Condorcet's Theory of Voting H

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 statistically 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 mathematician 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 alternative 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 better 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 contradiction 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. Here we shall content ourselves Table 1. First we are to choose the three with Condorcet's statement of the solu- propositions having a majority, namely b tion. > c with eleven votes, a > b with eight First some terminology is required. In votes, and c > a with seven votes. Since Condorcet's lexicon, an "opinion" is a these three propositions form a cycle, series of pairwise comparisons on the however, we delete the proposition with alternatives. Each pairwise comparison is smallest plurality, namely c > a. This called a "proposition" and written a > b, leaves the combination b > c and a > b etc. An opinion is said to be "impossible," (and implicitly a > c), which implies the "contradictory," or "absurd" if some of ranking a > b > c. the propositions composing it form a The rule is clear enough in the case of cycle, such as a > b, b > c, c > a. Nor- three alternatives. However, in the mally, each individual voter is able to general case some ambiguities arise. The rank all of the candidates in a consistent difficulty lies in the phrase "successively order. This is not necessarily true of the deletes" in step 3. Taken literally, it means aggregate opinion, however, as Table 1 the following: If there are cycles in the demonstrate^.^ To break such cyclic ma- propositions selected in step 2, delete first jorities, Condorcet proposed the follow- the proposition having the least number ing method? of votes in its favor. (This proposition American Political Science Review Vol. 82

Table 2. A Voting Matrix with In the above example the resulting 25 Voters and Four Alternatives opinion is abed. Unfortunately, this answer does not square with the rest of Condorcet's argument. His stated intention is to choose "the most probable combination of opinions," that is, the set of propositions that contains no cycles and is supported by the largest number of pairwise votes. The total number of votes supporting the ranking abed is 89. (That is, the sum of the votes for a over b, a might or might not be contained in a over c, a over d, b over c, b over d, and c cycle.) If any cycles remain, delete next over d equals 89.) But this is not a maxi- the proposition with the next lowest num- mum. The ranking baed is supported by a ber of votes in its favor, and so forth. total of 90 pairwise votes. This ranking is Continue until all cycles are eliminated. obtained by reversing exactly one major- There are several reasons why this is ity proposition, d > b. Note that this probably not what Condorcet meant to is not the proposition with smallest say. In the first place, the procedure may majority. yield an ambiguous ordering of the candi- At this point we might be tempted to dates. Consider Table 2, with twenty-five throw up our hands and declare, as pre- voters and four candidates. vious commentators have, that "the gen- In step 2 of Condorcet's algorithm one eral rules for the case of any number of would select the six propositions having candidates as given by Condorcet are greatest majorities. In descending size of stated so briefly as to be hardly intelligible majority, these are c >d, a >d, b >c, a . . . and as no examples are given it is > c, d >b, b > a. According to a literal quite hopeless to find out what Condorcet reading of step 3, one would first delete meant" (E. J. Nanson as quoted in Black the proposition b > a, as it has the small- 1958, 175). Even the influential mathe- est majority in its favor. But this does not matician Isaac Todhunter found Con- result in an "opinion" because one cycle dorcet's argument completely exasperat- still remains: b >c, c > d, d >b. There- ing: "The obscurity and self-contradiction fore one would delete the proposition d > are without any parallel, so far as our b, as it has the next-smallest majority in experience of mathematical works extends its favor. All cycles are now eliminated. . . .no amount of examples can convey an But there is a difficulty: in the resulting adequate impression of the evils" (Tod- partial order both a and b are undomi- hunter 1949, 352). nated. Either one of them could be inter- In fact, Condorcet's overall intention is preted as the top-ranked candidate, so the quite clear. It is to find the most probable outcome is indeterminate. combination of opinions. In describing his It seems more likely that Condorcet method he seems to have used the term meant to reverse, rather than simply "successively deletes" rather loosely as a delete, the weakest propositions. Accord- heuristic for computing the s~lution.~(He ing to this interpretation one would first may also have believed, mistakenly, that reverse the proposition with the least successive deletion always does give the majority (thus b > a becomes a > b), most probable combination.) But there then reverse the proposition with the appears to be little doubt about his objec- next-least majority (thus d > b becomes b tive. Hence I propose to amend Con- > d) and so forth until no cycles remain. dorcet's statement of the rule as follows: - - Condorcet's Theory

3'. One forms an opinion from those n(n - 1)/2 Consider again the voting matrix in propositions that agree with the most voices. If Table 1. Suppose that the true ranking is this opinion is among the n(n - 1) . . .2possible opinions, one regards as elected the subject to abc-a first, b second, c third. A vote for whom this opinion accords the preference. If this a over b will occur with probability 1- opinion is among the 2n(n-1)'2- n(n - 1) .. .2 p, whereas a vote for b over a will occur impossible opinions, then one reverses in that with the lesser probability 1 - p. The impossible opinion the set of propositions that combined probability of observing the 39 have the least combined plurality and one adopts the opinion from those that remain. painvise votes in Table 1is therefore It seems reasonably likely that this is what Condorcet meant to say. In any case, it is the only interpretation that is consistent with his goal of finding the ranking that is most likely to be correct.

Condorcet's Method As a Statistical Hypothesis Test Similarly, the probability of observing the From a historical standpoint, Con- results in Table 1if the true ranking is acb dorcet's theory is particularly interesting would be because it represents one of the earliest applications of what today would be called "statistical hypothesis testing." Based on a sample of observations, one wishes to infer which of several under- lying but unobservable states of nature is most likely to be true. This approach is known as maximum likelihood estima- tion. All that is required is a model of how the observed quantities depend proba- The coefficient is the same for all six rank- bilistically on the unobservable state of ings, so their relative likelihoods are as in nature. In Condorcet's scheme of things a Table 3. state of nature corresponds to the true or For each ranking the exponent of p is correct ordering of the candidates. This the sum of all pairwise votes that agree datum is essentially unobservable. What with the ranking, and the exponent of (1 can be observed are the votes of the repre- - p) is the sum of all painvise votes that sentatives who are attempting to read the disagree with it. Assuming that p > 1/2, true state of nature. Condorcet assumed, first, that in any painvise comparison each voter will Table 3. Likelihoods of Six Rankings choose the better candidate with some Based on Table 1 Data fixed probability p, where 1/2 < p < 1 and p is the same for all voter^.^ Second, Ranking Likelihood every voter's judgment on every pair of abc pZ5(1- p)" candidates is independent of his or her abc pIb(l - pI23 judgment on every other pair. Finally, bac p2'(1 - p)" each voter's judgment is independent of bca p2j(l - p)I6 cab p17(l - p)~' the other voters' judgments (a significant cba p14(1- p)25 assumption). American Political Science Review Vol. 82 which seems to be a credibly optimistic pertise in measurement, he served on the view of human judgment, it follows that committee for determining the standard the most likely ranking is the one with the weights and measures in the new metric highest exponent of p. In the above exam- system (Mascart 1919). ple the answer is abc. In 1770, some 15years prior to the pub- In general, suppose that there are n lication of Condorcet's treatise on elec- voters and m candidates or alternatives tions, Borda read a paper to the academy al, ...,a,. Let nijbe the total number of on the design of voting procedures (see votes that aireceives over aj. Assume that Borda 1784). Exactly what prompted every voter registers an opinion on every Borda to sail in these uncharted waters is pair of candidates, so that nij + nji = n unclear. Possibly he was concerned with for all i # j. Let R be a complete ranking reforming the election procedures in the of the candidates, where aiR ajmeans that academy itself. He began by pointing out ai is preferred to aj. The argument out- the defects with the customary "first-past- lined above is easily generalized to show the-post" system in which the candidate that a ranking R has maximum likelihood with the largest plurality is elected. When if and only if it maximizes Enij, the sum there are many candidates, this procedure over all pairs ij such that aiRai. That is, can easily elect someone who is endorsed the maximum likelihood ranking is the by only a small minority of the electorate. one that agrees with the maximum num- Instead, Borda suggested, the election ber of pairwise votes, summed over all procedure should be based on each voter's pairs of candidates. This is equivalent to entire rank ordering of the candidates. the statement of Condorcet's rule given in The correct candidate to choose is the one 3'. Note that it does not depend on the that on average stands highest in the value of the success rate p as long as all voters' lists. voters have the same success rate, which Operationally, the method works as is larger than one half. This rule is very follows: Each voter submits a list in which natural and, not surprisingly, has been all of the candidates are rank-ordered. In rediscovered by other writers. In particu- each list a candidate who is ranked last lar it was proposed in a different guise by receives a score of zero, a candidate who John Kemeny (1959; see also Kemeny and is ranked next-to-last receives a score of Snell1960) and is sometimes known in the one, a candidate who is ranked next literature as "Kemeny's rule" (Young and above that receives a score of two, and so Levenglick 1978). forth. Thus the score of a candidate in any particular list is equal to the number of Condorcet versus Borda candidates ranked lower on that list. The scores of the candidates are summed over Another pioneer in the theory of elec- all voter lists and the candidates with tions was the Chevalier Jean Charles de highest total score is elected. Borda. Like Condorcet, Borda was a If Borda's objective was practical re- member of the French Academy of Sci- form, he appears to have been successful, ences and an important figure in official for something like his method was subse- scientific circles. Unlike Condorcet, he quently adopted for the election of new was an applied scientist with a practical members to the a~ademy.~ bent. He was noted for his work in Condorcet was certainly aware of hydraulics, mechanics, optics, and Borda's work. Indeed he introduced the especially for the design of advanced method in the Essai with the following navigational instruments. He also cap- remark: "As the celebrated Geometer to tained maritime vessels. Because of his ex- whom one owes this method has pub- Condorcet's Theorv lished nothing on this subject, I have Table 4. A Voting Matrix with taken the liberty of citing it here" (1785, 60 Voters and Three Candidates clx~ix).~Condorcet repeatedly refers to Borda, not by name, but as "the celebrated Geometer." 'The tone is ironical. From his private correspondence it is clear that Condorcet held Borda's work in low esteem and considered him a personal enemy within the academy. For example, in a letter to Turgot (1775), he gave this assessment: "[M. Malesherbes] makes a great case for Borda, not because of his Not necessarily. Candidate c will be best memoirs, some of which suggest talent if two propositions hold: namely, if c is (although nothing will ever come of t~hem, better than a (c > a) and cis better than b and no one has ever spoken of them or (c > b). We are therefore interested in the ever will) but because he is what one calls joint probability of these two statements a good academician, that is to say because being true. If c > a, then the probability he speaks in meetings of the Academy and of observing the voting pattern in Table 4 asks for nothing better than to waste his (31 votes for c versus 29 for a) is time doing prospectuses, examining (60!/31!29!)p31(1 - p)29. If the contrary machines, etc., and above all because, holds (a > c), then the probability of feeling eclipsed by other geometers, he, observing 31 votes for c versus 29 for a is like d'Arcy, has abandoned geometry for (60!/31!29)pZ9(1- p)31.Assume that a > petty physics" (quoted in Henry 1883). c and c > a are a priori equally likely. The contempt with which Condorcet Then the probability of observing the viewed Borda's scientific contributions vote 31 for c, 29 for a is may help to clarify an abrupt shift in Con- dorcet's argument. As we have already seen, Condorcet's initial objective was to design a method for estimating the "true" ranking of the candidates. But he also By Bayes' rule1" it follows that the poster- recognized that the problem is often one ior probability of c > a, given the ob- of determining which single candidate is served vote, is most likely to be best (1785, lxi-lxv, 122- Pr(c > alvote) = 23). At first glance the solution would appear to be obvious: first rank all of the Pr(vote1c > a)Pr(c > a) candidates and then choose the one that is Pr (vote) top-ranked. But this is not necessarily the most convincing answer. that is, Consider the following example of Pr(c > alvote) = Condorcet (1785, Ixiii) with 60 voters and three candidates. Here c has a simple majority over both a and b and b has a simple majority over a. So there are no cycles, and the ranking by Condorcet's Similarly, rule is cba. That is, cba is the ranking most likely to be correct. But is c the candidate that is most likely to be the best (i.e., top-ranked in the true ranking)? American Political Science Review Vol. 82

Therefore the posterior probability that c approximately as follows: is best is a: (1/2)lZ0+ (1/2)119(23 - 37 + 29 - 31)~ b: (1/2)12" + (1/2)119(37- 23 + 29 - 31)~ c: (1/2)lZ0+ (1/2)119(31- 29 + 31 - 29)~.

In each case the coefficient of E is the sum The posterior probability that b is best is of all painvise votes for the candidate minus the sum of all pairwise votes against the candidate. Since the total number of votes (for and against) is the same for all candidates, the coefficient of E will be maximized for that candidate that receives the most painvise votes over all And the posterior probability that a is others. In this case the answer is b. best is This argument is quite general. For any number of voters and any number of Pr(a1vote) = alternatives, if p is sufficiently close to pz3(l - p)37p29(1- P)~' 1/2, then the candidate with the highest probability of being best is the one that [pz3(l- p)37 + ~ ~- ~ (X 1 receives the most pairwise votes. This is + p3l(1- p)2911. precisely the Borda candidate. The reason Which of a, b, or c has the highest is that the Borda score of a candidate in posterior probability? The answer de- each individual voter's list is equal to the pends on the assumed value of the com- number of candidates ranked below it, petence level p. If p is close to one, then that is, the number that it beats. So a can- the controlling factor is the size of the didate's total Borda score is its row sum in exponent of 1 - p. The most probable the voting matrix.12 candidate will be the one whose weakest Borda's method is therefore one solu- comparison is as strong as possible.ll In tion to Condorcet's problem. In particu- this case the answer would be c, because lar, a candidate that beats every other by the lowest number of votes that c receives simple majority does not necessarily have in any pairwise contest is 31, while a's the strongest claim to being the best can- weakest showing is 23 and b's is 29. didate; the Borda winner may be Suppose instead that p is close to 1/2, stronger.13 Condorcet apparently realized that is, p = 1/2 + E for some small E > 0. this implication of his argument. Under- In this case the relative probabilities de- standably, he would be loath to draw pend essentially only on the numerators much attention to it, given his feelings of the above three expressions. Substitut- toward Borda. In addition, Condorcet ing in p = 1/2 + E we find that the rela- may have become disillusioned with his tive probabilities of a, b, and c can be approach because of its failure to provide approximated as follows: a simple and definitive solution independently of the competence level p. a: (1/2 + e)13(1/2 - e)j7(1/2 + ~)~~(l/2- e)jl For whatever reason, Condorcet b: (112 + ~)37(1/2 ~)~~(1/2~)~~(1/2 - + - abruptly changed course at this point in c: (112 + ~)~l(1/2- ~(~~(1/2~)~l(1/2 - + the Essai (1785, Ixiii-lxv). He abandoned If we expand these expressions using the the statistical framework that he had so binomial formula and drop all powers of E painstakingly built up and fell back on a higher than the first, then the relative more "straightforward line of reasoning. probabilities of the three candidates are In doing so he opened up a whole new Condorcet's Theorv

approach that has had enormous influ- While Condorcet's solution is attrac- ence on the modern theory of social tive, Borda's rule still has advantages that choice. To illustrate Condorcet's new- cannot be dismissed easily. First, it is con- found argument, consider the example in siderably simpler to compute than Con- Table 4. If a candidate, such as c, obtains dorcet's ranking rule. Second, if one a simple majority over every other candi- accepts Condorcet's basic premises, then date then, Condorcet argued, that candi- the Borda winner is in fact a better esti- date is the only reasonable choice. Indeed, mate of the best candidate provided that p candidate a has no reasonable claim to is close to 1/2. Third, if p is not close to being best, therefore a should be set aside. 1/2, then it is still very likely that the So the contest is between b and c. But c Borda winner is the best candidate, even defeats b, so the choice must come down though strictly speaking it may not be the on the side of c. optimum estimate of the best candidate.14 More generally, Condorcet proposed If there are a large number of voters and p that whenever a candidate obtains a sim- is not very close to 1/2, then the probabil- ple majority over every other candidate, ity is very high that the truly best candi- then that candidate is presumptively the date will be selected by any reasonable "best." This decision rule is now known as choice rule (i.e., with high probability it "Condorcet's criterion," and such a candi- will be the majority winner and the Borda date (if it exists) is a "Condorcet win- winner at the same time). The critical case ner" or a "majority candidate" (Fishburn is precisely when p is near 1/2, and then 1973). Borda's rule is definitely optimal. Thus we In advancing this new approach, Con- must either jettison the whole idea of dorcet was not abandoning his basic selecting the "best" candidate with high premise that the object of voting is to probability or admit that Borda's rule is a make the "correct" choice. He merely good way of estimating which candidate shifted his ground and argued for adopt- that is. ing a simple estimate of the best choice On the other hand, when the problem is rather than a complex one. Furthermore, to rank a set of alternatives, Condorcet's this "new" approach is entirely consistent rule is undoubtedly better than Borda's. with Condorcet's earlier solution of the Furthermore, even if one does not accept ranking problem, Indeed, if a majority the specific probability model on which candidate exists, then it must be ranked this conclusion is based, Condorcet's rule first in a Condorcet ranking. The reason is has an important stability property that simple. Suppose that x is a Condorcet Borda's rule lacks, as I shall now show. candidate but that this candidate is not ranked on top in the Condorcet ranking. Then it must be ranked just below some Local Stability and other candidate y. By assumption, x Nonmanipulability defeats y by a simple majority. Therefore, if we switch only the order of x and y in One of the most important concepts in the ranking, then we obtain a new rank- the theory of social choice is independ- ing that is supported by more pairwise ence: a social decision procedure should votes. But this contradicts the definition depend only on the voters' preferences for of the Condorcet ranking as the one sup- the "relevant" alternatives, not on their ported by the maximum number of pair- preferences for alternatives that are in- wise votes. Thus Condorcet's choice rule feasible or outside the domain of dis- is consistent with his ranking rule when- course. Independence has been given sev- ever a majority candidate exists. eral distinct formulations (Arrow 1963; American ~oliticalScience Review Vol. 82

Table 5. A Voting Matrix on Six Alternatives and 100 Voters

a b c d e f Borda Scores

Nash 1950). The one that I shall adopt ones. Consider the following example: A here is closest in spirit to Nash's version: if legislature of 100 members is considering the set of alternatives shrinks, then the re- three alternative decisions a, b, and c (see duced set of alternatives should be ranked Table 5). Other alternatives might in in the same way that they were within the theory be considered, such as d, e, and f, larger ranking. There are at least two rea- but they are weaker than a, b, and c in the sons why independence is desirable. First, sense that any one of them would be de- it says that decisions cannot be manipu- feated by any one of a, b, or c. lated by introducing extraneous alterna- The question is whether the exclusion tives. Second, it allows sensible decisions of the nominally "inferior" alternatives d, to be made without requiring an evalua- e, f affect the choice between a, b, and c. tion of all possible decisions. In practice, Put differently, can an inferior alternative society cannot consider all possible be introduced strategically in order to choices in a situation. It cannot, or at least manipulate the choice between the usually does not, even consider all of the "superior" alternatives? This is surely a best choices. In choosing a candidate for situation that we would like to avoid. Un- public office, for example, it is impractical fortunately Borda's rule suffers from this to consider all eligible citizens. In con- defect. To see this, observe that the total sidering a piece of legislation, it is impos- Borda score of each alternative is simply sible to hold votes on all possible amend- the sum of the row entries corresponding ments. Independence assures that a deci- to that alternative (e.g., the right-hand sion made from a limited agenda of alter- column of Table 5). These scores imply natives is valid relative to that agenda. the ranking cabdef. Now consider just the Enlarging the agenda may introduce bet- three alternatives a, b, and c. The cor- ter possibilities, but it should not cause us responding voting matrix is the one en- to revise the relative ranking of the old closed by dashed lines. The Borda scores possibilities. for this three-alternative situation are 117 Unfortunately, it is impossible to design for b, 105 for a, and 78 for c. Hence when any reasonable social decision rule with just the top three alternatives are con- this property. I claim, however, that sidered, they would be ordered bac, independence can be satisfied if we restrict which is the reverse of how the electorate ourselves to agendas that are sufficiently would have ordered them in the context "connected." By a connected agenda I of all alternatives. Introducing the inferior mean, roughly speaking, a subset of close- alternatives changes the outcome from b ly related alternatives rather than an arbi- to c. trary collection of unrelated or extreme I claim that Condorcet's rule is immune Condorcet's Theory to this type of instability. To compute the tion that abcdfe is supported by the maxi- Condorcet solution observe first that a mum number of pairwise votes. A similar has a simple majority over every other argument applies if the last two alterna- alternative, hence (as was demonstrated tives are dropped, the last four, or indeed in the previous section) Condorcet's rule any bottom segment of the ranking. Simi- must rank a first. Since each of the alter- larly, the ordering of the bottom part of natives a, b, and c defeats each of the the list will be preserved if we remove the alternatives d, e, and f by a majority, it top alternative, the top two alternatives, also follows that the former three alter- or any top segment of the ranking. natives must be ranked above the latter In general, a Condorcet ranking has the three. For if this were not the case, then property that the ordering of the alter- some alternative would be ranked im- natives does not change whenever we mediately below another alternative that restrict attention to an interval of the full it defeats, which is impossible in a Con- ordering.15 This property will be called dorcet ranking. To maximize agreement local stability. Intervals tend to consist of with the pairwise voting data, a must be alternatives that are relatively closely ranked first, b second, and c third. Posi- related. For example, an interval might tions four, five, and six in the ranking will consist of a set of candidates that occupy be occupied by d, e, and f in some order. a certain segment of the political spectrum To determine which order, it suffices to (e.g., "centrist"). Or it might consist of a apply Condorcet's criterion to these three series of minor modifications or amend- alternatives alone; that is, they should be ments to a proposed piece of legislation. ordered so as to agree with the maximum I shall show that Condorcet's method is number of pairwise votes that they the only plausible ranking procedure that receive among themselves. The reader is locally stable. To do so we must recall may verify that the ordering that accom- several properties that are standard in the plishes this objective is dfe. Hence the social choice literature. Let F be a social total ordering implied by Condorcet's rule ranking rule, that is, a rule for strictly is abcdfe. This is quite different from the ordering any set of alternatives given the Borda ordering. preference data of the voters. We shall I have worked out the details of this assume that the input data to F is given in example to show that Condorcet's rule is the form of a voting matrix: the number not necessarily difficult to compute even of votes for and against each pair of alter- when a sizable number of alternatives is natives. All voters are weighted equally involved (though it is certainly more diffi- because only the number of votes counts, cult than Borda's method). It also illus- not the identities of the individuals who trates why Condorcet's method satisfies a cast the votes. This property is known in certain form of independence. For exam- the literature as anonymity. F is neutral if ple, if the bottom three alternatives are it is not "biased with respect to one or dropped from consideration, then Con- another alternative: if the names of the dorcet's rule (unlike Borda's) must pre- alternatives are permuted, then their serve the ordering abc. Indeed, if it did ranking is similarly permuted. Taken not, then the interval abc could be re- together, anonymity and neutrality rearranged into a new ordering that is sup- quire that F sometimes yields ties.l6 A ported by more painvise votes. But then third standard condition in the social abc could also be rearranged within the choice literature is that Fbe unanimous: if full ordering abcdfe to obtain a new all voters cast their votes the same way on ordering that is supported by more pair- every pair, then the ordering of the candi- wise votes. This contradicts the assump- dates agrees with all of the votes. American Political Science Review Vol. 82

Finally we shall require a degree of con- known, contribution by Condorcet to the sistency in the way that F aggregates theory of group decision making. Con- individual opinions into group opinions. dorcet proposed a decision rule that ap- Consider a bicameral system such as the plies to any voting situation whether U.S. Congress. To pass, a decision must cyclic majorities are present or not. This receive a majority of both chambers. This rule can be interpreted as a best guess much is clear when a decision involves about the "true" ordering of the alterna- just two choices. Now consider a decision tives. The rule can also be justified on involving more than two alternatives. entirely diffeent grounds; namely, it is Suppose that both chambers use the same unbiased with respect to both individuals decision rule F, and that they consider the and alternatives and it aggregates indi- same agenda. Let each chamber separate- vidual opinions in a "consistent" way. ly rank all of the alternatives. If both This axiomatic justification differs signifi- chambers happen to rank them in exactly cantly from the probabilistic one in that the same way, then we may say that this it places emphasis on the operational ranking is approved by both chambers. If properties of the method rather than on a a ranking is approved by both chambers, model about how the world works. Of then it stands to reason that it would also course one may still question whether be approved if the two chambers were to these are the "right" properties, just as one be considered as a single body (i.e., to may question whether Condorcet had the cast their votes in plenary session). F satis- "right" probabilistic model. But at least fies reinforcement if whenever a ranking the nature of the choice is clarified. Fur- is approved by two separate groups of thermore, one must admit that the specific voters, then it would also be approved probabilistic model by which Condorcet when the votes of the two groups are reached his conclusions is almost certainly poo1ed.17 not correct in its details. The plausibility The condition of "local stability" can of his solution must therefore be subjected now be formally stated as follows: Let A to other tests, as I have attempted to do. be an agenda of alternatives, let V be a I conclude that there is a strong case for voting matrix on A, and let F(V) = R be Condorcet's rule on two counts. First, in the (unique) consensus ranking. F is local- situations where voters are primarily con- ly stable if for every interval B of the rank- cerned with the accuracy of a decision (as ing R, F(VB) = R where VBandRBrepre- in jury trials, expert opinion pooling, and sent V and R restricted to the subset B. In in some types of public referenda), Con- case of ties, the condition says that for dorcet's method gives the ranking of the every R E F(V), every interval B of R, RB€ outcomes that is most likely to be correct. F(VB), and if R'B E F(VB)for some R'B z Second, where voters are motivated pri- RBIthen RfBcanbe substituted for RBinR marily by their subjective preferences to obtain a new ranking R' E F(V). about outcomes, Condorcet's rule is re- It follows from a theorem of myself and sponsive to changes in individual pref- Arthur Levenglick (1978) that Condor- erences, is unbiased with respect to voters cet's rule is the unique ranking rule that and alternatives, and (unlike Borda's rule) satisfies anonymity, neutrality, unanim- it is relatively stable against manipulation ity, reinforcement, and local stability.18 by the introduction of unrealistic or in- Note that Borda's rule satisfies all of these ferior alternatives. In short, we need not properties except local stability. resolve the question whether voters attempt to judge which decision is in the Conclusion public interest or whether they merely register their subjective opinions on an The purpose of this article has been to issue. Both of these elements are probably draw attention to a major, but little- present in most decisions. In either case, Condorcet's Theorv

Condorcet's method is a rational way of 10. Condorcet arrived at this answer intuitively, aggregating individual choices into a col- not by an explicit application of Bayes' rule. 11. This "minimax" rule was proposed in modem lective preference ordering. times by Kramer (1977). 12. Borda himself gave this alternative statement of his method. 13. This idea was conjectured independently by Grofman (1981). The distinction between the major- Notes ity alternative and the alternative most likely to be best has also been discussed by Baker (1975) and This work was supported by the Office of Naval Urken and Traflet (1984). Research under contract N00014-86-K-0586 at the 14. The following argument was suggested by University of Maryland. The author is indebted to Bernard Grofman. Bernard Grofman and William Riker for very help- 15. This property of Condorcet's method (which ful comments on the manuscript. is known in the literature as the "median proce- 1. Condorcet did not explicitly say that he was dure") was noted by Jacquet-Lagreze (1969). For addressing Rousseau's problem, but Rousseau's other results on the median procedure see Michaud ideas would have been much discussed in Condor- and Marcotorchino (1978), Barthelemy and Mon- cet's circles. Baker (1975, 229) also suggests that jardet (1981), Michaud (1985), and Barthelemy and Condorcet drew his inspiration from Rousseau. McMorris (1986). Grofman and Feld (1988) explore the connections 16. For example, if there are an even number of between Condorcet's and Rousseau's ideas in greater voters and every pairwise vote is a tie, then anony- depth. mity and neutrality imply that all linear orderings of 2. Poisson (1837) studied a similar set of ques- the alternatives are equally valid. Thus Fmust some- tions using probabilistic methods. See also Gelfand times take on multiple values. and Solomon (1973) for recent work in the Poisson 17. We require furthermore that if another rank- tradition. ing is approved by only one of the two groups, then 3. A related problem is to rank teams or players it is not tied with a ranking that is approved by both based on their win-loss records (as in baseball, ten- of them. nis, or chess). Condorcet's model can be used to 18. Let F satisfy the five conditions of the theo- estimate the most likely true ordering of the players rem. When there are exactly two alternatives, it (see Batchelder and Bershad 1979; Good 1955; Jech follows that F is simple majority rule (Young 1974). 1983). Now consider the case of more than two alterna- 4. The following preference data would yield the tives. Let R be the social order. Local stability im- painvise votes in Table 1: six voters have preference plies that whenever two alternatives are ranked one abc, five voters have preference bca, and two voters immediately above another in R, then their ordering have preference cab. It bears emphasizing, however, remains invariant when F is restricted to those two that Condorcet did not think of the voters' opinions alternatives alone. Since F is simple majority rule on as expressing their personal preferences. Rather, they every two alternatives, it follows that in the social represent the voters' best judgments as to which can- order R every alternative must defeat (or tie) the didate in each pair is the better one. alternative ranked immediately below it. Further- 5. In this translation I have modified the mathe- more, if the two alternatives are tied, then they may matical notation slightly to conform with modern be switched to obtain a social ordering tied with R. It conventions. can be shown by induction on the number of alter- 6. Michaud (1985) comes to a similar conclu- natives that this property, together with neutrality, sion. unanimity, and reinforcement implies that F is Con- 7. I exclude the case p = 1, because this would dorcet's ranking rule (Young and Levenglick 1978). imply that all voters agree with probability one. The case where the voters have different a priori probabilities of being right is treated in Young 1986 (see References also Nitzan and Paroush [I9821 and Shapley and Grofman [1984]). Arrow, Kenneth J. 1963. Social Choice and Indi- 8. For reasons that remain obscure, Borda's vidual Values. 2d ed. New York: John Wiley. method was jettisoned by Napoleon after his own Baker, Keith M. 1975. Condorcet. Chicago: Univer- election to membershp (Mascart 1919). sity of Chicago Press. 9. In a footnote, Condorcet adds that his own Barthelemy, J. P., and F. R. McMorris. 1986. "The treatise "had been printed in its entirety before I had Median Procedure for n-Trees." lournal of Clas- any acquaintance with this method [of Borda], apart sification 3:329-34. from the fact that I had heard several people speak Barthelemy, J. P., and Bernard Monjardet. 1981. of it" (1785, clxxix). "The Median Procedure in Cluster Analysis and American Political Science Review Vol. 82

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H. P. Young is Professor of Public Policy and Applied Mathematics, School of Public Affairs, University of Maryland, College Park, MD 20742.