Condorcet, Preference, and Judgement Introduction While Nicholas de Condorcet is primarily known in the Anglophone world today for his work on social choice theory, he did much more. Prior to Duncan Black’s rediscovery of social choice in the 1950s, Condorcet was probably better known for his magnum opus, Sketch of a Historical Picture of the Progress of the Human Mind (1795). Sketch is exactly that, a sketch for what was supposed to be a much larger work. But Condorcet died in the French Revolution before he could write Historical Picture, which was to be the grand defense of his philosophical system. What I try to do here is show Condorcet’s transition from preference aggregation to judgment aggregation between 1785 and 1794, while emphasizing the connections between Condorcet’s social choice theory and political philosophy. One of the most puzzling things about Condorcet’s social choice work is that initially (circa 1785), while he was supporting judgment aggregation, he was more expressly in favor of preference aggregation in order to increase the probability of making correct social decisions. However during the middle period between 1785 and 1794, perhaps due to the difficulties of intransitive majority preference and of calculating the most probable preference order, Condorcet seems to abandon using elections to increase the probability of making correct social decisions. But in the last years of his life (circa 1793-1794), he passionately revives his interest in using elections to make better social decisions, while at the same time, putting greater emphasis on judgment aggregation than he had before. Examining Condorcet seems especially relevant now given contemporary debates. First, there has been a recent spike of interest in epistemic democratic theory (EDT). While there are 1

many recent varieties of EDT (e.g. (Cohen, 1986), (List and Goodin, 2001), and (Estlund, 2008)), the basic argument contends that democracy is the best form of government because it does a better job of pooling information to make good social decisions than do other forms of government. EDT was first championed by Condorcet, and any serious historical work on EDT will have to address Condorcet. Second, with Balinski and Laraki’s publication of Majority Judgment (2010), there has been a significant increase of interest in judgment aggregation in the social choice literature. Some of what is being restated in current social choice debates about judgment aggregation was already stated by Condorcet over 200 years ago. Re-examining his works will be profitable for contemporary social choice. Biographical Sketch of Condorcet Nicholas de Condorcet lived from 1743 to 1794. Professionally, Condorcet was a mathematician who made significant contributions to integral calculus, though perhaps his most enduring mathematical “contribution” is the curly-d notation for partial differentiation (Cajori, 1929, p. 225). In 1785, he basically founded social choice, with the publication of Essay on the Application of Analysis to the Probability of Plurality Decisions (1785) which introduced Condorcet’s paradox and his jury theorems. Sadly there is no complete English translation of this text from the French original. In Essay (1785), Condorcet made three major discoveries which are of relevance to us: his judgment jury theorem, his preference jury theorem, and his paradox about the intransitivity of majority preference. All three are readily demonstrable, though it is useful to explicitly state the assumptions of each of the jury theorems to clarify their differences for future reference.

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Table 1: Condorcet Judgment Jury Theorem Assumption 1: There exist n voters Assumption 2a: There exists a single alternative, namely a1 Assumption 3a: The alternative a1 can be in one of two states: s xor ¬s Assumption 4a: Any given voter, vi, has an identical and independently distributed probability p > ½ of correctly determining the state of a1 Conclusion 1a: If a majority of the n voters determine that the alternative is in state s, then this is more probably true than ¬s Conclusion 2a: As n approaches infinity, the probability of the majority of voters being correct about the state approaches 1.

Table 2: Condorcet Preference Jury Theorem Assumption 1: There exist n voters Assumption 2b: There exist two alternatives, namely a1 and a2 Assumption 3b1: The relation between a1 and a2 is either a1 > a2 xor a2 > a1 Assumption 4b: Any given voter, vi, has an identical and independently distributed probability p > ½ of correctly determining the relation between a1 and a2 Conclusion 1b: If a majority of the n voters determine that a1 > a2, then this is more probably true than a2 > a1 Conclusion 2b: As n approaches infinity, the probability of the majority of voters being correct about the relation approaches 1.

1

Two notes. First, technically, we should say that with respect to some dimension, “either a1 > a2 xor a2 > a1”. Second, though in later works, Condorcet allowed for indifference, in Essay, he assumed voters could not assume indifference, most probably because this allowed for calculation of probabilities using the binomial distribution, which greatly simplified calculations. 3

In Essay, Condorcet argued that through voting, people were more likely to make correct group decisions. However, Condorcet’s paradox, the intransitivity of majority preference, created a problem in his argument in that it could prevent majorities from conclusively and fairly making correct decisions.2 After Essay (1785), possibly due to the discovery of this paradox, Condorcet moves away from probability justifications for voting. For example, we see this in Essay on the Constitution and Functions of Provincial Assemblies (1788).3 But nevertheless, we see he returns to probability justifications in his last work on voting, On Elections (1793) (McLean and Hewitt 1994, 47-48). Condorcet was very active in the French Revolution. He was a leading figure commonly associated with the Girondins faction, along with Madame Roland, Jacques Pierre Brissot, and Thomas Paine. The Girondins were one of the two leading anti-royalist factions during the French Revolution. The other major anti-royalist faction was the Jacobins faction, which would be primarily responsible for the Reign of Terror that killed tens of thousands of Frenchpersons. Condorcet basically wrote the Girondins’ draft for the French constitution. After the Jacobins became more powerful than the Girondins, they issued a warrant for Condorcet’s arrest. Thus, Condorcet went into hiding in the Parisian home of Mme. Vernet; and it was during his several months of hiding that Condorcet wrote his last testament, an essay of advice for his toddler daughter who he never saw again, and most famously his Sketch of a Historical Picture of the Progress of the Human Mind (1795). In Sketch, Condorcet lays out his optimistic understanding

2

Condorcet’s paradox is his famous “paradox” about the intransitivity of majority preference that even if every voter has a complete and transitive preference order, it can be the case with three or more alternatives that a majority prefer, for example a1 over a2, a2 over a3, and a3 over a1. 3 Condorcet’s Essay on the Constitution and Functions of Provincial Assemblies has been called Condorcet’s most comprehensive work on political theory. (Baker 1976, xxiv) But sadly, like his Essay (1785), it too has not had a complete English translation. 4

of human history as a story of progress where truth and reason eventually resolve humanity’s problems. Sketch was hastily written as Condorcet possibly believed that it was unlikely he would survive to write a full defense of his beliefs; thus, writing a sketch, however hasty and incomplete, would be the best manner in which to circulate his ideas. A few months after he goes into hiding, Condorcet is caught and dies under mysterious circumstances in prison. Condorcet and Human Rights Condorcet is an unusual political thinker for his times, as he held several beliefs that are now either universally or commonly held about slavery, women, sexuality, colonialism, citizen rights, and suffrage.4 Condorcet was an abolitionist and was very involved in the anti-slavery movement (McLean and Hewitt 1994, 341-363). Regarding women, Condorcet believed that because women are rational and sentient beings like men, they should have exactly the same rights as men (Baker 1976, 97-104). With respect to sexuality, Condorcet believed there was nothing wrong with homosexuality between consenting persons (McLean and Hewitt 1994, 56). Additionally, Condorcet’s stance on non-Europeans was also relatively unusual. Condorcet argued that colonialism was wrong because it treated natives as having less rights than European colonists, though they both ought to have the same rights (Sketch, 144). Condorcet’s Declaration of Rights (1792) includes freedom of expression, freedom of religion, right of all citizens to vie for any public office or position, right to equal education, and the right to public aid (McLean and Hewitt 1994, 280-283). Perhaps most remarkably for an 18th century thinker, Condorcet believed in universal suffrage, because in his view, an election is only legitimate if it has the consent of

4

Like many people, Condorcet’s positions on issues changed over time. What I present here are the ones he held towards the end of his life. 5

the citizens; therefore, every citizen, whether White or non-White, woman or man, has the right to vote (McLean and Hewitt 1994, 170). Existing Literature on Condorcet’s Political Philosophy Arguably, the thinker most responsible for minimizing Condorcet’s standing as a political philosopher was Thomas Malthus. In fact, the full title of the first edition of Malthus’ masterpiece was Essay on the Principle of Population, as it affects the Future Improvement of Society with remarks on the Speculations of Mr. Godwin, M. Condorcet, and Other Writers (bold added). Malthus satirized Condorcet. For example, he says “… Condorcet’s [Sketch] was written… under the pressure of cruel proscription which terminated in his death.*…. [Sketch] is a singular instance of the attachment of a man to principles which every day’s experience was so fatally for himself contradicting” (Malthus 1993, 62-63). In the footnote corresponding to the asterisk in the above quote, Malthus sarcastically mentioned “… I refer the reader to the work [Sketch] itself, which will amuse, if it does not convince him” (Malthus 1993, 63). Like most examinations of Sketch prior to the late 20th century, Malthus does not seem to be aware of Condorcet’s work on voting and elections, or at least he fails to mention them. (See (Frankel 1948) for another example where the connection between the two works is unmentioned.) It was only in the late twentieth century after Duncan Black rediscovered Condorcet’s work on voting and elections that the connection between his works on voting and Sketch began to bubble, though it has never truly come to a full boil. Since the mid-20th century, there has been some excellent historical and interpretative work on Condorcet’s political philosophy that addresses both his jury results and political philosophy, such as (Baker 1976) and (McLean and Hewitt 1994).

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(Grofman and Feld 1988) is possibly the first work to aggressively use Condorcet’s jury results to Condorcet’s political philosophy. Grofman and Feld argue that Condorcet’s jury results could be interpreted as a way of clarifying Rousseau’s general will. (McLean and Hewitt 1994, 37-38) agrees with this claim. Furthermore, Condorcet greatly admired Rousseau’s contributions to political thought: …[A]ll [persons] have an equal right to be informed on all that concerns them, and that none of the authorities established by men over themselves has the right to hide them one single truth. These principles, which the noble [Algernon] Sydney paid for with his blood and on which Locke set the authority of his name, were later developed by Rousseau with greater precision, breadth and energy, and he deserves renown for having established them among the truths…. (Condorcet 1955, 129-130)

Arguably, the Anglophone work that most closely examines Condorcet’s political philosophy with the tools of social choice and probability theory is (Young 1988). Young argued that Condorcet sought to use voting to create a transitive rank ordering of alternatives based on how likely the rank orderings are to be true. In Young’s view, Condorcet was himself not able to do this, but that by a change of text in Condorcet’s Essay (1785), one could infer that Condorcet’s problem with majority cycles could have been overcome with the Kemeny-Young method. The basic assumptions Young had and the relevant conclusions he arrived at, are summarized in the following figure.

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Table 35: Generalizing Condorcet’s Preference Jury Theorem to Multiple Alternatives Assumption 1: There exist n voters Assumption 2c: There exist m alternatives, namely a1, a2,…, am Assumption 3c6: The relations between all alternatives can be expressed by a complete, transitive, and strict preference order, without loss of generality, of the form a1 > a2 >…> am Assumption 4c: Any given voter, vi, has an identical and independently distributed probability p > ½ of correctly determining the relation between any two alternatives ax and ay

Conclusion 1c: The Kemeny-Young method chooses the complete, transitive, and strict preference that has the greatest probability of being true Conclusion 2c: The Kemeny-Young method does not guarantee that the top preference of the preference order it selects has the greatest probability of being the true top preference

Note from the figure that the Kemeny-Young method does not guarantee that the top preference of the Kemeny-Young winning preference order has the greatest probability of truly being the one with the most of the desired quality. For example, suppose that after some election under the conditions mentioned in table 3, that attempts to pool the information of knowledgeable music fans to determine rank ordering of three Memphis musicians by height: Johnny Cash, B.B. King, and Elvis Presley. Each fan submits a strict and complete rank ordering of the three Memphis musicians from tallest to shortest. Suppose that the Kemeny-Young method selected the preference order Presley > Cash > King as the most probably correct ordering of the three musicians from tallest to shortest. While it would be true that the ranking ordering Cash > King > Presley has the most probability of being the correct ordering (given only the information of 5 6

All of the information in this table comes from (Young 1988) Again, note that by a1 > a2, we mean that on some given dimension, a1 is greater than a2. 8

the votes from the fans), it does not necessitate that Presley has the greatest probability of being the tallest (given only the information of the votes from the fans).7 It can be inferred from (Young 1988) that if we only know that p > ½, this is insufficient information for any voting system based only on ordinal preferences to guarantee that the top preference has the greatest amount of the measured quality. This is a huge problem, especially if the goal of voting is to choose a single winner to enact. Additionally, it has been shown that the Kemeny-Young method is NP-hard, which can make computation of the winning preference order take inordinately long times (e.g. years, decades, centuries, and even longer) (Conitzer, Davenport, Kalagnanam 2006). (List and Goodin 2001) emphasized the relevance of Condorcet’s ideas to contemporary political philosophy and provided an impressive generalization of CJT by generally extending CJT to voting systems that use ordinal preferences. But as Young’s work demonstrated, no voting system based on ordinal preferences can guarantee that the top social preference has the greatest probability of being in the desired state even if it were true that all voters have the same p > ½ and p was not specified. Condorcet and Epistemic Democracy Like Plato and Hobbes before him, Condorcet offered an original solution to what we would today call the collective action problem via his magnum opus, Sketch. Condorcet does not present the argument for his solution in a straightforward manner, but this makes sense given that he wrote Sketch while he was in hiding during the Reign of Terror. Furthermore in Sketch, Con-

7

Peyton Young gives a more detailed example of this phenomenon (Young, 1998, p. 12361239). By the way, Cash is the tallest, followed by Presley, followed by King. 9

dorcet never directly refers to his work on voting and elections. So we will have to tease out an argument by connecting his social choice writings and Sketch. The original solution Condorcet offered to the collective action problem is that through voting, humans are able to aggregate their reason and knowledge in a manner that makes them more accurate and allows them to make better decisions. These better decisions improve the lot of all humanity such that it is better for any human to live in such a society that aggregates individual opinions (e.g. a democracy) than to go it alone. In what follows, I try to lay out an argument for this proposed solution from Condorcet’s writings.

In Sketch, Condorcet contends that discovery of the truth through reason is the basis of all human progress in all realms (including, but not limited to, the fine arts, social progress, philosophy, and science) (Condorcet 1955). So presumably, if any human wants any sort of improvement, then pursuit of the truth leads to it. But how can a human find truth? This is where Condorcet’s famous judgment jury result becomes important. First, he showed that if there were n voters confronted with a proposition, which is in exactly one of two states (e.g. either true xor false), and each voter had an independent probability p > ½ of correctly determining the state of x, then the state which the majority of voters believed the alternative was in has the greatest probability of being the state the alternative is actually in. Second, he showed that if p > ½ and n approaches infinity, then the probability that the majority of voters have the correct belief rapidly approaches 1 (i.e. absolutely certain correctness). It is worth examining some of the assumptions of Condorcet’s judgment jury theorem. First, Condorcet believed that there are objective moral and empirical truths in the world (McLean and Hewitt 1994, 32). Thus for Condorcet, voting is not done to aggregate the tastes of 10

citizens, but rather to determine the truth. And given that Condorcet was living in a pre-manyvalued logic world, we have no reason to doubt he was using bivalent logic, where either a statement is true xor false. (Recall assumption 3a). Second, voters don’t have to have perfect knowledge; they just need to be sufficiently knowledgeable in order for individual voters to be more likely to be correct than otherwise. This can help explain why Condorcet felt it was important everybody had access to free education, since this would presumably improve their p to better than .5 (McLean and Hewitt 1994, 22-33). (Recall assumption 4a). Third, each voter has the same independent probability of correctly determining the state of the alternative. This is clearly a simplifying assumption by Condorcet to make his math easier. That said, it is still consistent with Condorcet because he believed in universal equality in human rights and allowing each person’s own reason to be their only guide in decision making (McLean and Hewitt 1994, 37-63); because if it were assumed that different voters had different p-values, then some voters would greater affect the group decision than others and this would be unequal.8 In other words, if the goal of voting is to maximize the probability of determining the truth, then assuming that all voters have the same p is a way of ensuring a probabilistic anonymity condition, as it means that no voter’s knowledge is given more weight than any other voter’s knowledge in determining the group decision. Now in order to complete Condorcet’s argument, we have to infer an assumption from his writing: the prior probability of a given alternative being true (i.e. q) is the same regardless of the alternative. (Note this is the prior probability, before we have any information about the al-

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One might contend that in fact some voters choose correctly more often, but determination of these voters cannot be made prior to empirical observations of their decision-making. By asserting that p is equal for all voters, this allows one to say, using a French revolution era example, that the input of a nobleperson is not more valuable than the input of merchant commoner. 11

ternative). Given that we are uncertain about the truth or falsity of an alternative, any q greater than 0 and less than 1 will do. However, we know that Condorcet, using an equiprobability assumption, assumed q = ½ (Grofman, Owen, Feld, 1983, p. 264). With this, we can proceed to generalize Condorcet’s judgment jury theorem to multiple alternatives. Table 49

Generalizing Condorcet’s Judgment Jury Theorem to Multiple Alternatives

Assumption 1

There exist n alternatives

Assumption 2c Bivalence Assumption Probabilistic Anonymity Assumption

There exist m alternatives, namely a1, a2,…, am

Probabilistic Neutrality Assumption

Each of the m alternatives has an identical and independently distributed proba-

Conclusion 1d

If each voter approves each alternative she believes is in state s and does not

For any of the m alternatives, say ax, it is in one of two states: s xor ¬s Any given voter, vi, has an identical and independently distributed probability p > ½ of correctly determining the state of any given alternative ax

bility q = ½ of being in state s

approve any alternative she believes is in state ¬s, then the complete and transitive preference order resulting from approval voting will rank order the alternatives from most probably in state s to least probably in state s Corollary 1

If s is truth and a majority of the n voters believe that ax is true, then as n approaches infinity, the probability that ax is true equals 1

Corollary 2

If s is truth, then an alternative that is ranked below no other alternatives in such an approval voting generated rank ordering will have the greatest probability of being true

9

The proof is simple. It is described in (Prasad, 2011) and the appendix. 12

If the generalization in table 4 were true, then society could use voting to determine which of several alternatives has the greatest probability of being true. This is Condorcet’s original contribution and solution to the collective action problem. This interpretation is further enforced by comments by Condorcet on other views of how to solve the collective action problem. In Sketch, Condorcet clearly rejected the Platonic solution that humanity was divided by an enlightened elite that had to give noble lies to the masses in order to get the masses to do that which was in society’s best interests (Condorcet 1959, 129). He also rejected the social contract as a solution for two reasons: first, in his view, such a contract was easy to annul via a defection of just one party to the contract; second, he believed it was absurd that people would be required to abide by a contract into which they were born (Condorcet 1959, 129). It is worth noting at this point second possible criticisms of Condorcet’s solution. First, is Condorcet’s optimistic solution, that humanity would get closer and closer to perfection as it got closer to the truth, invalid? While it is probably overly optimistic and unsound, it is not invalid. It is clear, that in an argument that presages the analogous argument for a technological singularity by over 200 years, Condorcet believed that as human knowledge of the truth increased, this snowballed into better and better accuracies of determining the truth, in a manner that asymptotically, but at an accelerating pace, approached perfect truth and the achievement of all rationally desired human endeavors (Condorcet 1959, 173-202). Though in Sketch, Condorcet never refers to his work on voting, one can be easily infer an argument for this claim based on the jury theorem. Recall that either an increase in individual probability of correct judgment or increase in the number of voters correctly voting would lead to a greater aggregate probability of correct judgment. But when voters discover a new truth, their individual decision skills and their individual 13

probability of correct judgment presumably increase. This in turn increases their ability to make aggregate judgments that are more probable to be correct. This autocatalytic process continues in a manner that rapidly increases the accuracy of group decision making. Thus, by combining his work on voting with Sketch, there is a clear and valid argument that a Condorcet could possibly provide to demonstrate that discovery of truth asymptotically progresses towards perfection.

A second more serious criticism is whether Condorcet actually made this interpretation himself. We will never know for sure because Condorcet wrote Sketch very hastily under great duress while hiding. Furthermore, he did not mention his work on voting explicitly in Sketch. However, I think there are at least three reasons to believe that Condorcet was aware of the results in table 4, even if he never explicitly stated it. However, by connecting Condorcet’s philosophical work with his social choice work, we can explain Condorcet’s mysterious rekindled interest in using voting systems to discover truth if we assert that Condorcet discovered the multiple alternatives extension of his judgment jury theorem. Condorcet’s last writing on voting and elections, On Elections (1793), seems to show that while not making a clean break from preference based voting systems, Condorcet was moving away from a comparison based preference judgments towards approval based absolute judgments. I am not the only to have come to this conclusion. McLean and Hewitt say “The [On Elections] manuscript suggests that Condorcet was moving away from rank-ordering procedures to approval-votes ones…” (McLean and Hewitt 1994, 48). In On Elections, Condorcet makes a distinction between absolute judgments (i.e. judgments) and preference judgments (i.e. preferences) (McLean and Hewitt 1994, 241-243). For example, for a judgment, a voter would look at each of the n alternatives individually and ask if the given candidate is capable of fulfilling the office’s responsibilities? For preferences, for example, a voter would look at each of the (n2-n)/2 14

possible pairs of candidates individually and ask for a given pair which of the two candidates is more capable of fulfilling the office’s responsibilities? In fact, Condorcet clearly thought about the probabilities for judgments. For example, he states “But we can also say in general, that for the majority of voters, an absolute judgment has a greater probability than a preference judgment. This is yet another reason for disregarding the latter” (McLean and Hewitt 1994, 242). It is not clear what exactly this means, but it is clear he thought about probabilities of absolute judgments, and may have discovered the connection between approval voting and CJT, though we will never really know barring new textual discoveries. I don’t want to overstate the case and make it seem like Condorcet completely abandoned preference judgments. Even in On Elections, Condorcet states that the ideal voting procedure is the Essay (1785) solution described earlier after the candidates have been vetted to ensure through absolute judgments that each candidate is competent (McLean and Hewitt 1994, 238). But this is with respect to discovering which of several alternatives is most competent. Truth for Condorcet was an absolute judgment, as in a claim cannot be more true than another. It is either true xor not. Thus, Condorcet’s resurgent interest in using elections to determine the truth make much more sense when we look at On Elections (1793) with Sketch, which was written in 1794. When using the approval voting method under the conditions of the multiple alternatives extension of Condorcet’s judgment jury theorem, a transitive and complete rank ordering of alternatives based on their probability of truth is the outcome. In addition, it guarantees that the first ranked alternative has the greatest probability of being true, the second ranked alternative has the second greatest, and so forth. These are not true of voting systems based on ordinal preferences, and this failure of preference based voting systems perhaps caused Condorcet to abandon interest using vot15

ing to determine truth during the late 1780s. But if Condorcet were to have discovered the multiple alternatives extension of his judgment jury theorem, then it makes sense how Sketch so confidently contends that it is through reason and truth that all human progress is achieved. Conclusion In this paper, I have discussed many things. Three things I did not address were extensions of results to different p-values for different voters, to non-independence assumptions, and to insincere voters. After learning the results of an earlier draft of this paper, (Brams and Kilgour 2011) showed that even if p-values differed among voters, so long as the mean p of all voters was greater than 0.5 and voters still voted independently and expressively sincerely, then the approval winner has the greatest expected probability of being the correct alternative. (Ladha 1992) showed that Condorcet’s jury theorem could be extended to correlated votes, but it is yet to be shown that it can be extended in the same way (or any other way) for this extension with approval voting. (Austen-Smith and Banks 1996) demonstrated that even if all Condorcet jury voters had the same goals, voters still have incentives to be insincere about their beliefs. The extent to which this affects this approval voting extension is still unknown. But those are issues of relevance to other research, as this paper was primarily focused on understanding Condorcet’s political thought. McLean and Hewitt had presented a puzzle in understanding Condorcet: After an initial burst of enthusiasm for voting as a means to discover truth during the mid-1780s, Condorcet seemed to lose interest in using voting in this manner during the late-1780s, only to revive this interest prior to his tragic death. But when we connect Condorcet’s social choice research with his political philosophy, the move makes much more sense.

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Condorcet originally tried to discover truth through preference aggregation during the mid-1780s. However, this proved untenable for two reasons. First, it was extremely hard to calculate when there were several alternatives to consider. This is not due to lack of genius on Condorcet’s part, since it is an NP hard question and we still don’t know how to guarantee a solution to this problem for many alternatives in a reasonable amount of time. Second, the highest ranked alternative in the Kemeny-Young produced rank ordering is not guaranteed to have the greatest probability of being the best alternative. Given these two problems, Condorcet seems to have lost interest in using voting systems to discover truth during the late-1780s. But if he discovered the simple multiple alternatives extension to his judgment jury theorem in the last years of his life, then it would explain his beliefs expressed in Sketch. Recall that Sketch does not make reference to Condorcet’s social choice research. Thus, his claims of unending human progress in Sketch seem delusional and unfounded as Malthus pointed out. However, when we connect Condorcet’s social choice research with his philosophical work, we can see that Condorcet had offered an argument for a solution to the collective action problem of why individuals should cooperate as a society. By cooperating in a society, people were better able to arrive at the truth which led to progress in all its forms. And according to Condorcet, as people learned more and more truth, this would snowball into greater and greater accuracy in determining the truth. Armed with close approximations of the truth, people did better as a society than when they went it alone.

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Appendix 1. The Extension of Condorcet’s Jury Theorem to Multiple Alternatives Introduction Informally, we can say: If there are m alternatives, and if each voter has an independent probability p > ½ of correctly determining which of two states any given alternative is in, then approval voting guarantees the selection of an alternative that optimizes the probability of being in the desired state. In what follows, we will proceed through the proof formally. Assumptions There are m alternatives, where m is some integer greater than 1. Any given alternative x from the m alternatives is in one of two states: s xor ¬s. Any given alternative x has a probability 0 < q < 1 of being in state s. There are n voters. When confronted with any given alternative, say alternative x, any given voter will have one of two beliefs about x: the belief that x is in state s xor the belief x is in state ¬s. Note that a given voter can have different beliefs for different alternatives. Any given voter i from the n voters has a probability p of having the correct belief about the state of any given alternative x. The probability p is independently and identically distributed. Now assume that exactly k number of voters believe alternative x is in state s, and exactly l number of voters believe that alternative y is in state s, where 0 ≤ k < l ≤ n, (and k, l, and n are

18

all integers). We further assume that when voters vote, they express the beliefs which they believe. That is, voters are expressively sincere.10 Theorem If p > ½, then the alternative, which is approved (i.e. believed to be in state s) by the greatest number of voters, has the greatest probability of being s. Corollary If p > ½, then the rank ordering of alternatives based on their probability of being in the desired state is equivalent to the rank ordering of alternatives based on their number of approvals. Proof of Theorem and Corollary Note that given our assumptions, we can use the binomial distribution and multiplication rule to show that the total probability of (x being in state s) and (k voters believe x is in state s) is: [1]

q(n!/[k!(n-k)!])(pk)(1-p)n-k

Similarly, we can show that the total probability that (x is ¬s) and (k voters believe x is s) is: [2]

(1-q)(n!/[(n-k)!k!])(pn-k)(1-p)k

Thus, the total probability that (k voters believe x is s) is [1] + [2]: [3]

[q(n!/[k!(n-k)!])(pk)(1-p)n-k]+[(1-q)(n!/[(n-k)!k!])(pn-k)(1-p)k]

Therefore, the probability that (x is s) given that (k voters believe x is s) is [1]/[3]: 10

The notion of sincerity used here is not the same as that used in (Brams and Fishburn 2007). Here we are defining sincere voting as voters vote based on their beliefs about the state of the alternative. Brams and Fishburn define sincerity using preferences. 19

[4]

[q(pk)(1-p)n-k]/([q(pk)(1-p)n-k]+[(1-q)(pn-k)(1-p)k])

Similarly, we can show that the probability that (y is s) given that (l voters believe y is s) is: [5]

[q(pl)(1-p)n-l]/([q(pl)(1-p)n-l]+[(1-q)(pn-l)(1-p)l])

In order to prove the theorem, we must first prove the lemma that: [6]

If [5] ≤ [4], then p ≤ ½

In order to demonstrate this lemma, assume the lemma’s antecedent: [7]

[q(pl)(1-p)n-l]/([q(pl)(1-p)n-l]+[(1-q)(pn-l)(1-p)l]) ≤ [q(pk)(1-p)n-k]/([q(pk)(1-p)n-k]+[(1-q)(pn-k)(1-p)k])

Solving for p using algebra, we get: [8]

p≤½

This proves [6]. When we take the contrapositive of [6], we get: [9]

If p ≥ ½, then [5] ≥ [4]

What we have demonstrated so far is that if the number of voters that approve y is greater than the number of voters that approve x, then the probability that y is in state s is greater than the probability that x is in state s. It should also be self evident that if x and y have the same number of approvals, then the probability that x is s is the same as the probability that y is s. Since the number of voters that approve any given alternative is always some integer, and we know the integers are transitive, we can say that:

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[10]

If p > ½, then an alternative y, which is approved by a number of voters greater than or

equal to the number that approve any other given alternative x, has the greatest probability of being in state s. Furthermore, [11]

If p > ½, then the rank ordering of alternatives based on their probability of being in the

desired state is equivalent to the rank ordering of alternatives based on their number of approvals. QED 2. Computational Complexity of Approval Voting To construct a rank ordering of alternatives based on approval voting, one rank orders alternatives from most approvals to least approvals. Sorting alternatives based on their number of approvals can be done with an algorithm that is polynomial time with respect to the number of alternatives, such as the quicksort algorithm.

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