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Marie-Jean-Antoine-Nicolas Caritat
de Condorcet (1743-1794)
Gilbert Faccarello∗
Mathematics and Philosophy
Condorcet is considered as the last of the eighteenth-century French
philosophes who powerfully shaped the intellectual landscape in France
and Europe. Born on 17 September 1743 in Ribemont, in the province
of Picardie, he was first educated at the Jesuit school in Reims and
the celebrated Collège de Navarre in Paris. Possessed of a talent for
mathematics, he studied with the mathematician and philosophe Jean
Le Rond d’Alembert (1717–1783) — the co-editor, with Denis Diderot
(1713–1784), of the flagship of the French Enlightenment, the Ency-
clopédie, ou dictionnaire raisonné des sciences, des arts et des métiers
(1751–72). He quickly gained the reputation of a prominent géomètre,
∗Panthéon-Assas University, Paris. Email: gilbert.faccarello@u-paris2.fr. Home-
page: http://ggjjff.free.fr/. To be published in Gilbert Faccarello and Heinz D. Kurz
(eds), Handbook on the History of Economic Analysis, vol. 1, Cheltenham: Edward
Elgar, 2016.
1
Condorcet 2
his domains of predilection being integral calculus and probability the-
ory. But as many scientists and philosophes of the time he had an ency-
clopaedic mind, and he showed a great interest in the “sciences morales et
politiques” or “sciences sociales” (see, for example, Granger 1956; Baker
1975; Kintzler 1984; Crépel and Gilain 1989; McLean and Hewitt 1994).
During the 1760s and early 1770s, he became a disciple and friend of
Voltaire (1694–1778) and Turgot (1727–1781). He later published a cel-
ebrated Vie de M. Turgot (1786) — immediately translated into English
(1787) and much appreciated by the British reformers — and a Vie de
Voltaire (1789). A promising member of the clan of the Encyclopaedists,
he was quickly elected at the Académie des Sciences (1769) — of which he
became the secrétaire perpétuel in 1776 — and the Académie Française
(1782). Thanks to Turgot, he also held the official position of Inspecteur
des Monnaies from 1775 to 1791. His first publications in economics, such
as “Monopole et monopoleur” (1775) and Réflexions sur le commerce des
blés (1776), were made to support Turgot’s free trade program of reforms
during his ministry (August 1774–May 1776). After the fall of Turgot,
he turned back to mathematics and sciences but never abandoned his
political and philosophical concerns. This can be seen in particular in
Vie de M. Turgot, or in his attempts to apply mathematics and proba-
bility either to the traditional problems of insurance — for example, in
some 1784 texts for C.-J. Panckoucke’s Encyclopédie Méthodique: “Ab-
sent”, “Arithmétique politique (supplément)” or “Assurances maritimes”
— or to the fields of law (decisions to be taken by a panel of judges) and
elections. He shared Turgot’s project to transform the French political
system with a series of elected assemblies: he wished not only to define
their tasks but also sought the best way to organize ballots and decisions.
This last points were mainly developed in the voluminous and complex
Essai sur l’application de l’analyse à la probabilité des décisions rendues
à la pluralité des voix (1785), in Essai sur la constitution et les fonctions
des assemblées provinciales (1788), Lettres d’un bourgeois de New Haven
à un citoyen de Virginie, sur l’utilité de partager le pouvoir législatif entre
plusieurs corps (1788) or Sur la forme des élections (1789).
Condorcet 3
While not elected to the 1789 États Généraux du Royaume — the
convening of which is the emblematic starting point of the French Rev-
olution — he was an enthusiast supporter of the revolutionary process,
either as a member of the Commune de Paris or as a careful observer and
journalist in various newspapers. He was also the co-founder of a political
club, the Société de 1789, and two periodicals, Bibliothèque de l’homme
public in 1790 and Journal d’instruction sociale in 1793. Finally elected
to the Assemblée Législative in 1791 and to the Convention Nationale
in 1792, he demanded the deposition of the King and the proclamation
of the Republic after a failed attempt of the Royal family to leave the
country. His activities encompassed a wide range of subjects: money,
finance, taxes, public debt, public instruction and the new constitution
— continuing also his former fights in favour of the equality of men and
women and the abolition of slavery. As regards political economy proper,
the most significant texts from this period are “Sur l’impôt progressif”
and “Tableau général de la science, qui a pour objet l’application du cal-
cul aux sciences politiques et morales”, both published in 1793 in Journal
d’instruction sociale. After he refused to vote for the death sentence for
the King — he was against capital punishment — and criticized the Ja-
cobins in power, the Convention decreed his arrest. While hiding, he
wrote his philosophical testament, Esquisse d’un tableau historique des
progrès de l’esprit humain, posthumously published in 1795, which pro-
voked T.R. Malthus’s Essay on the Principle of Population (1798) and
formed the starting point of some nineteenth-century developments in
political philosophy. After having been arrested, Condorcet died in jail,
probably on 30 March 1794.
The large number of Condorcet’s writings on mathematics, philoso-
phy, politics and economics present a problem of interpretation (for a
brief history of some reactions, see Faccarello 1989). Commentaries gen-
erally referred to a vague theory of evolution and progress associated
with the 1795 Esquisse, and, after the Second World War, to his ideas
on elections to which Georges-Théodule Guilbaud (1952), Gilles-Gaston
Granger (1956), Duncan Black (1958) and Kenneth Arrow (1963) drew
Condorcet 4
attention — they had been almost forgotten for some 150 years, with the
exceptions of Edward John Nanson (1882 [1907]) and Charles Lutwidge
Dodgson (alias Lewis Carroll 1876 [1958]). While much is still to be done,
recent research has made it possible to identify a quite different intellec-
tual stature of Condorcet. Leaving aside the widely commented Esquisse
— which is a small part of a wider project, the Tableau historique des
progrès de l’esprit humain proper (Condorcet 2004), of which it was sup-
posed to be the Prospectus — it is first necessary to understand the main
characteristics of his approach.
Sensationism, Knowledge and Probability
Probability and the nature of knowledge
Unlike many of his contemporaries, including d’Alembert, but like Tur-
got, Condorcet thought that progress was possible in the new moral and
political sciences and that it was also possible to reach there the same
degree of “certainty” than in the more traditional fields of, for example,
physics, chemistry or astronomy (see, for example, the first pages of the
1785 Essai). This conviction, however, ought to be understood properly.
While it implies that the nature of knowledge is basically the same in all
fields of inquiry, this nature is such that nowhere is it possible to find
propositions that are absolutely certain. This is not only because no sci-
ence has achieved, or could ever reach, its highest degree of perfection.
The reason lies with the nature of knowledge itself. Following Locke
and Turgot’s sensationist philosophy, and insisting on the importance of
Turgot’s entry “Existence” in the Encyclopédie, Condorcet stressed that
any knowledge of the existence and properties of objects comes from our
senses and our ability to think about our sensations and combine them.
While it is also based on the idea that there exist constant laws for
the various observable phenomena, this constancy is only an hypothesis
and, by nature, this knowledge can never produce any absolute certainty,
whatever the field of inquiry — mathematics included because this hy-
Condorcet 5
pothesis also concerns the human understanding, and not only external
phenomena. It only leads to a more or less strong confidence that these
phenomena, in the same circumstances, will happen again in the future.
This is the reason why, when Condorcet speaks of “certainty”, he does
so only metaphorically to express a great degree of assurance — the word
“assurance” is, in his view, more adapted in this context (Condorcet 1785:
xvi, 1994a: 523), and a better choice than the ambiguous phrase “certi-
tude morale” (moral certainty). It is also why, in sciences and in everyday
life, this assurance is called by Condorcet a probability — founded on
past experience and measuring a “motif de croire” (reason to believe).
“The knowledge that we call certain is .. . nothing else than a knowledge
based on a very high probability” (1994a: 602) that in most cases it is
meaningless to calculate (Condorcet 1785: xiv). Hence his statement
that all propositions “belong to this part of the calculus of probability
where one judges the future order of unknown events on the basis of the
order of known events” (Condorcet 1994a: 291) and the parallel explicitly
made with a classical example in probability theory:
The reason to believe that, from ten million white balls and
one black, it is not the black one that I will pick up at the
first go, is of the same nature as the reason to believe that
the sun will not fail rising tomorrow, and these two opinions
only differ as to their lower or higher probability. (Condorcet
1785: xi)
However, Condorcet did not follow the sceptical tradition (for fur-
ther developments, see Rieucau 2003). He believed in the progress and
usefulness of knowledge, and he often denounced “the absurdity of abso-
lute scepticism” (Condorcet 1994a: 602). The systematic collectioning of
data and the organization of accurate experiences permit an undisputable
progress in sciences, and what happened in physics or astronomy will also
happen in the new fields concerning society. Politics or political economy,
with time, and with the knowledge of human nature based on sensation-
ist philosophy, are liable to approach the same degree of assurance in the
truths they establish.
Condorcet 6
Probability and the conduct of life
Condorcet’s probabilistic approach has also important consequences on
a more practical level. In all fields of life where decisions are to be
taken, Condorcet stresses, people almost always have to face uncertainty.
D’Alembert did not see that
in the sciences the aim of which is to teach how to act, as in
the conduct of life, man can content himself with higher or
lower probabilities, and that .. . the right method consists
less in searching rigorously proven truths than in choosing
among probable propositions, and above all in knowing how
to estimate their degree of probability. (Condorcet 1994a:
544)
In this perspective probability theory is an indispensable tool for es-
timating in an accurate way the data of the problems and the outcomes
of alternative choices, and this theory was precisely developing since the
seventeenth century (see Hacking 1975; Daston 1988; Hald 1990). Lively
controversies never ceased about the meaning of the main concepts of the
theory (about mathematical expectation, for example) and their use or
abuse in various applied fields, and one prominent critic was d’Alembert
himself. In defence of probability theory — and of the use of mathemat-
ical expectation — Condorcet developed an important reflection on the
nature and significance of its concepts, especially in his 1784–87 “Mé-
moire”, his 1785 Essai and in various other texts and manuscripts, such
as Éléments du calcul des probabilités. In his view, while the probabil-
ity of an event is a “purely intellectual consideration” (Condorcet 1994a:
289) that “does not pertain to the real order of things” (ibid.: 291) and
does not predict its occurrence — the contrary event can happen — nev-
ertheless “we judge of all the things of life from this probability and it
rules our conduct” (ibid.). This probability is the measure of our reason
to believe in the occurrence of this event.
It is finally to be noted that Condorcet also calls “probability” the
number of votes in favour or against a candidate or a proposal in an elec-
Condorcet 7
tion or decision-making process, particularly in his 1785 Essai. While
this could seem confusing, it is not in the perspective of voting as judge-
ment aggregation: in absence of any other usable evidence regarding the
relevant qualities of two candidates, the number of votes may be taken
as the best probabilistic indicator of those qualities.
From Political Arithmetic to “Social Mathematic”
The main question is how to use the calculus of probability in a legiti-
mate way. Calculation should be handled cautiously because it can be
dangerous in the hands of “charlatans” (Condorcet 1994a: 337): in pol-
itics, it is so easy to impress people with the use of some numbers in
order to influence their opinions and choices. Some “ridiculous” appli-
cations of calculus to political questions have also been made, but “how
many applications, just as ridiculous, have not been made in each part
of physics?” (Condorcet 1785: clxxxix). In his 1771–72 correspondence
with Piero Verri (Condorcet 1994a: 68–74), Condorcet had also criticized
Verri’s attempt, with the aid of the mathematician Paolo Frisi, to formal-
ize economic theory in the sixth edition of his Meditazioni sulla Econo-
mia Politica. He considered that this was a complex undertaking which
could not be achieved with simple and careless solutions — the same
erroneous method was still to be applied less than three decades later
by two géomètres, Nicolas-François Canard in his Principes d’économie
politique and above all Charles-François de Bicquilley in his Théorie élé-
mentaire du commerce, both works presented at the Institut in 1799 and
respectively published in 1801 and 1804 (Crépel 1998).
All these critiques notwithstanding, the use of calculus could no longer
be dispensed with. Condorcet stressed that with its help it is possible
to reason in a more precise way, to go further than “what reason alone
can do”, and avoid the negative influence of vague impressions due to im-
perfect knowledge, prejudices, interests or passions. In this perspective,
he had the ambitious project to develop “political arithmetic” into a sys-
tematic science — this field being defined as “the application of calculus
Condorcet 8
to political sciences” in his 1784 eponymous entry for the Encyclopédie
méthodique. In his eyes, the first attempts by William Petty or John
Graunt were almost insignificant. Serious things only started with the
works of Jan De Witt and, above all, Jakob Bernoulli (1654–1705) (Ars
Conjectandi, posthumously published in 1714) and his nephew Nicolas I
Bernoulli (1687–1759) (Dissertatio Inauguralis Mathematico-juridica, de
Usu Artis Conjectandi in Jure, 1709). Probability theory was used there
to solve economic and juridical questions of marine insurance, life an-
nuities, calculation of interest or the problem of the “absent” (after how
many years can an absent person be considered as dead with a sufficient
probability?). However, the science was new, and all remained to be
done. It is this same science that Condorcet, in an even more ambitious
way than before, called “social mathematic” in his 1793 “Tableau général”
(on the different editions of this important text and their shortcomings,
see Crépel and Rieucau 2005):
I prefer the word mathematic, although now no longer used
in the singular, to arithmetic, geometry or analysis because
these terms refer to particular areas of mathematics . . .
whereas we are concerned .. . with the applications in which
all these methods can be used. . . . I prefer the term so-
cial to moral or political because the sense of these words is
less broad and less precise. (Condorcet 1994b: 93–4, original
emphasis)
Condorcet died the following year and could not accomplish his pro-
gramme. But he had already some outstanding achievements to his
credit.
Economic Behaviour in the Face of Uncertainty and
Risk
In the first field of “social mathematic”, that is, at the individual level,
Condorcet’s developments were mainly in line with those of the Bernoullis.
But he went further, especially in the questions related to the problem
Condorcet 9
of the absent or marine insurance (Crépel 1988, 1989). In particular,
generalizing his analysis of the behaviour of both a merchant and his
insurer facing uncertainty and risk in maritime trade, he conceived of
any economic activity as an uncertain and risky undertaking — “under-
takings in which men expose themselves to losses in view of a profit”
(Condorcet 1994a: 396) — and used probability theory to describe the
entrepreneurs’ decisions to invest (Rieucau 1998). A parallel is made
with the traditional analysis of “fair” games of chance, in which a fair
stake is equal to the mathematical expectation of gain: but Condorcet
explains that, in economic activity, additional constraints and calcula-
tions arise because the analogy between a gambler and an entrepreneur
is somewhat misleading.
When a merchant makes a conjecture [fait une spéculation]
implying a significant risk, it is not enough that his profit be
such that the mean value of his expectations be equal to his
stake [sa mise] plus the interest that a riskless trade would
have brought him. In addition he must have .. . a very high
probability that he would not suffer a loss in the long run.
To submit this kind of project to calculus, one should thus
determine, for the funds that each trader could successively
employ in such a risky trade, what is the excess of profit that
he must obtain in order either to have a sufficient probability
not to lose his entire funds, or to lose only part of them, or
to just get them back, or to get them back with a profit.
(Condorcet 1986: 561–2)
However a second field for “social mathematic” is related to the col-
lective level (public economics, social choice). Here Condorcet is clearly
continuing Turgot’s analysis of public economics, who had already made
some definite advances concerning the political organization of a modern
state in a free society based on the respect of human rights, and the
nature of public interventions in markets — for example, the definition
and classification of public goods, a reflection on the nature of taxation
from a quid pro quo perspective, the taking into account of externalities
and the free rider problem (Faccarello 2006). Of particular interest are
Condorcet’s ideas on taxation and decision-making processes.
Condorcet 10
Rules for a Just and Optimal Taxation
Two significant developments on taxation are to be found in the 1793
paper “Sur l’impôt progressif” — they do not explicitly use mathematics
but they entail an implicit formalization. The first consists in providing
a theoretical proof of the fact that a progressive income tax complies
with justice: this is done in the “equal absolute sacrifice” perspective, as-
suming a decreasing marginal utility of wealth (one of Daniel Bernoulli’s
hypotheses) and an elasticity of the utility of marginal income with re-
spect to income greater than unity (Faccarello 2006: 26–30).
The second 1793 development on taxation consists in the determina-
tion of what is called today the optimal volume of public expenses and
taxation, with a reasoning that is probably the first to refer to an equilib-
rium at the margin. A question debated at that time was that a theory of
public finance cannot be limited to the affirmation that the state should
not spend too much and that the normal financing of its expenditure
should be made through taxation in a quid pro quo perspective. It is
also important to determine what public goods and services should be
produced, and in which quantities. The list of the goods and services
useful to society could be long and it is generally impossible to provide
them at once. Choices must be made, and, in a given period, a criterium
to determine the optimal volume of public spending is needed. The
essence of Condorcet’s answer is the following (Faccarello 2006: 19–21).
Amounts of public expenses may be classified according to the decreas-
ing order of utility they produce. One might then imagine (although
Condorcet does not do so explicitly) a plan in which one would have,
as abscissa, the successive volumes of public spending, and as ordinate,
the levels of utility engendered by each supplementary volume of expense
(the curve of decreasing “marginal” utility of public spending). But pub-
lic spending must be financed by taxes, taxation meaning a diminution
of the disposable income. As Condorcet accepted Bernoulli’s hypothesis
of a diminishing marginal utility of wealth, successive increases in public
spending necessarily entail an increasing marginal disutility of taxation.
Condorcet 11
As a consequence, it is also possible to imagine an increasing “marginal”
disutility curve for public expenditure; in the same schema as before,
this disutility is shown along the ordinate while the successive volumes
of taxation (equal to those of public spending) is represented along the
abscissa. The two curves cross. Public expenses “have a limit: the point
where the utility of the expense becomes equal to the evil generated by
the tax” (Condorcet 1793, in 1847-49, XII: 629). In other words, their
volume is determined by the point at which their marginal utility is equal
to the marginal disutility they entail, the “margins” being here broadly
defined.
But in a modern state, all these decisions about public expenses and
taxation are taken by an elected assembly. How to choose its members
and which decision-making process should they follow in order to take
just and true decisions?
“Social Mathematic” and Social Choices
The most spectacular example of “mathématique sociale” concerns elec-
tions 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 deci-
sions 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 exam-
ple, 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 inter-
ests and passions, from factions or lobbies. The “general will”, Rousseau
stressed, was to be distinguished from the “will of all”:
Condorcet 12
[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.) — prob-
ably 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 So-
cial. 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
Condorcet 13
should be adapted according to the degree of enlightenment of its mem-
bers (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).
The Jury Theorem
The first point concerns what has been called Condorcet’s “jury problem”
(Black 1958) or “jury theorem”. Let v(vfor “vérité”, that is, truth) be the
probability for each voter to make the right choice, and e(efor error) the
probability of being mistaken: e= (1−v). Suppose a dichotomous choice
situation (for example, is a person guilty or not guilty of a crime?) in
which the number of voters is nand qis the required majority expressed
in terms of a number of votes. For Condorcet, two questions are of
particular importance: (1) before the vote, what is the probability p
to obtain a decision complying with truth? (2) Once the decision is
taken, what is, for an external observer, the probability p∗that this
decision complies with the truth? In modern parlance (see, for example,
Granger 1956: 105–6), probability pis found using Bernoulli’s binomial
distribution. It is the sum, for all x,q≤x≤n, of the probability
vx(1 −v)(n−x)that a decision is true when it obtains xvotes, multiplied
by the possible number of occurrences n
x=n!
x!(n−x)! of this event:
p=
n
X
x=qn
xvx(1 −v)(n−x)
Probability p∗is found using the Bayes-Laplace theorem and is given
by:
Condorcet 14
p∗=vq
vq+ (1 −v)q
From the first equation, p→1when n→ ∞ if v > 0.5, but p→0
in the opposite case. (Note that in case v= 0.5,p= 0.5for all n.)
This is the “jury theorem”: in an assembly in which the probability for
each voter to make the right choice is greater than 0.5, the probability
for the outcome to be true increases with the number of voters — and
conversely, when v < 0.5, the probability of the outcome to be true is
a decreasing function of this number (Condorcet 1785: xxiii–xxiv, 6–9).
From the second equation — in which the number of voters plays no role
— it is possible to conclude that, all other things being equal, p∗is an
increasing function of vand q.
These are both positive and negative results. The positive side of the
story is the proposition that — under the very restrictive conditions noted
above — an assembly could collectively have a degree of wisdom superior
to its individual members, and that, if v > 0.5, this degree increases
with the number of voters. This is the kind of statement already made
by Aristotle when, examining the different possible political regimes, he
declared that it is possible that many individuals, of whom no one is
“virtuous”, are collectively better when they are assembled than the best
ones among them (Politics, III, 11, 1281-a). Condorcet’s theorem could
thus be taken as a powerful argument in favour of democracy.
The negative aspect arises if v < 0.5. Then the opposite conclusion
applies: “it could be dangerous to give a democratic constitution to an un-
enlightened people: a pure democracy could even only suit a people much
more enlightened, much more freed of prejudices than is any of those we
know in history” (Condorcet 1785: xxiv). In these circumstances, never-
theless, a pure democracy would be acceptable if decisions are “limited to
what regards the maintaining of safety, liberty and property, all objects
on which a direct personal interest can enlighten everybody” (ibid.; see
also ibid.: 135) — these being precisely among the “general” or “univer-
sal” objects in Rousseau’s approach. Otherwise the assembly, to decide
Condorcet 15
on an issue, could designate a committee composed of its most enlight-
ened members and then judge, not the decision itself, but whether the
decision does not hurt justice or some of the fundamental human rights
(ibid.: 7).
However, while aware of the novelty and complexity of his develop-
ments on the forms of elections or choices made in the various parts of
the book, Condorcet in the end relativized the importance of the choice
to be made between the different possible devices. For him, the key vari-
able remains the probability for each voter to be right or wrong; hence
his tireless action in favour of public instruction.
[T]he happiness of men depends less on the form of assem-
blies that decide their fate than on the enlightenment of those
who compose them, or, in other words, . . . the progress of
reason affects more their happiness than the form of political
constitutions. (1785: 136; see also ibid.: lxx)
The Condorcet Effect
What happens when there is more than one alternative? Voters, Con-
dorcet states, must rank them following a procedure of pairwise com-
parisons. 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 at-
tention 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,Band C, through pairwise comparisons (Con-
dorcet 1785: 120–21). For each voter, there are a priori eight possibilities
(“XY ” meaning “Xis preferred to Y”): (1) AB,AC ,BC; (2) AB,AC,
Condorcet 16
CB; (3) AB,CA,BC; (4) AB,C A,CB; (5) BA,AC,BC; (6) BA,AC,
CB; (7) BA,CA,BC; and (8) BA,C A,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 tran-
sitive 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
Condorcet 17
these occasions, thanks to reason and science, there exists a truth, never
imposed from above but which could be known provided those who de-
cide 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 ex-
amination 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 in-
terests”: the result would be the “will of all”, not the “general will”. Two
different conceptions of democracy and the role of the State are here at
stake.
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 accord-
ing 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)
See also:
Daniel Bernoulli; Formalization and Mathematical Modelling; French Enlight-
enment; Social Choice; Anne-Robert-Jacques Turgot; Uncertainty and Infor-
mation.
Condorcet 18
Note
After the death of Condorcet, his widow Marie-Louise Sophie de Grouchy
edited the Œuvres complètes de Condorcet with the collaboration of A.-A.
Barbier and the idéologues P.J.G. Cabanis and D.J. Garat (Condorcet 1804).
This edition, by no means complete, was followed four decades later by an-
other edition, the Œuvres de Condorcet, by his daughter Elisa, his son-in-law,
Arthur O’Connor, and the scientist and republican François Arago (Condorcet
1847–49). Nor is this edition complete: many important texts, like the 1785
Essai, are missing, as well as his entries for Encyclopédie méthodique or his
writings on mathematics and probability — for example, “Mémoire sur le cal-
cul des probabilités” (published by instalments, 1784–87) or Éléments du calcul
des probabilités et son application aux jeux de hasard, à la loterie et aux juge-
mens des hommes (1789–90, posthumously published in 1805). Moreover, in
both editions, a huge amount of manuscripts were disregarded: it was only
recently that they started to be explored systematically (see, for example,
Condorcet 1994a, 2004, two models of edition). His correspondence is now
also re-examined (Rieucau 2014).
References and further reading
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Barry, B. (1964), ‘The public interest’, Proceedings of the Aristotelian Society,
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—– (1965), Political Argument, London: Routledge and Kegan Paul, New
York: Humanities Press.
Condorcet 19
Black, D. (1958), The Theory of Committees and Elections, Cambridge: Cam-
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