Part 4: Voting Topics - Continued

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Part 4 Voting Topics - Continued

• Problems with Approval Voting
• Arrows Impossibility Theorem
• Condorcets Voting Paradox
• Condorcet

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Problems with Approval Voting

• Approval voting does not satisfy
• the Majority Criterion.
• the Condorcet Winner Criterion.
• the Pareto criterion.

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Problems with Approval Voting

• Approval voting does not satisfy the majority
  criterion

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.
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Approval Voting fails the Condorcet Winner
Criterion

• To demonstrate another problem with approval
  voting, consider this example...

The Condorcet winner of this election would be A
This is because 67 voters prefer A over B, and
66 voters prefer A over C. That is, A beats
the others one-on-one.
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.
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Arrows 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.

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Arrows Impossibility Theorem

• Our textbook provides the beginning of a proof of
  a simplified version of Arrows 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.

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Condorcets 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.
• Condorcets 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

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.
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Condorcets 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.

Not having a winner is not a paradox by itself.
In fact, there is no mention in Condorcets
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.
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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).

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Marquis de Condorcet Terms in Voting Theory

• Things to know
• 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.
• The Voting Paradox of Condorcet The preferences
  of a group may not be transitive even if
  individual preferences are assumed to be
  transitive.
• 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.
• 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.