A Mathematical View of Our World

A Mathematical View A Mathematical View of Our Worldof Our World

11stst ed. ed.

Parks, Musser, Trimpe, Parks, Musser, Trimpe, Maurer, and MaurerMaurer, and Maurer

Chapter 3Chapter 3

Voting and ElectionsVoting and Elections

Section 3.1Section 3.1Voting SystemsVoting Systems

• GoalsGoals• Study voting systemsStudy voting systems

• Plurality methodPlurality method• Borda count methodBorda count method• Plurality with elimination methodPlurality with elimination method• Pairwise comparison methodPairwise comparison method

• Discuss tie-breaking methodsDiscuss tie-breaking methods

3.1 Initial Problem3.1 Initial Problem• The city council must select among 3 The city council must select among 3

locations for a new sewage treatment plantlocations for a new sewage treatment plant..• A majority of city councilors say they prefer site A A majority of city councilors say they prefer site A

to site B.to site B.• A majority of city councilors say they prefer site A A majority of city councilors say they prefer site A

to site C.to site C.• In the vote site B is selected.In the vote site B is selected.

• Did the councilors necessarily lie about their Did the councilors necessarily lie about their preferences before the election?preferences before the election?• The solution will be given at the end of the section.The solution will be given at the end of the section.

Voting SystemsVoting Systems

• The following voting methods will be The following voting methods will be discussed:discussed:• Plurality methodPlurality method• Borda count methodBorda count method• Plurality with elimination methodPlurality with elimination method• Pairwise comparison methodPairwise comparison method

Plurality MethodPlurality Method

• When a candidate receives more than half of When a candidate receives more than half of the votes in an election, we say the candidate the votes in an election, we say the candidate has received a has received a majoritymajority of the votes. of the votes.

• When a candidate receives the greatest When a candidate receives the greatest number of votes in an election, but not more number of votes in an election, but not more than half, we say the candidate has received than half, we say the candidate has received a a pluralityplurality of the votes. of the votes.

Question:Question:

Suppose in an election, the vote totals are Suppose in an election, the vote totals are as follows. Andy gets 4526 first-place as follows. Andy gets 4526 first-place votes. Lacy gets 1901 first-place votes. votes. Lacy gets 1901 first-place votes. Peter gets 2265 first-place votes.Peter gets 2265 first-place votes.

Choose the correct statement.Choose the correct statement.

a. Andy has a majority. a. Andy has a majority. b. Andy has a plurality only.b. Andy has a plurality only.

Plurality Method, cont’dPlurality Method, cont’d• In the In the plurality methodplurality method::

• Voters vote for one candidate.Voters vote for one candidate.• The candidate receiving the most votes The candidate receiving the most votes

wins.wins.• This method has a couple advantages:This method has a couple advantages:

• The voter chooses only one candidate.The voter chooses only one candidate.• The winner is easily determined.The winner is easily determined.

Plurality Method, cont’dPlurality Method, cont’d• The plurality method is used:The plurality method is used:

• In the United States to elect senators, In the United States to elect senators, representatives, governors, judges, and representatives, governors, judges, and mayors.mayors.

• In the United Kingdom and Canada to In the United Kingdom and Canada to elect members of parliament.elect members of parliament.

Example 1Example 1• Four persons are running for student body Four persons are running for student body

president. The vote totals are as follows:president. The vote totals are as follows:• Aaron: 2359 votesAaron: 2359 votes• Bonnie: 2457 votesBonnie: 2457 votes• Charles: 2554 votesCharles: 2554 votes• Dion: 2288 votesDion: 2288 votes

• Under the plurality method, who won the Under the plurality method, who won the election?election?

Example 1, cont’dExample 1, cont’d• Solution: With 2554 votes, Charles has Solution: With 2554 votes, Charles has

a plurality and wins the election. a plurality and wins the election. • Note that there were a total of 9658 votes Note that there were a total of 9658 votes

cast. cast. • A majority of votes would be at least 4830 A majority of votes would be at least 4830

votes. Charles did not receive a majority votes. Charles did not receive a majority of votes. of votes.

Example 2Example 2• Three candidates ran for Attorney General in Three candidates ran for Attorney General in

Delaware in 2002. The vote totals were as Delaware in 2002. The vote totals were as follows:follows:• Carl Schnee: 103,913 votesCarl Schnee: 103,913 votes• Jane Brady: 110,784Jane Brady: 110,784• Vivian Houghton: 13,860Vivian Houghton: 13,860

• What percent of the votes did each What percent of the votes did each candidate receive and who won the candidate receive and who won the election?election?

Example 2, cont’dExample 2, cont’d• Solution: A total of 228,557 votes were Solution: A total of 228,557 votes were

cast.cast. • Schnee received Schnee received

• Brady receivedBrady received

• Houghton receivedHoughton received

• Brady received a plurality and is the winner.Brady received a plurality and is the winner.

110784 48.5%228557

103913 45.5%228557

13860 6.1%228557

Borda Count MethodBorda Count Method• In the In the Borda count methodBorda count method::

• Voters rank all of the Voters rank all of the mm candidates. candidates.• Votes are counted as follows:Votes are counted as follows:

• A voter’s last choice gets 1 point.A voter’s last choice gets 1 point.• A voter’s next-to-last choice gets 2 points.A voter’s next-to-last choice gets 2 points.• ……• A voter’s first choice gets A voter’s first choice gets mm points. points.

• The candidate with the most points wins.The candidate with the most points wins.

Borda Count Method, cont’dBorda Count Method, cont’d

• The main advantage of the Borda The main advantage of the Borda count method is that it uses more count method is that it uses more information from the voters.information from the voters.

• A variation of the Borda count method A variation of the Borda count method is used to select the winner of the is used to select the winner of the Heisman trophy.Heisman trophy.

Example 3Example 3• Four persons are running for student body Four persons are running for student body

president. Voters rank the candidates as shown president. Voters rank the candidates as shown in the table below. in the table below.

• Under the Borda count method, who is elected?Under the Borda count method, who is elected?

Example 3, cont’dExample 3, cont’d

• Solution: Convert the votes to points.Solution: Convert the votes to points.

Example 3, cont’dExample 3, cont’d• Solution: Total the points for each person:Solution: Total the points for each person:

• Aaron: 9436 + 4104 + 5572 + 3145 = 22,257 Aaron: 9436 + 4104 + 5572 + 3145 = 22,257 • Bonnie: 9828 + 10,497 + 4948 + 1228 = Bonnie: 9828 + 10,497 + 4948 + 1228 =

26,50126,501• Charles: 10,216 + 7101 + 3468 + 3003 = Charles: 10,216 + 7101 + 3468 + 3003 =

23,78823,788• Dion: 9152 + 7272 + 5328 + 2282 = 24,034Dion: 9152 + 7272 + 5328 + 2282 = 24,034

• Bonnie has the most points and is the Bonnie has the most points and is the winner.winner.

Example 3, cont’dExample 3, cont’d• Note that in this same election: Note that in this same election:

• Charles won using the plurality method Charles won using the plurality method because he had more first place votes because he had more first place votes than any other candidate. than any other candidate.

• Bonnie won using the Borda count Bonnie won using the Borda count method because her point total was method because her point total was highest, due to having many second-highest, due to having many second-place votes.place votes.

Plurality with Elimination MethodPlurality with Elimination Method

• In the In the plurality with elimination methodplurality with elimination method::• Voters choose one candidate.Voters choose one candidate.• The votes are counted.The votes are counted.

• If one candidate receives a majority of the If one candidate receives a majority of the votes, that candidate is selected.votes, that candidate is selected.

• If no candidate receives a majority, eliminate If no candidate receives a majority, eliminate the candidate who received the fewest votes the candidate who received the fewest votes and do another round of voting. and do another round of voting.

Plurality with Elimination, cont’dPlurality with Elimination, cont’d• Cont’d:Cont’d:

• This process is repeated until someone This process is repeated until someone receives a majority of the votes and is receives a majority of the votes and is declared the winner.declared the winner.

• The plurality with elimination method is The plurality with elimination method is used:used:• To select the location of the Olympic To select the location of the Olympic

games.games.• In France to elect the president. In France to elect the president.

Plurality with Elimination, cont’dPlurality with Elimination, cont’d

• Rather than needing to potentially Rather than needing to potentially conduct multiple votes, the voters can conduct multiple votes, the voters can be asked to rank all candidates during be asked to rank all candidates during the first election.the first election.

• A A preference tablepreference table is used to display is used to display these rankings.these rankings.

Example 4Example 4• Four persons are running for department Four persons are running for department

chairperson. The 17 voters ranked the chairperson. The 17 voters ranked the candidates 1candidates 1stst through 4 through 4thth..

• Under plurality with elimination, who is the Under plurality with elimination, who is the winner? winner?

Example 4, cont’dExample 4, cont’d• Solution: Some voters had the same preference Solution: Some voters had the same preference

ranking. Identical rating have been grouped to ranking. Identical rating have been grouped to form the preference table below. form the preference table below. • The number at the top of each column indicates the The number at the top of each column indicates the

number of voters who shared that ranking.number of voters who shared that ranking.


• Solution, cont’d: The first-place votes for Solution, cont’d: The first-place votes for each candidate are totaled: each candidate are totaled: • Alice: 6; Bob: 4; Carlos: 4; Donna: 3Alice: 6; Bob: 4; Carlos: 4; Donna: 3• No candidate received a majority, 9 votes.No candidate received a majority, 9 votes.• Donna, who has the fewest first-place votes, is Donna, who has the fewest first-place votes, is

eliminated. eliminated.

Example 4, cont’dExample 4, cont’d• Solution, cont’d: A new preference table, without Solution, cont’d: A new preference table, without

Donna, must be created. Donna, must be created. • Donna is eliminated from each column.Donna is eliminated from each column.• Any candidates ranked below Donna move up.Any candidates ranked below Donna move up.


• Solution, cont’d: The first-place votes for Solution, cont’d: The first-place votes for each candidate are totaled: each candidate are totaled: • Alice: 7; Bob: 4; Carlos: 6Alice: 7; Bob: 4; Carlos: 6• No candidate received a majority.No candidate received a majority.• Bob, who has the fewest first-place votes, is Bob, who has the fewest first-place votes, is

eliminated. eliminated.

Example 4, cont’dExample 4, cont’d• Solution, cont’d: A new preference table, Solution, cont’d: A new preference table,

without Bob, must be created. without Bob, must be created. • Bob is eliminated from each column.Bob is eliminated from each column.• Any candidates ranked below Bob move up.Any candidates ranked below Bob move up.


• Solution, cont’d: The first-place votes Solution, cont’d: The first-place votes for each candidate are totaled: for each candidate are totaled: • Alice: 9; Carlos: 8Alice: 9; Carlos: 8• Alice received a majority and is the Alice received a majority and is the

winner.winner.

Pairwise Comparison MethodPairwise Comparison Method• In the In the pairwise comparison methodpairwise comparison method::

• Voters rank all of the candidates.Voters rank all of the candidates.• For each pair of candidates For each pair of candidates XX and and YY, ,

determine how many voters prefer determine how many voters prefer XX to to YY and vice versa.and vice versa.

• If If XX is preferred to is preferred to YY more often, more often, XX gets 1 point. gets 1 point.• If If YY is preferred to is preferred to XX more often, more often, YY gets 1 point. gets 1 point.• If the candidates tie, each gets ½ a point.If the candidates tie, each gets ½ a point.

• The candidate with the most points wins.The candidate with the most points wins.

Pairwise Comparison, cont’dPairwise Comparison, cont’d

• The pairwise comparison method The pairwise comparison method is also called the is also called the Condorcet Condorcet methodmethod..

Example 5Example 5• Three persons are running for department chair. Three persons are running for department chair.

The 17 voters rank all the candidates, as shown The 17 voters rank all the candidates, as shown in the preference table below.in the preference table below.

• Under the pairwise comparison method, who wins Under the pairwise comparison method, who wins the election? the election?

Example 5, cont’dExample 5, cont’d• Solution: There are 3 pairs of candidates to Solution: There are 3 pairs of candidates to

compare:compare:• Alice vs. BobAlice vs. Bob• Alice vs. CarlosAlice vs. Carlos• Bob vs. CarlosBob vs. Carlos

• For each pair of candidates, delete the third For each pair of candidates, delete the third candidate from the preference table and candidate from the preference table and consider only the two candidates in consider only the two candidates in question.question.

Example 5, cont’dExample 5, cont’d• Solution, cont’d:Solution, cont’d:

• Alice receives 10 first-place votes, while Alice receives 10 first-place votes, while Bob only receives 7.Bob only receives 7.• We say Alice is preferred to Bob 10 to 7.We say Alice is preferred to Bob 10 to 7.• Alice receives one point.Alice receives one point.


• Alice is preferred to Carlos 9 to 8, so Alice Alice is preferred to Carlos 9 to 8, so Alice receives another point. receives another point.


• Carlos is preferred to Bob 10 to 7, so Carlos is preferred to Bob 10 to 7, so Carlos receives one point. Carlos receives one point.


• Solution, cont’d: The final point totals Solution, cont’d: The final point totals are:are:• Alice: 2 pointsAlice: 2 points• Bob: 0 pointsBob: 0 points• Carlos: 1 pointCarlos: 1 point

• Alice wins the election.Alice wins the election.

Question:Question:Candidate B is the winner of an election with the Candidate B is the winner of an election with the following preference table .following preference table .

What voting method could have been used to What voting method could have been used to determine the winner?determine the winner?

a. Plurality methoda. Plurality methodb. Borda count methodb. Borda count methodc. Plurality with elimination methodc. Plurality with elimination methodd. Pairwise comparison methodd. Pairwise comparison method

77 1212 44 99 6611stst CC BB AA CC AA22ndnd AA AA BB AA BB3rd3rd BB CC CC BB CC

Voting Methods, cont’dVoting Methods, cont’d• The four voting systems studied here The four voting systems studied here

can produce different winners even can produce different winners even when the same voter preference table when the same voter preference table is used.is used.

• Any of the four methods can also Any of the four methods can also produce a tie between two or more produce a tie between two or more candidates, which must be broken candidates, which must be broken somehow. somehow.

Tie BreakingTie Breaking• A tie-breaking method should be chosen A tie-breaking method should be chosen

before the election.before the election.• To break a tie caused by perfectly balanced To break a tie caused by perfectly balanced

voter support, election officials may:voter support, election officials may:• Make an arbitrary choice.Make an arbitrary choice.

• Flipping a coinFlipping a coin• Drawing strawsDrawing straws

• Bring in another voter.Bring in another voter.• The Vice President votes when the U. S. Senate is The Vice President votes when the U. S. Senate is

tied.tied.

3.1 Initial Problem Solution3.1 Initial Problem Solution• A majority of city councilors said they A majority of city councilors said they

preferred site A to site B and also site A to preferred site A to site B and also site A to site C. If B won the election, did they site C. If B won the election, did they necessarily lie? necessarily lie?

• Solution: Solution: • The councilors would not have to lie in The councilors would not have to lie in

order for this to happen. This situation order for this to happen. This situation can occur with some voting methods.can occur with some voting methods.

Initial Problem Solution, cont’dInitial Problem Solution, cont’d• For example, this situation could occur if the For example, this situation could occur if the

voting method used was plurality with voting method used was plurality with elimination.elimination.• Suppose 11 councilors ranked the sites as shown in the Suppose 11 councilors ranked the sites as shown in the

table below.table below.

Initial Problem Solution, cont’dInitial Problem Solution, cont’d

• Notice that in this scenario:Notice that in this scenario:• Site A is preferred to site B 7 to 4.Site A is preferred to site B 7 to 4.• Site A is preferred to site C 7 to 4.Site A is preferred to site C 7 to 4.

• However, in the vote count:However, in the vote count:• Site A, with the fewest first-place votes, is Site A, with the fewest first-place votes, is

eliminated.eliminated.• In the second round of voting, site B wins.In the second round of voting, site B wins.

Section 3.2Section 3.2Flaws of the Voting SystemsFlaws of the Voting Systems

• GoalsGoals• Study fairness criteriaStudy fairness criteria

• The majority criterionThe majority criterion• Head-to-head criterionHead-to-head criterion• Montonicity criterionMontonicity criterion• Irrelevant alternatives criterionIrrelevant alternatives criterion

• Study fairness of voting methodsStudy fairness of voting methods• Arrow impossibility theoremArrow impossibility theorem• Approval votingApproval voting

3.2 Initial Problem3.2 Initial Problem• The Compromise of 1850 averted civil war The Compromise of 1850 averted civil war

in the U.S. for 10 years.in the U.S. for 10 years.• Henry Clay proposed the bill, but it was Henry Clay proposed the bill, but it was

defeated in July 1850.defeated in July 1850.• A short time later, Stephen Douglas was able to A short time later, Stephen Douglas was able to

get essentially the same proposals passed.get essentially the same proposals passed. • How is this possible?How is this possible?

• The solution will be given at the end of the The solution will be given at the end of the section.section.

Flaws of Voting SystemsFlaws of Voting Systems

• We have seen that the choice of voting We have seen that the choice of voting method can affect the outcome of an method can affect the outcome of an election.election.

• Each voting method studied can fail to Each voting method studied can fail to satisfy certain criteria that make a satisfy certain criteria that make a voting method “fair”.voting method “fair”.

Fairness CriteriaFairness Criteria• The The fairness criteriafairness criteria are properties that are properties that

we expect a good voting system to we expect a good voting system to satisfy. satisfy.

• Four fairness criteria will be studied:Four fairness criteria will be studied:• The majority criterionThe majority criterion• The head-to-head criterionThe head-to-head criterion• The monotonicity criterionThe monotonicity criterion• The irrelevant alternatives criterionThe irrelevant alternatives criterion

The Majority CriterionThe Majority Criterion

• If a candidate is the first choice of a If a candidate is the first choice of a majority of voters, then that candidate majority of voters, then that candidate should be selected. should be selected.

Question:Question:

Candidate A won an election with Candidate A won an election with 3000 of the 8500 votes. Was the 3000 of the 8500 votes. Was the majority criterion necessarily majority criterion necessarily violated?violated?

a. yesa. yesb. nob. no

The Majority Criterion, cont’dThe Majority Criterion, cont’d• For the majority criterion to be violated:For the majority criterion to be violated:

• A candidate must have more than half of the A candidate must have more than half of the votes.votes.

• This same candidate must not win the election.This same candidate must not win the election. • Note:Note:

• This criterion does not say what should happen This criterion does not say what should happen if no candidate receives a majority.if no candidate receives a majority.

• This criterion does not say that the winner of an This criterion does not say that the winner of an election must win by a majority.election must win by a majority.

The Majority Criterion, cont’dThe Majority Criterion, cont’d• If a candidate is the first choice of a If a candidate is the first choice of a

majority of voters, then that candidate majority of voters, then that candidate will alwayswill always win using: win using: • The plurality method.The plurality method.• The plurality with elimination method.The plurality with elimination method.• The pairwise comparison method.The pairwise comparison method.

• In both of these methods any candidate with In both of these methods any candidate with more than half the vote will always winmore than half the vote will always win. .

The Majority Criterion, cont’dThe Majority Criterion, cont’d

• If a candidate is the first choice of a If a candidate is the first choice of a majority of voters, then that candidate majority of voters, then that candidate might notmight not win using: win using: • The Borda count method.The Borda count method.

• The candidate with the most points may The candidate with the most points may not be the candidate with the most first-not be the candidate with the most first-place votes.place votes.

Example 1Example 1• Four cities are being considered for an Four cities are being considered for an

annual trade show. The preferences of the annual trade show. The preferences of the organizers are given in the table.organizers are given in the table.

Example 1, cont’dExample 1, cont’da)a) Which site has a majority of first-place votes?Which site has a majority of first-place votes?

b)b) Which site wins using the Borda count Which site wins using the Borda count method?method?


• Solution: There are 9 votes, so a Solution: There are 9 votes, so a majority would be 5 or more votes.majority would be 5 or more votes.

a)a) The first place vote totals are:The first place vote totals are:• Chicago: 5; Seattle: 3; Phoenix: 1; Boston: 0Chicago: 5; Seattle: 3; Phoenix: 1; Boston: 0• Chicago has a majority of first-place votes.Chicago has a majority of first-place votes.

Example 1, cont’dExample 1, cont’d• Solution, cont’d: Find the point totals Solution, cont’d: Find the point totals

for Borda count.for Borda count.b)b) The points are calculated as follows:The points are calculated as follows:

• Chicago: 5(4) + 0(3) + 2(2) + 2(1) = 26Chicago: 5(4) + 0(3) + 2(2) + 2(1) = 26• Seattle: 3(4) + 4(3) + 2(2) + 0(1) = 28Seattle: 3(4) + 4(3) + 2(2) + 0(1) = 28• Phoenix: 1(4) + 2(3) + 2(2) + 4(1) = 18Phoenix: 1(4) + 2(3) + 2(2) + 4(1) = 18• Boston: 0(4) + 3(3) + 3(2) + 3(1) = 18Boston: 0(4) + 3(3) + 3(2) + 3(1) = 18• Under the Borda count method, Seattle is Under the Borda count method, Seattle is

the winner.the winner.

Example 1, cont’dExample 1, cont’d• Note that Chicago had a majority of Note that Chicago had a majority of

first-place votes, but under the Borda first-place votes, but under the Borda count method Seattle was the winner.count method Seattle was the winner.

• This is an example of the Borda count This is an example of the Borda count method failing the majority criterion. method failing the majority criterion. In this case, we would say that the In this case, we would say that the Borda count method was unfair.Borda count method was unfair.

The Head-to-Head CriterionThe Head-to-Head Criterion

• If a candidate is favored when If a candidate is favored when compared separately with each of the compared separately with each of the other candidates, then the favored other candidates, then the favored candidate should win the election.candidate should win the election.

• This is also called the This is also called the Condorcet Condorcet criterioncriterion..

Head-to-Head Criterion, cont’dHead-to-Head Criterion, cont’d

• For the head-to-head criterion to be violated:For the head-to-head criterion to be violated:• A candidate must be preferred pairwise to every A candidate must be preferred pairwise to every

other candidate.other candidate.• This same candidate must not win the election. This same candidate must not win the election.

• Note:Note:• This criterion does not say what should happen This criterion does not say what should happen

if no candidate is preferred pairwise to every if no candidate is preferred pairwise to every other candidate. other candidate.


• If a candidate is favored pairwise to If a candidate is favored pairwise to every other candidate, then that every other candidate, then that candidate candidate will alwayswill always win using: win using: • The pairwise comparison method.The pairwise comparison method.

• This candidate will earn the most points This candidate will earn the most points from the pairwise comparisons.from the pairwise comparisons.


• If a candidate is favored pairwise to If a candidate is favored pairwise to every other candidate, then that every other candidate, then that candidate candidate might notmight not win using: win using: • The plurality method.The plurality method.• The plurality with elimination method.The plurality with elimination method.• The Borda count method.The Borda count method.

Example 2Example 2• Seven people are choosing an option for a Seven people are choosing an option for a

retirement party: catering, picnic, or restaurant. retirement party: catering, picnic, or restaurant. The preferences are shown in the table below.The preferences are shown in the table below.a)a) Which site is selected using the plurality method?Which site is selected using the plurality method?

b)b) Show that the head-to-head criterion is violated.Show that the head-to-head criterion is violated.


• Solution:Solution:a)a)The picnic has the most votes, 3, so The picnic has the most votes, 3, so

it is the winning option under the it is the winning option under the plurality method. plurality method.


b)b) The pairwise comparisons are made:The pairwise comparisons are made:• R is preferred to P 4 to 3.R is preferred to P 4 to 3.• R is preferred to C 5 to 2. R is preferred to C 5 to 2.

• R is preferred separately to every other R is preferred separately to every other candidate, but R did not win the election. candidate, but R did not win the election. This is a violation of the head-to-head This is a violation of the head-to-head criterion. criterion.

The Monotonicity CriterionThe Monotonicity Criterion• Suppose a particular candidate, X, wins an Suppose a particular candidate, X, wins an

election.election.• If, hypothetically, this election were redone If, hypothetically, this election were redone

and the only changes were that some and the only changes were that some voters switched X with the candidate they voters switched X with the candidate they had ranked one higher, then X should still had ranked one higher, then X should still win.win.

• This criterion is only used in special cases.This criterion is only used in special cases.

Monotonicity CriterionMonotonicity Criterion, , cont’dcont’d

• The monotonicity criterion The monotonicity criterion is alwaysis always satisfied by: satisfied by: • The plurality method.The plurality method.• The Borda count method.The Borda count method.• The pairwise comparison method.The pairwise comparison method.

Monotonicity CriterionMonotonicity Criterion, , cont’dcont’d

• The monotonicity criterion The monotonicity criterion is not alwaysis not always satisfied by: satisfied by: • The plurality with elimination method.The plurality with elimination method.

Example 3Example 3• Teachers are voting for a union president Teachers are voting for a union president

from the candidates Akst, Bailey, and from the candidates Akst, Bailey, and Chung. The preferences are shown in the Chung. The preferences are shown in the table below.table below.

Example 3, cont’dExample 3, cont’da)a) Who will win using the plurality with Who will win using the plurality with

elimination method?elimination method?• Solution: The first-place vote totals Solution: The first-place vote totals

are:are:• Akst: 14; Bailey: 12; Chung: 15.Akst: 14; Bailey: 12; Chung: 15.• No candidate received a majority of at No candidate received a majority of at

least 21 votes, so Bailey is eliminated.least 21 votes, so Bailey is eliminated.


• Solution, cont’d:Solution, cont’d:• After Bailey’s elimination, a new After Bailey’s elimination, a new

preference table is created.preference table is created.• Akst now has 26 first-place votes, and is Akst now has 26 first-place votes, and is

the winner.the winner.

Example 3, cont’dExample 3, cont’db)b) If 4 of the 5 teachers who ranked the If 4 of the 5 teachers who ranked the

candidates CAB changed to a ranking candidates CAB changed to a ranking of ACB, would this affect the outcome of ACB, would this affect the outcome of the election? of the election?

• Solution: Akst won the first election Solution: Akst won the first election and the only changes now are that 4 and the only changes now are that 4 teachers moved him from 2teachers moved him from 2ndnd to 1 to 1stst place.place.

Example 3, cont’dExample 3, cont’d• Solution, cont’d: The new preference table Solution, cont’d: The new preference table

is shown below.is shown below.


• Solution, cont’d: The first-place vote Solution, cont’d: The first-place vote totals are:totals are:

• Akst: 18; Bailey: 12; Chung: 11.Akst: 18; Bailey: 12; Chung: 11.• No candidate received a majority of at No candidate received a majority of at

least 21 votes, so Chung is eliminated.least 21 votes, so Chung is eliminated.


• Solution, cont’d:Solution, cont’d:• After Chung’s elimination, a new After Chung’s elimination, a new

preference table is created.preference table is created.• Bailey now has 22 votes, and is the Bailey now has 22 votes, and is the

winner.winner.


• The only changes in the preference table The only changes in the preference table were ones that favored Akst, who won were ones that favored Akst, who won the first election.the first election.

• However, Akst ended up losing the However, Akst ended up losing the modified vote to Bailey.modified vote to Bailey.

• This is a violation of the monotonicity This is a violation of the monotonicity criterion.criterion.

The Irrelevant Alternatives CriterionThe Irrelevant Alternatives Criterion

• Suppose a candidate, X, is selected Suppose a candidate, X, is selected in an election.in an election.

• If, hypothetically, this election were If, hypothetically, this election were redone with one or more of the redone with one or more of the unselected candidates removed from unselected candidates removed from the vote, then X should still win.the vote, then X should still win.

Irrelevant Alternatives Criterion, cont’dIrrelevant Alternatives Criterion, cont’d

• The irrelevant alternatives criterion The irrelevant alternatives criterion is is not alwaysnot always satisfied by any of the 4 satisfied by any of the 4 voting methods studied.voting methods studied.

Example 4Example 4• The 5 members of a book club are voting on The 5 members of a book club are voting on

what book to read next: a mystery, a what book to read next: a mystery, a historical novel, or a science fiction fantasy. historical novel, or a science fiction fantasy. The preference table is shown below.The preference table is shown below.


a)a) Which of the books is selected using the Which of the books is selected using the plurality with elimination method?plurality with elimination method?

• Solution: The first-place vote totals are:Solution: The first-place vote totals are:• M: 2; H: 1; S: 2.M: 2; H: 1; S: 2.• No book has a majority, so H is eliminated.No book has a majority, so H is eliminated.


• Solution, cont’d: After H is eliminated Solution, cont’d: After H is eliminated a new preference table is created.a new preference table is created.• Book M receives 3 first-place votes, and Book M receives 3 first-place votes, and

is the winner. The book club will read is the winner. The book club will read the mystery.the mystery.

Example 4, cont’dExample 4, cont’db)b) If the science fiction book is removed from If the science fiction book is removed from

the list, is the irrelevant alternatives the list, is the irrelevant alternatives criterion violated in the new election?criterion violated in the new election?

• Solution: A new preference table, without Solution: A new preference table, without S, is created.S, is created.


• Solution, cont’d:Solution, cont’d:• In the new table, M has 2 votes and H In the new table, M has 2 votes and H

has 3.has 3.• Book H is the new winner, violating the Book H is the new winner, violating the

irrelevant alternatives criterion.irrelevant alternatives criterion.

Fairness Criteria, cont’dFairness Criteria, cont’d

Arrow Impossibility TheoremArrow Impossibility Theorem

• The The Arrow Impossibility TheoremArrow Impossibility Theorem states that no system of voting will states that no system of voting will always satisfy all of the 4 fairness always satisfy all of the 4 fairness criteria.criteria.

• This fact was proved by Kenneth Arrow This fact was proved by Kenneth Arrow in 1951.in 1951.

Question:Question:

The results of an election using the The results of an election using the plurality method were analyzed. It plurality method were analyzed. It was found that the election did not was found that the election did not violate any of the four fairness violate any of the four fairness criteria. Does this contradict the criteria. Does this contradict the Arrow Impossibility Theorem?Arrow Impossibility Theorem?


Approval VotingApproval Voting• No voting system is always fair, but we No voting system is always fair, but we

can explore systems that are unfair less can explore systems that are unfair less often than others. One such system is often than others. One such system is called approval voting.called approval voting.

• In In approval votingapproval voting::• Each voter votes for all candidates he/she Each voter votes for all candidates he/she

considers acceptable.considers acceptable.• The candidate with the most votes is selected.The candidate with the most votes is selected.

Example 5Example 5• Three candidates are running for two Three candidates are running for two

positions. There are 9 voters and the positions. There are 9 voters and the votes are shown in the table below.votes are shown in the table below.

• Who is the winner under approval voting? Who is the winner under approval voting?


• Solution: The vote totals are:Solution: The vote totals are:• Ammee: 6 Ammee: 6 • Bonnie: 7Bonnie: 7• Celeste: 5Celeste: 5

• Bonnie and Ammee are selected for the two positions.Bonnie and Ammee are selected for the two positions.

3.2 Initial Problem Solution3.2 Initial Problem Solution• Henry Clay presented the Compromise Henry Clay presented the Compromise

of 1850 as one bill containing all the of 1850 as one bill containing all the proposals. proposals.

• Of the 60 senators, a majority would Of the 60 senators, a majority would not approve the bill because they not approve the bill because they disagreed on individual issues within disagreed on individual issues within the bill.the bill.


• Stephen Douglas presented each Stephen Douglas presented each proposal of the Compromise in a proposal of the Compromise in a separate bill. separate bill.

• A (different) majority of the senators A (different) majority of the senators passed each proposal and the passed each proposal and the Compromise of 1850 went into effect, Compromise of 1850 went into effect, although a majority never approved the although a majority never approved the measures as a whole.measures as a whole.

Section 3.3Section 3.3Weighted Voting SystemsWeighted Voting Systems

• GoalsGoals• Study weighted voting systemsStudy weighted voting systems

• CoalitionsCoalitions• Dummies and dictatorsDummies and dictators• Veto powerVeto power

• Study the Banzhaf power indexStudy the Banzhaf power index

3.3 Initial Problem3.3 Initial Problem

• A stockholder owns 17% of the shares of A stockholder owns 17% of the shares of a company.a company.

• Among the other 3 stockholders, no one Among the other 3 stockholders, no one owns more than 32% of the shares.owns more than 32% of the shares.

• Why will no one listen to the stockholder Why will no one listen to the stockholder with 17%?with 17%?• The solution will be given at the end of the section.The solution will be given at the end of the section.

Weighted Voting SystemsWeighted Voting Systems• In a In a weighted voting systemweighted voting system, an , an

individual voter may have more than individual voter may have more than one vote.one vote.

• The number of votes that a voter The number of votes that a voter controls is called the controls is called the weightweight of the voter. of the voter.• An example of a weighted voting system is An example of a weighted voting system is

the election of the U.S. President by the the election of the U.S. President by the Electoral College. Electoral College.

Weighted Voting Systems, cont’dWeighted Voting Systems, cont’d• The weights of the voters are usually listed The weights of the voters are usually listed

as a sequence of numbers between square as a sequence of numbers between square brackets.brackets.

• For example, the voting system in which For example, the voting system in which Angie has a weight of 9, Roberta has a Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11 is represented as Darrell has a weight of 11 is represented as [12, 11, 9, 8].[12, 11, 9, 8].

Weighted Voting Systems, cont’dWeighted Voting Systems, cont’d• The voter with the largest weight is The voter with the largest weight is

called the “first voter”, written Pcalled the “first voter”, written P11..• The weight of the first voter is The weight of the first voter is

represented by Wrepresented by W11..• The remaining voters and their weights The remaining voters and their weights

are represented similarly, in order of are represented similarly, in order of decreasing weights.decreasing weights.

Example 1Example 1• The voting system in which Angie has a The voting system in which Angie has a

weight of 9, Roberta has a weight of 12, weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a Carlos has a weight of 8, and Darrell has a weight of 11 was represented as [12, 11, 9, weight of 11 was represented as [12, 11, 9, 8]. 8].

• In this case, PIn this case, P1 1 = Roberta, P= Roberta, P2 2 = Darrell, P= Darrell, P3 3 = = Angie, and PAngie, and P4 4 = Carlos.= Carlos.

• Also, WAlso, W1 1 = 12, W= 12, W2 2 = 11, W= 11, W3 3 = 9, and W= 9, and W4 4 = 8.= 8.

Weighted Voting Systems, cont’dWeighted Voting Systems, cont’d• Yes or no questions are commonly Yes or no questions are commonly

called called motionsmotions..• A final decision of ‘No’ A final decision of ‘No’ defeatsdefeats the the

motion and leaves the status quo motion and leaves the status quo unchanged.unchanged.

• A final decision of ‘Yes” A final decision of ‘Yes” passespasses the the motion and changes the status quo.motion and changes the status quo.

Weighted Voting Systems, cont’dWeighted Voting Systems, cont’d• A A simple majoritysimple majority requirement means that a requirement means that a

motion must receive more than half of the motion must receive more than half of the votes to pass.votes to pass.

• A A supermajoritysupermajority requirement means that the requirement means that the minimum number of votes required to pass a minimum number of votes required to pass a motion is set higher than half of the total motion is set higher than half of the total weight.weight.• A common supermajority is two-thirds of the A common supermajority is two-thirds of the

total weight.total weight.

Weighted Voting Systems, cont’dWeighted Voting Systems, cont’d• The weight required to pass a motion is The weight required to pass a motion is

called the called the quotaquota..• Example: A simple majority quota for Example: A simple majority quota for

the weighted voting system [12, 11, 9, the weighted voting system [12, 11, 9, 8] would be 21.8] would be 21.• Half of the total weight is (12 + 11 + 9 + Half of the total weight is (12 + 11 + 9 +

8)/2 = 40/2 = 20. More than half of the 8)/2 = 40/2 = 20. More than half of the weight would be at least 21 ‘Yes’ votes.weight would be at least 21 ‘Yes’ votes.

Question:Question:

Given the weighted voting system Given the weighted voting system [10, 9, 8, 8, 5], find the quota for a [10, 9, 8, 8, 5], find the quota for a supermajority requirement of two-supermajority requirement of two-thirds of the total weight.thirds of the total weight.

a. 27a. 27b. 21b. 21c. 26c. 26d. 20d. 20

Weighted Voting Systems, cont’dWeighted Voting Systems, cont’d• The quota for a weighted voting The quota for a weighted voting

system is usually added to the list system is usually added to the list of weights.of weights.

• Example: For the weighted voting Example: For the weighted voting system [12, 11, 9, 8] with a quota of system [12, 11, 9, 8] with a quota of 21 the complete notation is 21 the complete notation is [21 : 12, 11, 9, 8].[21 : 12, 11, 9, 8].

Example 2Example 2

• Given the weighted voting systemGiven the weighted voting system

[21 : 10, 8, 7, 7, 4, 4], suppose P[21 : 10, 8, 7, 7, 4, 4], suppose P11, P, P33, , and Pand P5 5 vote ‘Yes’ on a motion. vote ‘Yes’ on a motion.

• Is the motion passed or defeated?Is the motion passed or defeated?


• Solution:Solution: • The given voters have a combined The given voters have a combined

weight of 10 + 7 + 4 = 21. weight of 10 + 7 + 4 = 21. • The quota is met, so the motion The quota is met, so the motion

passes.passes.

Example 3Example 3

• Given the weighted voting system Given the weighted voting system

[21 : 10, 8, 7, 7, 4, 4], suppose P[21 : 10, 8, 7, 7, 4, 4], suppose P11, P, P55, , and Pand P6 6 vote ‘Yes’ on a motion. vote ‘Yes’ on a motion.

• Is the motion passed or defeated?Is the motion passed or defeated?


• Solution: Solution: • The given voters have a combined The given voters have a combined

weight of 10 + 4 + 4 = 18. weight of 10 + 4 + 4 = 18. • The quota is not met, so the motion is The quota is not met, so the motion is

defeated.defeated.

CoalitionsCoalitions• Any nonempty subset of the voters in a Any nonempty subset of the voters in a

weighted voting system is called a weighted voting system is called a coalitioncoalition..• If the total weight of the voters in a coalition If the total weight of the voters in a coalition

is greater than or equal to the quota, it is is greater than or equal to the quota, it is called a called a winning coalitionwinning coalition..

• If the total weight of the voters in a coalition If the total weight of the voters in a coalition is less than the quota, it is called a is less than the quota, it is called a losing losing coalitioncoalition..

Question:Question:

Given the weighted voting system Given the weighted voting system [27: 10, 9, 8, 8, 5], is the coalition [27: 10, 9, 8, 8, 5], is the coalition {P{P11, P, P44, P, P55} a winning coalition or a } a winning coalition or a losing coalition?losing coalition?

a. winninga. winningb. losingb. losing

Example 4Example 4

• For the weighted voting system For the weighted voting system

[8: 6, 5, 4], list all possible coalitions [8: 6, 5, 4], list all possible coalitions and determine whether each is a and determine whether each is a winning or losing coalition. winning or losing coalition.

Example 4, cont’dExample 4, cont’d• Solution: Each coalition and its status is Solution: Each coalition and its status is

listed in the table below.listed in the table below.

Coalitions, cont’dCoalitions, cont’d• In a weighted voting system with In a weighted voting system with nn

voters, exactly 2voters, exactly 2nn - 1 coalitions are - 1 coalitions are possible.possible.

• Example: Example: • How many coalitions are possible in a How many coalitions are possible in a

weighted voting system with 7 voters?weighted voting system with 7 voters?• Solution: The formula tells us there are Solution: The formula tells us there are 227 7 - 1 = 128 – 1 = 127 coalitions.- 1 = 128 – 1 = 127 coalitions.

Example 5Example 5• The voting weights of EU members in a The voting weights of EU members in a

council in 2003 are shown in the table.council in 2003 are shown in the table.

Example 5, cont’dExample 5, cont’da)a)If resolutions must receive 71% of the If resolutions must receive 71% of the

votes to pass, what is the quota?votes to pass, what is the quota?

b)b)How many coalitions are possible? How many coalitions are possible?


• Solution:Solution:

a)a) There are 87 votes total. So the quota is There are 87 votes total. So the quota is 71% of 87, or approximately 62 votes.71% of 87, or approximately 62 votes.

b)b) There are There are nn = 15 members, so there are = 15 members, so there are 2215 15 – 1 = 32,767 coalitions possible.– 1 = 32,767 coalitions possible.

Dictators and DummiesDictators and Dummies• A voter whose presence or absence in any A voter whose presence or absence in any

coalition makes no difference in the coalition makes no difference in the outcome is called a outcome is called a dummydummy..

• A voter whose presence or absence in any A voter whose presence or absence in any coalition completely determines the coalition completely determines the outcome is called a outcome is called a dictatordictator..• When a weighted voting system has a When a weighted voting system has a

dictator, the other voters in the system are dictator, the other voters in the system are automatically dummies. automatically dummies.

Veto PowerVeto Power• In between the complete power of a In between the complete power of a

dictator and the zero power of a dummy is dictator and the zero power of a dummy is a level of power called veto power.a level of power called veto power.

• A voter with A voter with veto powerveto power can defeat a can defeat a motion by voting ‘No’ but cannot motion by voting ‘No’ but cannot necessarily pass a motion by voting ‘Yes’.necessarily pass a motion by voting ‘Yes’.• Any dictator has veto power, but a voter Any dictator has veto power, but a voter

with veto power is not necessarily a with veto power is not necessarily a dictator.dictator.

Example 6Example 6• Consider the weighted voting system Consider the weighted voting system

[12: 7, 6, 4].[12: 7, 6, 4].

a)a)List all the coalitions and determine List all the coalitions and determine whether each is a winning or losing whether each is a winning or losing coalition.coalition.

b)b)Are there any dummies or dictators?Are there any dummies or dictators?

c)c) Are there any voters with veto power? Are there any voters with veto power?

Example 6, cont’dExample 6, cont’d• Solution: Solution:

a)a)Each coalition and its status is listed in Each coalition and its status is listed in the table below.the table below.


• Solution, cont’d: Solution, cont’d:

b)b)Removing the third voter from any Removing the third voter from any coalition does not change the status of coalition does not change the status of the coalition. Pthe coalition. P33 is a dummy. is a dummy.



b)b)No voter has complete power to pass or No voter has complete power to pass or defeat a motion. There is no dictator.defeat a motion. There is no dictator.



c)c) If PIf P1 1 is not in a coalition, then it is a losing is not in a coalition, then it is a losing coalition. Pcoalition. P11 has veto power. has veto power.

Question:Question:

In the weighted voting system In the weighted voting system [27: 10, 9, 8, 8, 5], is P[27: 10, 9, 8, 8, 5], is P11 a: a:

a. dictatora. dictatorb. dummyb. dummyc. voter with veto powerc. voter with veto powerd. none of the above d. none of the above

Example 7Example 7

• Consider the weighted voting system Consider the weighted voting system [10: 10, 5, 4]. [10: 10, 5, 4].

• Are there any dummies, dictators, or Are there any dummies, dictators, or voters with veto power? voters with veto power?

Example 7, cont’dExample 7, cont’d• Solution: Solution:

• PP11 has enough weight to pass a motion has enough weight to pass a motion by voting ‘Yes’ no matter how anyone by voting ‘Yes’ no matter how anyone else votes.else votes.

• If PIf P11 votes ‘No’, the motion will not pass votes ‘No’, the motion will not pass no matter how anyone else votes.no matter how anyone else votes.

• PP11 is a dictator and thus all other voters is a dictator and thus all other voters are dummies. are dummies.

Critical VotersCritical Voters

• If a voter’s weight is large enough so that If a voter’s weight is large enough so that the voter can change a particular winning the voter can change a particular winning coalition to a losing coalition by leaving the coalition to a losing coalition by leaving the coalition, then that voter is called a coalition, then that voter is called a critical critical votervoter in that winning coalition. in that winning coalition.

Question:Question:

Given the weighted voting system Given the weighted voting system [27: 10, 9, 8, 8, 5], is the voter P[27: 10, 9, 8, 8, 5], is the voter P44 a a critical voter in the winning coalition critical voter in the winning coalition {P{P11, P, P22, P, P44, P, P55}?}?


Example 8Example 8

• Consider the weighted voting system Consider the weighted voting system [21 : 10, 8, 7, 7, 4, 4]. [21 : 10, 8, 7, 7, 4, 4].

• Which voters in the coalition {PWhich voters in the coalition {P22, P, P33, , PP44, P, P5 5 } are critical voters in that } are critical voters in that coalition? coalition?


• Solution: The weight in the winning Solution: The weight in the winning coalition is 26.coalition is 26.• If PIf P22 leaves, the weight goes down to leaves, the weight goes down to

26 – 8 = 18 < quota. 26 – 8 = 18 < quota.

• If PIf P33 leaves, the weight goes down to leaves, the weight goes down to

26 – 7 = 19 < quota.26 – 7 = 19 < quota.

Example 8, cont’dExample 8, cont’d• Solution cont’d:Solution cont’d:


26 – 7 = 19 < quota.26 – 7 = 19 < quota.


26 – 4 = 22 > quota.26 – 4 = 22 > quota.• The critical voters in this coalition are The critical voters in this coalition are

PP22, P, P33, and P, and P4.4.

The Banzhaf Power IndexThe Banzhaf Power Index• The more times a voter is a critical The more times a voter is a critical

voter in a coalition, the more power voter in a coalition, the more power that voter has in the system.that voter has in the system.

• The The Banzhaf powerBanzhaf power of a voter is the of a voter is the number of winning coalitions in which number of winning coalitions in which that voter is critical.that voter is critical.

Banzhaf Power Index, cont’dBanzhaf Power Index, cont’d• The sum of the Banzhaf powers of all The sum of the Banzhaf powers of all

voters is called the voters is called the total Banzhaf total Banzhaf powerpower in the weighted voting system. in the weighted voting system.

• An individual voter’s An individual voter’s Banzhaf power Banzhaf power indexindex is the ratio of the voter’s Banzhaf is the ratio of the voter’s Banzhaf power to the total Banzhaf power in power to the total Banzhaf power in the system.the system.• The sum of the Banzhaf power indices of all The sum of the Banzhaf power indices of all

voters is 100%.voters is 100%.

Banzhaf Power Index, cont’dBanzhaf Power Index, cont’d• An individual voter’s An individual voter’s Banzhaf power indexBanzhaf power index

is calculated using the following process:is calculated using the following process:1)1) Find all winning coalitions for the system.Find all winning coalitions for the system.

2)2) Determine the critical voters for each winning Determine the critical voters for each winning coalition.coalition.

3)3) Calculate each voter’s Banzhaf power.Calculate each voter’s Banzhaf power.

4)4) Find the total Banzhaf power in the system.Find the total Banzhaf power in the system.

5)5) Divide each voter’s Banzhaf power by the total Divide each voter’s Banzhaf power by the total Banzhaf power.Banzhaf power.

Example 9Example 9

• For the weighted voting system For the weighted voting system [18 : 12, 7, 6, 5], determine:[18 : 12, 7, 6, 5], determine:• The total Banzhaf power in the system.The total Banzhaf power in the system.• The Banzhaf power index of each voter.The Banzhaf power index of each voter.

Example 9, cont’dExample 9, cont’d• Solution Solution

Step 1: Step 1: Find all the Find all the winning winning coalitions.coalitions.


• Solution Step 2: Determine the critical Solution Step 2: Determine the critical voters for each winning coalition.voters for each winning coalition.• Remove each voter one at a time and Remove each voter one at a time and

check to see whether the resulting check to see whether the resulting coalition is still a winning coalition.coalition is still a winning coalition.

• This work is shown in the next slides.This work is shown in the next slides.

Example 9, cont’dExample 9, cont’d• Solution Step 3: Count the number of Solution Step 3: Count the number of

times each voter is a critical voter:times each voter is a critical voter:• PP11: 5 times: 5 times

• PP22: 3 times: 3 times

• PP33: 3 times: 3 times

• PP44: 1 time: 1 time

• Step 4: The total Banzhaf power in the Step 4: The total Banzhaf power in the system is 5 + 3 + 3 + 1 = 12system is 5 + 3 + 3 + 1 = 12

Example 9, cont’dExample 9, cont’d• Solution Step 5: Solution Step 5:

Divide each Divide each voter’s Banzhaf voter’s Banzhaf power by the power by the total Banzhaf total Banzhaf power to find power to find the Banzhaf the Banzhaf power indices.power indices.

3.3 Initial Problem Solution3.3 Initial Problem Solution

• One of the 4 stockholders owns 17% One of the 4 stockholders owns 17% of the shares of a company. Among of the shares of a company. Among the other 3, no one owns more than the other 3, no one owns more than 32%. Why will no one listen to the 32%. Why will no one listen to the stockholder with 17%?stockholder with 17%?


• We know one stockholder owns 32% We know one stockholder owns 32% and one owns 17%, so the other 51% and one owns 17%, so the other 51% is split between the remaining two is split between the remaining two stockholdersstockholders..• Suppose the other percents are 26% Suppose the other percents are 26%

and 25%.and 25%.

Initial Problem Solution, cont’dInitial Problem Solution, cont’d• The winning coalitions in this case are:The winning coalitions in this case are:

• {32%, 26%, 25%, 17%}{32%, 26%, 25%, 17%}• {32%, 26%, 25%}{32%, 26%, 25%}• {32%, 26%, 17%}{32%, 26%, 17%}• {32%, 25%, 17%}{32%, 25%, 17%}• {26%, 25%, 17%}{26%, 25%, 17%}• {32%, 26%}{32%, 26%}• {32%, 25%}{32%, 25%}• {26%, 25%}{26%, 25%}

Initial Problem Solution, cont’dInitial Problem Solution, cont’d• The voter with 17% is in several winning The voter with 17% is in several winning

coalitions, but removing that voter does not coalitions, but removing that voter does not cause any of them to become losing cause any of them to become losing coalitions.coalitions.

• The voter with 17% is not a critical voter in The voter with 17% is not a critical voter in any winning coalition. any winning coalition.

• The reason no one will listen to the voter is The reason no one will listen to the voter is that he or she is a dummy.that he or she is a dummy.

A Mathematical View of Our World

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