Old and new political arithmetic and “social mathematic”

As in the case of population - the field where the progress in techniques of analysis was the most striking - calculations in the commercial and financial fields also assumed a more theoretical flavour when they started to use the new probability theory.

Probability theory and socio-economic themes

Reasoning on risk and uncertain statements already had a long history,^[135] but system­atic reasoning was not developed until the second half of the seventeenth century with Blaise Pascal (1623-1662), Christiaan Huygens and Jakob Bernoulli (1654­1705)^[136] in particular. It was developed during the eighteenth century by Abraham de Moivre (1667-1754) - a French protestant who fled to England after Louis XIV's revocation of the Edict of Nantes in 1685 - Thomas Bayes (1702-1761) and Rich­ard Price in England,^[137] and Pierre Remond de Montmort (1678-1719), Condorcet and Laplace in France. These authors first dealt with calculations of the probability of winning in “games of chance” (“alea” being the Latin word for “game of dice”, or “chance”) or questions linked to these games: what is a fair stake to participate in a game, or how to divide the stakes in a fair way if the game is interrupted before its end, etc. Currently used today to cover the entire field, the word “probability” came in fact from the judicial field and referred, for example, to already widely discussed questions like the estimation of the veracity of a testimony, the guilt or innocence of a person, or commercial and inheritance problems. Jakob Bernoulli used the phrase “art of conjecturing” and proposed to call it “stochastics”. The first books to be pub­lished were Jakob Bernoulli’s Ars Conjectandi (in Latin, published posthumously in 1713) and Montmort’s 1708 Essay d’analyse sur les jeux de hazard (second edition:

Montmort 1713).

The mathematician goes only from the true to the true or from the false to the true by ab absurdo arguments. They do not know that middle which is the probable, the more or less probable. In this respect, there is not more or less in mathematics.

([1720-55] 1964, 957; [1720-55] 2012, 54)

As in the case of insurance, the fact that the developments of probability theory were initially linked to games of chance - an invention of the Devil according to some theologians, and therefore considered with hostility by the Church - did not at first favour its moral reputation. “Games of dice and of cards... where the gain principally depends on chance, are not only dangerous recreations”, Franςois de Sales wrote in his influential Introduction a la vie devote, they are “bad and blame­worthy, and... prohibited by civil and ecclesiastical laws”. The gain made in this way is reprehensible: “The gain, which should be the price of industry, is made the price of luck, which does not deserve any price, because it does not depend on us” (Sales [1608] 1730, 362-3). Moreover, these kinds of game are more than simple “recreation”: they totally monopolise the attention of the players and are thus “vio­lent occupations”; and the winner’s joy is unfair because it implies the loss and displeasure of somebody else ([1608] 1730, 363-4). It is significant that in 1738, in the dedication^[138] of the second edition of The Doctrine of Chance, Abraham de Moivre still had to rebut the presumption of amorality levelled at probability theory.

There are many People in the World who are prepossessed with an Opinion, that the Doctrine of Chances has a Tendency to promote Play, but they soon will be undeceived, if they think fit to look into the general Design of this Book....

Later in the century, Condorcet felt obliged to stress that, if Pascal and Pierre de Fermat (1607-1665) chose games of chance for the application of the theory of combinations to contingent events, this is only because it was the easiest way to proceed, and not because they were frivolous people (Condorcet 1994a, 339). In his 1786 “Discours sur l’astronomie et le calcul des probabilites”,^[139] he insisted that the only utility of this first application was

to prove how futile are all the expectations, of which those who give them­selves over to these games are too often the dupes and the victims: perhaps a mathematician, showing the ridiculousness of their speculations, would have a greater impact than a moralist stating their disastrous consequences.

(1994a, 602-3)

This kind of charge against probabilities, however, faded away as the theory became more and more complex on the mathematical level and showed its useful­ness in practical life. Jakob Bernoulli’s intention - in the fourth and unfinished part of Ars Conjectandi titled “Usum & applicationem precedentis doctrine in civilibus, moralibus & rnconomicis” (“The use and application of the preceding doctrine in civil, moral and economic matters”) - was to apply the mathematical developments made for games of chance to social and economic matters, and he considered it as the main part of his work. In the same perspective, his nephew Nikolaus Bernoulli (1687-1759) wrote his 1709 thesis (Dissertatio inauguralis mathematico-juridica de usu artis conjectandi in jure) and published an abridged version of it (“Specimina artis conjectandis, ad quaestiones juris applicatae”, 1711) to show how probability theory could deal with legal and economic questions such as the reliability of witnesses and of suspicions, marine insurance, the probability of human life, life annuities or the problem of the “absent” (after how many years can an absent person be considered as dead?) - all fields where the logic of the probable was already present.

If some objections to the probabilistic approach still developed later in the cen­tury, they had nothing to do with the former ones and were simply part of the usual scientific debate. The most interesting were expressed by the mathematician and philosopher Jean Le Rond d’Alembert (1717-1783),^[141] a dissenting voice who induced his disciples Condorcet and Laplace to find new solutions.

But could probability theory play a role in the development of political economy proper, that is, go beyond calculations in specific fields of social and economic life dealing with a great number of data (population, mortality tables, insurances, quantification of risks, etc.)? Laplace was sceptical. In 1795, in his last lesson at the short-lived Ecole normale, devoted to probability, he asked whether probability theory could also be applied to political economy “and improve it”. His answer was rather negative as far as a priori reasoning is concerned, and he subsequently remained vague and cautious about this subject.^[142]

The questions raised by this science are so complicated and depend on so many unmeasurable or unknown elements that it is impossible to solve them a priori.

(Laplace 1795 [1800], 73)

Condorcet, political arithmetic and “social mathematic”

With Condorcet, the question of the use of probability theory took on a new dimen­sion and received an ambitious answer. For him, the use of probability theory in moral sciences is legitimate, as there is no difference in nature between these sciences and those which had been successfully developed since the seventeenth century. Tur­got, Condorcet wrote approvingly in the introduction to his Essai sur l'application de l'analyse a la probability des decisions rendues a la pluralite des voix, “was con­vinced that the truths of the moral and political sciences are likely to have the same certainty as those which form the system of the physical sciences” - astronomy being, for example, “close to mathematical certainty” (1785, i). This conviction, however, needs to be understood properly. While it implies that the nature of knowledge is basically the same in all fields of inquiry, it is such that nowhere it is possible to find propositions that are absolutely certain. Insisting on the importance of Turgot’s entry “Existence” in the Encyclopedie, 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. The idea that there exist constant laws for the vari­ous observable phenomena is only a hypothesis: by nature, this knowledge can never produce any absolute certainty, whatever the field of inquiry - mathematics included, because the hypothesis also concerns human understanding, and not only external phenomena.

This is the reason why, when he speaks of “certainty”, Condorcet does so only metaphorically to express a great degree of “assurance” - the word “assurance” was judged by him more suitable in this context (Condorcet 1785, xvi, 1994a, 523) and a better choice than the ambiguous phrase “certitude morale” (moral certainty). “The knowledge that we call certain is... nothing other than knowledge based on a very high probability” (1994a, 602) which is, moreover, meaningless to calculate in most cases (1785, xiv).^[143] Hence his statement that all propositions belong to this part of the calculus of probability “where one estimates the future order of events on the basis of their past order or, to be more precise, the future order of unknown events on the basis of... known events” (1994a, 291), and his reference to the classical urn model 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 to rise 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 (Rieucau 2003). He believed in the progress and usefulness of knowledge, and he often denounced “the absurdity of absolute scepticism” (1994a, 602). The systematic collection of data and the organisation of accurate experiences permitted undisputable progress in the sciences, and what happened in physics or astronomy would also happen in the new sciences of society. Over time, politics or political econ­omy were likely to approach the same degree of assurance in the truths they establish.

The question then is how to use probability theory and mathematics in a legiti­mate way. Calculation should be handled cautiously because it can be dangerous in the hands of “charlatans” (Condorcet 1994a, 337): in politics, it is so easy to impress people with the use of numbers in order to influence their opinions and choices. Some “ridiculous” applications of calculation to political questions have also been made, but “how many applications, just as ridiculous, have not been made in each part of physics?” (1785, clxxxix). All this notwithstanding, the use of probability could no longer be dispensed with. With its help, it is possible to reason in a more precise way and to avoid the negative influence of vague impressions due to imperfect knowledge, prejudices, interests or passions:

Almost everywhere [in the Essai] one will find results which comply with what the simplest reason would dictate; but it is so easy to blur reason with sophisms and vain subtleties that I would nevertheless feel happy if all I have done is to support a single useful truth with the authority of a mathematical proof.

(1785, ii)

But it is possible to go beyond “what reason alone can do.”^[144] In this perspective, he planned to develop “political arithmetic” into a systematic science. The first attempts by Petty or Graunt were almost insignificant, he states, and serious things only started with the works of Jan De Witt - “the illustrious and unfortunate Jan De Witt sensed that political economy, as all science, must be subjected to the princi­ples of philosophy and to the precision of calculation” ([1794] 2004, 380) - and, above all, Jakob and Nikolaus Bernoulli. Hence, the wide and ambitious defini­tion Condorcet gave of political arithmetic in the supplement he wrote in 1784 to the Diderot entry “Arithmetique politique” on the occasion of its republication in the first volume of the series Mathematiques of the Encyclopedie methodique: “Political arithmetic, in its wider sense, is the application of calculation to politi­cal sciences” (1994a, 483). “The application of the calculation of combinations and probabilities to these sciences [the ‘social art’]... is the only means to confer upon their results an almost mathematical precision and to estimate their degree of certainty or likelihood” ([1794] 2004, 447).

It is this same science that Condorcet called “social mathematic” in his 1793 “Tableau general de la science qui a pour objet l’application du calcul aux sciences politique et morales”:

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 social to moral

or political because the sense of these words is less broad and less precise.

(1994b, 93-4)

Condorcet used probability theory in many fields, especially those where decisions are to be taken in the face of uncertainty and imperfect information, to estimate the outcomes of alternative choices. These applications, in line with those of the Bernoullis, first regard the traditional fields of insurance, life annuities, tontines or the problem of the “absent”, for example, (Crepel 1988, 1989) but also the activity of any entrepreneur who - as Richard Cantillon already insisted in his Essai sur la nature du commerce en general - always acts in an uncertain world. Generalising his analysis of the behaviour of both a merchant and his insurer facing uncertainty and risk in maritime trade, Condorcet conceived of any economic activity as an uncertain and risky undertaking - “undertakings in which men expose themselves to losses in view of a profit” (1994a, 396) - and used probability to describe the entrepreneur’s 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: Condorcet explains that, as additional constraints and calculations arise in economic activity, the analogy between a gambler and an entrepreneur is somewhat misleading. As he wrote in his posthumously published Elements du calcul desprobabilites:

When a merchant makes a conjecture [fait une speculation] 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 calculation, 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.

(1805, 117)

But this statement concerning entrepreneurs in general mainly remained program­matic. By contrast, a second field for “social mathematic”, related to the collective level (public economics, social choice), is more developed.^[145] Of particular inter­est are Condorcet’s ideas about decision-making processes, which originate in his discussions with Turgot, especially on juries and more generally on any assembly susceptible to making decisions through voting and some majority rule. At the Academie royale des sciences, the question of the organisation of votes had already been discussed by Jean-Charles de Borda (1733-1799) in a memoir presented on 16 June 1770 - of which we have no record unless it is the one presented and pub­lished by the Academie in 1784 (Borda 1784) - on the occasion of a discussion of the rules to adopt for the election of academicians. Borda criticised the usual way of organising a vote. Suppose, he wrote, three candidates A, B and C, and 21 voters. A gets 8 votes, B 7 and C 6. With a simple relative majority rule, A is thus elected. But suppose further that the 13 electors who voted in favour of B and C also prefer, respectively, C and B to A: in this case, the former result contradicts the judgement based on these preferences (Borda 1784, 657-8). This is the reason why a good form of election “must give the electors the means of pronouncing on the merits of each candidate [‘sujet’] and to compare them successively with the merits of each of his competitors” (1784, 658-9).

This criticism is “very important and absolutely novel”, Condorcet wrote in 1784 in his presentation of Borda’s memoir in Histoire de l'Academie royale des sciences (in Condorcet 1994a, 359; see also 1785, 119). But while accepting the criticism, he found Borda’s proposal of a new method unsatisfactory. In a nutshell, the Borda count consists in giving marks to candidates standing for an election or running for office. In the presence of n candidates, each voter gives a mark of n to the one arriving first in his or her preference, (n - 1) to the second, and so on, ending with a mark of 1 for the least preferred. The winner is the candidate who gets the highest mark once all voters’ choices have been added up.^[146] Condorcet’s opinion is that this method could sometimes suffer from the same defect as the usual method it is sup­posed to correct (Condorcet 1785, clxxvii-clxxix, 1788, I, “Note premiere”).

“Social mathematic” and social choices

In dealing with how to take decisions in any kind of assembly, be it a political assembly or a tribunal, Condorcet’s aim was to develop some ideas presented by Jean-Jacques Rousseau (1712-1778) in Du contrat social (1762) - a treatise Turgot himself had praised - and to clarify Rousseau’s concept of “general will” (see, for example, 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 distance themselves from their own interests and passions, from factions or lobbies. The “general will”, Rousseau stressed, must 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 deliberations of the people 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)

Condorcet’s 1785 ambitious Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix deals with the various ways to organ­ise a vote, to fix the majority needed for the decision, and to estimate their relative advantages - building, as G.-G. Granger (1956, Chapter 3) called it, a model of “homo suffragans”. The cases studied are numerous, and here too, Condorcet’s pro­ject was only partly achieved: starting with a set of strong simplifying hypotheses, the analysis becomes mainly programmatic when some of these hypotheses are relaxed. In the first part of the book (1785, xxi-xxii, 3), Condorcet supposes 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) and (3) have only the public good in mind, ignoring their own private interests.

Rousseau had already set out these hypotheses in Contrat Social. Condorcet’s approach is, however, more detailed and systematic, with some significant differences:

(1) the object of the vote can be any decision which needs to be taken in the public or private sphere, and not necessarily a “general object” (a law) as in Rousseau; (2) the outcome of the voting process must not only be “right” and honest because emanat­ing from the assembly of virtuous citizens (nor must it deal with liberty and utility, as in the Ancient peoples whose concern, in assemblies, was only to counterbalance the passions and the interests of different groups in society): it must also comply with “truth” - the voting process is a collective quest for “truth”, on the model of a jury in a tribunal, what was called an epistemic approach to democracy; (3) in the political sphere, Condorcet was 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; (4) in this perspective, Condorcet intro­duced an additional and central variable, the probability for each voter of making the “true” choice, and an additional simplifying assumption: this probability is the same for all.^[147] Needless to say, he was conscious that all these hypotheses were strong theoretical assumptions and he progressively tried to relax some of them. On this basis, neverthe­less, two results are particularly striking (1785, 3-11): in the literature, they are known as the “jury theorem” and the “Condorcet effect” or “paradox of voting”.

The jury theorem

Let v (v for “verite”, that is, truth) be the probability for each voter of making the right choice, and e (e for “erreur” or error) the probability of being mistaken: e = (1 - v). Suppose a dichotomous choice (for example, is a person guilty or not guilty of a crime?) in which the number of voters is n and q is the required major­ity in terms of a number of votes. Condorcet asked two questions: (1) before the vote, what is the probability p of obtaining a decision complying with the truth?

(2) once the decision is made, what is, for an external observer, the probability p* that this decision complies with the truth?^[148] In modern parlance (see, for exam­ple, Granger 1956, 105-106), probability p is found using Bernoulli’s binomial distribution. It is the sum, for all x, q ≤ x ≤ n, of the probability v^x (1 - v)” ^x that a decision is true when it obtains x votes, multiplied by the possible number of occurrences of this event:

that is,

Probabilityp* is found using the Bayes-Laplace theorem and is given by

From Equation (1), p → 1 when n → ∞ if v > 0.5; p → 0 if v < 0.5; and in case v = 0.5, p = 0.5 for all n. This is the “jury theorem”: in an assembly in which the probability for each voter of making the right choice is greater than 0.5, the prob­ability of the outcome being true increases with the number of voters. Conversely, when v < 0.5, the probability of the outcome being true is a decreasing function of this number (1785, xxiii-xxiv, 6-9). From Equation (2) - in which the number of voters plays no role - and all other things being equal, p* is an increasing function of v and q.

The positive side of the story is the proposition that - under the very restrictive conditions noted earlier - an assembly could collectively have a degree of wisdom superior to that of its individual members, and, 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 than the best ones among them when they are assembled (Politics, III, 11, 1281-a). Condorcet’s theorem could thus be understood as a powerful argument in favour of democracy. However, if v < 0.5, the opposite conclusion applies:

it could be dangerous to give a democratic constitution to an unenlightened people: a pure democracy could even only suit a people much more enlight­ened, much more free of prejudice than any of those we know in history.

(1785, xxiv)

A pure democracy would nevertheless be acceptable if decisions are “limited to that which regards the maintaining of safety, liberty and property, all objects on which a direct personal interest can enlighten everybody” (1785, xxiv; see also 135) - these topics belonging precisely to the “general” or “universal” objects in Rousseau’s approach. Alternatively, to decide on an issue, the assembly could des­ignate a committee composed of its most enlightened members and then judge, not the decision itself, but whether the decision is detrimental to justice or to certain fundamental human rights (1785, 7). Condorcet, however, relativised the impor­tance of the choice to be made between the different possible devices. For him, the key variable remains the probability for each voter of being right or wrong: hence his tireless action in favour of public instruction and science.

[T]he happiness of men depends less on the form of assemblies that decide their fate than on the enlightenment of those who compose them, or, in other words,... the progress of reason affects their happiness more than the form of political constitutions does.

(1785, 136; see also lxx)

The Condorcet effect, or paradox of voting

The second main point which attracted attention in the 1785 Essai is the possible intransitivity of social choices resulting from the aggregation of individual choices made by rational voters. Suppose, Condorcet wrote, that voters have to rank three candidates or proposals A, B and C, through pairwise comparisons (1785, 120-21). For each voter, there are a priori eight possibilities (“XY” meaning “X is preferred to Y^,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. Choices (3) and (6) are not transitive and will not be chosen by a rational voter. But, at the aggregate level, outcomes (3) and (6) are possible. Imagine that, among 31 voters, 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 cyclic result ABCA.

Of course, this outcome has significant consequences for any social choice theory based on the aggregation of individual choices, and Arrow’s so-called impossibility theorem is well known, stating 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” (social choice should reflect individual choices in some minimal way). But this is not Condorcet’s approach: he 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 of having a “Condorcet effect” quickly increases with them. He did not get locked in a logical dilemma, but proposed solutions to the impasse.^[149] In particular, in a three-alternative case, one simple solution (Condorcet 1785, 122) consists in referring to the total number of votes that each candidate or proposal obtains against the two others. In the earlier 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.

In this respect, it is important to emphasise that Condorcet’s approach is aimed at discovering “the truth”, even in decisions which do not deal with justice but with choosing the right proposal or candidate. He was convinced that on these occasions, thanks to reason and science, there exists a truth, never imposed from above but which can 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 distance themselves from their private or partisan interests. The same is true with Condorcet. He is dealing with judgements. For him, contrary to Arrow later, the problem does not consist in aggregating individual preferences and obtain­ing social choices respecting the “particular wills” or “private interests” of voters because in this case the result would be the “will of all”, not the “general will”. Two different conceptions of democracy and the role of the State are in play 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 must therefore not behave according to what I believe to be reasonable, but according to what all, distancing themselves, like me, from their opinion, must consider as complying with reason and truth.

(Condorcet 1785, cvii)

Condorcet’s ideas were subsequently discussed by two authors before the theme sank into oblivion for decades: in 1794 by the Swiss mathematician Simon Antoine Jean Lhuillier (1750-1840), and by Pierre Claude Franςois Daunou (1761-1840) who presented a memoir on this theme in 1800 at the Classe des sciences morales et poli- tiques of the Institut national des sciences et des arts (Lhuilier 1794; Daunou 1803).^[150]

3.