More Voting Methods

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More Voting Methods

• MAT 105 Spring 2008

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More Methods

• There are many more methods for determining the
  winner of an election with more than two
  candidates
• We will only discuss a few more
• sequential pairwise voting
• Hare system
• plurality runoff

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Sequential Pairwise Voting

• Idea We like pairwise voting (where we can use
  majority rule), but if we look at all pairwise
  elections, we sometimes dont get a winner
• In sequential pairwise voting, we put the
  candidates in order on a list, called the agenda

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How It Works

• We pit the first two candidates on the agenda
  against each other. The winner moves on to face
  the next candidate on the list, and so on. The
  candidate remaining at the end is the winner.
• This process resembles a tournament bracket, and
  has the advantage that, unlike Condorcets
  method, we always get a winner

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An Example

• Lets use sequential pairwise voting with this
  profile and the agenda A, B, C, D

Voters Preference Order
4 A gt B gt D gt C
3 C gt A gt B gt D
3 B gt D gt C gt A
A
A beats B, 7-3
A
C beats A, 6-4
B
C
D beats C, 7-3
C
D
D
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Problems

• If we look closely at thisagenda, we notice
  thatevery single voter prefersB over D, and yet
  D wasour winner!
• In fact, by cleverly choosing the right agenda,
  we could make any of the four candidates win this
  election
• Sequential pairwise voting does not satisfy the
  Pareto condition

Voters Preference Order
4 A gt B gt D gt C
3 C gt A gt B gt D
3 B gt D gt C gt A
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The Pareto Condition

• If every voter prefers one candidate over
  another, then the latter candidate should not be
  among the winners of the election
• Named for Vilfredo Pareto (1848-1923), Italian
  economist
• Does plurality satisfy the Pareto condition?

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Another Method The Hare System

• Also known as Instant Runoff Voting, this system
  is used for various elections in the US, Canada,
  the UK, Ireland, and Australia
• Repeatedly delete candidates that are least
  preferred in the sense of being at the top of
  the fewest ballots. If there is a tie, eliminate
  all of the tied candidates, until there is no one
  left to eliminate

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An Example
Voters Preference Order
5 A gt B gt C
4 C gt B gt A
3 B gt C gt A
2 B gt A gt C

• In this example, A has 5first-place votes, B has
  5first-place votes, and C has4 first-place
  votes, so C iseliminated
• Now A has 5 first-placevotes, and B has 9, so A
  iseliminated
• B is the only candidate left,so B is the winner

Voters Preference Order
5 A gt B
4 B gt A
3 B gt A
2 B gt A
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Another Example
Voters Preference Order
5 A gt B gt C
4 C gt B gt A
3 B gt C gt A
1 B gt A gt C

• This time, A has 5 first-placevotes, and B and C
  are tiedwith 4, so B and C are botheliminated
  at the same time
• This leaves only A to win the election

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Problems

• Now lets modify the profile from the previous
  example, so that the 1 voter with preference B gt
  A gt C now has preference A gt B gt C
• Notice that this change moves the winner higher
  on that voters ballot

Voters Preference
6 A gt B gt C
4 C gt B gt A
3 B gt C gt A
Voters Preference
6 A gt C
4 C gt A
3 C gt A
C wins!
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Why is this a problem?

• A was the winner of the original election, and
  one of the voters changed his ballot to move A
  higher, causing A to lose
• This shows that the Hare system is not monotone

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Monotone

• A voting system is monotone if whenever a
  candidate is a winner, and a new election is held
  where the only change is for some voter to move
  that winner higher on his/her ballot, then the
  original winner should remain the winner
• The Hare system is not monotone, but despite this
  drawback it is one of the more common alternative
  voting systems in use today

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One More Method Plurality Runoff

• Hold a plurality election, but if no candidate
  receives a majority, we hold a runoff election
• The runoff election is between the two candidates
  who received the most first-place votes in the
  original election
• In case of ties, there might be more than two
  candidates with the most first-place votes, so we
  use plurality to decide a winner between those
  candidates only

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Example
Voters Preference Order
5 A gt B gt C
4 C gt B gt A
3 B gt C gt A
2 B gt A gt C

• In this profile, A gets 5first-place votes, B
  gets5 first-place votes, and Conly gets 4
• The runoff is between A and B
• B wins the runoff 9 votes to 5

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Another Example
Voters Preference Order
4 A gt B gt C gt D
3 C gt D gt B gt A
3 B gt C gt D gt A
2 D gt B gt A gt C

• In this profile, A has 4first-place votes, B has
  3,C has 3, and D has 2
• The runoff is between A, B, and C
• We use plurality to decide the winner keep in
  mind that the 2 voters who like D best get to
  vote in the runoff!
• B wins the runoff with 5 votes