The Condorcet Effect
What happens when there is more than one alternative? Voters, Condorcet states, must rank them following a procedure of pairwise comparisons. What has been called the “Condorcet winner” is the proposal or candidate who would win a two- candidate election against each of the other proposals or candidates (for a possible tension between Condorcet’s probabilistic and social choice approach, see Black 1958: ch 18; Young 1988).
In this context, the second main point which attracted the attention in the 1785 Essai is what G.-T. Guilbaud called “the Condorcet effect” and K. Arrow called the “paradox of voting”, which expresses the possible intransitivity of social choices resulting from the aggregation of individual choices made by rational voters.Suppose that voters have to express their preferences among three candidates or proposals A, B and C, through pairwise comparisons (Condorcet 1785: 120-21). For each voter, there are a priori eight possibilities (“XY” meaning “X is preferred to Y”): (1) AB, AC, BC; (2) AB, AC, CB; (3) AB, CA, BC; (4) AB, CA, CB; (5) BA, AC, BC; (6) BA, AC, CB; (7) BA, CA, BC; and (8) BA, CA, CB. A rational voter will never choose choices
(3) and (6) which are not transitive. But, at the social level, outcomes (3) and (6) are possible. Among 31 voters, imagine that nine vote for (1), two for (2), seven for (4), four for (5), six for (7) and three for (8). Eighteen voters prefer AB against 13, 19 BC against 12, and 16 CA against 15, with the “cycling” result ABCA.
This outcome has significant consequences for any social choice theory based on an aggregation of individual choices. The logic of the problem has been made explicit in the general framework of Arrovian social choice theory: Arrows’s so-called impossibility theorem shows that there is no procedure for the aggregation of individual choices guaranteeing a transitive social ranking, while at the same time respecting some seemingly mild axioms expressing “individualistic concerns” (that is, that the social choice should reflect individual choices at least in some minimal way).
Condorcet, however, did not think that the paradox of voting was such an important problem, even when the numbers of alternatives and voters grow - and it has been shown that the probability to have a Condorcet effect quickly increases with them. He did not get locked in a logical dilemma, but proposed solutions out of the impasse (Black 1958: ch. 18; Young 1988, 1995; Monjardet 2008), which, in modern terms, are the maximum likelihood estimation, Kemeny’s rule or the search for a median in a metric space. In particular, in the three-alternative cases dealt with above, one simple solution (Condorcet 1785: 122) consists in respecting the total number of votes that each candidate or proposal obtains against the two others. In the above example, AB and AC obtain together 18 + 15 = 33 votes, BA and BC 13 + 19 = 32 votes and CA and CB 16 + 12 = 28 votes. The winner is A.
To conclude, an essential aspect of Condorcet’s thought must again be emphasized. All his developments are aimed at discovering “the truth”, even in decisions that do not deal with justice but with choosing the right proposal or candidate in an assembly. He was convinced that on all these occasions, thanks to reason and science, there exists a truth, never imposed from above but which could be known provided those who decide are enlightened enough and follow the right procedure. As Rousseau had already insisted, a member of an assembly, when voting, must not express his own preferences but decide whether the proposal under examination does or does not comply with the common good. The “will of all” can differ from the “general will” whenever individuals are unable to abstract from their particular or partisan interests. The same is true with Condorcet. Hence, while Arrow’s impossibility theorem can take as a starting point the Condorcet cycle, there is a fundamental difference between the problems Condorcet and Arrow are concerned with. The distinction between preference and judgement is concerned - and the recent developments of the theory of judgement aggregation, in a way initiated by Guilbaud (Mongin and Dietrich 2010, Mongin 2012), while more faithful to Condorcet, do not cancel the difference.
For Condorcet, the problem does not consist in aggregating individual preferences and obtaining social choices respecting the “particular wills” or “private interests”: the result would be the “will of all”, not the “general will”. Two different conceptions of democracy and the role of the State are at stake here.When he [a man] submits himself to a law which is contrary to his opinion, he must say to himself: It is not here a question of myself alone, but of all; I thus must not behave according to what I believe to be reasonable, but according to what all, abstracting, like me, from their opinion, must consider as complying with reason and truth. (Condorcet 1785: cvii, emphasis in the original)
Gilbert Faccarello
See also:
Daniel Bernoulli (I); Formalization and mathematical modelling (III); French Enlightenment (II); Social choice (III); Anne-Robert-Jacques Turgot (I); Uncertainty and information (III).