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Condorcet winners and the paradox of voting: probability calculations for weak preference orders.

As conventionally understood, democracy implies a process of reaching social decisions via the principle of majority rule. Political decisions are putatively made in accordance with the preferences of citizens as a collective entity. Collective preferences, in turn, are determined by an aggregation of individual preferences, wherein the collective choice is that option which is majority-preferred to all other options in pairwise comparisons; such a majority choice is called a Condorcet winner. More than anything else, it is this emphasis on majority rule that underlies both the liberal (e.g. Locke) and radical (e.g. Rousseau) interpretations of democracy.

Unfortunately, this simple vision of majority rule is plagued by deep and seemingly ineradicable theoretical difficulties. As the social choice literature documents, Condorcet winners need not exist (Arrow 1963).(1) Simply put, there need be no alternative that is majority-preferred. If so, then voting games would appear capable of producing results that are not consistent with the majoritarian principle. This phenomenon has been known for centuries. In fact, awareness of the problem has affected practical politics, from Pliny the Younger's manipulation of the sentencing agenda in the case of the murdered consul Afranius Dexter (Farquharson 1969) to tantalizing speculations that Thomas Jefferson and James Madison may have been influenced by Condorcet's work on social choice in their views of governmental structure (McLean and Urken 1992; Miller and Hammond 1989).(2)

Of course, the various impossibility theorems about group choice suggest not that Condorcet winners never exist, but only that they are not guaranteed. In other words, majority-preferred alternatives are not assured, though they may sometimes exist. Given that fact, a great deal of attention has been paid to the question of the frequency of Condorcet winners.(3) One approach has been to search for empirical examples of Condorcet winners in real-world situations, such as presidential elections (Niemi and Wright 1987; Radcliff 1994), legislatures (Riker 1958), university elections (Niemi 1970), and professional associations (Chamberlain, Cohen, and Coombs 1984). Another has been to examine the a priori probability of majority winners (devoid of any particular institutional or empirical setting) by making certain simplifying assumptions about the preferences of individuals (DeMeyer and Plott 1970; Garman and Kamien 1968; Klahr 1966; Niemi and Weisberg 1968).

While the latter studies deservedly have become classics within the public choice literature, each suffers from the fact that they consider only "strong" preferences. In other words, they begin with the assumption that individuals have strict preferences over each pair of alternatives, such that indifference relations are not admissible. The difficulty is that this assumption is purely analytic: it is not driven by any empirical evidence on the structure of preferences but rather by the daunting complexity of the problem. Since that complexity increases dramatically as the number of possible (transitive) preference orders increases, the appeal of strong orders as a simplifying assumption is obvious. The need to reduce complexity was particularly compelling, given the technological limitations of the period in which much of this work was completed (e.g., DeMeyer and Plott 1970; Klahr 1966).

[TABULAR DATA FOR TABLE 1 OMITTED]

In the last decade, there has been enormous progress in both computer hardware and the conceptual tools of computational modelling (Taber and Timpone 1994). In short, it is now feasible to relax the assumption of strong preferences.(4) Equally important, recent empirical evidence suggests that indifference relations are quite common, at least in the context of some mass voting games (Brady and Ansolabehere 1989; Radcliff 1993). Given that we thus have tools that render the strong-order assumption unnecessary, as well as evidence that it is empirically unrealistic, we provide results of computer simulations of the structure of collective preferences that do not depend upon strong individual preferences.

PRIOR RESEARCH

Given n voters (n [greater than or equal to] 3) and m alternatives (m [greater than or equal to] 3), there are two quantities of interest: (1) the probability of a Condorcet winner, Q(m, n), and (2) the probability of a transitive, strongly ordered social preference structure, P(m, n).(5) Two research strategies have been used to estimate these probabilities. The first and most obvious is to derive analytic expressions and compute the resulting values for a given m and n (e.g., DeMeyer and Plott 1970). While the advantage of the analytic solution is obvious, it quickly becomes intractable for large values of rn or n. This problem is doubly pronounced when abandoning the assumption of strong preferences, in that the number of possible preference profiles becomes extremely large as rn increases.

Accordingly, we have turned to the second approach: computer simulation (e.g., Klahr 1966). With this method, one generates a preference profile for each of the n individuals (selected according to a uniform probability distribution from the set of all possible transitive preference profiles for the m alternatives), stores each individual's preferences for all pairwise comparisons over the given alternatives, and computes the resulting social order. Given x iterations of this procedure, one may estimate Q(m, n) to be the proportion of the x simulated elections that yield a Condorcet winner, and P(m, n) to be the proportion that produce a fully transitive strong social order. Given the emphasis on the absence of these outcomes, the values (1 - Q(m, n)) and (1 - P(m, n)) are typically reported.

Table 1, which replicates and extends prior findings, reports the probability of no Condorcet winner (1 - Q(m, n)) for strong orders (for comparison to prior work, see Ordeshook 1986, 58; Riker 1982, 122). As is apparent, Condorcet winners become less likely as (1) the number of alternatives increases and (2) the number of individuals increases. As one reads either down each column (meaning more voters for a given number of alternatives) or across each row (meaning more alternatives for a set number of voters), probabilities uniformly increase. Observing that "there is an apparent limit of one as n and m approach infinity," Ordeshook reasonably concludes that "in large societies confronted with numerous alternatives, the likelihood that [Condorcet winners do not occur] is great" (1986, 58).

Estimates of (1 - P(m, n)), or the probability of an intransitive or weak social order, are provided in Table 2 (again, limiting our analysis to strong individual orders). Qualitatively, these results repeat those just described. But, as one would expect, the probability of an intransitive social ordering is always greater than or equal to the probability of no Condorcet [TABULAR DATA FOR TABLE 2 OMITTED] winner, in that we now consider all intransitivities rather than merely top cycles. The former quantity is also more sensitive to increases in m and n, such that the probability of a strong transitive aggregation of individual preferences rapidly approaches zero as the size of the electorate and the number of options increase.

NEW ESTIMATES FOR WEAK ORDERS

Our new simulation follows this same logic but assigns to voters a preference order chosen equiprobably from all transitive profiles, including those containing indifference relations.(6) For example, with three alternatives there are 13 possible orders rather than the six when considering only strong orders. The simulation program is described in an Appendix.(7)

Table 3 reports the new estimates for the probability of no Condorcet winner, denoted (1 - [Q.sup.*](m, n)). Immediately we can see that one of the classic findings remains unchanged: Condorcet winners still become less likely as the number of alternatives increases. But the second classic result - that Condorcet winners uniformly become less likely as the number of voters increases - does not remain when we allow individual indifference. Indeed, there are several important differences between the estimates in Tables 1 and 3. First, as might be expected, the probability of no majority winner is often greater when we allow indifference, but only for small n and m. For three voters and three alternatives, the probability increases by a factor of six (from .056 to .336); for seven voters and five alternatives, a Condorcet winner is almost twice as unlikely when indifference is involved (.418 vs. .216).

Significantly, the difference decreases as the number of individuals increases, suggesting a second and more interesting finding. In Table 1, which assumes strong preferences, the probability of no Condorcet winner increases monotonically with the number of individuals. Table 3 shows, however, that when we allow weak preferences, the probability is at a maximum when the number of alternatives equals the number of individuals.(8) For any number of options, the probability increases as the number of voters increases, but only until n equals m. It then falls monotonically.

It thus appears that the probability of no majority winner is quite large for a small number of voters (especially with many alternatives) but decreases considerably as the number of voters increases relative to the number of options. Thus, Condorcet winners are more likely when the ratio of individuals to alternatives is great (as in mass elections). However, in smaller groups (such as committees), there is much less chance that a majority winner will be attained.

In sum, relaxing the assumption of strong preferences radically alters our deductions about the general probability of Condorcet winners, at least in regard to the size of the electorate. With the assumption of strong orders, the probability of no majority winner increases with the number of individuals or alternatives, though it remains fairly small for few alternatives. Following these findings for strongly ordered individuals, one might conclude that the issue is most pronounced in "large societies confronted with numerous alternatives" (Ordeshook 1986, 58). By contrast, allowing individual indifference, the probability of no majority winner is largest for committees or small electorates but decreases substantially with larger electorates, eventually becoming smaller than when we assume strong preferences. Figure 1, which directly compares the trends [TABULAR DATA TABLE 3 OMITTED] in Tables 1 and 3 for m = 3, illustrates this point. Based on these trends, we might suggest that the problem is actually more relevant to small groups than large societies.(9)

Thus, when n is relatively small (less than 501 for m [less than] 7),

(1 - [Q.sup.*](m, n)) [greater than] (1 - Q(m, n)),

but when n is relatively large (greater than 501 for m [less than] 7),

(1 - [Q.sup.*](m, n)) [less than] (1 - Q(m, n)).

For fewer than seven alternatives, these trend lines cross before we reach 501 voters. Moreover, this result probably holds for larger numbers of alternatives, but computational constraints force us to extrapolate from the data for this conclusion.(10) Increasing the number of alternatives increases both sets of estimates, though those based on strong orders appear to be more sensitive to changing rn than those based on weak preferences.(11)

Comparing Tables 2 and 4 suggests a similar pattern for (1 - [P.sup.*](m, n)) and (1 - P(m, n)). The probability of weak or intransitive social orders based on strong individual preferences is smaller than those allowing individual indifference for small n, but the former increases with larger n while the latter decreases.(12)

To this point, we have treated all preference profiles as equiprobable in both sets of simulations. But since there are more possible weak than strong preference orders (e.g., 7/13 orders are weak for m = 3, and the discrepancy grows with m), this assumption may actually overweight the probability of indifference. At a minimum, we must check to see how sensitive our basic findings are to this problem. More generally, it might be substantively interesting to explore the impact of varying the amount of indifference in the electorate.

We conducted a third simulation analysis in which we manipulated the probability of selecting a strong or weak profile, denoted W. The set of possible preference orders for a given m was divided into a strong subset and a weak subset. Each individual was then assigned to either a "strong" or "weak" condition, according to a probability W. Those in the strong category (i.e., they have preferences across all pairs of alternatives) were then assigned a preference order from the strong subset according to the uniform probability distribution. Those in the weak category were assigned preferences from the set of possible weak orderings, also according to the uniform probability distribution.(13) Note that this procedure allows us to control the amount of indifference present in the [TABULAR DATA FOR TABLE 4 OMITTED] electorate because for n sufficiently large, the individual probability W translates into a proportion of voters w who have weakly ordered preferences. We do not control how much indifference a voter in the weak condition has, however, allowing this to depend on random assignment. Completely indifferent voters are least likely, because there are more possible combinations with fewer indifference relations. Of the seven weakly ordered preference patterns when m = 3, for example, only one is totally indifferent (x = y = z), while six have one indifference relation (e.g., x = y [greater than] z). Table 5 gives the results from this analysis for three, four, and five alternatives, with estimates based on one hundred thousand iterations. Note that column 1, where W = 0, reports (essentially) the same results as the strong preference estimates in Table 1 and that W = .54 generally recreates our weak results from Table 3 for m = 3 (7 of 13 alternatives are weak, which translates to W = .54). Finally, as described earlier, though 100% of the voters who contribute to the last columns of Table 5 are weakly ordered, only a minority of them have no preference.

For three voters, Condorcet winners become increasingly unlikely as W increases, reaching a maximum probability of .389 when all voters have weak orders. More interesting, however, is the finding that these trends are nonmonotonic for n [greater than] 3, reaching some maximum before W = 1.00 and then declining. Moreover, this maximum estimate for 1 - [Q.sup.*](m, n) is reached for an ever smaller value of W as n grows larger. With three alternatives and an electorate of 31, for example, Condorcet winners are least likely when each voter has between .10 and .20 chance of being indifferent between at least one pair of alternatives. But 101 voters choosing across three alternatives are least likely to get a Condorcet winner when fewer than 5% of voters are weakly ordered (when W = .05, 1 - [Q.sup.*](3, 101) = .143). The chance of getting a majority winner increases steadily from this point as more voters' preference orders contain indifference. And we also find this basic pattern for m = 4 or m = 5.

Mass electorates (n [greater than or equal to] 501) do not have to be unreasonably indifferent for our basic conclusion to hold. When 40% or more of an electorate of 1,001 have weakly ordered preferences, they are more likely to get a Condorcet winner than if all have strong preferences. In general, for large n, highly indifferent electorates are most likely to find Condorcet winners,(14) but a lot of indifference is not necessary for majority winners to be more probable than for electorates with strongly ordered voters. This is true for three, four, or five alternatives. On the other hand, committees (n relatively small) with indifferent members are less likely to find Condorcet winners than those with strongly ordered members.

DISCUSSION

In terms of committees (where the number of voters tends to be relatively small), Condorcet winners are much less likely when abandoning the assumption of strong preferences. If committee members tend to be indifferent among some of the alternatives, the probability of achieving a nonarbitrary outcome becomes relatively small. When many alternatives are available, committees will have an even more difficult time finding Condorcet winners. This result suggests that institutions that rely heavily on committee decision making may require special mechanisms designed to create stable and fair decision outcomes (Miller and Hammond 1989; Shepsle 1979).

In terms of mass elections (where the number of voters is by definition extremely large), our results show that Condorcet winners are more likely than when using the strong-order assumption. Grofman (1993), interestingly enough, comes to a similar conclusion, [TABULAR DATA TABLE 5 OMITTED] though following a different argument. Observing that empirical examples of majority cycles in mass electorates are nearly as rare as sightings of the Loch Ness monster, he suggests that social choice theorists greatly underestimate the probability of majority winners because they assume that preferences are equally likely, when in fact the distribution of preferences is skewed by political culture and the two-party system.(15) Though we agree that the uniformity assumption is highly suspect, we find that relaxing a different assumption - that of strong preference orders - produces theoretical trends in keeping with his empirical observation. To paraphrase Grofman, when we shot a million Arrow's theorems into the body politic, very few hit anything (1993, 1547)! In other words, fewer simulation runs over mass electorates produced cycles when we allowed for individual indifference.(16) These findings clearly suggest that the democratic method is more likely to function as predicted in majoritarian theory when large numbers participate.

Curiously, that conclusion is quite close to the traditional argument that large numbers of citizens are more likely to make "good" decisions than smaller numbers. Aristotle makes the classical case for this contention in the Politics, arguing that the summing of mass opinion may be superior to that of a more select group, even if the elite is, person for person, more knowledgeable or sophisticated than the mass. Similar arguments can also be found in Rousseau's Social Contract and Condorcet's "jury theorems" (for discussions in social choice terms, see Grofman and Feld 1988; Young 1988). While our results say nothing about the quality of decisions that the analogy of the jury implies, they do suggest that larger groups are more likely to produce Condorcet winners than smaller ones. Thus the structure of social preferences, at least, is more likely to conform to theoretical expectations when more rather than less people take part in voting.

We close with two caveats. First, and most obviously, we do not pretend that the results from this simulation are empirically "realistic" (for a discussion of these limitations in computational modelling, see Taber and Timpone 1994). Like prior scholars of the subject, we recognize that preference profiles are not really uniformly distributed in the electorate, though we still make that simplifying assumption (only partly relaxed when we manipulate the probability of indifference). While this assumption limits the applicability of the findings to real-world situations, it does allow us to determine the a priori propensity for cycling and, more importantly, to better understand how that propensity varies with the number of people choosing and the range of options available to them.

Second, while our work paints quite another picture than did prior studies that only considered strong preferences, we do not mean to imply that ours is categorically superior. The opposing results simply reflect the difference in the enabling assumption regarding the admissibility of indifference relations. The relative utility of the results reported here versus prior findings thus depends entirely upon the empirical question, peculiar to any particular institutional setting, of whether most individuals have strict preferences over each pair of alternatives. Still, though the assumption that indifference relations do not occur may not be genuinely "heroic" in the worst sense of that term, it nonetheless remains empirically suspect and theoretically unnecessary. We thus suggest, in the absence of any empirical reason to believe otherwise, that our estimates are likely to prove more useful.

APPENDIX

Social Choice 4.0 (SC4) and Weak Social Choice 5.0 (WKS) are computer programs (written in C and compiled for an IBM-compatible personal computer) that estimate the probabilities of getting a majority winner (Q(m, n)) and of getting a transitive social ordering (P(m, n)) for different electorate sizes (n) and numbers of vote alternatives (m). SC4 assumes that all voters have strong preference patterns, while WK5 allows voters to be indifferent between pairs of alternatives. The probabilities are estimated as the proportion of times no Condorcet winner emerges or the proportion of times the social ordering is intransitive over a large number of simulated elections.

Each program consists of five subprocesses and two loops, one loop embedded inside the other. SC4 and WK5 differ in the assignment of individual preferences only, so they will be discussed together.

Input. The first subprocess simply defines the parameters (m, n, and the number of iterations, x) for the simulation run. Once these parameters have been defined, two loops are set up to simulate the x elections and the choices of the n voters within each election.

Individual Preferences. If we only allow strong preferences (SC4), there are m! possible preference orders. Assuming that each voter is equally likely to have any of these possible preference patterns, we can "discover" each voter's preference pattern over the m alternatives by selecting without replacement according to the uniform probability distribution. Relaxing the assumption of strong preferences, the problem becomes somewhat more complex. For example, instead of 6 transitive orders for 3 alternatives, there are now 13 distinct preference patterns. (There are actually [2.sup.m-1]m! different preference patterns, but many are redundant; for example, x [greater than] y = z is equivalent to x [greater than] z = y.) The logic of individual choice remains the same, however. WK5 determines the admissible profiles, including all strongly and weakly ordered preference patterns and chooses from this set uniformly.

Individual Vote. Each individual's vote must be tallied. This is generated through a pairwise comparison of the m alternatives for each voter, keeping track of the voter's pairwise preferences across all the alternatives. Each individual's "votes" on all pairings of alternatives are stored in a large matrix (VOTE) summing individual pairwise preferences.

Social Choice. Once we have tallied the preferences of all the voters in pairwise comparisons over the rn alternatives, a social choice must be made. If the VOTE matrix for the simulated election shows an alternative that is preferred to all others, a CONDORCET counter increments. Similarly, if the matrix shows a completely transitive social order, a TRANSITIVE counter increments. The program then loops to the next simulated election.

Output. After all x elections have been simulated, frequency-based estimates of Q(m, n), [Q.sup.*](m, n), P(m, n), and [P.sup.*](m, n) are output. These estimates are equal to CONDORCET/x and TRANSITIVE/x, respectively. Following convention, we report 1 - Q(m, n), 1 - P(m, n), 1 - [Q.sup.*](m, n), and 1 - [P.sup.*](m, n).

Notes

The order of authors was determined by the tyranny of the alphabet. We are grateful to Joel Kaji and Jeff Segal for their helpful suggestions. This research was supported by the National Science Foundation under award no. SES-9102901.

1. For an accessible review, see Riker 1982; for a discussion of later developments, see Schofield, Grofman, and Feld 1988.

2. Though it seems clear that Jefferson was influenced by Condorcet, the case for Madison is more tenuous, and whether either truly understood his work on social choice has been questioned (McLean and Urken 1992).

3. Grofman (1993) foreshadows one of our findings when he asserts that social choice theorists greatly overestimate the probability of cycling majorities in mass elections (see also Feld and Grofman 1992).

4. Another questionable assumption of earlier work - that all preference orders are equally likely to be selected by each individual - has not been relaxed in the work reported here, though empirical work also undermines this assumption (Grofman 1993; Niemi and Wright 1987).

5. P(m, n) is a measure of the "robustness" of the social choice process in the sense that it measures the likelihood of attaining a collective order with a majority first, second, . . . , and mth preference over the remaining alternatives. Such a social order contains no ambiguity in the form of cycles or indifferences. For strong individual preferences (containing no indifference), a "strong" collective ordering is assured, though it could be intransitive. When we allow individual indifference, however, collectively transitive orders can contain indifference. So P(m, n) and [P.sup.*](m, n) are defined to be the probabilities of getting a fully transitive and strongly ordered collective preference order.

6. Indifference relations formally have three properties: reflexive, symmetric, and transitive. (For fuller treatment, see Bonner 1986, chap. 3.) Preference relations are irreflexive, asymmetric, and transitive. We also manipulate the probability of indifference in other simulations reported later.

7. For either the C code or a working copy, compiled for IBM-compatible personal computers, contact Charles Taber, Department of Political Science, State University of New York, Stony Brook, Stony Brook, NY 11794-4392.

8. We have no theoretical explanation for this finding.

9. We think this result adds to the significance of the structure-induced equilibria literature (see Shepsle 1979). This literature emerged as a response to the findings of McKelvey (1976, 1979) showing that in an n-dimensional, majority-rule system, if a majority winner is not produced, the majority-rule relation "breaks down completely" (Ordeshook 1986, 81). Shepsle (1979) demonstrated that institutional arrangements (or structural arrangements) in voting systems such as committee jurisdiction and germaneness of amendments can dilute the pervasiveness of cyclical majorities or prevent them from occurring at all. While we recognize that our probability estimates presented here are not derived in a spatial preference model, we believe that our findings implicitly bear on the importance of structure-induced equilibria. If non-Condorcet winners are much more likely for small groups than for large societies (as is the case when indifference relations are allowed), then the institutional arrangements that preclude cyclical majorities under the traditional assumption of strong preference orderings are extremely important when indifference relations are considered. In short, institutional arrangements in small decision-making bodies (i.e. committees) really matter.

10. Even with today's power, simulating an election with m = 6 and n = 1,001 can take a week of computer time!

11. Notice that [Q.sup.*](3, n) approaches .056 for very large n. Oddly, this is the oft-computed value of Q(3, 3). Since we can think of no explanation, we must assume this is a coincidence.

12. Values were also calculated for even n, but for simplicity, they are not reported. In general, 1 - Q(m, n) is larger for even than odd n, supporting the work of Kelly (1974). The difference between odd and even electorates, though large for small n, rapidly decreases as n grows large. Kelly's (1986) conjecture that these theorems should extend to the case of weakly ordered individual preferences was also supported, though the difference was extremely small. As we have seen, Kelly's further conjecture that the general behavior of Q(m, n) would be repeated for weak preference orders was not supported.

13. We do not directly manipulate the "amount" of indifference because we do not control how indifferent an individual is. Once an individual is categorized "weak," they are as likely to be x = y = z as x [greater than] y = z. A different approach would vary the probability of indifference in each binary comparison, which would directly control how indifferent people are likely to be. We prefer our closer analog to the classical analysis, because we treat profiles as the unit for selection.

14. If they are all completely indifferent, however, any election would end in a meaningless tie.

15. Radcliff (1993, 1994) makes a similar point, arguing that ideology constrained the preferences of the electorate in the 1972-84 presidential elections.

16. Note, however, that not many of Arrow's theorems find their mark for strong preferences either. But for strong preferences, the trend monotonically increases with the number of voters, which is Grofman's real point and the major traditional finding that our results amend.

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Bradford Jones is Assistant Professor of Political Science, University of Arizona, Tucson, AZ 85721.

Benjamin Radcliff is Assistant Professor of Government, University of Notre Dame, Notre Dame, IN 46556-0368.

Charles Taber and Richard Timpone are Assistant Professors of Political Science, State University of New York at Stony Brook, Stony Brook, NY 11794-4392.
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Date:Mar 1, 1995
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