¹ Well, maybe somewhat more than a “little less important,” but not trivial by comparison: for evidence in a very practical argument, consider Servan-Schreiber, J. J., The American Challenge (New York: Atheneum, 1968), pp. 170–178 Google Scholar.

² This limitation is illustrated by Abraham Lincoln's remark: “Unanimity is impossible; the rule of a minority as a permanent arrangement, is wholly inadmissable; so that, rejecting the majority principle, anarchy or despotism in some form is all that is left.” Cited in Mayo, Henry B., An Introduction to Democratic Theory (New York: Oxford University Press, 1960), p. 178 Google Scholar, and in Thorson, Thomas Landon, The Logic of Democracy (New York: Hold, Rinehart and Winston, 1962), p. 142 Google Scholar.

³ Rules with a value equal to or less than n over 2, of course, present special liabilities, for they permit the formation of more than one winning coalition on a single issue. But for some classes of issues this is not a serious difficulty, since multiple policies may be effected. An example is provided by certiorari proceedings in the U.S. Supreme Court: in these instances, the so-called “rule of four” (k = 4, n = 9) is applied, so that any four justices may require that a writ be granted. This produces no difficulties, since the granting of one writ does not preclude the granting of another—multiple policies are permissible.

⁴ The term “political committee” is used here in its broadest sense—to include all bodies which vote on collective policies. This use is illustrated by Black's, Duncan book, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1963)Google Scholar.

⁵ This assumption is less innocuous than it may-appear: complete indifference is not improbable even among the very active (viz. committee members).

⁶ Here, I am putting aside the theoretically important “problem of intensity.” In effect, the value assumption implies that all issues evoke equally intense preferences. This decision rests on theoretical intractability of the problem. Some of the major difficulties are outlined in, Kendall, Willmoore and Carey, George, “The ‘Intensity’ Problem and Democratic Theory,” this Review, 62 (1968), 5–24 Google Scholar, and in Rae, Douglas W. and Taylor, Michael, “Some Ambiguities in the Concept of ‘Intensity,’” Polity, 1 (03, 1969)CrossRefGoogle Scholar.

⁷ See Dahl, Robert A., A Preface to Democratic Theory (Chicago: University of Chicago Press, 1956), pp. 67, 84 Google Scholar.

⁸ This requirement is closely related to K. O. May's condition of “anonymity,” under which the decision-rule does not depend in any way upon the identification of individual members. See his “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision,” Econometrica, 20 (1952), 680–684 CrossRefGoogle Scholar. May's paper shows that majority rule alone satisfies “anonymity” and three other conditions simultaneously.

⁹ Reimer, Neal, “The Case for Bare Majority Rule,” Ethics, 62 (1951–1952), 16–32 CrossRefGoogle Scholar. The cited statement is from p. 17.

¹⁰ Baty, Thomas, “The History of Majority Rule,” The Quarterly Review, 216 (1912), 1–28 Google Scholar See especially pp. 22–23. The Council also was tri-cameral: within each chamber, a two-thirds rule prevailed; between chambers the rule of consensus was followed. This is indeed a restrictive decision-rule, and a fine hedge against doctrinal error, if we assume the existing doctrine is altogether correct.

¹¹ The Social Contract, translated by Cole, G. D. H. (New York: E. P. Dutton, 1950), p. 107 Google Scholar. Rousseau's second rule was that “… the more the matter in hand calls for speed, the smaller the prescribed difference in the number of votes may be allowed to become: where an instant decision has to be reached, a. majority of one should be enough.” Condorset actually devised a truth-function theory using probabilistic assumptions similar to those found in Part II below. See Essai sur l'application de l'analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix (Paris, 1785)Google Scholar.

¹² Olson, Mancur, The Logic of Collective Action (Cambridge, Massachusetts: Harvard University Press, 1965)Google Scholar; and Buchanan, James and Tullock, Gordon, The Calculus of Consent (Ann Arbor: University of Michigan Press, 1962)Google Scholar.

¹³ Buchanan and Tullock, op. cit., p. 29. The quotation is one of two interpretations which these authors give to the more general criterion of utility maximization.

¹⁴ Ibid., pp. 64, 68.

¹⁵ Ibid., p. 70. “For a given activity the fully rational individual, at the time of constitutional choice, will try to choose that decision-making rule which will minimize the present value of the expected costs that he must suffer. He will do so by minimizing the sum of the expected external costs and expected decision-making costs….” It is not clear that these costs can be added on any single scale. For a commentary on these cost functions see, Kiesling, Herbert J., “Potential Costs of Alternative Decision-Making Rules,” Public Choice, 4 (1968), 49–66 CrossRefGoogle Scholar.

¹⁶ Although Buchanan and Tullock produce no rigorous evaluations of decision-rules, two of their speculations seem especially relevant to the present analysis. First, they point out that majority-rule holds no special a priori promise as a cost minimization (p. 81). Second, they suggest that “… the low-cost point on the aggregate ‘cost curve’ would tend to be represented by simple majority voting,” under circumstances which appear comparable to those presented in Part II of this paper (p. 129). No strict demonstration of this tendency is offered, and care must be taken not to confuse the frequency curves employed here with the cost-functions which these authors consider. If any determinate relation exists, and I am not sure one does, it is between the frequencies usd here and “external costs” in their analysis.

¹⁷ “An Individualistic Theory of Political Process,” in Easton, David (ed.), Varieties of Political Theory (Englewood Cliffs: Prentice-Hall, 1966), pp. 25–37 Google Scholar, quotation from p. 29. For a related statement, see also Elinor Ostrom, “Constitutional Decision-Making,” unpublished paper.

¹⁸ The effect of this assumption is to discard an event in which every member “has his way” since all would vote no (i.e., Event D.) This simply makes the prediction slightly less optimistic than it would otherwise have been, but it has no effect on the location of the A + B minimum in which we are interested.

¹⁹ This follows from the rule of combinations: the number of combinations of size f which may be drawn from n objects equals n!/f!(n – f)!. If we allow f to take all values 0 … n, the resulting sum is 2ⁿ. Assumption III eliminates the one case in which f = 0 (i.e. n!/n! = 1), so the result is 2ⁿ – 1.

²⁰ This follows from Assumptions I and II. The combinations are compounded of independent, equally probable events (i.e., individual decisions), and are therefore themselves equally probable events. Since there are 2ⁿ – 1 combinations (Deduction I), and each is equally probable, each has a probability of 1/2ⁿ – 1.

²¹ This formula is derived from the rule of combinations and our set definitions. Let us begin with the rule of combinations: The number of support sets of size f is:

Now, we wish to find the number of combinations of any size (f) which contain a generalized member (Ego), in accordance with our set definition.

We may accomplish this by saying “If Ego belongs to a set of size f, there will remain n – 1 members to be combined in a set of size f – 1 in order to bring our set to size f with Ego included.” Accordingly, we subtract 1 from n and f in the combinatory formula, obtaining:

This may be simplified by letting the –1 terms in the lower right hand bracket cancel each other:

This tells us how many support sets of size f will contain Ego. Now, we also need to know the sum of these values for all instances in which the set's size (f) is less than k. This is accomplished by letting the values be summed from f = 1 to f = k – 1. This accounts for summation term:

And the result is the formula given:

²² This formula is also derived from the combination rule. In this instance, we wish to “count” the support sets of size f which do not include Ego and for which f is at least k. We may begin by saying, “If Ego is not to be included in a set of size f, there will be n − 1 members to be combined in sets of size f.” We therefore subtract one from n, and leave f unaltered, obtaining:

This gives the number of sets of size f which exclude Ego. We wish to sum the value of this fraction for all values of f at or above the value of k. Hence, the summation term:

The upper limit is set at n − 1 for two reasons: (1) we know that no sets belonging to B may be of size n, and (2) this extreme case can produce a minus factorial term, which is nonsensical. The result is the formula given:

²³ This upper limit occurs because there is only a 5 probability that Ego will vote yes (or that he will vote no.) These expected frequencies must not, then, be confused with the conditional probability that A or B will occur once Ego has voted yes or no.

²⁴ This is because all more restrictive rules allow negative minorities to outvote positive majorities, while less restrictive rules allow positive minorities to outvote negative majorities. Under all such rules, it is therefore possible that Event A or Event B will occur for more than half the members on a single vote: under majority-rule (and under k = n/2 in committees of even size) this is not possible.

²⁵ And so long as we do not consider “positional preferences,” as discussed in Part III below.

²⁶ “A Proof for Rae's Majority-Rule Theorem,” forthcoming, Behavioral Science (Summer, 1969).

²⁷ In “Conservatism as an Ideology.” this Review, Vol. 51, Huntington describes conservatism as a “positional ideology,” leading its proponent to an “… articulate, systematic, theoretical resistence to change.” (p. 461.) This suggests that a conservative would be apt to weight Event B (unwanted change) more heavily than Event A (unwanted stability).

₂₈ This is an extrapolation from Madison's, James Journal of the Federal Convention, Vol. 2 (Chicago: Albert Scott and Co., 1893) p. 500 Google Scholar. Since the premises of the argument are inexplicit, one cannot be sure that any interpretation is historically correct; the interpretation offered here is, however, at least as plausible as any alternative.

²⁹ The following example states the problem in the simplest possible way, and it is just one of many sorts of examples which might be devised. If one considers more than two factions, and partial degrees of conflict, the argument is vastly more complex, but the essential conclusion of this section is probably unaffected: the prospect of factional exploitation is not a unique attribute of majority-rule.

³⁰ One useful analysis of this concept is offered by Axelrod, Robert, Conflict of Interest: Theory and Political Applications (unpublished Ph.D. thesis, Yale University, 1968)Google Scholar.

³¹ In the terms of this analysis, we might say that partition introduces a special class of events in which policies which Ego favors, and which would have been adopted before the partition, are rejected because the required resources are no longer available. These events are analogous to Event A—the rejection of proposals which Ego favors. These events are singularly relevant to proposals for separation of minorities, as with French Quebec or Black America, since the minority communities would operate with vastly restricted resources.

³² Taylor, op. cit.

³³ This suggests an incentive for the formation of minimum winning coalitions, even if we assume that preferences are independently arrived at so that there is no a priori basis for group cohesion. This is because the expected frequency for events A and B is lower, for any given k, in small committees than in large committees. By forming a coalition of k members, and operating under the analogous decision-rule internally, members can always have their way slightly more often than they would by voting independently. In a committee of five, with independent voting, members can expect to have their way 11/16ths of the time if k = 3; by forming a coalition of three, however, it is possible to raise this expectation to 12/16ths, if k = 2 is adopted by the coalition. This suggestion is not taken into account by our simple model, which is based on sincere individual voting. This suggestion is an interesting parallel to William Riker's prediction of minimum winning coalitions on the basis of a quite different model. See The Theory of Political Coalitions (New Haven: Yale University Press, 1962 Google Scholar.)