Social Choice
Chapter 9
Social Choice
• How Groups arrive at decisions
– Find the will of the people
– Turn individual preferences into a single
choice by the whole group
.
Some Definitions and Assumptions
• Preference List Ballot – a ranking of the
candidates (vertical list) with the most
preferred candidate on the top and the
least preferred on the bottom
• Number of Voters – throughout this
course, we will assume that the umber of
voters is always odd.
Majority Rules
• Two candidates
• Algorithm:
– All voters get one vote each
– Candidate with a majority of the votes wins
May’s(Kenneth) Theorem
•
If:
1. The number of voters is odd
2. The required properties are satisfied
•
Then: Majority voting is a perfect way to
find the social choice of the group .
Majority Voting
•
Required Properties
1. All voters treated equally
2. Both Candidates treated equally
3. It is monotone.
If there was a new election and only one
voter changes his vote from the loser to
the winner – there should still be the
same winner .
Plurality Voting
• Algorithm
– All participants get one vote
– If there is a preference list – only first place
votes are counted
– Candidate with Most votes wins .
Marquis de Condorcet (1743)
Condorcet Winner Criteria CWC)
For the election to be “valid”
1. There is no condorcet winner
2. The Condorcet winner is the winner of
the election .
Marquis de Condorcet (1743)
Condorcet’s Method
• Candidates should defeat each other in a
head-to-head election
Charles Borda
• Goal – arrive at group ranking of all
candidates that best expresses the
desires of the voters
– Class ranking
– Hall of Fame elections
– Track meets .
Borda Method
• Algorithm
– 1. Assign points to each position on the
preference list
– First place = N-1 points
– Last place = 0
– 2. Sum Points for each candidate
– 3. Winner has the most points
Borda
3
2
1
0
A
B
C
D
3
A
D
B
C
1
A
B
C
D
1
B
C
D
A
1
B
C
A
D
1
C
B
D
A
1
C
D
B
A
1
D
C
B
A
3(3)+ 3(1)+ 0(1)+ 1(1)+ 0(1)+ 0(1)+ 0(1)=
1(3)+ 2(1)+ 3(1)+ 3(1)+ 2(1)+ 1(1)+ 1(1)+
0(3)+ 1(1)+ 2(1)+ 2(1)+ 3(1)+ 3(1)+ 2(1)+
2(3)+ 0(1)+ 1(1)+ 0(1)+ 1(1)+ 2(1)+ 3(1)+
13
15
13
13
Independence of Irrelevant
Alternatives
• It should be impossible for “B” to move
from a non-winner to a winner unless at
least one voter reverses the order in
which he/she had “B” and the winner
ranked .
Charles Borda’s response to IIA
“My scheme is only intended
for honest men” .
Sequential Pairwise Voting
• Algorithm:
– 1. Start with an agenda
– 2. First on the agenda vs Second on the
agenda in a one-to-one contest (majority
voting)
– 3. Winner takes on the next on the agenda
(one-to-one)
– 4. Continue through entire agenda – one
remaining at the end is the winner .
Sequential Pairwise Voting
• Agenda – A listing of all the candidates
(a horizontal list e.g. B,C,A,D,E) .
Pareto Condition
• If everyone prefers one candidate (B) to
another candidate (D) then the latter (D)
should not be the winner!! .
Thomas Hare - 1861
• Arrive at a winner by repeatedly
deleting the “least preferred” candidate.
• Hare Method
• Algorithm:
– 1. Every participant get one vote
– 2. Candidate with fewest votes is
eliminated
– 3. Repeat steps 1 and 2 until winner(s) is
found .
Monotonicity
• Hare does not satisfy monotonicity
(Hare is not monotone)
• “If a candidate is the winner and in a new
election, the only change is favorable to
the winner – He should win again!! .
Plurality Runoff Method
• Runoff – new election using the same
ballots
• Algorithm:
– Take the two candidates receiving the
most first-place votes
– Using the same preference list find the
winner between the two remaining
candidates
– If there are more than two first (or second)
place winners that are tied the runoff is
among all the tied candidates
Plurality Runoff Method
4
3
3
2
1
First
A
B
C
D
E
Second
B
A
A
B
D
Third
C
C
B
C
C
Fourth
D
D
D
A
B
Fifth
E
E
E
E
A
A=4
B=3
C=3
D=2
E=1
A=4
B=3+2=5
C=3+1=4
Plurality Runoff Method
4
3
3
2
1
First
A
B
C
B
E
Second
B
A
A
D
D
Third
C
C
B
C
C
Fourth
D
D
D
A
B
Fifth
E
E
E
E
A
A=4
B=5
C=3
E=1
A=4+3=7
B=5+1=6
Plurality Runoff Method
• Problem:
– The plurality runoff method (like the Hare
Method) is not monotone
– “If a candidate is the winner and in a new
election, the only change is favorable to
the winner – He should win again!!
Kenneth Arrow
• Arrow’s Impossibility TheoremWhen there are three or more
candidates, There does not exist, and never
will, A Perfect Social Choice Method.
• (All Social Choice methods for three or
more choices have flaws.) .
Approval Voting
• Algorithm:
1. Each voter gives one vote to as many
candidates as they find acceptable.
2. Show disapproval by not voting for a
candidate
3. The candidate(s) with the most votes wins
4. Appropriate when more than one candidate
can win .
Baseball Hall of Fame
• 420 ballots go out to Sportswriters
• Requirements for nomination
– Retired for at least 5 years
– Nominated by 50 sportswriters
– Receive 75% of votes cast
• Stay on list until:
– 15 years and not elected
– Receive < 5% of the votes .
Approval Voting -example
7
A
B
C
X
A
B
C
7+
8
9
9
X
X
X
X
9+
3
X
X
9+
9+
8+
6
X
X
X
X
X
3+
3+
1=
1=
1=
6+
6+
1
2
23
21
19
Crowds
• Charles Mackay – “Men, it has been well said,
think in herds,”
• In the popular imagination groups tend to make
people either dumb or crazy, or both.
• Bernard Baruch – “anyone taken as an individual
is tolerable sensible and reasonable – as a
member of a crowd, he at once becomes a
blockhead.”
• Friedrich Nietzsche – “ Madness is the exception
in the individual and the rule in a group” .
Wisdom
• Most people believe that valuable
knowledge is concentrated in a very few
hands. We assume that the key to
solving problems is finding the one right
person
• The argument of the book is that chasing
the expert is a mistake. We should stop
hunting and ask the crowd instead. .
Francis Galton -1906
• Went to the county fair where 800 people took a
chance at guessing the weight of an ox
• Galton wanted to prove that the average voter
was capable of very little
• Ran statistical tests on the tickets
• Mean of the groups guesses = the wisdom of the
crowd
• Crowd guessed 1,197 pounds.
• Ox weighed 1,198 pounds .
Francis Galton - cont
• Under the right circumstances, groups are
remarkably intelligent and are often smarter than
the smartest people in the group
• Even if most of the people within the group are
not especially well-informed or rational, it can still
reach a collectively wise decision
• When imperfect judgments are aggregated in the
right way, collective intelligence is often excellent
.
Who wants to be a Millionaire?
•
Lifelines
1. Ask the audience (the group)
2. Call an expert
•
Results
–
–
Audience right – 91%
Expert right – 65% .
Scorpion
• In May 1968, the U.S. submarine Scorpion
disappeared on its way back to Newport News.
The possible area was a circle twenty miles wide
and thousands of feet deep.
• John Craven drew up a series of scenarios for
what might have happened to the sub.
• He assembled a team of men with a wide range
of knowledge, including mathematicians,
submarine specialists, and salvage men .
Scorpion - cont
•
He asked each of them to offer his best
guess about how likely each of the
scenarios was – to keep it interesting,
the guesses were in the form of wagers,
with bottles of Chivas Regal as the
prizes. The bets:
1. Why the submarine ran into trouble
2. Its speed and heading
3. The steepness of its descent .
Scorpion - results
• Applying Bayer’s theorem to all the guesses he
estimated the Scorpion’s final location
• He had the group’s collective estimate of where
the sub was
• The sub was found 220 yards from where the
group said it would be!!
• While no one in the group knew where it was –
the group as a whole knew everything .
It’s all Math
• The answer comes from a mathematical truism –
it you ask a large enough group of diverse,
independent people to make a prediction or
estimate a probability, and then average those
estimates, the errors each of them makes in
coming up with an answer will cancel themselves
out.
• Each person’s guess has two components:
information and error. Subtract the error, and
you are left with the information. .
The Challenger Disaster
• At 11:32 a.m. on January 28, 1986 the
space Shuttle Challenger blew up
• Within minutes, investors started dumping
the stocks of the four major contractors
participating in the Challenger launch.
• Morton Thiokol’s stock was hit hardest
– 12% in 10 minutes .
Challenger - cont
• By the end of the day the other
contractors stock was within 2% of the
start of the day. Thiokol was still –12%
• The stock market knew almost
immediately who was responsible. .
Challenger – why?
•
Conditions for Wise Crowds
1.
Diversity of opinion – each person had private
information
Independence – people’s opinions are not
determined by others
Decentralization – people can specialize and
draw on local knowledge
Aggregation – mechanism exists to take
private judgments and turn them into a
collective decision .
2.
3.
4.
Types of Problems
1.
2.
3.
Cognition problems – problems that have a
definitive solution e.g. who will win the Super
Bowl
Coordination problems – require members of
a group to figure out how to coordinate their
behavior with each other – stock market
Cooperation problems – involve the challenge
of getting self-interested people to work
together e.g. paying taxes, dealing with
pollution .
Requirements for Success
• Diversity – need people with difference
views, skills, and backgrounds
• Independence – the members cannot be
influenced by outside forces Or members
of the group
• Aggregation- there has to be a person or
institution to take the the views and
combine them .