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8.3 Fair Division: What is a fair share? (7 of 32)

Items are then transferred until an item changes the point leader.

First the jewelry is transferred to me: my expended points is 60 and my sister’s is 70.

The next item, the cash, is the Critical Item; it changes the point leader. Just enough of the critical item is transferred to make the score come out even.

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8.3 Fair Division: What is a fair share? (8 of 32)

The equation for dividing the critical item is as follows:

My score+𝑝 percent of my cash bid=Sister’s score –𝑝 percent of her cash bid

This gives the equation: 60+20𝑝=70–30𝑝 or 𝑝=0.20.

So I get the guitar, the library, the jewelry, and 20% of the cash.

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8.3 Fair Division: What is a fair share? (9 of 32)

My sister gets the car and the remaining 80% of the cash.

The value scores are:

My total points: 35+10+15+20×0.20=64

My sister’s total points: 40+30×0.80=64

Both are 64% satisfied.

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8.3 Fair Division: What is a fair share? (10 of 32)

The Adjusted Winner Procedure: Each of two people makes a bid totaling 100 points on a list of items to be divided.

Step 1: Initial division of items: Each item goes to the higher bidder. Tied items are held for now. The higher score is the point leader, the lesser score is the trailer.

Step 2: Tied items: Tied items go to the leader.

Step 3: Calculate leader/trailer ratios: For each item belonging to the leader, calculate the ratio:

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8.3 Fair Division: What is a fair share? (11 of 32)

Step 4: Transference of some items from leader to trailer: Transfer items from leader to trailer in order of increasing ratios as doing so does not change the lead. The item to change the lead is the critical item.

Step 5: Division of the critical item: Give p percent of the critical item to the trailer from leader with the following equation:

Trailer’s score + 𝑝 × Trailer’s bid = Leader’s score – 𝑝 × Leader’s bid

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8.3 Fair Division: What is a fair share? (12 of 32)

Example: Divide the following items from an inheritance:

Solution: Initially John gets the Hawk, the silver, and the cap and gown. Faye gets the condominium.

Faye is the leader; her condo is the critical item so is considered for the transfer.

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8.3 Fair Division: What is a fair share? (13 of 32)

John’s score+𝑝×John’s bid=Faye’s score –𝑝×Faye’s bid

John takes 13% of the condo from Faye.

Faye gets 87% ownership of the condo.

John gets the Hawk, the silver, the cap/gown, and 13% ownership of the condo.

The divided ownership of the condo could be satisfied by splitting use of the condo up with John getting 7 weeks and Faye getting 45 weeks each year.

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8.3 Fair Division: What is a fair share? (14 of 32)

Example: Use the adjusted winner procedure to fairly divide the following items:

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8.3 Fair Division: What is a fair share? (15 of 32)

Solution: Initially Anne gets the laptop, the MP3 player, and the tablet for 87 points. Becky gets the TV for a point total of 40.

Anne is the leader so she gets the tied item, the DVDs, bringing her points to 92.

Anne’s items are compared against Becky’s:

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8.3 Fair Division: What is a fair share? (16 of 32)

The DVDs and MP3 player are transferred. Anne’s score is now 81 and Becky’s is 50. The laptop is the critical item.

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8.3 Fair Division: What is a fair share? (17 of 32)

Becky gets the DVDs, the tapes, the MP3 player, the TV, and 34% of the laptop. Anne gets the tablet and the remaining 66% of the laptop.

This is as fair as it can be for a party of two, considering of course that neither party changes their mind about point allocation.

There are some problems with this procedure, for example, parties of three or more, critical items being non-dividable, etc.

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8.3 Fair Division: What is a fair share? (18 of 32)

The Knaster procedure is a method for dividing items among several parties, and it does not require dividing ownership of items. The procedure is based on having the parties assign monetary value to each item, their bid. The bidding is without knowledge of others’ bids.

Example: Assume there is one item to be divided, an automobile. The four people involved are each entitled to 1/4 of the car. Each person places a bid and each will end up with a value of at least 1/4 of his bid. Suppose the bids are as follows:

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8.3 Fair Division: What is a fair share? (19 of 32)

Example:

The highest bidder, Abe, gets the car.

But that is the entire estate and he is only entitled to 1/4, which is $7500. So he puts the remaining 3/4, $22,500, into a kitty.

Each of the others withdraws 1/4 of what they assigned as the value of the car:

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8.3 Fair Division: What is a fair share? (20 of 32)

Ben takes 1/4 of $23,000=$5750

Caleb withdraws 1/4 of $28,000=$7000

Dan takes 1/4 of $25,000=$6250

The total withdrawn is $5750+$7000+$6250=$19,000

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8.3 Fair Division: What is a fair share? (21 of 32)

This leaves a kitty of $3500 to be divided equally among the four people. Each person receives ¼ of the value he assigned to the car plus .

The following table summarizes the transaction:

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8.3 Fair Division: What is a fair share? (22 of 32)

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8.3 Fair Division: What is a fair share? (23 of 32)

The Knaster Procedure, or method of sealed bids:

Step 1: Each person bids a secret dollar amount for each item to be divided (sealed bids).

Step 2: For each item separately:

The item is given to the person who bids the most.

The winner contributes the difference between their vote and his or her share of the item to a kitty.

Those who don’t win the item withdraw their share of each of their bids from the kitty.

Step 3: The money left in the kitty is divided equally among the bidders.

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8.3 Fair Division: What is a fair share? (24 of 32)

Example: A family farm is to be divided; parts of the farm are to be bid on by three family members.

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8.3 Fair Division: What is a fair share? (25 of 32)

Determine the fair division of the farm using the Knaster procedure.

Solution: Each part of the farm is given to the highest bidder.

Julie gets the farmhouse.

Anne gets the wheat field and the cattle operations.

Steve will not get any of the farm, but he will get cash.

The divisions of each piece of the farm and its cash value follow.

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8.3 Fair Division: What is a fair share? (26 of 32)

Farmhouse: Julie gets the farmhouse. She bid $120,000 but is entitled to 1/3 of that, $40,000, so she puts $80,000 in the kitty. Anne and Steve are entitled to 1/3 of their respective bids.

Anne takes 1/3 of $60,000=$20,000.

Steve takes 1/3 of $75,000=$25,000.

The total removed from the kitty is $20,000+$25,000=$45,000.

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8.3 Fair Division: What is a fair share? (27 of 32)

This leaves in the kitty.

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8.3 Fair Division: What is a fair share? (28 of 32)

Wheat fields: Anne gets the wheat fields. She bids $450,000 but is entitled to 1/3 of that, $150,000, so she puts $300,000 in the kitty. Julie and Steve are entitled to 1/3 of their respective bids.

Julie takes 1/3 of $210,000=$70,000.

Steve takes 1/3 of $390,000=$130,000.

The total removed from the kitty is $70,000+$130,000=$200,000.

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8.3 Fair Division: What is a fair share? (29 of 32)

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8.3 Fair Division: What is a fair share? (30 of 32)

Cattle ops.: Anne gets the cattle operations. She bids $750,000 but is entitled to 1/3 of that, $250,000, so she puts $500,000 in the kitty. Julie and Steve are entitled to 1/3 of their respective bids.

Julie takes 1/3 of $300,000=$100,000

Steve takes 1/3 of $600,000=$200,000

The total removed from the kitty is $100,000+$200,000=$300,000

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8.3 Fair Division: What is a fair share? (31 of 32)

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8.3 Fair Division: What is a fair share? (32 of 32)

After the item distribution, there is $335,000 left in the kitty to be divided among the three.

335,000/3=$111,667 each

Julie gets the farmhouse and cash totaling:

−$80,000+$70,000+$100,000+$111,667=$201,667

Anne gets the wheat fields, the cattle operations, and cash totaling:

$20,000−$300,000 –$500,000+$111,667=−$668,333

That is, Anne needs to pay out that much.

Steve gets no property but cash totaling:

$25,000+$130,000+$200,000+$111,667=$446,667

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8.4 Apportionment: Am I represented? (1 of 39)

Apportionment: To find the size of the ideal district for a House representative you use the formula:

The ideal district size is also known as the Standard divisor. This number is used to determine each state’s quota, or its share of the House of Representatives:

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8.4 Apportionment: Am I represented? (2 of 39)

Example: For the U.S. population of 6,584,255 in 1810, with the House having members: