“Social Mathematic” and Social Choices
The most spectacular example of “mathematique sociale” concerns elections or, more generally, social choices: it deals with the way in which to take decisions in any kind of assembly, be it a political assembly or a tribunal.
The subject was of foremost importance because Condorcet shared Turgot’s ideas of political reforms and because of the discussions Condorcet had with Turgot and Voltaire about the problem of the decisions of justice. But the subject was also important because Condorcet’s aim was to develop some ideas presented by Jean-Jacques Rousseau in Du contrat social (1762), a treatise Turgot himself had praised, and in particular to clarify Rousseau’s concept of “general will” (see, for example, Barry 1964, 1965: 292-3; Baker 1975: 229-31; Grofman and Feld 1988; Estlund et al. 1989). It was not clear how this “general will” could be known, especially when voters could not abstract from their own interests and passions, from factions or lobbies. The “general will”, Rousseau stressed, was to be distinguished from the “will of all”:[T]he general will is always right [droite] and always tends toward the public utility. But it does not follow that the people’s deliberations always have the same rectitude.... There is often a considerable difference between the will of all and the general will. The latter considers only the common interest, while the former considers private interest and is merely a sum of particular wills. (Rousseau 1762 [2012], II, iii: 182)
Moreover Rousseau’s statement that if we remove “from these same wills the pluses and minuses, which mutually cancel each other out,... the remaining sum of the differences is the general will” (ibid.) - probably alluding to differential and integral calculus (Philonenko 1986) - was puzzling.
Condorcet’s 1785 Essai deals with the various ways to organize a vote, to fix the majority needed for the decision, and to estimate their relative advantages - building, as G.-G. Granger (1956: ch.
3) called it, a model of “homo suffragans”. The cases studied are numerous, and in this also Condorcet’s project was realized only in part: starting with a set of strong simplifying hypotheses, the analysis becomes only programmatic when some of these hypotheses are relaxed. In the first part of the book (Condorcet 1785: xxi-xxii, 3), it is supposed that voters (1) are equally enlightened, (2) try honestly to answer the question asked (nobody tries deliberately to influence others, there are no lobbies, no parties), (3) have only the public good in mind and abstract from their own interests.All these hypotheses Rousseau had already invoked in Contrat Social. Condorcet’s approach is however more detailed and systematic, with some significant differences:
(1) the object of the vote must not necessarily be a “general object”, that is, a law, but also any decision which needs to be taken in the public or private sphere; (2) the outcome of the voting process must comply with “truth” (the voting process is a collective quest for “truth”) and not only be “right” and honest because emanating from the assembly of virtuous citizens; (3) in the political sphere, Condorcet is in favour of a representative assembly: the most important thing is the truth of the decision, and the size of the assembly should be adapted according to the degree of enlightenment of its members (below); (4) in this perspective, Condorcet introduces an additional and central variable, the probability for each voter to make the “true” choice, and an additional simplifying assumption: this probability is the same for all.
Note that Condorcet also formulated the condition of independence of irrelevant alternatives (Young 1988; McLean 1995). It is in this context that the attention focused on two main points, stated for the most simple case in the first pages of the Essai (1785: 3-11).