Monday, April 02, 2018

Condorcet’s Paradox


When I made a walk through Paris, two weeks ago, with the intention to take pictures of statues of philosophers, Condorcet was not on my list. I passed his statue by chance, simply because I had turned into the wrong street. But once I saw the statue, of course, I knew that he was an important Enlightenment philosopher from the 18th century. His ideas were and are still modern. He stood for a liberal economy; he was an advocate of human rights, especially women’s rights and Black’s rights (he actively worked for the abolition of slavery); he opposed the death penalty; he stressed the importance of education and wished free public education for all citizens, including women; and he strived for a constitutional republican political system. When Condorcet criticized the new French constitution of 1793, he was considered a traitor and he had to flee. In 1794 he was caught and he died in mysterious circumstances in prison.
Marquis de Condorcet was not only an important and progressive political thinker, he was also a mathematician. Combining both interests, he developed a voting system, which came to be known as the Condorcet Method, but he also discovered that this method can sometimes lead to what is now known as Condorcet’s Paradox or the Paradox of Voting. Let me concentrate on the paradox.
In an electoral system based on the Condorcet Method the voters vote for candidates by arranging them in their order of preference. For keeping it simple, let me assume that there are three voters, namely X, Y and Z, and three candidates for a certain political function, namely A, B and C. Look how the voters vote:
X prefers    A to B and B to C
Y prefers    B to C and C to A
 Z prefers    C to A and A to B.
Let’s compare the candidates pairwise:
A > B (= A is preferred to B) by two voters, namely X and Z, against one (Y).
B > C (= B is preferred to C) by two voters, namely X and Y, against one (Z).
This would make that A > B > C, or A is preferred to B and B to C.
However, C has also received two votes, for
C > A (= C is preferred to A) by Y and Z, while only X prefers A to C.
This would make that A > B > C > A, which is not possible, of course, since nobody can be preferred to himself at the cost of himself (resp. herself). Voilà the paradox: Everybody becomes first but no one wins. Currently nowhere in the world a Condorcet Method of voting is used in government elections. However, some private organisations do. I suppose that usually these elections function well, but nevertheless the risk remains that everybody wins, although everybody is a loser.

Sources: Wikipedia and Mário Filipe Pinhal, “Condorcet’s Paradox”, on http://www.cs.uu.nl/docs/vakken/ig/archive/presentations/2007/IG%202007%20-%20Mario%20Filipe%20Pinhal%20-%20Introductory%20-%20Condorcets%20Paradox.pdf

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