IMPOSSIBILITY AND
MANIPULABILITY
Section 9.3 and Chapter 10
Review of Conditions and Criteria

There are many conditions and criteria that are
used to determine if an election is “fair”

These criteria often refer to voters changing their
ballots in some way, and the result of the election
changing (or not)
Condorcet Winner Criterion


A voting system is said to satisfy CWC provided
that, for every possible sequence of preference list
ballots, either (1) there is no Condorcet winner or
(2) the voting system produces exactly the same
winner for this election as does Condorcet’s
method.
Plurality does not satisfy this criterion, but
Condorcet’s method does
Independence of Irrelevant
Alternatives


A voting system is said to satisfy IIA if it is
impossible for a candidate A to move from
nonwinner status to winner status unless at least
one voter reverses the order in which he or she
had A and the winning candidate ranked.
The Borda count does not satisfy IIA, but
Condorcet’s method does
The Pareto Condition


If everyone prefers one candidate (say, B) to
another candidate (say, D), then this latter
candidate (D) should not be among the winners of
the election.
Plurality satisfies the Pareto condition, but
sequential pairwise voting does not
Monotone


If a candidate is a winner, and a new election is
held in which the only ballot change made is for
some voter to move the former winning candidate
higher on his or her ballot, then the original winner
should remain a winner.
Plurality is monotone, but the Hare system is not
The Search for a Perfect Voting System

All of the methods we have discussed are flawed in
some way

Why didn’t I just tell you about the “best” voting
system in the first place?

Recall May’s Theorem says that majority rule is the
“best” method for deciding the winner of an
election with two candidates
Arrow’s Impossibility Theorem

Named after Kenneth Arrow, an American
economist

Essentially, the theorem states that there is no
perfect voting method

It doesn’t just say that we haven’t thought of a
perfect system yet; it says that we can never create
one
Arrow’s Impossibility Theorem

With three or more candidates and any number of
votes, there does not exist (and will never exist) a
voting system that:
 always
produces a winner
 satisfies the Pareto condition
 satisfies the independence of irrelevant alternatives
condition
 is not a dictatorship
Proving the Theorem

Arrow’s Theorem is hard to prove (he earned a
Nobel prize in 1972 for his work in this area)

We will prove a weaker version of his theorem

To prove that it is impossible to create a voting
system, we will assume that we have created such
a system

This assumption will lead to an impossibility
Weak Version of Arrow’s Theorem

With three or more candidates, there does not
exist (and will never exist) a voting system that:
 satisfies
the Condorcet winner criterion
 satisfies the independence of irrelevant alternatives
condition
 always produces at least one winner
A Hypothetical Voting System

Let’s assume that we have a hypothetical voting
system that
 satisfies
the Condorcet winner criterion
 satisfies the independence of irrelevant alternatives
condition
 always produces at least one winner

In other words, we’re assuming that we have
exactly the kind of voting system that Arrow’s
Theorem says should not exist
A Problematic Profile

Consider this voter profile
Voters
Preference
1
A>B>C
1
B>C>A
1
C>A>B

Since our system “always produces at least one
winner,” we might wonder who the winner should be

We will show that A is not the winner
A New Profile


What about this profile?
Voters
Preference
1
A>B>C
1
C>B>A
1
C>A>B
Since C is the Condorcet winner, and our
hypothetical system satisfies the Condorcet winner
criterion, C must be the winner using our
hypothetical system also
Modifying the Profile



Now we’ll modify the second profile to turn it into the first
one:
Voters
Preference
Voters
Preference
1
A>B>C
1
A>B>C
1
C>B>A
1
B>C>A
1
C>A>B
1
C>A>B
Notice that the only change was for the second voter to
change from C > B > A to B > C > A
B is irrelevant to the question of A versus C, so since C was
the previous winner and A was a previous non-winner, IIA
means that A must continue to be a non-winner
The Problematic Profile




So we have just shown that for this profile, A is not a
winner
Voters
Preference
1
A>B>C
1
B>C>A
1
C>A>B
A similar argument shows that B and C are also nonwinners
But we assumed that our hypothetical system always
finds a winner
Therefore our hypothetical system can’t exist
Summary of Chapter 9

One “best” way to determine the winner of an
election with two candidates: majority rule

Many ways to determine the winner of an election
with more than two candidates

All of these methods are “unfair” in some way

We use criteria to be very specific about the ways
in which the methods are “unfair”

It is impossible to find a completely fair system
Chapter 10: Manipulability

Sometimes, in order to achieve the election result
you prefer, you submit a ballot that misrepresents
your actual preferences

This type of strategic voting is called manipulation,
and the misrepresented ballot is referred to as an
insincere or disingenuous ballot
An Example

Consider this voter profile, with just two voters
Voter
Preference
#1
A>B>C>D
#2
B>C>A>D

If we use the Borda count to determine the winner, B
wins with 5 points

Assuming that Voter #1 knew the ballot that Voter #2
was going to submit, could Voter #1 have submitted
her ballot so that A wins?
A Manipulated Outcome

What if Voter #1 changes her ballot like this:
Voter
Preference
#1
A>D>C>B
#2
B>C>A>D

This ballot is insincere: #1 likes B better than C or D,
but she has ranked B last to try to change the result

Using this new ballot, A is now the Borda count winner

Voter #1 prefers this outcome according to her original
ballot
Manipulability

A voting system is said to be manipulable if there
are two sets of ballots and a voter (we’ll call him
“Bob”) such that
 neither
election ends in a tie
 the only difference between the two sets of ballots is
Bob’s ballot
 Bob prefers (according to his actual preferences as
expressed in the first election) the outcome of the
second election to that of the first
Majority Rule

In an election with two candidates (A and B), is
majority rule manipulable?

In order to manipulate the election, A would have to
be the winner, but your true preference would have to
be B > A

The only change you can make to your ballot is to
change it to A > B

Since May’s Theorem guarantees that majority rule is
monotone, changing your vote from a vote for the
loser to a vote for the winner cannot change the
outcome
Condorcet’s Method

Condorcet’s Method is also non-manipulable

If you prefer B, but the winner is A, then A beats B
head-to-head even with your vote preferring B
over A

No matter how you change your ballot, A will still
beat B head-to-head
The Perfect System?

Condorcet’s Method has some very nice properties

elections never result in ties (assuming the number of voters is
odd)

satisfies the Pareto condition

non-manipulable

not a dictatorship

However, Condorcet’s Method also sometimes doesn’t
produce a winner

Is there a “perfect” system that satisfies all of these
properties and always gives a winner?
The Gibbard-Satterthwaite Theorem

With three or more candidates and any number of
voters, there does not exist (and never will exist) a
voting system that always produces a winner,
never has ties, satisfies the Pareto condition, is
non-manipulable, and is not a dictatorship.