CONDORCET'S THEORY OF VOTING H. P. YOUNG University of Maryland Condcrcet's criterion states that an alternative that defeats every other by a simple majority is the socially optimal choice. Condorcet argued that if the object of voting is to determine the "best" decision for society but voters sometimes make mistakes in their judgments, then the majority alternative (if it exists) is statistical- ly most likely to be the best choice. Strictly speaking, this claim is not true; in some situations Borda's rule gives a sharper estimate of the best alternative. Nevertheless, Condorcet did propose a novel and statistically correct rule for finding the most likely ranking of the alternatives. This procedure, which is sometimes known as "Kemeny's rule," is the unique social welfare function that satisfies a variant of independence of irrelevant alternatives together with several other standard properties. [Our aim is] to inquire by mere reasoning, what own prevails, this proves only that I have degree of confidence the judgment of assemblies made a mistake, and that what I believed deserves, whether large or small, subject to a high or low plurality, split into several different to be the general will was not so" (1962, bodies or gathered into one only, composed of 153). men more or less wise. This ambiguous and slightly disquieting -Condorcet idea was given a more satisfactory expres- sion some twenty years later by the math- ematician and social philosopher Marie A central problem Jean Antoine Nicolas Caritat, Marquis de in democratic theory is to justify the prin- C0ndorcet.l In his remarkable work, ciple of majority rule. Why should the Essai sur l'application de l'analyse h la opinion of a majority of citizens or of probabilith des dhcisions rendues h la elected representatives bind the rest of pluralith des voix (1785), Condorcet pro- society? One of the earliest and most cele- posed the following argument for decision brated answers to this question was at- by majority: Enlightened voters honestly tempted by Rousseau in The Social Con- attempt to judge what decision will best tract (1762). The opinion of the majority serve society. They may occasionally is legitimate, said Rousseau, because it judge wrongly. But assuming that they expresses the "general will." "When a law are more often right than wrong, the is proposed in the people's assembly, majority opinion will very likely be "cor- what is asked of them is not precisely rect." This is just a property of large num- whether they approve of the proposition bers and is intuitively obvious when there or reject it, but whether it is in conformity are only two possible decisions. Con- with the general will . each by giving dorcet rigorously demonstrated this prop- his vote gives his opinion on this ques- osition using the newly developed cal- tion, and the counting of votes yields a culus of probabilities. He then proceeded declaration of the general will. When, to show how the idea could be elaborated therefore, the opinion contrary to my into a whole series of results about the AMERICAN POLITICAL SCIENCE REVIEW VOL. 82 NO. 4 DECEMBER 1988 American Political Science Review Vol. 82 reliability of decisions of a voting body, object of voting is not merely to balance depending on its size, the competence of subjective opinions; rather, it is a collec- its members, and the number of alterna- tive quest for truth. In any field of candi- tives under consideration. These "jury dates (or agenda of alternatives) there is theorems" have recently attracted con- some truly best candidate, a truly second- siderable attention (Grofman and Owen best candidate, and so forth. What is 1986; Grofman, Owen, and Feld 1983; meant here by truly best? Ultimately it is Urken and Traflet 1984).2 an undefined term, a notion that is postu- It has not been widely recognized, how- lated in order to make the theory work- ever, that Condorcet showed how the like the ether in classical physics. To be same arguments could be extended to the sure, in some situations the notion of case of more than two decisions. He was, "best" decision is quite realistic. An exam- of course, the first to realize the complica- ple is trial by jury, where the group deci- tions that arise in this case, namely, the sion problem is to determine guilt or inno- possibility of cyclic majorities. But he cence. Another is the pooling of expert went on to suggest-or at least came very opinions about an observable event, like close to suggesting-a definite rule for the state of the economy or tomorrow's. ranking any number of alternatives even weather.3 when cyclic majorities are involved. For the present let us grant Condorcet's Admittedly, Condorcet's argument is dif- hypothesis that the notion of a "correct" ficult to follow and full of inconsistencies. decision also applies to political decision Moreover, the rule that he proposed is at making. Thus, voters evaluate candidates variance with the algorithm that he pro- for political office in terms of their ability posed for computing it. Nevertheless, his to lead. Legislators consider bills in terms stated goal is clear: to find the social deci- of their potential contribution to the pub- sion that is most likely to be "correct." lic welfare. Granted that voters are at First I summarize Condorcet's proposal least sometimes mindful of the public in- and dispose of the claims made by some terest and vote accordingly, Condorcet's commentators that Condorcet gave no argument proceeds aIong the following solution for the general case. I then show lines. People differ in their opinions be- that Condorcet's method can be inter- cause they are imperfect judges of which preted as a statistical procedure for esti- decision really is best. If on balance each mating the ranking of the candidates that voter is more often right than wrong, is most likely to be correct. If only one however, then the majority view is very candidate is to be selected, however, then likely to identify the decision that is objec- an earlier method proposed by Con- tively best. In other words, the majority dorcet's rival Borda often gives better opinion gives the best estimate, in a statis- results. In the final section I show that tical sense, of which candidate is most Condorcet's decision rule can also be jus- likely to be best. One of Condorcet's tified in modern social choice terms. It is major contributions was to show how this the unique ranking procedure that satis- idea can be generalized to the case of fies a variant of independence of irrele- many candidates. vant alternatives together with several Consider a list of candidates (or other standard conditions in social choice decisions) that are voted upon pair- theory. wise. Assume that each voter, when faced with a comparison of the form, "Is a bet- ter than b or b better than a?" chooses Condorcet's Model correctly with some fixed probability For Condorcet, as for Rousseau, the greater than 1/2. Each voter judges every Condorcet's Theory pair of candidates and the results are tal- Table 1. Voting Matrix with lied in a voting matrix. Table 1shows a Three Alternatives and 11 Voters voting matrix with 13 voters and three candidates. Each entry in the matrix gives the total number of votes obtained by the row candidate over the column candidate. For example, the comparison a versus b yields eight votes in favor of a, five in favor of b. The comparison b versus c 1. All possible opinions that do not imply a con- tradiction reduce to an indication of the order of yields eleven votes for b and two for c. merit that one judges to exist among the candi- The comparison a versus c yields six for ,a dates ... therefore for n candidates one would and seven for c. Notice that this vote have n(n - 1) . .2 possible opinions. entails the cyclical majority a >b, b >c, 2. Each voter having thus given his or her c >a. opinion by indicating the candidates' order of worth, if one compares them two by two, one Given a series of pairwise votes as in will have in each opinion n(n - 1)/2 proposi- Table 1, the problem posed by Condorcet tions to consider separately. Taking the number is to find the ranking of all the candidates of times that each is contained in the opinion of that is most likely to be correct, that is, one of the q voters, one will have the number of voices who are for each proposition. the ranking that would have produced the 3. One forms an opinion from those n(n - observed votes with highest probability. 1)/2 propositions that agree with the most At first blush it might appear that this voices. If this opinion is among the n(n - 1) ... problem is insoluble unless the compe- 2 possible opinions, one regards as elected the subject to whom this opinion accords the pref- tency levels of the voters are known. This erence. If this opinion is among the 2n(n-1)'2- difficulty can be circumvented if one n(n - 1)...2 impossible opinions, then one suc- makes the simplifying assumption (which cessively deletes from that impossible opinion Condorcet did) that the voters all have the the propositions that have the least plurality, and same competence. In this case the answer one adopts the opinion from those that remain. turns out to be independent of the level of (1785, 125-26) competence, as I shall show in the next Let us apply this rule to the situation in section.