Part 4: Voting Topics - Continued
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Problems with Approval Voting
Arrow’s Impossibility Theorem
Condorcet’s Voting Paradox
Condorcet
Problems with Approval Voting
Approval voting does not satisfy
• the Majority Criterion.
• the Condorcet Winner Criterion.
• the Pareto criterion.
Problems with Approval Voting
Approval voting does not satisfy the majority criterion …
Number of Voters (5 total)
3
1
1
1st
A
B
C
2nd
B
A
A
3rd
C
C
B
In this preference schedule, A has a majority of first place
votes, however, by approval voting, the winner is B (with 4
approval votes, versus A who has only 3.)
Note: This example assumes that even though voters will vote only for the
candidates they approve of, they can still rank those for which they approve.
Approval Voting fails the Condorcet Winner Criterion
To demonstrate another problem with approval voting, consider this
example...
Number of Voters
(100 total)
The Condorcet winner of this election
would be A …
33
33
34
1st
A
B
C
This is because 67 voters prefer A
over B, and 66 voters prefer A over C.
2nd
B
A
A
3rd
C
C
B
That is, A beats the others one-onone.
However, B is the winner of this election by approval voting. That is, the
Condorcet winner was not elected and hence approval voting has been
shown to violate the Condorcet Winner Criterion.
Again, the assumption is made that voters can still rank the candidates they
approve of.
Arrow’s Impossibility Theorem
• In 1951 Kenneth Arrow proved the following remarkable theorem:
There is no voting method (nor will there ever be) that will satisfy a
reasonable set of fairness criteria when there are three or more
candidates and two or more voters.
• We have considered many different voting methods in these lecture
notes. Every method has been shown to fail at least one of the criteria
given at the beginning of the chapter.
• Kenneth Arrow has shown mathematically that all voting methods
must fail at least one of those criteria.
• His theorem implies that when there are two or more voters and three
or more candidates there are no perfect voting methods and there
never will be any perfect voting method.
Arrow’s Impossibility Theorem
• Our textbook provides the beginning of a proof of
a simplified version of Arrow’s Impossibility
theorem. That simplified version states “There is
no voting method that will satisfy both the CWC
and IIA criteria.”
• The theorem and proof use the version of IIA
given in the book, not the version of IIA given in
these notes. The version of IIA given in these
notes is the more common version.
Condorcet’s Voting Paradox
•
We can assume individual voter preferences are transitive. That is, we can
assume that if an individual voter prefers candidate A over B and prefers
candidate B over C, then it is reasonable to assume that same voter prefers
candidate A over C.
•
Condorcet’s Voting Paradox is the fact that societal preferences are not
necessarily transitive even when individual voter preferences are transitive.
•
For example, consider the following preference schedule…
Number of Voters
(3 total)
1
1
1
1st
A
B
C
2nd
B
C
A
3rd
C
A
B
In this preference schedule, even assuming
that individual preferences are transitive, it
becomes apparent that the group preferences
are not transitive…
For example, A is preferred over B by 2 to 1
B is preferred over C also by a vote of 2 to 1
and yet we see that C is preferred over A also
by a vote of 2 to 1.
Condorcet’s Voting Paradox
• CAREFUL – our textbook seems to suggest that
the Condorcet Paradox is simply the fact that, in
the preference table below, there is no winner.
This is not a paradox. The paradox is the fact
that group preferences may not be transitive
even if we assume individual preferences are
transitive.
Number of Voters
(3 total)
1
1
1
1st
A
B
C
2nd
B
C
A
3rd
C
A
B
Not having a winner is not a paradox by
itself. In fact, there is no mention in
Condorcet’s paradox as to what method of
voting is being used, so no winner need be
established. To repeat, what is paradoxical
is this…
Suppose the group prefers A over B and
prefers B over C. It would be expected, that
the group would prefer A over C (that would
mean the preferences are transitive.) But as
can be seen from the table, C is preferred
over A.
Marquis de Condorcet – Notes from Wikipedia
• Marie Jean Antoine Nicolas Caritat, marquis de Condorcet
(September 17, 1743 - March 28, 1794) was a French philosopher,
mathematician, and early political scientist.
• Unlike many of his contemporaries, he advocated a liberal economy,
free and equal public education, constitutionalism, and equal rights
for women and people of all races. His ideas and writings were said
to embody the ideals of the Age of Enlightenment and rationalism,
and remain influential to this day.
• Condorcet took a leading role when the French Revolution swept
France in 1789, hoping for a rationalist reconstruction of society, and
championed many liberal causes.
• He died a mysterious death in prison after a period of being a fugitive
from French Revolutionary authorities. The most widely accepted
theory is that his friend, Pierre Jean George Cabanis, gave him a
poison which he eventually used. However, some historians believe
that he may have been murdered (perhaps because he was too loved
and respected to be executed).
Marquis de Condorcet – Terms in Voting Theory
Things to know:
1. The Condorcet Winner – The candidate, if there is one, that beats
all other candidates in one-on-one comparisons. With some
preference schedules, there is no Condorcet winner.
2. The Voting Paradox of Condorcet – The preferences of a group
may not be transitive even if individual preferences are assumed
to be transitive.
3. The Condorcet Method – The winner of the election is the
Condorcet winner. Note that this is not a valid method of voting
with three or more candidates because there may be no
Condorcet winner and so no winner of the election.
4. The Condorcet Winner Criterion – The condition that if there is a
Condorcet winner, that candidate should be the winner of the
election by whichever voting method is actually being used.