Title: A Mathematical View of Our World
1A Mathematical View of Our World
- 1st ed.
- Parks, Musser, Trimpe, Maurer, and Maurer
2Chapter 3
3Section 3.1Voting Systems
- Goals
- Study voting systems
- Plurality method
- Borda count method
- Plurality with elimination method
- Pairwise comparison method
- Discuss tie-breaking methods
43.1 Initial Problem
- The city council must select among 3 locations
for a new sewage treatment plant. - A majority of city councilors say they prefer
site A to site B. - A majority of city councilors say they prefer
site A to site C. - In the vote site B is selected.
- Did the councilors necessarily lie about their
preferences before the election? - The solution will be given at the end of the
section.
5Voting Systems
- The following voting methods will be discussed
- Plurality method
- Borda count method
- Plurality with elimination method
- Pairwise comparison method
6Plurality Method
- When a candidate receives more than half of the
votes in an election, we say the candidate has
received a majority of the votes. - When a candidate receives the greatest number of
votes in an election, but not more than half, we
say the candidate has received a plurality of the
votes.
7Question
- Suppose in an election, the vote totals are as
follows. Andy gets 4526 first-place votes. Lacy
gets 1901 first-place votes. Peter gets 2265
first-place votes. - Choose the correct statement.
- a. Andy has a majority.
- b. Andy has a plurality only.
8Plurality Method, contd
- In the plurality method
- Voters vote for one candidate.
- The candidate receiving the most votes wins.
- This method has a couple advantages
- The voter chooses only one candidate.
- The winner is easily determined.
9Plurality Method, contd
- The plurality method is used
- In the United States to elect senators,
representatives, governors, judges, and mayors. - In the United Kingdom and Canada to elect members
of parliament.
10Example 1
- Four persons are running for student body
president. The vote totals are as follows - Aaron 2359 votes
- Bonnie 2457 votes
- Charles 2554 votes
- Dion 2288 votes
- Under the plurality method, who won the election?
11Example 1, contd
- Solution With 2554 votes, Charles has a
plurality and wins the election. - Note that there were a total of 9658 votes cast.
- A majority of votes would be at least 4830 votes.
Charles did not receive a majority of votes.
12Example 2
- Three candidates ran for Attorney General in
Delaware in 2002. The vote totals were as
follows - Carl Schnee 103,913 votes
- Jane Brady 110,784
- Vivian Houghton 13,860
- What percent of the votes did each candidate
receive and who won the election?
13Example 2, contd
- Solution A total of 228,557 votes were cast.
- Schnee received
- Brady received
- Houghton received
- Brady received a plurality and is the winner.
14Borda Count Method
- In the Borda count method
- Voters rank all of the m candidates.
- Votes are counted as follows
- A voters last choice gets 1 point.
- A voters next-to-last choice gets 2 points.
-
- A voters first choice gets m points.
- The candidate with the most points wins.
15Borda Count Method, contd
- The main advantage of the Borda count method is
that it uses more information from the voters. - A variation of the Borda count method is used to
select the winner of the Heisman trophy.
16Example 3
- Four persons are running for student body
president. Voters rank the candidates as shown
in the table below. - Under the Borda count method, who is elected?
17Example 3, contd
- Solution Convert the votes to points.
18Example 3, contd
- Solution Total the points for each person
- Aaron 9436 4104 5572 3145 22,257
- Bonnie 9828 10,497 4948 1228 26,501
- Charles 10,216 7101 3468 3003 23,788
- Dion 9152 7272 5328 2282 24,034
- Bonnie has the most points and is the winner.
19Example 3, contd
- Note that in this same election
- Charles won using the plurality method because he
had more first place votes than any other
candidate. - Bonnie won using the Borda count method because
her point total was highest, due to having many
second-place votes.
20Plurality with Elimination Method
- In the plurality with elimination method
- Voters choose one candidate.
- The votes are counted.
- If one candidate receives a majority of the
votes, that candidate is selected. - If no candidate receives a majority, eliminate
the candidate who received the fewest votes and
do another round of voting.
21Plurality with Elimination, contd
- Contd
- This process is repeated until someone receives a
majority of the votes and is declared the winner. - The plurality with elimination method is used
- To select the location of the Olympic games.
- In France to elect the president.
22Plurality with Elimination, contd
- Rather than needing to potentially conduct
multiple votes, the voters can be asked to rank
all candidates during the first election. - A preference table is used to display these
rankings.
23Example 4
- Four persons are running for department
chairperson. The 17 voters ranked the candidates
1st through 4th. - Under plurality with elimination, who is the
winner?
24Example 4, contd
- Solution Some voters had the same preference
ranking. Identical rating have been grouped to
form the preference table below. - The number at the top of each column indicates
the number of voters who shared that ranking.
25Example 4, contd
- Solution, contd The first-place votes for each
candidate are totaled - Alice 6 Bob 4 Carlos 4 Donna 3
- No candidate received a majority, 9 votes.
- Donna, who has the fewest first-place votes, is
eliminated.
26Example 4, contd
- Solution, contd A new preference table, without
Donna, must be created. - Donna is eliminated from each column.
- Any candidates ranked below Donna move up.
27Example 4, contd
- Solution, contd The first-place votes for each
candidate are totaled - Alice 7 Bob 4 Carlos 6
- No candidate received a majority.
- Bob, who has the fewest first-place votes, is
eliminated.
28Example 4, contd
- Solution, contd A new preference table, without
Bob, must be created. - Bob is eliminated from each column.
- Any candidates ranked below Bob move up.
29Example 4, contd
- Solution, contd The first-place votes for each
candidate are totaled - Alice 9 Carlos 8
- Alice received a majority and is the winner.
30Pairwise Comparison Method
- In the pairwise comparison method
- Voters rank all of the candidates.
- For each pair of candidates X and Y, determine
how many voters prefer X to Y and vice versa. - If X is preferred to Y more often, X gets 1
point. - If Y is preferred to X more often, Y gets 1
point. - If the candidates tie, each gets ½ a point.
- The candidate with the most points wins.
31Pairwise Comparison, contd
- The pairwise comparison method is also called the
Condorcet method.
32Example 5
- Three persons are running for department chair.
The 17 voters rank all the candidates, as shown
in the preference table below. - Under the pairwise comparison method, who wins
the election?
33Example 5, contd
- Solution There are 3 pairs of candidates to
compare - Alice vs. Bob
- Alice vs. Carlos
- Bob vs. Carlos
- For each pair of candidates, delete the third
candidate from the preference table and consider
only the two candidates in question.
34Example 5, contd
- Alice receives 10 first-place votes, while Bob
only receives 7. - We say Alice is preferred to Bob 10 to 7.
- Alice receives one point.
35Example 5, contd
- Alice is preferred to Carlos 9 to 8, so Alice
receives another point.
36Example 5, contd
- Carlos is preferred to Bob 10 to 7, so Carlos
receives one point.
37Example 5, contd
- Solution, contd The final point totals are
- Alice 2 points
- Bob 0 points
- Carlos 1 point
- Alice wins the election.
38Question
- Candidate B is the winner of an election with
the following preference table . - What voting method could have been used to
determine the winner? - a. Plurality method
- b. Borda count method
- c. Plurality with elimination method
- d. Pairwise comparison method
7 12 4 9 6
1st C B A C A
2nd A A B A B
3rd B C C B C
39Voting Methods, contd
- The four voting systems studied here can produce
different winners even when the same voter
preference table is used. - Any of the four methods can also produce a tie
between two or more candidates, which must be
broken somehow.
40Tie Breaking
- A tie-breaking method should be chosen before the
election. - To break a tie caused by perfectly balanced voter
support, election officials may - Make an arbitrary choice.
- Flipping a coin
- Drawing straws
- Bring in another voter.
- The Vice President votes when the U. S. Senate is
tied.
413.1 Initial Problem Solution
- A majority of city councilors said they preferred
site A to site B and also site A to site C. If B
won the election, did they necessarily lie? - Solution
- The councilors would not have to lie in order for
this to happen. This situation can occur with
some voting methods.
42Initial Problem Solution, contd
- For example, this situation could occur if the
voting method used was plurality with
elimination. - Suppose 11 councilors ranked the sites as shown
in the table below.
43Initial Problem Solution, contd
- Notice that in this scenario
- Site A is preferred to site B 7 to 4.
- Site A is preferred to site C 7 to 4.
- However, in the vote count
- Site A, with the fewest first-place votes, is
eliminated. - In the second round of voting, site B wins.
44Section 3.2Flaws of the Voting Systems
- Goals
- Study fairness criteria
- The majority criterion
- Head-to-head criterion
- Montonicity criterion
- Irrelevant alternatives criterion
- Study fairness of voting methods
- Arrow impossibility theorem
- Approval voting
453.2 Initial Problem
- The Compromise of 1850 averted civil war in the
U.S. for 10 years. - Henry Clay proposed the bill, but it was defeated
in July 1850. - A short time later, Stephen Douglas was able to
get essentially the same proposals passed. - How is this possible?
- The solution will be given at the end of the
section.
46Flaws of Voting Systems
- We have seen that the choice of voting method can
affect the outcome of an election. - Each voting method studied can fail to satisfy
certain criteria that make a voting method fair.
47Fairness Criteria
- The fairness criteria are properties that we
expect a good voting system to satisfy. - Four fairness criteria will be studied
- The majority criterion
- The head-to-head criterion
- The monotonicity criterion
- The irrelevant alternatives criterion
48The Majority Criterion
- If a candidate is the first choice of a majority
of voters, then that candidate should be
selected.
49Question
- Candidate A won an election with 3000 of the
8500 votes. Was the majority criterion
necessarily violated? - a. yes
- b. no
50The Majority Criterion, contd
- For the majority criterion to be violated
- A candidate must have more than half of the
votes. - This same candidate must not win the election.
- Note
- This criterion does not say what should happen if
no candidate receives a majority. - This criterion does not say that the winner of an
election must win by a majority.
51The Majority Criterion, contd
- If a candidate is the first choice of a majority
of voters, then that candidate will always win
using - The plurality method.
- The plurality with elimination method.
- The pairwise comparison method.
- In both of these methods any candidate with more
than half the vote will always win.
52The Majority Criterion, contd
- If a candidate is the first choice of a majority
of voters, then that candidate might not win
using - The Borda count method.
- The candidate with the most points may not be the
candidate with the most first-place votes.
53Example 1
- Four cities are being considered for an annual
trade show. The preferences of the organizers
are given in the table.
54Example 1, contd
- Which site has a majority of first-place votes?
- Which site wins using the Borda count method?
55Example 1, contd
- Solution There are 9 votes, so a majority would
be 5 or more votes. - The first place vote totals are
- Chicago 5 Seattle 3 Phoenix 1 Boston 0
- Chicago has a majority of first-place votes.
56Example 1, contd
- Solution, contd Find the point totals for Borda
count. - The points are calculated as follows
- Chicago 5(4) 0(3) 2(2) 2(1) 26
- Seattle 3(4) 4(3) 2(2) 0(1) 28
- Phoenix 1(4) 2(3) 2(2) 4(1) 18
- Boston 0(4) 3(3) 3(2) 3(1) 18
- Under the Borda count method, Seattle is the
winner.
57Example 1, contd
- Note that Chicago had a majority of first-place
votes, but under the Borda count method Seattle
was the winner. - This is an example of the Borda count method
failing the majority criterion. In this case, we
would say that the Borda count method was unfair.
58The Head-to-Head Criterion
- If a candidate is favored when compared
separately with each of the other candidates,
then the favored candidate should win the
election. - This is also called the Condorcet criterion.
59Head-to-Head Criterion, contd
- For the head-to-head criterion to be violated
- A candidate must be preferred pairwise to every
other candidate. - This same candidate must not win the election.
- Note
- This criterion does not say what should happen if
no candidate is preferred pairwise to every other
candidate.
60Head-to-Head Criterion, contd
- If a candidate is favored pairwise to every other
candidate, then that candidate will always win
using - The pairwise comparison method.
- This candidate will earn the most points from the
pairwise comparisons.
61Head-to-Head Criterion, contd
- If a candidate is favored pairwise to every other
candidate, then that candidate might not win
using - The plurality method.
- The plurality with elimination method.
- The Borda count method.
62Example 2
- Seven people are choosing an option for a
retirement party catering, picnic, or
restaurant. The preferences are shown in the
table below. - Which site is selected using the plurality
method? - Show that the head-to-head criterion is violated.
63Example 2, contd
- Solution
- The picnic has the most votes, 3, so it is the
winning option under the plurality method.
64Example 2, contd
- Solution, contd
- The pairwise comparisons are made
- R is preferred to P 4 to 3.
- R is preferred to C 5 to 2.
- R is preferred separately to every other
candidate, but R did not win the election. This
is a violation of the head-to-head criterion.
65The Monotonicity Criterion
- Suppose a particular candidate, X, wins an
election. - If, hypothetically, this election were redone and
the only changes were that some voters switched X
with the candidate they had ranked one higher,
then X should still win. - This criterion is only used in special cases.
66Monotonicity Criterion, contd
- The monotonicity criterion is always satisfied
by - The plurality method.
- The Borda count method.
- The pairwise comparison method.
67Monotonicity Criterion, contd
- The monotonicity criterion is not always
satisfied by - The plurality with elimination method.
68Example 3
- Teachers are voting for a union president from
the candidates Akst, Bailey, and Chung. The
preferences are shown in the table below.
69Example 3, contd
- Who will win using the plurality with elimination
method? - Solution The first-place vote totals are
- Akst 14 Bailey 12 Chung 15.
- No candidate received a majority of at least 21
votes, so Bailey is eliminated.
70Example 3, contd
- Solution, contd
- After Baileys elimination, a new preference
table is created. - Akst now has 26 first-place votes, and is the
winner.
71Example 3, contd
- If 4 of the 5 teachers who ranked the candidates
CAB changed to a ranking of ACB, would this
affect the outcome of the election? - Solution Akst won the first election and the
only changes now are that 4 teachers moved him
from 2nd to 1st place.
72Example 3, contd
- Solution, contd The new preference table is
shown below.
73Example 3, contd
- Solution, contd The first-place vote totals
are - Akst 18 Bailey 12 Chung 11.
- No candidate received a majority of at least 21
votes, so Chung is eliminated.
74Example 3, contd
- Solution, contd
- After Chungs elimination, a new preference table
is created. - Bailey now has 22 votes, and is the winner.
75Example 3, contd
- Solution, contd
- The only changes in the preference table were
ones that favored Akst, who won the first
election. - However, Akst ended up losing the modified vote
to Bailey. - This is a violation of the monotonicity criterion.
76The Irrelevant Alternatives Criterion
- Suppose a candidate, X, is selected in an
election. - If, hypothetically, this election were redone
with one or more of the unselected candidates
removed from the vote, then X should still win.
77Irrelevant Alternatives Criterion, contd
- The irrelevant alternatives criterion is not
always satisfied by any of the 4 voting methods
studied.
78Example 4
- The 5 members of a book club are voting on what
book to read next a mystery, a historical novel,
or a science fiction fantasy. The preference
table is shown below.
79Example 4, contd
- Which of the books is selected using the
plurality with elimination method? - Solution The first-place vote totals are
- M 2 H 1 S 2.
- No book has a majority, so H is eliminated.
80Example 4, contd
- Solution, contd After H is eliminated a new
preference table is created. - Book M receives 3 first-place votes, and is the
winner. The book club will read the mystery.
81Example 4, contd
- If the science fiction book is removed from the
list, is the irrelevant alternatives criterion
violated in the new election? - Solution A new preference table, without S, is
created.
82Example 4, contd
- Solution, contd
- In the new table, M has 2 votes and H has 3.
- Book H is the new winner, violating the
irrelevant alternatives criterion.
83Fairness Criteria, contd
84Arrow Impossibility Theorem
- The Arrow Impossibility Theorem states that no
system of voting will always satisfy all of the 4
fairness criteria. - This fact was proved by Kenneth Arrow in 1951.
85Question
- The results of an election using the plurality
method were analyzed. It was found that the
election did not violate any of the four fairness
criteria. Does this contradict the Arrow
Impossibility Theorem? - a. yes
- b. no
86Approval Voting
- No voting system is always fair, but we can
explore systems that are unfair less often than
others. One such system is called approval
voting. - In approval voting
- Each voter votes for all candidates he/she
considers acceptable. - The candidate with the most votes is selected.
87Example 5
- Three candidates are running for two positions.
There are 9 voters and the votes are shown in the
table below. - Who is the winner under approval voting?
88Example 5, contd
- Solution The vote totals are
- Ammee 6
- Bonnie 7
- Celeste 5
- Bonnie and Ammee are selected for the two
positions.
893.2 Initial Problem Solution
- Henry Clay presented the Compromise of 1850 as
one bill containing all the proposals. - Of the 60 senators, a majority would not approve
the bill because they disagreed on individual
issues within the bill.
90Initial Problem Solution, contd
- Stephen Douglas presented each proposal of the
Compromise in a separate bill. - A (different) majority of the senators passed
each proposal and the Compromise of 1850 went
into effect, although a majority never approved
the measures as a whole.
91Section 3.3Weighted Voting Systems
- Goals
- Study weighted voting systems
- Coalitions
- Dummies and dictators
- Veto power
- Study the Banzhaf power index
923.3 Initial Problem
- A stockholder owns 17 of the shares of a
company. - Among the other 3 stockholders, no one owns more
than 32 of the shares. - Why will no one listen to the stockholder with
17? - The solution will be given at the end of the
section.
93Weighted Voting Systems
- In a weighted voting system, an individual voter
may have more than one vote. - The number of votes that a voter controls is
called the weight of the voter. - An example of a weighted voting system is the
election of the U.S. President by the Electoral
College.
94Weighted Voting Systems, contd
- The weights of the voters are usually listed as a
sequence of numbers between square brackets. - For example, the voting system in which Angie has
a weight of 9, Roberta has a weight of 12, Carlos
has a weight of 8, and Darrell has a weight of 11
is represented as 12, 11, 9, 8.
95Weighted Voting Systems, contd
- The voter with the largest weight is called the
first voter, written P1. - The weight of the first voter is represented by
W1. - The remaining voters and their weights are
represented similarly, in order of decreasing
weights.
96Example 1
- The voting system in which Angie has a weight of
9, Roberta has a weight of 12, Carlos has a
weight of 8, and Darrell has a weight of 11 was
represented as 12, 11, 9, 8. - In this case, P1 Roberta, P2 Darrell, P3
Angie, and P4 Carlos. - Also, W1 12, W2 11, W3 9, and W4 8.
97Weighted Voting Systems, contd
- Yes or no questions are commonly called motions.
- A final decision of No defeats the motion and
leaves the status quo unchanged. - A final decision of Yes passes the motion and
changes the status quo.
98Weighted Voting Systems, contd
- A simple majority requirement means that a motion
must receive more than half of the votes to pass. - A supermajority requirement means that the
minimum number of votes required to pass a motion
is set higher than half of the total weight. - A common supermajority is two-thirds of the total
weight.
99Weighted Voting Systems, contd
- The weight required to pass a motion is called
the quota. - Example A simple majority quota for the weighted
voting system 12, 11, 9, 8 would be 21. - Half of the total weight is (12 11 9 8)/2
40/2 20. More than half of the weight would be
at least 21 Yes votes.
100Question
- Given the weighted voting system 10, 9, 8, 8,
5, find the quota for a supermajority
requirement of two-thirds of the total weight. - a. 27
- b. 21
- c. 26
- d. 20
101Weighted Voting Systems, contd
- The quota for a weighted voting system is usually
added to the list of weights. - Example For the weighted voting system 12, 11,
9, 8 with a quota of 21 the complete notation is
21 12, 11, 9, 8.
102Example 2
- Given the weighted voting system
- 21 10, 8, 7, 7, 4, 4, suppose P1, P3, and P5
vote Yes on a motion. - Is the motion passed or defeated?
103Example 2, contd
- Solution
- The given voters have a combined weight of 10 7
4 21. - The quota is met, so the motion passes.
104Example 3
- Given the weighted voting system
- 21 10, 8, 7, 7, 4, 4, suppose P1, P5, and P6
vote Yes on a motion. - Is the motion passed or defeated?
105Example 3, contd
- Solution
- The given voters have a combined weight of 10 4
4 18. - The quota is not met, so the motion is defeated.
106Coalitions
- Any nonempty subset of the voters in a weighted
voting system is called a coalition. - If the total weight of the voters in a coalition
is greater than or equal to the quota, it is
called a winning coalition. - If the total weight of the voters in a coalition
is less than the quota, it is called a losing
coalition.
107Question
- Given the weighted voting system 27 10, 9, 8,
8, 5, is the coalition P1, P4, P5 a winning
coalition or a losing coalition? - a. winning
- b. losing
108Example 4
- For the weighted voting system
- 8 6, 5, 4, list all possible coalitions and
determine whether each is a winning or losing
coalition.
109Example 4, contd
- Solution Each coalition and its status is listed
in the table below.
110Coalitions, contd
- In a weighted voting system with n voters,
exactly 2n - 1 coalitions are possible. - Example
- How many coalitions are possible in a weighted
voting system with 7 voters? - Solution The formula tells us there are
- 27 - 1 128 1 127 coalitions.
111Example 5
- The voting weights of EU members in a council in
2003 are shown in the table.
112Example 5, contd
- If resolutions must receive 71 of the votes to
pass, what is the quota? - How many coalitions are possible?
113Example 5, contd
- Solution
- There are 87 votes total. So the quota is 71 of
87, or approximately 62 votes. - There are n 15 members, so there are 215 1
32,767 coalitions possible.
114Dictators and Dummies
- A voter whose presence or absence in any
coalition makes no difference in the outcome is
called a dummy. - A voter whose presence or absence in any
coalition completely determines the outcome is
called a dictator. - When a weighted voting system has a dictator, the
other voters in the system are automatically
dummies.
115Veto Power
- In between the complete power of a dictator and
the zero power of a dummy is a level of power
called veto power. - A voter with veto power can defeat a motion by
voting No but cannot necessarily pass a motion
by voting Yes. - Any dictator has veto power, but a voter with
veto power is not necessarily a dictator.
116Example 6
- Consider the weighted voting system
- 12 7, 6, 4.
- List all the coalitions and determine whether
each is a winning or losing coalition. - Are there any dummies or dictators?
- Are there any voters with veto power?
117Example 6, contd
- Solution
- Each coalition and its status is listed in the
table below.
118Example 6, contd
- Solution, contd
- Removing the third voter from any coalition does
not change the status of the coalition. P3 is a
dummy.
119Example 6, contd
- Solution, contd
- No voter has complete power to pass or defeat a
motion. There is no dictator.
120Example 6, contd
- Solution, contd
- If P1 is not in a coalition, then it is a losing
coalition. P1 has veto power.
121Question
- In the weighted voting system
- 27 10, 9, 8, 8, 5, is P1 a
- a. dictator
- b. dummy
- c. voter with veto power
- d. none of the above
122Example 7
- Consider the weighted voting system 10 10, 5,
4. - Are there any dummies, dictators, or voters with
veto power?
123Example 7, contd
- Solution
- P1 has enough weight to pass a motion by voting
Yes no matter how anyone else votes. - If P1 votes No, the motion will not pass no
matter how anyone else votes. - P1 is a dictator and thus all other voters are
dummies.
124Critical Voters
- If a voters weight is large enough so that the
voter can change a particular winning coalition
to a losing coalition by leaving the coalition,
then that voter is called a critical voter in
that winning coalition.
125Question
- Given the weighted voting system 27 10, 9, 8,
8, 5, is the voter P4 a critical voter in the
winning coalition P1, P2, P4, P5? - a. yes
- b. no
126Example 8
- Consider the weighted voting system 21 10, 8,
7, 7, 4, 4. - Which voters in the coalition P2, P3, P4, P5
are critical voters in that coalition?
127Example 8, contd
- Solution The weight in the winning coalition is
26. - If P2 leaves, the weight goes down to
- 26 8 18 lt quota.
- If P3 leaves, the weight goes down to
- 26 7 19 lt quota.
128Example 8, contd
- Solution contd
- If P4 leaves, the weight goes down to
- 26 7 19 lt quota.
- If P5 leaves, the weight goes down to
- 26 4 22 gt quota.
- The critical voters in this coalition are P2, P3,
and P4.
129The Banzhaf Power Index
- The more times a voter is a critical voter in a
coalition, the more power that voter has in the
system. - The Banzhaf power of a voter is the number of
winning coalitions in which that voter is
critical.
130Banzhaf Power Index, contd
- The sum of the Banzhaf powers of all voters is
called the total Banzhaf power in the weighted
voting system. - An individual voters Banzhaf power index is the
ratio of the voters Banzhaf power to the total
Banzhaf power in the system. - The sum of the Banzhaf power indices of all
voters is 100.
131Banzhaf Power Index, contd
- An individual voters Banzhaf power index is
calculated using the following process - Find all winning coalitions for the system.
- Determine the critical voters for each winning
coalition. - Calculate each voters Banzhaf power.
- Find the total Banzhaf power in the system.
- Divide each voters Banzhaf power by the total
Banzhaf power.
132Example 9
- For the weighted voting system 18 12, 7,
6, 5, determine - The total Banzhaf power in the system.
- The Banzhaf power index of each voter.
133Example 9, contd
- Solution Step 1 Find all the winning coalitions.
134Example 9, contd
- Solution Step 2 Determine the critical voters
for each winning coalition. - Remove each voter one at a time and check to see
whether the resulting coalition is still a
winning coalition. - This work is shown in the next slides.
135Example 9, contd
136Example 9, contd
137Example 9, contd
- Solution Step 3 Count the number of times each
voter is a critical voter - P1 5 times
- P2 3 times
- P3 3 times
- P4 1 time
- Step 4 The total Banzhaf power in the system is
5 3 3 1 12
138Example 9, contd
- Solution Step 5 Divide each voters Banzhaf
power by the total Banzhaf power to find the
Banzhaf power indices.
1393.3 Initial Problem Solution
- One of the 4 stockholders owns 17 of the shares
of a company. Among the other 3, no one owns
more than 32. Why will no one listen to the
stockholder with 17?
140Initial Problem Solution, contd
- We know one stockholder owns 32 and one owns
17, so the other 51 is split between the
remaining two stockholders. - Suppose the other percents are 26 and 25.
141Initial Problem Solution, contd
- The winning coalitions in this case are
- 32, 26, 25, 17
- 32, 26, 25
- 32, 26, 17
- 32, 25, 17
- 26, 25, 17
- 32, 26
- 32, 25
- 26, 25
142Initial Problem Solution, contd
- The voter with 17 is in several winning
coalitions, but removing that voter does not
cause any of them to become losing coalitions. - The voter with 17 is not a critical voter in any
winning coalition. - The reason no one will listen to the voter is
that he or she is a dummy.