Impossibility and Manipulability

1
Impossibility and Manipulability

• Section 9.3 and Chapter 10

2
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)

3
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
  Condorcets method.
• Plurality does not satisfy this criterion, but
  Condorcets method does

4
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
  Condorcets method does

5
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

6
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

7
The Search for a Perfect Voting System

• All of the methods we have discussed are flawed
  in some way
• Why didnt I just tell you about the best
  voting system in the first place?
• Recall Mays Theorem says that majority rule is
  the best method for deciding the winner of an
  election with two candidates

8
Arrows Impossibility Theorem

• Named after Kenneth Arrow, an American economist
• Essentially, the theorem states that there is no
  perfect voting method
• It doesnt just say that we havent thought of a
  perfect system yet it says that we can never
  create one

9
Arrows 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

10
Proving the Theorem

• Arrows 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

11
Weak Version of Arrows 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

12
A Hypothetical Voting System

• Lets 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, were assuming that we have
  exactly the kind of voting system that Arrows
  Theorem says should not exist

13
A Problematic Profile

• Consider this voter profile
• 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

Voters Preference
1 A gt B gt C
1 B gt C gt A
1 C gt A gt B
14
A New Profile

• What about this profile?
• 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

Voters Preference
1 A gt B gt C
1 C gt B gt A
1 C gt A gt B
15
Modifying the Profile

• Now well modify the second profile to turn it
  into the first one
• Notice that the only change was for the second
  voter to change from C gt B gt A to B gt C gt 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

Voters Preference
1 A gt B gt C
1 B gt C gt A
1 C gt A gt B
Voters Preference
1 A gt B gt C
1 C gt B gt A
1 C gt A gt B
16
The Problematic Profile

• So we have just shown that for this profile, A is
  not a winner
• A similar argument shows that B and C are also
  non-winners
• But we assumed that our hypothetical system
  always finds a winner
• Therefore our hypothetical system cant exist

Voters Preference
1 A gt B gt C
1 B gt C gt A
1 C gt A gt B
17
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

18
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

19
An Example

• Consider this voter profile, with just two voters
• 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?

Voter Preference
1 A gt B gt C gt D
2 B gt C gt A gt D
20
A Manipulated Outcome

• What if Voter 1 changes her ballot like this
• 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

Voter Preference
1 A gt D gt C gt B
2 B gt C gt A gt D
21
Manipulability

• A voting system is said to be manipulable if
  there are two sets of ballots and a voter (well
  call him Bob) such that
• neither election ends in a tie
• the only difference between the two sets of
  ballots is Bobs 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

22
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 gt A
• The only change you can make to your ballot is to
  change it to A gt B
• Since Mays 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

23
Condorcets Method

• Condorcets 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

24
The Perfect System?

• Condorcets 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, Condorcets Method also sometimes
  doesnt produce a winner
• Is there a perfect system that satisfies all of
  these properties and always gives a winner?

25
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.