Arrow’s Theorem
The search for the perfect election
decision procedure
The story so far…
• We have studied several election decision
procedures designed to produce one or more
winners from a slate of 3 or more candidates.
• Each procedure has had some desirable
features and some undesirable ones (‘quirks’).
• We’ve even seen that these methods can give
different winners using exactly the same set of
ordinal ballots!
Enter Kenneth Arrow
• Arrow, an economist, wanted to find a
completely ‘fair’ election decision
procedure.
• He began by making a list of a few basic
properties that he believed any good
election decision procedure should have:
Arrow’s Properties
• Universality—The decision procedure
must be able to process any set of ordinal
ballots to produce a winner, and must be
able to compare any two alternatives.
• Non-dictatorship (no one voter can
determine the outcome)
• Independence-of-Irrelevant-Alternatives
Criterion (Binary Independence)
• Pareto Criterion
IIA or Binary Independence
• It is impossible for an alternative B to
move from non-winner to winner unless at
least one voter reverses the order in
which he/she had ranked B and the
winning alternative.
• In other words, whether A or B wins
should depend only on how the voters
compare A to B, and not on how other
alternatives are ranked relative to A or B.
Pareto Criterion
• If every voter prefers alternative X to
alternative Y, then the decision procedure
should rank X above Y.
Asking the Impossible
• In 1951 Arrow published a book Social
Values and Individual Choice in which he
proved that there does not exist an
election procedure which ranks for
society 3 or more candidates based on
individual preferences and which
satisfies the fairness criteria we have
listed.
Manipulability
• Manipulability is voting insincerely in order
to influence the election result.
Example: You feel your top choice has no
chance to win, so you vote insincerely by
voting for your second choice, which you
prefer over any other alternative.
(Also called tactical or strategic voting.)
Let’s look at a simple example…
Manipulability at work
Look what happens if the 3 C
supporters vote for B!
49 47
3
A
B
C
B
C
B
C
A
A
Activity 8, Exercise 1,2
12
Aerobics
5
10
Badminton Football
11
Softball
Badminton Football
Badminton Badminton
Football
Softball
Softball
Football
Softball
Aerobics
Aerobics
AerobicsA
More Impossibility!
• Gibbard-Satterthwaite Impossibility Theorem
“For elections with at least 3 alternatives and
ordinal ballots as in Arrow’s Theorem, any nondictatorial election procedure can be
manipulated.”
It is impossible to design a procedure where it is
never in the voter’s best interest to vote
insincerely (strategically)!!
Approval Voting
• You give every candidate that you find
acceptable a vote.
• The candidate with the most votes wins.
• Works well for elections with several
candidates and where more than one
candidate can be elected (board of
directors, hall of fame, etc.)
A defect in Approval Voting
• It is uncomfortably easy for any candidate
to win
100 voters hate Bush, but Gore, Nader ok
100 voters hate Gore, but Bush, Nader ok
200 voters prefer Nader but Bush, Gore ok
Vote totals?