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 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 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) 1 st ChoiceNo BillAmendmentOriginal Bill 2 nd ChoiceAmendmentOriginal BillNo Bill 3 rd choiceOriginal BillNo BillAmendment

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) 1 st ChoiceNo BillAmendmentOriginal Bill 2 nd ChoiceAmendmentOriginal BillNo Bill 3 rd choiceOriginal BillNo BillAmendment Now consider the results of all these voters together…

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. Number of voters Republicans (203) Northern Democrats (116) Southern Democrats (116) 1 st ChoiceNo BillAmendmentOriginal Bill 2 nd ChoiceAmendmentOriginal BillNo Bill 3 rd choiceOriginal BillNo BillAmendment

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”.