The Voting Paradox of
Condorcet
Marquis de Condorcet
• Marie Jean Antoine Nicolas de Caritat was the name of a French
philosopher, mathematician and political scientist who was born in
1743.
• He was given the title Marquis de Condorcet, a title of nobility,
meaning that he was a part of the aristocracy in France in a period
right before the French Revolution.
• The French Revelution began in 1789 when Condorcet was 46. In
spite of his position in the aristocracy, Condorcet advocated equality
among men and women, and among people of all races. He
believed in a constitutional form of government and the rule of law,
instead of a monarchy and aristocracy.
• He was arrested for those beliefs. And eventually died at the age of
51 because of those beliefs. He may have been murdered while in
jail because he was too popular to be publicly excecuted. Another
theory is that he took his own life rather than be excuted.
• During his life, Condorcet made many contributions to both
mathematics and political science.
• One contribution is the discovery of a paradox in voting with groups of
voters. The paradox is now called “The Voting Paradox of
Condorcet.”
• To understand this paradox, one must understand the concept of
transitivity.
• In algebra, transitivity is quite simple. The transitivity of the “greater
than” inequality states: If A > B and B > C then A > C.
• In voting, we can refer to the transitivity of voter preferences: If a
certain voter prefers candidate A more than B and he prefers B more
than C than one can reasonably assume that this voter prefers
candidate A more than C.
• The Voting Paradox of Condorcet is the following:
• Even if we assume that the preferences of individual voters are
transitive, this does not imply that the preferences of a group are
transitive.
• The following is a “real-life” example from a vote in Congress in
1956.
• In 1956, the breakdown of the 435 members of House of
Representatives was 232 Democrats and 203 Republicans.
• However, at that time there was a significant difference between the
116 Northern Democrats and 116 Southern Democrats.
• A vote was to be held regarding federal funding for schools
throughout the nation.
• An amendment to proposed legislation was offered that would
provide funding only to districts that favored integration of blacks
and whites in the schools.
• The Republicans were against the legislation from the beginning
but preferred promoting integration over unconditional funding.
• The Northern Democrats liked the idea of supporting integration
through the funding but still wanted to provide the funding.
• The Southern Democrats wanted the school funding but were
completely against the federal interference in local control over
integration of the schools.
The following table lists the preferences of the voters.
Number of
voters
Republicans
(203)
Northern
Democrats
(116)
Southern
Democrats
(116)
1st Choice
No Bill
Amendment
Original Bill
2nd Choice
Amendment
Original Bill
No Bill
3rd choice
Original Bill
No Bill
Amendment
• We can reasonably assume that the preferences of voters in each
group are transitive.
• For example, we can assume that if a voter prefers the amendment
over the original bill and prefers the original bill over no bill at all,
then that voter would prefer the amendment over no bill at all.
Number of
voters
Republicans
(203)
Northern
Democrats
(116)
Southern
Democrats
(116)
1st Choice
No Bill
Amendment
Original Bill
2nd Choice
Amendment
Original Bill
No Bill
3rd choice
Original Bill
No Bill
Amendment
• Now consider the results of all these voters together…
Number of
voters
Republicans
(203)
Northern
Democrats
(116)
Southern
Democrats
(116)
1st Choice
No Bill
Amendment
Original Bill
2nd Choice
Amendment
Original Bill
No Bill
3rd choice
Original Bill
No Bill
Amendment
With these preferences, we find the amendment is preferred over the
original bill by both Republicans and Northern Democrats and
therefore, comparing those two options only, the amendment would
win 319 to 116. However, we also notice that the original bill is
preferred over “no bill” by all Democrats and would therefore win, by a
vote of 232 to 203, if only those options were considered.
Finally, notice that “no bill” is preferred over the amendment by a vote
of 319 (from Southern Democrats and Republicans) to 116 (from the
Northern Democrats.
• A strange thing has happened here. A paradox has occurred –
following a logical process with logical assumptions, we have a
result that is not logical.
• We have just found that, in this group of voters, the amendment is
preferred over the original bill, the original bill is preferred over
no bill and yet no bill is preferred to the amendment.
• Visually, the preferences create a never ending cycle. Suppose A is
the amendment, O is the original bill and N is “no bill”.