1. Part 4: Voting Topics - Continued
Problems with Approval Voting
Arrow’s Impossibility Theorem
Condorcet’s Voting Paradox
Condorcet

2. Problems with Approval Voting
Approval voting does not satisfy
the Majority Criterion.
the Condorcet Winner Criterion.
the Pareto criterion.

3. Problems with Approval Voting
Approval voting does not satisfy the majority criterion …
Number of Voters (5 total) 3 1
1st A B C
2nd
3rd
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.

4. 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.
Number of Voters (100 total) 33 34
1st A B C
2nd
3rd
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.

5. 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.

6. 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.

7. 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…
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.
Number of Voters
(3 total) 1
1st A B C
2nd
3rd

8. 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.
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.
Number of Voters
(3 total) 1
1st A B C
2nd
3rd

9. Marquis de Condorcet – Notes from Wikipedia
Marie Jean Antoine Nicolas Caritat, marquis de Condorcet (September 17, 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).

10. 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.