Chapter 9: Social Choice

1
Chapter 9 Social Choice
2
How do we choose the winner of an election?

• Harder than you might think
• There are many examples in history where the
  results were disputed

3
2000 Presidential Election

• Al Gore vs. George W. Bush
• There were two other candidates Ralph Nader
  (Green Party) and Pat Buchanan (Reform Party)
• The result of the popular vote was
• Gore 48.4
• Bush 47.9
• Nader 2.7
• Buchanan 0.4

4
Controversy

• Even though Al Gore won the popular vote, George
  W. Bush won the electoral college
• The result of Florida was in dispute for several
  weeks, but eventually Floridas electoral votes
  were given to Bush after the Supreme Court
  ordered a stop to recounts
• The official margin of victory for Bush in
  Florida was 537 votes, out of 5.8 million votes
  cast

5
1998 Minnesota Gubernatorial Election

• Three main candidates
• Norm Coleman (R)
• Hubert Humphrey (D)
• Jesse Ventura (Reform Party)
• The results of the vote were
• Ventura (37)
• Coleman (34)
• Humphrey (28)

6
Why were these results controversial?

• Few people expected the former professional
  wrestler to win the election
• Most of the people who voted for Coleman or
  Humphrey probably had Ventura as their last
  choice
• That means that Ventura was elected governor even
  though 63 of the voters would have ranked him
  last!

7
How do we choose the winner of an election?

• There are many different systems, as we will
  learn
• The most common system used in US elections is
  the plurality system
• The candidate who gets more votes than any other
  candidate is said to receive a plurality
• A candidate receives a majority if they earn
  more than half of the total number of votes
• Al Gore won a plurality of the popular vote in
  2000
• Jesse Ventura won a plurality of the vote in
  1998
• Neither of these candidates won a majority

8
Preference Lists

• In most US elections, voters can only cast a
  single ballot for the candidate he or she likes
  the best
• However, most voters will have preference
  lists a ranking of the candidates in order of
  most preferred to least preferred
• For example, many (but not all) of the people who
  voted for Ralph Nader in 2000 would have had Al
  Gore as their second choice

9
An Example Milk, Soda, and Juice

• Suppose a class of children is trying to decide
  what drink to have with their lunch
• The choices are milk, soda, and juice
• Each child votes for their top choice
• The results are
• Milk 6
• Soda 5
• Juice 4
• Milk wins a plurality of the votes

10
Considering Preferences

• What if we examine the full voting preferences of
  the children?
• 6 have the preference Milk gt Soda gt Juice
• 5 have the preference Soda gt Juice gt Milk
• 4 have the preference Juice gt Soda gt Milk
• Is the outcome fair? If we choose Milk as the
  winner of this election, 9 of the 15 students are
  stuck with their last choice

11
Assumptions

• We will not allow ties on individual preference
  lists, though some methods will result in a tie
• All candidates must be listed in a specific order
• We will sometimes assume that the number of
  voters is odd to avoid ties (remember we will
  think about applying these methods to situations
  where we have thousands or millions of voters)

12
Two Candidates

• Well start off simple and only consider the case
  where we have two candidates
• There are only two preferences A gt B and B gt A
• Voters with preference A gt B vote for A
• Voters with preference B gt A vote for B
• The candidate with the most votes wins
• This method is called majority rule

13
Majority Rule

• Notice that one of the two candidates will
  definitely get a majority (they cant both get
  less than half of the votes)
• Majority rule has three desirable properties
• All voters are treated equally
• Both candidates are treated equally
• Monotone

14
All voters treated equally

• If any two voters exchange (marked) ballots
  before submitting them, the outcome of the
  election does not change

15
Both candidates treated equally

• If a new election were held and every voter
  reversed their vote (people who voted for A now
  vote for B, and vice versa), then the outcome of
  the election is also reversed

16
Monotone

• If a new election were held and a single voter
  were to change his or her ballot from being a
  vote for the loser of the previous election to
  being a vote for the winner of the previous
  election, and everyone else voted exactly as
  before, then the outcome of the new election
  would be the same as the outcome of the previous
  election.

17
Other Methods

• Majority rule is not the only way to determine
  the winner of an election with two candidates
• Mays Theorem states that majority rule is the
  only method for determining the winner of an
  election with two candidates that treats all
  voters equally, treats both candidates equally,
  and is monotone