Social Choice 2 alternatives

1
Social Choice 2 alternatives

• Elections with only two alternatives
• Majority Rule each voter indicates a preference
  for one of the 2 candidates, and the candidate
  with the most votes wins.
• Desirable Properties
• All voters treated equally
• Both candidates treated equally
• If new election held single voter changed
  ballot from being a vote for loser of previous
  election to being a vote for the winner of the
  previous election, everyone else voted exactly
  as before, the outcome of new election would be
  same as previous election.

2
Social Choice 2 alternatives

• Elections with only two alternatives
• Property 1 not satisfied by dictatorship i.e. all
  ballots except that of dictator are ignored
• Property 2 not satisfied by imposed rule i.e.
  candidate X wins regardless of who votes for whom
• Property 3 not satisfied by minority rule i.e.
  candidate with the fewest votes wins
• Mays Theorem If of voters is odd, we are
  interested only in voting systems that never
  result in a tie, then majority rule is the only
  voting system for 2 alternatives that satisfies
  the 3 properties.

3
Social Choice 3 or more alternatives

• Plurality Voting
• Borda Count
• Sequential Pairwise Voting
• Hare System
• Approval Voting
• Summary of Drawbacks

4
Social Choice - Plurality

• Elections with three or more alternatives
• Plurality Voting only 1st place votes
  considered. Candidate with most 1st place votes
  wins.
• Condorcet Winner looking at all possible
  pairings, it is the candidate that would have
  defeated every other candidate in a head to head
  race.
• Condorcet Winner Criterion (CWC) for every
  possible sequence of preference lists, either
• a) there is no Condorcet winner
• b) there is a unique Condorcet winner

5
Social Choice Plurality - Example
Preference List
Plurality winner is Friday since 12 students
voted it as 1st choice
Compare all possible pairings to find Condorcet
Winner
6
Social Choice Borda Count

• Elections with three or more alternatives
• Borda Count (rank method) assigns points in a
  non-increasing manner to each voters ranking and
  then sums up the points to arrive at a groups
  final ranking.
• Independence of Irrelevant Alternatives (IIA)
  an alternative B cannot move from nonwinner
  status to winner status unless at least 1 voter
  reverses the order in which he/she had B and the
  winning alternative ranked.

7
Social Choice Borda Count - Example
Preference List
Assign points to 1st, 2nd, and 3rd choices 2,1,0
respectively Compute point totals. Winner has the
most points.
Friday (2)(12)(1)(0)(0)(18)240024 Wednesday
(2)(10)(1)(20)(0)(0)202040 Thursday
(2)(8)(1)(10)(0)(12)1610026
Winner
In-Class Exercise Thursday Friday are
nonwinners. Suppose the 12 students in the first
column of the table changed their votes by
switching the order in which they ranked Thur and
Fri . Does this change winner?
8
Social Choice Borda Count - Example
Preference List
Assign points to 1st, 2nd, and 3rd choices 2,1,0
respectively Compute point totals. Winner has the
most points.
Friday (2)(0)(1)(0)(0)(30)0 Wednesday
(2)(10)(1)(20)(0)(0)202040 Thursday
(2)(20)(1)(10)(0)(0)4010050
Winner
Note that the winner has changed from Wednesday
to Thursday
9
Social Choice Sequential Pairwise Voting

• Elections with three or more alternatives
• Sequential Pairwise Voting starts with an agenda
  and pits the 1st alternative against the 2nd in a
  one-on-one contest. The winner (or both, if they
  tie) then moves on to confront the 3rd
  alternative in the list, one-on-one. Losers are
  deleted. This process continues throughout the
  entire agenda, and those remaining at the end are
  the winners.
• An agenda must be given in this voting system !
• Pareto Condition If everyone prefers one
  alternative to another alternative, then the
  latter alternative is not among the winners.

10
Social Choice Sequential Pairwise Voting -
example
Preference List
Use this agenda to find the winner Thursday,
Friday, Wednesday
Thurs vs Fri Thurs 810 18 Fri 12
Thurs wins, delete Fri
Thurs vs Wed Thurs 8 Wed 1210 22 Wed
wins, delete Thurs
Wednesday is declared the winner!
11
Social Choice Hare System

• Elections with three or more alternatives
• Hare System alternatives are successively
  eliminated in an order based on the number of 1st
  place votes.
• Monotonicity if an alternative is a winner, and
  a new election is held in which the only ballot
  change made is for some voter to exchange that
  winning alternative with the one immediately
  above it on his ballot, then the original winner
  should remain a winner.

12
Social Choice Hare System - example
Preference List
Round 1 (1st place votes) Fri (12)
Thur (8 ) Wed (10)
Thrus has only 8 votes so Thurs is deleted. Begin
round 2 by Replacing Thurs with the entry below
it in the table.
Round 2 Wed now has 18 1st place votes while
Fri has only 12. Wed is declared the winner.
13
Social Choice Hare System example cont.
Suppose in our original example, 1 of the 12
students were to exchange Wed with list. Our
new preference list would be given below
Round 1 (1st place votes) Fri (11) Thur (8)
Wed (11 )
Thur is eliminated. Wed Fri are now tied for
the winner !
14
Social Choice Approval Voting

• Elections with three or more alternatives
• Approval Voting A method of electing one or
  more candidates from a field of several in which
  each voter submits a ballot that indicates which
  candidates he or she approves of. Winning is
  determined by the total number of approvals a
  candidate obtains.

5 people preferred Wed
7 people preferred Thu
5 people preferred Fri
Thursday is declared the winner since it got the
most approval votes.
15
Social Choice Summary of Drawbacks
16
Credits

• COMAP, For All Practical Purposes, 5th ed