Marie-Jean-Antoine-Nicolas Caritat, Marquis de Condorcet
Condorcet was born in 1743 in Ribemont near Saint-Quentin (North-East of Paris). He was also educated by the Jesuits, first privately and then at the College des Jesuites in Reims.
He studied mathematics in Paris and at 26 he entered the Academie Royale des Sciences. He was a friend of Turgot and, as such, was interested in economics but also in politics and in the theory of elections. He was elected to the Legislative Assembly in 1791, and then at the Convention in 1792. He contested in vain the Assembly’s right to judge the King and voted against capital punishment. He wrote the famous Esquisse d’un Tableau Historique des Progres de l,Esprit Humain while he was hiding. Finally, he was arrested and died on 7 April 1794, of poisoning or exhaustion.The 1785 Essai sur l’application de l’analyse a la probability des decisions rendues a la pluralite des voix is an impressive piece of work. It has nearly 500 pages if we include the “Discours preliminaire”. While it is the most important book Condorcet devoted to elections, it is not the only one. Other important works include, among others, the Lettres d’un bourgeois de New Heaven a un citoyen de Virginie, sur l’inutilite departager lepouvoir legislatif entre plusieurs corps, and the Essai sur la constitution et les fonctions des assem- blees provinciales, both published in 1788 (see Condorcet 1986). Most scholars have focused their attention on the “Discours preliminaire” for several reasons, an obvious one being that some pages of the main text are covered with long probability calculations that are very similar, at first sight, to expressions we find today in works about the probability of pathologies for specific voting rules (see, for instance, Gehrlein and Lepelley 2011). However, according to Bernard Bru and Pierre Crepel (1994), the main part of the Essai cannot be eschewed.
In particular, according to them, how could we explain why some crucial parts gave rise to contradictory interpretations?The basic theme of the 1785 Essai concerns the probability of taking a correct decision. This is the now famous Condorcet’s jury theorem, where we have members of a jury for whom the probability to have the correct opinion is given by v and to be in error is given by e (= 1 - v). If v is greater than 0.5, majorities are more likely to select the correct opinion and this likelihood will increase with the number of voters. However, Condorcet was not certain that v > e, and, since with e > v, the result would be inversed, he was rather prudent. He wrote:
The assumption that e > v is not absurd. For many important questions either complex or under the influence of prejudices or passions, it is likely that a poorly educated man will have an erroneous opinion. There are, consequently, a great number of points for which, the more we increase the number of voters, the more we can fear to obtain, with plurality, a decision in contradiction with truth so that a purely democratic constitution would be the worst of all for all these objects on which the people would not know the truth. (English translation of Condorcet 1785: 6-7)
Condorcet then recommended that only enlightened men be attributed the prerogatives to make proposals of law. The popular assemblies would not be asked to vote on whether the law is useful or dangerous, but only if it is against justice or against the primary rights of men. A “pure” democracy could only be good for a very well educated people, so well educated that there had never been such a people.
Condorcet is most known now for the example showing that majority rule could generate a cycle. Suppose there are 60 voters and three candidates A, B and C. The rankings are given by the following - 23 voters: ABC, meaning A ranked first, B, second and C third; 17 voters: BCA; 2 voters: BAC; 10 voters: CAB; 8 voters: CBA.
A majority of voters (33) can be seen to prefer A to B, a majority (35) prefer C to A and a majority (42) prefer B to C. Of course one can also obtain a cycle very simply with three voters whose rankings are respectively ABC, BCA and CAB.In the main text of the Essai, Condorcet proposed a method to deal with this problem. This method, rather obscure in Condorcet’s words, has been the object of a reconstruction by, among others, Young (1988) and Monjardet (1990).
Condorcet also alluded to Borda and gave an example showing that Borda’s rule could select another candidate than the Condorcet winner. Suppose 81 voters have the following rankings over three candidates A, B and C- 30 voters: ABC; 1 voter: ACB; 10 voters: CAB; 29 voters: BAC; 10 voters: BCA; 1 voter: CBA.
Candidate A is a Condorcet winner (he beats B only by 41 against 40), but B is the Borda winner. That A is a better candidate than B seems obvious to Condorcet. This was the beginning of a long debate which still goes on today (see, for instance, Dummett 1984, 1997; Saari 1995, 2006; Risse 2005; Emerson 2007). Of course, again, a very simple example is possible, for instance, with 19 voters. 10 voters: ABC and 9 voters: BCA. Candidate A is a Condorcet winner (and a plurality winner) but B is the Borda winner. On page clxxix, Condorcet (1785) alluded to some kind of strategic voting indicating that Borda’s rule is not immune to this possible voters’ behaviour.
In Lettres d’un bourgeois de New Heaven, Condorcet proposed that the Condorcet winner be selected if there is one, and, if there is none, he proposed to select the candidate that won the most pair-wise confrontations (again, this is Copeland’s method suggested by Lull long before). For the selection of committees of k members to be chosen in a set of 3k candidates, Condorcet recommended in 1792 that each voter partitions the set of candidates in three sets of k candidates and ranks the three sets (a set of k most-preferred candidates, a set of k intermediately-preferred candidates, and a set of k least preferred candidates). Each voter indicates his k most-preferred candidates and, as a supplementary list, the k “intermediate” candidates. If at least k candidates obtain a majority, this is done by selection of the k candidates who have obtained the most votes. If not, one considers the supplementary lists. It is at the same time original and unorthodox, but still to be formally studied.
The works of Condorcet prompted a number of studies by Pierre-Simon Laplace, Simon Antoine Jean Lhuillier, Sylvestre-Franςois Lacroix, Jose Isidore Morales and Pierre-Claude-Franςois Daunou (see McLean and Urken 1995).