Quantitative literacy thinking between the lines 3rd edition answers

Quantitative Literacy: Thinking Between the Lines

Third Edition

Chapter 8

Voting and Social Choice

© 2018 W. H. Freeman and Company

Lesson Plan

Measuring voting power: Does my vote count?

Voting systems: How do we choose a winner?

Fair division: What is a fair share?

Apportionment: Am I represented?

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (1 of 31)

Voting Coalition: A group of voters who vote the same way

Winning Coalition: A set of voters with enough votes to determine the outcome of an election; otherwise it is a Losing Coalition

Quota: The number of votes necessary to win the election, in a voting system

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (2 of 31)

Example: Suppose there are three delegates to a county convention: Abe has 4 votes from his precinct, Ben has 3 votes, and Condi has 1 vote. A simple majority of the votes wins.

What is the quota?

Make a table listing all of the coalitions of voters. Designate which of them are winning coalitions.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (3 of 31)

Solution:

There are 8 votes, so the quota for simple majority is 5 votes.

The following table shows all possible coalitions and votes each coalition controls. The last column indicates whether the coalition controls a majority of the votes and is a winning coalition.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (4 of 31)

Number of Votes: 4 Number of Votes: 3 Number of Votes: 1 Total Votes Winning Coalition?
Abe Ben Condi 8 Yes
Abe Ben 7 Yes
Abe Condi 5 Yes
Abe 4 No
Ben Condi 4 No
Ben 3 No
Condi 1 No
© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (5 of 31)

Critical voter: A member of a winning coalition is a critical voter if the coalition becomes a losing coalition when that voter is removed.

Example: The table below contains information from the three winning coalitions in the previous example.

Number of Votes: 4 Number of Votes: 3 Number of Votes: 1 Total Votes Winning
Abe Ben Condi 8 Yes
Abe Ben 7 Yes
Abe Condi 5 Yes
© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (6 of 31)

In the first coalition: Abe is the critical voter.

In the second coalition: both are critical voters.

In the third coalition: both are critical voters.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (7 of 31)

The previous information can be summarized in the following coalition table:

Number of Votes: 4 Number of Votes: 3 Number of Votes: 1 Total Votes Winning Coalition? Critical Voters
Abe Ben Condi 8 Yes Abe
Abe Ben 7 Yes Abe, Ben
Abe Condi 5 Yes Abe, Condi
Abe 4 No Not applicable
Ben Condi 4 No Not applicable
Ben 3 No Not applicable
Condi 1 No Not applicable
© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (8 of 31)

Winning Coalition and Critical Voters

A set of voters with enough votes to determine the outcome of an election is a winning coalition. A voter in a winning coalition is critical for that coalition if the coalition is no longer a winning one when that voter is removed. We can summarize the essential information about coalitions in a coalition table.

Counting coalitions: Number of coalitions

For n voters, there are 2n - 1 possible coalitions (each of which includes at least one voter).

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (9 of 31)

Example: If there are 7 voters, there are 27 possibilities for voters to be in or not in a coalition. This includes the possibility of all the voters not being in any coalition, so there are 27 - 1possibilities.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (10 of 31)

Banzhaf power index: The number of times a voter is critical in a winning coalition divided by the total number of instances in which any voter is critical. Expressed as a fraction or percentage.

Example: Use coalition Table 8.1 to compute the Banzhaf index for each county convention delegate described in the previous example.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (11 of 31)

Solution: From the coalition table: overall there were 5 instances in which any voter was critical. So the Banzhaf power index of a voter is the number of times that voter is critical divided by 5.

Abe was critical 3 of the 5 times, so he has a Banzhaf power index of 3/5 or 60%.

Ben was critical I of the 5 times, so he has a Banzhaf power index of 1/5 or 20%.

Condi was critical I of the 5 times, so she has a Banzhaf power index of 1/5 or 20%.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (12 of 31)

Example: In the first round of voting at the 1932 Democratic National Convention, Franklin D. Roosevelt received 666.25 votes, Al Smith received 201.75 votes, John Nance Garner received 90.25 votes, and the other seven candidates combined received 195.75 votes. These are referred to as voting blocs. It is common for the nomination to be decided by negotiations among the various blocs. There were 770 votes required to win the nomination.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (13 of 31)

What is the quota?

Determine the winning coalitions.

Determine the critical voting blocs in each winning coalition.

Determine the Banzhaf index of each bloc.

© 2018 W. H. Freeman and Company

8.1 Measuring voting power: Does my vote count? (14 of 31)

Solution:

Because 770 votes were required to win the nomination, that is the quota. Note that this number is considerably more than a simple majority of the 1154 votes cast.

© 2018 W. H. Freeman and Company

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

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

666.25 201.75 195.75 90.25 Total votes Winning coalition
R S O G 1154 Yes
R S O 1063.75 Yes
R S G 958.25 Yes
R S 868 Yes
R O G 952.25 Yes
R O 862 Yes
R G 756.5 No
R 666.25 No
© 2018 W. H. Freeman and Company

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

666.25 201.75 195.75 90.25 Total votes Winning coalition
S O G 487.75 No
S O 397.5 No
S G 292 No
S 201.75 No
O G 286 No
O 195.75 No
G 90.25 No
© 2018 W. H. Freeman and Company

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

Votes: 666.25 Votes: 201.75 Votes: 195.75 Votes: 90.25 Total votes Critical voters
R S O G 1154 R
R S O 1063.75 R
R S G 958.25 R,S
R S 868 R,S
R O G 952.25 R,O
R O 862 R,O
© 2018 W. H. Freeman and Company

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%.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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

Order of Voters Order of Voters Order of Voters Swing Voter
Ireland (7) Cyprus (4) Malta (3) Cyprus
Ireland (7) Malta (3) Cyprus (4) Malta
Cyprus (4) Ireland (7) Malta (3) Ireland
Cyprus (4) Malta (3) Ireland (7) Ireland
Malta (3) Ireland (7) Cyprus (4) Ireland
Malta (3) Cyprus (4) Ireland (7) Ireland
© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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%.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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

States States States States Swing
VA (13) WI (10) CO (9) IA (6) WI
VA (13) WI (10) IA (6) CO (9) WI
VA (13) CO (9) WI (10) IA (6) CO
VA (13) CO (9) IA (6) WI (10) CO
VA (13) IA (6) WI (10) CO (9) WI
VA (13) IA (6) CO (9) WI (10) CO
WI (10) VA (13) CO (9) IA (6) VA
WI (10) VA (13) IA (6) CO (9) VA
WI (10) CO (9) VA (13) IA (6) VA
WI (10) CO (9) IA (6) VA (13) IA
WI (10) IA (6) VA (13) CO (9) VA
WI (10) IA (6) CO (9) VA (13) CO
© 2018 W. H. Freeman and Company

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

States States States States Swing
CO (9) VA (13) WI (10) IA (6) VA
CO (9) VA (13) IA (6) WI (10) VA
CO (9) WI (10) VA (13) IA (6) VA
CO (9) WI (10) IA (6) VA (13) IA
CO (9) IA (6) VA (13) WI (10) VA
CO (9) IA (6) WI (10) VA (13) WI
IA (6) VA (13) WI (10) CO (9) WI
IA (6) VA (13) CO (9) WI (10) CO
IA (6) WI (10) VA (13) CO (9) VA
IA (6) WI (10) CO (9) VA (13) CO
IA (6) CO (9) VA (13) WI (10) VA
IA (6) CO (9) WI (10) VA (13) WI
© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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%.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

Candidate Votes
Clinton 2,546,870
Dole 2,244,536
Perot 483,870
Others 28,518
Total votes cast 5,303,794
© 2018 W. H. Freeman and Company

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?

© 2018 W. H. Freeman and Company

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?

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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:

Rank 4 Voters 4 Voters 2 Voters
First Choice Alfred Gabby Betty
Second Choice Betty Alfred Gabby
Third Choice Gabby Betty Alfred
Determine the winner under the elimination runoff system.

© 2018 W. H. Freeman and Company

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.

Rank 4 Voters 4 Voters 2 Voters
First Choice Alfred Gabby Betty
Second Choice Betty Alfred Gabby
Third Choice Gabby Betty Alfred
With Betty eliminated, the table is now as follows:

Rank 4 Voters 4 Voters 2 Voters
First Choice Alfred Gabby Gabby
Second Choice Gabby Alfred Alfred
© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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:

© 2018 W. H. Freeman and Company

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

Pizza Tacos Burgers
Ballot 1 2 1 0
Ballot 2 2 1 0
Ballot 3 2 1 0
Ballot 4 0 2 1
Ballot 5 0 2 1
Did one of the foods receive a majority for first-choice?

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

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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

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

Player First-place votes Second-place votes Third-place votes
Lamar Jackson (Louisville) 526 251 64
Deshaun Watson (Clemson) 269 302 113
Baker Mayfield (Oklahoma) 26 72 139
Determine the Borda counts and the Borda winner.

© 2018 W. H. Freeman and Company

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

Solution:

Player First-place votes Second-place votes Third-place votes
Lamar Jackson (Louisville) 526 251 64
Deshaun Watson (Clemson) 269 302 113
Baker Mayfield (Oklahoma) 26 72 139
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.

© 2018 W. H. Freeman and Company

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:

Rank 28 votes 25 votes 24 votes 23 votes
1st Choice A B C D
2nd Choice D C D C
3rd Choice B D B B
4th Choice C A A A
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?

© 2018 W. H. Freeman and Company

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:

Rank 28 Votes 25 Votes 24 Votes 23 Votes
Adjusted 1st Choice A B B B
Adjusted 2nd Choice B A A A
In this runoff, B has 72 first-choice votes; this is clearly a majority, so B is the winner.

© 2018 W. H. Freeman and Company

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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:

Rank 28 votes 25 votes 24 votes 23 votes
1st Choice A B C C
2nd Choice B C B B
3rd Choice C A A A
Now B has the fewest votes and is eliminated:

Rank 28 votes 25 votes 24 votes 23 votes
1st Choice A C C C
2nd Choice C A A A
C wins in this runoff by a majority of 72 votes.

© 2018 W. H. Freeman and Company

51

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

The Borda count incorporated into the original table:

Rank Borda Value 28 votes 25 votes 24 votes 23 votes
1st Choice 3 A B C D
2nd Choice 2 D C D C
3rd Choice 1 B D B B
4th Choice 0 C A A A
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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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:

Preferences 3 voters 2 voters 2 voters
1st Choice A C C
2nd Choice B B A
3rd Choice C A B
Is there a Condorcet winner? If so, which candidate?

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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:

Voter: 1 2 3 4 5 6 7
1st Choice A A B C D A C
2nd Choice B D A B B D B
3rd Choice C B C A A B A
4th Choice D C D D C C D
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?

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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:

© 2018 W. H. Freeman and Company

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

My Points Item Sister’s Points
35 Guitar 10
10 Jewelry 10
20 Car 40
15 Library 10
20 Cash 30
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.

© 2018 W. H. Freeman and Company

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:

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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

© 2018 W. H. Freeman and Company

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

Example: Divide the following items from an inheritance:

John Item Faye
50 Vacation condominium 65
20 Red 1962 GT Hawk 15
20 Family silver set 15
10 Dad’s Yale cap/gown 5
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.

© 2018 W. H. Freeman and Company

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.

© 2018 W. H. Freeman and Company

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

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

Anne Item Becky
50 Laptop 40
5 DVDs 5
6 MP3 player 5
31 Tablet 10
8 TV 40