Chapter 10: The
Manipulability of Voting Systems
Chapter 11: Weighted Voting
Systems
Presented by: Katherine Goulde
Chapter 10 Outline





Introduction and example
Majority Rule and Condorcet’s Method
Voting Systems for 3 or more candidates
 Borda Count
 Sequential Pairwise Voting
 Plurality Voting
Impossibility- The Gibbard- Satterthwaite Theorem
The Chair’s Paradox
Manipulability & the Borda Count

The Borda count assigns point values to the candidates
and the winner is the candidate with the most points

Voter 1
A
B
C
D
Voter 2
B
C
A
D
Candidate A has a score of 4
Candidate B has a score of 5
Candidate C has a score of 3
Candidate D has a score of 0
Therefore… B wins!
What if Voter 1 wants to manipulate the election???
Voter 1
A
B
C
D
Voter 2
B
C
A
D
Voter 1
A
D
C
B
Voter 2
B
C
A
D

Original Ballot where B wins
the election

However, Voter 1 wants A to
win. How can Voter 1 ensure
that A wins?

In this second ballot, A has a
Borda count of 4, B has 3, C
has 3, and D has 2.
Therefore A is the winner.
Is there any other way to
obtain this result?

Unilateral Change- A change (in ballot) by a voter while every other
voter keeps his or her ballot exactly as it was
- “single-voter manipulation”
A voting system is manipulable if
• there are two sequences of preference list ballots and a Voter so that
•Neither election results in a tie
•The only ballot change is by the Voter
•The Voter prefers the outcome of the second election to that
of the first election.
•Take the two-candidate case with majority rule, and recall that it is
monotone
•In this instance, nonmanipulability is the same thing as
monotonicity
Majority Rule and Condorcet’s Method

May’s Theorem for Manipulability:
 Among
all two-candidate voting systems that never
result in a tie, majority rule is the only one that treats
all voters equally, treats both candidates equally, and
is nonmanipulable

The Nonmanipulability of Condorcet’s Method:
 Condorcet’s
method is nonmanipulable in the sense
that a voter can never unilaterally change an election
result from one candidate to another candidate that
her or she prefers
Cordorcet’s Voting Paradox and
Manipulability
Election 1
Voter 1
A
C
B
Voter 2
B
C
A
Election 2
Voter 3
C
A
B
In this example, C is the
Condorcet winner
Voter 1
A
B
C
Voter 2
B
C
A
Voter 3
C
A
B
In this example, there is no
Condorcet winner at all
Back to the Borda Count

The Nonmanipulability of the Borda Count with exactly 3
candidates:



With exactly 3 candidates, the Borda count cannot be
manipulated in the sense of a voter unilaterally changing an
election outcome from one single winner to another single
winner that he prefers
Why?
Imagine B is the Borda winner, but you prefer A. Consider 3
cases:



A>B>C
C>A>B
A>C>B

The Manipulability of the Borda Count with Four or More
Candidates:

With four or more candidates and two or more voters, the
Borda count can be manipulated in the sense that there exists
an election in which a voter can unilaterally change the election
outcome from one single winner to another single winner that he
prefers

We’ve covered the example of 4 candidates and 2 voters.


1) Any candidates in addition to the 4 can be placed below those on
every ballot
2) The rest of the voters can be paired off with the members of each
pair holding ballots that rank the candidates in exactly opposite
orders
Sequential Pairwise Voting
Voter 1
A
B
D
C




Voter 2
C
A
B
D
Voter 3
B
D
C
A
Assume we are able to set the order.
Choose the ‘winner’, and place the candidate last
Look for the others that would beat that candidate one
on one.
Using this, we can arrange for any of the candidates to
be the winner.
Plurality Voting and Group
Manipulability

Plurality voting cannot be manipulated by a single
individual. However, it is group manipulable in the
sense that there are elections in which a group of voters
can change their ballots so that the new winner is
preferred to the old winner by everyone in the group

Real-world election: third party candidate acts as a
‘spoiler’
Impossibility: the G-S Theorem

Cordorcet’s theorm:








1) Elections never result in ties
2) Satisfies the Pareto condition
3) Is nonmanipulable
4) Isn’t a dictatorship
Can we extend this so that there is always a winner??
The Gibbard- Satterthwaite Theorem: With three or
more candidates and any number of voters, there
doesn’t exist a voting system that always produces a
winner, never has ties, satisfies the Pareto condition, is
nonmanipulable, and is not a dictatorship.
for proof click here
Weaker extension: Any voting system for 3 candidates
that agrees with Condorcet’s Method whenever there is
a winner is manipulable.
The Chair’s Paradox

The fact that with three voters and three candidates, the voter
with tie-breaking power (the ‘chair’) can, if all 3 voters act
rationally in their own self-interest, end up with her or his leastpreferred candidate as the election winner
Chair
A
B
C


You
B
C
A
Me
C
A
B
Each voter gets to vote for one of the candidates. If a candidate gets
2 or more, he or she wins. If each candidate receives one vote, then
whichever person the chair voted for wins.
Each voter will choose the best strategy given what the others might
do.
The Chair’s Paradox
Chair
A
B
C

You
B
C
A
Me
C
A
B
The chair will vote for A. ‘Me’ will vote for
C. ‘You’ will also vote for C.
Chapter 11 Outline


Introduction and definitions
The Shapley- Shubik Power Index
3

voters, 4 voters, a committee
The Banzhaf Power Index
 Critical

voters, winning & blocking, combinations
Comparing Voting Systems
3
voters, using minimal winning coalitions
Introduction and Definitions

Weighted voting system: a voting system in which
each participant is assigned a voting weight . A quota is
specified, and if the sum of the voting weights of the
voters supporting a motion is at least = the quota, the
motion is approved

Weight: the number of votes assigned to a voter

Quota: the minimum number of votes necessary to pass
a measure in a weighted voting system

Notation for Weighted voting systems

[q: W1, W2, …, Wn] where there are n voters, q quota, and
voting weights W1, W2, …, Wn
Introduction and Definitions

Dictator: a participant who can pass or block any issue
even if all other voters oppose it


Dummy Voter: a participant who has no power, is never
critical, and is never the pivotal voter


[8: 5, 3, 1]
Veto power: had by a voter if no issue can pass without
his vote. (a voter with veto power is a one-person
blocking coalition)


[10: 7, 13]
[6: 5, 3, 1] or [8: 5, 3, 1]
Power index: a numerical measure of an individual
voter’s ability to influence a decision; the individual’s
voting power
The Shapley-Shubik Power Index


1954- Lloyd Shapley and Martin Shubik
This index is defined in terms of permutations (a
permutation of voters in an ordering of all of the voters
in a voting system)



1) Voters are ordered in accordance with their commitment to an
issue (from most favorable to those most against)
2)The first voter in a permutation who, when joined by those
coming before her, would have enough voting weight to win is
the pivotal voter in that particular permutation.
Examples: animal rights, environmentalism
The Shapley-Shubik Power Index

This power index is computed by
 1)
counting the number of permutations in which that
voter is pivotal
 2) divide this number by the total number of possible
permutations

If there are n voters, the total number of
possible permutations is n!

Example: [6: 5, 3, 1].
 Result:
A= 4/6, B,C = 1/6
How to compute the S-S Power Index



If all the voters have the same voting weight, then each
has the same share of power.
If all but one of two voters have equal power, we can still
easily calculate the S-S power index
Example: 7-person committee with the voting system [5:
3, 1, 1, 1, 1, 1, 1]







CMMMMMM
MCMMMMM
MMCMMMM
MMMCMMM
MMMMCMM
MMMMMCM
MMMMMMC
How to compute the S-S Power Index

The chair is the pivotal voter 3 of the 7
times, so his S-S power index is 3/7.

The remaining 4/7 is split among the six
other voters (since all have the same
weight), so each has (4/7)/6 = 2/21 as their
S-S power index.
The Banzhaf Power Index

Based on the count of coalitions in which a voter is
critical

Coalition: a set of voters who are prepared to vote for,
or to oppose, a motion.


Winning coalition: favors the motion & has enough votes to
pass it
Blocking coalition: opposes the motion & has enough votes to
block it

Losing coalition: set of voters that does not have the
votes to have its way

Critical voter: a member of the winning (or blocking)
coalition whose vote is essential for the coalition to win
(or block) a measure
The Banzhaf Power Index

To determine the B. Power index of voter A, count all the
possible winning and blocking coalitions of which A is a
member and casts a critical vote



The weight of a winning coalition must be great than or equal to
q (where q is the quota)
The weight of a blocking coalition must be big enough to block
the ‘yes’ voters the q votes they need to win. So it must be at
least n-q+1 (where n is number of voters)
Extra Votes Principle:

A winning coalition with total weight w has w-q ‘extra votes’. A
blocking coalition with weight w has w-(n-q +1) extra votes. The
critical voters are those whose weight is more than the coalition’s
extra votes. These are the voters the coalition can’t afford to
lose.
Calculating the Banzhaf Index
Take the voting
system [3:2,1,1]
 Winning coalitionhave a weight of
3 or 4

Win.
Coalit.
Weight
Extra
votes
A
(c.v)
B
(c.v.)
C
(c.v)
{A,B}
3
0
1
1
0
{A,C}
3
0
1
0
1
{A,B,C}
4
1
1
0
0
Totals
3
1
1
A has 3 critical votes,
B and C both have1
Calculating the Banzhaf Index
Block.
Weight
Extra
A (c.v)
B (c.v)
C (c.v)
{A}
2
0
1
0
0
{B,C}
2
0
0
1
1
{A,B}
3
1
1
0
0
{A,C}
3
1
1
0
0
{A,B,C}
4
2
0
0
0
Totals: 3
1
1
Calculating the Banzhaf Index
In the blocking coalitions, A is critical in 3
and B and C are both critical in 1 each
 So, taking the blocking coalitions and
winning coalitions together,

A
has an index of 6
 B has an index of 2
 C has an index of 2
Comparing Voting Systems

Two voting systems are equivalent if there is a way for
all of the voters of the first system to exchange places
with the voters of the second system and preserve all
winning coalitions.


[50: 49, 1] and [4: 3, 3] - unanimous support
[2: 2, 1] and [5: 3, 6] – dictator

Every 2-voter system is equivalent to a system with a
dictator or one that needs consensus

Minimal winning coalition: a winning coalition in which
each voter is a critical voter
Minimal Winning Coalitions




Take the voting system [6: 5, 3, 1] where the
respective voters are A, B, C.
The 3 winning coalitions are {A,B}, {A,C} and
{A,B,C}.
Which coalitions are minimal?
Only {A,B} and {A,C}, but not {A,B,C} since only
A is critical
Minimal Winning Coalitions

Instead of using weights and quotas to describe a voting
system, one can describe it by using its minimal winning
coalitions.

The following conditions must be satisfied



1) The list can’t be empty (otherwise there is no way to
approve a motion)
2) There can’t be one minimal coalition that contains
another one
3)Every pair of coalitions in the list must overlap- otherwise
two opposing motions could pass.
3-Voter Systems & Minimal Winning
Coalitions



Make a list of all voting systems with 3 voters
The 3 voters are A, B, C
1) Suppose the M.W.C is {A}


2) Suppose the M.W.C is {A,B,C}


A clique where C is the dummy voter
4) Suppose the M.W.C. are {A,B} and {A,C}


Consensus rule
3) Suppose the M.W.C. is {A,B}


Dictatorship
A has veto power- the chair veto
All 2-member coalitions are M.W.C

Majority rule
3-Voter Systems & Minimal Winning
Coalitions
System
Min. W.
Coaltions
Weights
Banzhaf
Index
Dictator
{A}
[3: 3, 1, 1]
(8, 0, 0)
Clique
{A, B}
[4: 2, 2, 1]
(4, 4, 0)
Majority
(4, 4, 4)
Chair Veto
{A,B} {A,C}, [2: 1, 1, 1]
{B,C}
{A,B} {A, C} [3: 2, 1, 1]
Consensus
{A, B, C}
(2, 2, 2)
[3: 1, 1, 1]
(6, 2, 2)
Discussion

Chapter 10



Chapter 11




Where do we see manipulation of voting systems?
Are there any political elections that stand out in your mind?
What are some applications of weighted voting systems?
How would you describe a jury as a weighted voting system?
What might advantages/disadvantages of certain types of
weighted voting systems?
Homework:


Chapter 10 pg 387 # 9
Chapter 11 pg 425 # 7