51
Voting Theory 57
22. Show that when there is a Condorcet winner in an election, it is impossible for a
single voter to manipulate the vote to help a different candidate become a Condorcet
winner.
23. The Pareto criterion is another fairness criterion that states: If every voter prefers
choice A to choice B, then B should not be the winner. Explain why plurality, instant
runoff, Borda count, and Copeland’s method all satisfy the Pareto condition.
24. Sequential Pairwise voting is a method not commonly used for political elections, but
sometimes used for shopping and games of pool. In this method, the choices are
assigned an order of comparison, called an agenda. The first two choices are
compared. The winner is then compared to the next choice on the agenda, and this
continues until all choices have been compared against the winner of the previous
comparison.
a. Using the preference schedule below, apply Sequential Pairwise voting to
determine the winner, using the agenda: A, B, C, D.
Number of voters 10 15 12
12
1st choice
C A B
B
2nd choice
A B D
D
3rd choice
B D C
C
4th choice
D C A
A
b. Show that Sequential Pairwise voting can violate the Pareto criterion.
c. Show that Sequential Pairwise voting can violate the Majority criterion.
25. The Coombs method is a variation of instant runoff voting. In Coombs method, the
choice with the most last place votes is eliminated. Apply Coombs method to the
preference schedules from questions 5 and 6.
26. Copeland’s Method is designed to identify a Condorcet Candidate if there is one, and
is considered a Condorcet Method. There are many Condorcet Methods, which vary
primarily in how they deal with ties, which are very common when a Condorcet
winner does not exist. Copeland’s method does not have a tie-breaking procedure
built-in. Research the Schulze method, another Condorcet method that is used by the
Wikimedia foundation that runs Wikipedia, and give some examples of how it works.
27. The plurality method is used in most U.S. elections. Some people feel that Ross Perot
in 1992 and Ralph Nader in 2000 changed what the outcome of the election would
have been if they had not run. Research the outcomes of these elections and explain
how each candidate could have affected the outcome of the elections (for the 2000
election, you may wish to focus on the count in Florida). Describe how an alternative
voting method could have avoided this issue.
28. Instant Runoff Voting and Approval voting have supporters advocating that they be
adopted in the United States and elsewhere to decide elections. Research
comparisons between the two methods describing the advantages and disadvantages
of each in practice. Summarize the comparisons, and form your own opinion about
whether either method should be adopted.
21
58
29. In a primary system, a first vote is held with multiple candidates. In some states, each
political party has its own primary. In Washington State, there is a "top two"
primary, where all candidates are on the ballot and the top two candidates advance to
the general election, regardless of party. Compare and contrast the top two primary
with general election system to instant runoff voting, considering both differences in
the methods, and practical differences like cost, campaigning, fairness, etc.
30. In a primary system, a first vote is held with multiple candidates. In some many
states, where voters must declare a party to vote in the primary election, and they are
only able to choose between candidates for their declared party. The top candidate
from each party then advances to the general election. Compare and contrast this
primary with general election system to instant runoff voting, considering both
differences in the methods, and practical differences like cost, campaigning, fairness,
etc.
31. Sometimes in a voting scenario it is desirable to rank the candidates, either to
establish preference order between a set of choices, or because the election requires
multiple winners. For example, a hiring committee may have 30 candidates apply,
and need to select 6 to interview, so the voting by the committee would need to
produce the top 6 candidates. Describe how Plurality, Instant Runoff Voting, Borda
Count, and Copeland’s Method could be extended to produce a ranked list of
candidates.
52
Weighted Voting 59
© David Lippman
Creative Commons BY-SA
Weighted Voting
In a corporate shareholders meeting, each shareholders’ vote counts proportional to the
amount of shares they own. An individual with one share gets the equivalent of one vote,
while someone with 100 shares gets the equivalent of 100 votes. This is called weighted
voting, where each vote has some weight attached to it. Weighted voting is sometimes used
to vote on candidates, but more commonly to decide “yes” or “no” on a proposal, sometimes
called a motion. Weighted voting is applicable in corporate settings, as well as decision
making in parliamentary governments and voting in the United Nations Security Council.
In weighted voting, we are most often interested in the power each voter has in influencing
the outcome.
Beginnings
We’ll begin with some basic vocabulary for weighted voting systems.
Vocabulary for Weighted Voting
Each individual or entity casting a vote is called a player in the election. They’re often
notated as P
1
, P
2
, P
3
, … ,P
N
, where N is the total number of voters.
Each player is given a weight, which usually represents how many votes they get.
The quota is the minimum weight needed for the votes or weight needed for the
proposal to be approved.
A weighted voting system will often be represented in a shorthand form:
[q: w
1
, w
2
, w
3
, … , w
n
]
In this form, q is the quota, w
1
is the weight for player 1, and so on.
Example 1
In a small company, there are 4 shareholders. Mr. Smith has a 30% ownership stake in the
company, Mr. Garcia has a 25% stake, Mrs. Hughes has a 25% stake, and Mrs. Lee has a
20% stake. They are trying to decide whether to open a new location. The company by-
laws state that more than 50% of the ownership has to approve any decision like this. This
could be represented by the weighted voting system:
[51: 30, 25, 25, 20]
Here we have treated the percentage ownership as votes, so Mr. Smith gets the equivalent of
30 votes, having a 30% ownership stake. Since more than 50% is required to approve the
decision, the quota is 51, the smallest whole number over 50.
35
60
In order to have a meaningful weighted voting system, it is necessary to put some limits on
the quota.
Limits on the Quota
The quota must be more than ½ the total number of votes.
The quota can’t be larger than the total number of votes.
Why? Consider the voting system [q: 3, 2, 1]
Here there are 6 total votes. If the quota was set at only 3, then player 1 could vote yes,
players 2 and 3 could vote no, and both would reach quota, which doesn’t lead to a decision
being made. In order for only one decision to reach quota at a time, the quota must be at
least half the total number of votes. If the quota was set to 7, then no group of voters could
ever reach quota, and no decision can be made, so it doesn’t make sense for the quota to be
larger than the total number of voters.
Try it Now 1
In a committee there are four representatives from the management and three representatives
from the workers’ union. For a proposal to pass, four of the members must support it,
including at least one member of the union. Find a voting system that can represent this
situation.
A Look at Power
Consider the voting system [10: 11, 3, 2]. Notice that in this system, player 1 can reach
quota without the support of any other player. When this happens, we say that player 1 is a
dictator.
Dictator
A player will be a dictator if their weight is equal to or greater than the quota. The
dictator can also block any proposal from passing; the other players cannot reach quota
without the dictator.
In the voting system [8: 6, 3, 2], no player is a dictator. However, in this system, the quota
can only be reached if player 1 is in support of the proposal; player 2 and 3 cannot reach
quota without player 1’s support. In this case, player 1 is said to have veto power. Notice
that player 1 is not a dictator, since player 1 would still need player 2 or 3’s support to reach
quota.
Veto Power
A player has veto power if their support is necessary for the quota to be reached. It is
possible for more than one player to have veto power, or for no player to have veto
power.
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62
Weighted Voting 61
With the system [10: 7, 6, 2], player 3 is said to be a dummy, meaning they have no
influence in the outcome. The only way the quota can be met is with the support of both
players 1 and 2 (both of which would have veto power here); the vote of player 3 cannot
affect the outcome.
Dummy
A player is a dummy if their vote is never essential for a group to reach quota.
Example 2
In the voting system [16: 7, 6, 3, 3, 2], are any players dictators? Do any have veto power?
Are any dummies?
No player can reach quota alone, so there are no dictators.
Without player 1, the rest of the players’ weights add to 14, which doesn’t reach quota, so
player 1 has veto power. Likewise, without player 2, the rest of the players’ weights add to
15, which doesn’t reach quota, so player 2 also has veto power.
Since player 1 and 2 can reach quota with either player 3 or player 4’s support, neither player
3 or player 4 have veto power. However they cannot reach quota with player 5’s support
alone, so player 5 has no influence on the outcome and is a dummy.
Try it Now 2
In the voting system [q: 10, 5, 3], which players are dictators, have veto power, and are
dummies if the quota is 10? 12? 16?
To better define power, we need to introduce the idea of a coalition. A coalition is a group
of players voting the same way. In the example above, {P
1
, P
2
, P
4
} would represent the
coalition of players 1, 2 and 4. This coalition has a combined weight of 7+6+3 = 16, which
meets quota, so this would be a winning coalition.
A player is said to be critical in a coalition if them leaving the coalition would change it
from a winning coalition to a losing coalition. In the coalition {P
1
, P
2
, P
4
}, every player is
critical. In the coalition {P
3
, P
4
, P
5
}, no player is critical, since it wasn’t a winning coalition
to begin with. In the coalition {P
1
, P
2
, P
3
, P
4
, P
5
}, only players 1 and 2 are critical; any other
player could leave the coalition and it would still meet quota.
Coalitions and Critical Players
A coalition is any group of players voting the same way.
A coalition is a winning coalition if the coalition has enough weight to meet quota.
A player is critical in a coalition if them leaving the coalition would change it from a
winning coalition to a losing coalition.
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49
62
Example 3
In the Scottish Parliament in 2009 there were 5 political parties: 47 representatives for the
Scottish National Party, 46 for the Labour Party, 17 for the Conservative Party, 16 for the
Liberal Democrats, and 2 for the Scottish Green Party. Typically all representatives from a
party vote as a block, so the parliament can be treated like the weighted voting system:
[65: 47, 46, 17, 16, 2]
Consider the coalition {P
1
, P
3
, P
4
}. No two players alone could meet the quota, so all three
players are critical in this coalition.
In the coalition {P
1
, P
3
, P
4
, P
5
}, any player except P
1
could leave the coalition and it would
still meet quota, so only P
1
is critical in this coalition.
Notice that a player with veto power will be critical in every winning coalition, since
removing their support would prevent a proposal from passing.
Likewise, a dummy will never be critical, since their support will never change a losing
coalition to a winning one.
Dictators, Veto, and Dummies and Critical Players
A player is a dictator if the single-player coalition containing them is a winning
coalition.
A player has veto power if they are critical in every winning coalition.
A player is a dummy if they are not critical in any winning coalition.
Calculating Power: Banzhaf Power Index
The Banzhaf power index was originally created in 1946 by Lionel Penrose, but was
reintroduced by John Banzhaf in 1965. The power index is a numerical way of looking at
power in a weighted voting situation.
Calculating Banzhaf Power Index
To calculate the Banzhaf power index:
1. List all winning coalitions
2. In each coalition, identify the players who are critical
3. Count up how many times each player is critical
4. Convert these counts to fractions or decimals by dividing by the total times any
player is critical
151
Weighted Voting 63
Example 4
Find the Banzhaf power index for the voting system [8: 6, 3, 2].
We start by listing all winning coalitions. If you aren’t sure how to do this, you can list all
coalitions, then eliminate the non-winning coalitions. No player is a dictator, so we’ll only
consider two and three player coalitions.
{P
1
, P
2
}
Total weight: 9. Meets quota.
{P
1
, P
3
}
Total weight: 8. Meets quota.
{P
2
, P
3
}
Total weight: 5. Does not meet quota.
{P
1
, P
2
, P
3
} Total weight: 11. Meets quota.
Next we determine which players are critical in each winning coalition. In the winning two-
player coalitions, both players are critical since no player can meet quota alone. Underlining
the critical players to make it easier to count:
{P
1
, P
2
}
{P
1
, P
3
}
In the three-person coalition, either P
2
or P
3
could leave the coalition and the remaining
players could still meet quota, so neither is critical. If P
1
were to leave, the remaining players
could not reach quota, so P
1
is critical.
{P
1
, P
2
, P
3
}
Altogether, P
1
is critical 3 times, P
2
is critical 1 time, and P
3
is critical 1 time.
Converting to percents:
P
1
= 3/5 = 60%
P
2
= 1/5 = 20%
P
3
= 1/5 = 20%
Example 5
Consider the voting system [16: 7, 6, 3, 3, 2]. Find the Banzhaf power index.
The winning coalitions are listed below, with the critical players underlined.
{P
1
, P
2
, P
3
}
{P
1
, P
2
, P
4
}
{P
1
, P
2
, P
3
, P
4
}
{P
1
, P
2
, P
3
, P
5
}
{P
1
, P
2
, P
4
, P
5
}
{P
1
, P
2
, P
3
, P
4
, P
5
}
Counting up times that each player is critical:
P
1
= 6
P
2
= 6
P
3
= 2
P
4
= 2
P
5
= 0
Total of all: 16
176
64
Divide each player’s count by 16 to convert to fractions or percents:
P
1
= 6/16 = 3/8 = 37.5%
P
2
= 6/16 = 3/8 = 37.5%
P
3
= 2/16 = 1/8 = 12.5%
P
4
= 2/16 = 1/8 = 12.5%
P
5
= 0/16 = 0 = 0%
The Banzhaf power index measures a player’s ability to influence the outcome of the vote.
Notice that player 5 has a power index of 0, indicating that there is no coalition in which they
would be critical power and could influence the outcome. This means player 5 is a dummy,
as we noted earlier.
Example 6
Revisiting the Scottish Parliament, with voting system [65: 47, 46, 17, 16, 2], the winning
coalitions are listed, with the critical players underlined.
{P
1
, P
2
}
{P
1
, P
2
, P
3
}
{P
1
, P
2
, P
4
}
{P
1
, P
2
, P
5
}
{P
1
, P
3
, P
4
}
{P
1
, P
3
, P
5
}
{P
1
, P
4
, P
5
}
{P
2
, P
3
, P
4
}
{P
2
, P
3
, P
5
}
{P
1
, P
2
, P
3,
P
4
}
{P
1
, P
2
, P
3,
P
5
}
{P
1
, P
2
, P
4,
P
5
}
{P
1
, P
3
, P
4
, P
5
}
{P
2
, P
3
, P
4
, P
5
}
{P
1
, P
2
, P
3
, P
4
, P
5
}
Counting up times that each player is critical:
Interestingly, even though the Liberal Democrats party has only one less representative than
the Conservative Party, and 14 more than the Scottish Green Party, their Banzhaf power
index is the same as the Scottish Green Party’s. In parliamentary governments, forming
coalitions is an essential part of getting results, and a party’s ability to help a coalition reach
quota defines its influence.
Try it Now 3
Find the Banzhaf power index for the weighted voting system [36: 20, 17, 16, 3].
District
Times critical Power index
P
1
(Scottish National Party)
9
9/27 = 33.3%
P
2
(Labour Party)
7
7/27 = 25.9%
P
3
(Conservative Party)
5
5/27 = 18.5%
P
4
(Liberal Democrats Party)
3
3/27 = 11.1%
P
5
(Scottish Green Party)
3
3/27 = 11.1%
81
Weighted Voting 65
Example 7
Banzhaf used this index to argue that the weighted voting system used in the Nassau County
Board of Supervisors in New York was unfair. The county was divided up into 6 districts,
each getting voting weight proportional to the population in the district, as shown below.
Calculate the power index for each district.
Translated into a weighted voting system, assuming a simple majority is needed for a
proposal to pass:
[58: 31, 31, 28, 21, 2, 2]
Listing the winning coalitions and marking critical players:
There are a lot of them! Counting up how many times each player is critical,
It turns out that the three smaller districts are dummies. Any winning coalition requires two
of the larger districts.
District
Times critical Power index
Hempstead #1
16
16/48 = 1/3 = 33%
Hempstead #2
16
16/48 = 1/3 = 33%
Oyster Bay
16
16/48 = 1/3 = 33%
North Hempstead 0
0/48 = 0%
Long Beach
0
0/48 = 0%
Glen Cove
0
0/48 = 0%
{H1, H2}
}
{H1, OB}
{H2, OB}
{H1, H2, NH}
{H1, H2, LB}
{H1, H2, GC}
{H1, H2, NH, LB}
{H1, H2, NH, GC}
{H1, H2, LB, GC}
{H1, H2, NH, LB. GC}
{H1, OB, NH}
H}
{H1, OB, LB}
{H1, OB, GC}
{H1, OB, NH, LB}
{H1, OB, NH, GC}
{H1, OB, LB, GC}
{H1, OB, NH, LB. GC}
{H2, OB, NH}
{H2, OB, LB}
{H2, OB, GC}
{H2, OB, NH, LB}
B}
{H2, OB, NH, GC}
{H2, OB, LB, GC}
{H2, OB, NH, LB, GC}
{H1, H2, OB}
{H1, H2, OB, NH}
{H1, H2, OB, LB}
{H1, H2, OB, GC}
{H1, H2, OB, NH, LB}
{H1, H2, OB, NH, GC}
{H1, H2, OB, NH, LB, GC}
C}
District
Weight
Hempstead #1
31
Hempstead #2
31
Oyster Bay
28
North Hempstead 21
1
Long Beach
2
Glen Cove
2
95
66
The weighted voting system that Americans are most familiar with is the Electoral College
system used to elect the President. In the Electoral College, states are given a number of
votes equal to the number of their congressional representatives (house + senate). Most
states give all their electoral votes to the candidate that wins a majority in their state, turning
the Electoral College into a weighted voting system, in which the states are the players. As
I’m sure you can imagine, there are billions of possible winning coalitions, so the power
index for the Electoral College has to be computed by a computer using approximation
techniques.
Calculating Power: Shapley-Shubik Power Index
The Shapley-Shubik power index was introduced in 1954 by economists Lloyd Shapley and
Martin Shubik, and provides a different approach for calculating power.
In situations like political alliances, the order in which players join an alliance could be
considered the most important consideration. In particular, if a proposal is introduced, the
player that joins the coalition and allows it to reach quota might be considered the most
essential. The Shapley-Shubik power index counts how likely a player is to be pivotal.
What does it mean for a player to be pivotal?
First, we need to change our approach to coalitions. Previously, the coalition {P
1
, P
2
} and
{P
2
, P
1
} would be considered equivalent, since they contain the same players. We now need
to consider the order in which players join the coalition. For that, we will consider
sequential coalitions – coalitions that contain all the players in which the order players are
listed reflect the order they joined the coalition. For example, the sequential coalition
<P
2
, P
1
, P
3
> would mean that P
2
joined the coalition first, then P
1
, and finally P
3
. The angle
brackets < > are used instead of curly brackets to distinguish sequential coalitions.
Pivotal Player
A sequential coalition lists the players in the order in which they joined the coalition.
A pivotal player is the player in a sequential coalition that changes a coalition from a
losing coalition to a winning one. Notice there can only be one pivotal player in any
sequential coalition.
Example 8
In the weighted voting system [8: 6, 4, 3, 2], which player is pivotal in the sequential
coalition <P
3
, P
2
, P
4
, P
1
>?
The sequential coalition shows the order in which players joined the coalition. Consider the
running totals as each player joins:
P
3
Total weight: 3
Not winning
P
3
, P
2
Total weight: 3+4 = 7
Not winning
P
3
, P
2
, P
4
Total weight: 3+4+2 = 9
Winning
P
3
, P
2
, P
4
, P
1
Total weight: 3+4+2+6 = 15
Winning
Since the coalition becomes winning when P
4
joins, P
4
is the pivotal player in this coalition.
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