¹ This use of the word democracy is of course a narrow one. Note especially that the associated concept of liberal freedom is omitted, but remember, as Berlin reminds us, that negative freedom is consistent with some kinds of autocracy, and logically independent of democracy. See Berlin, Isaiah, Two Concepts of Liberty (Oxford: The Clarenden Press, 1958), pp. 14ffGoogle Scholar. It must also be clear that there exists no deductive relationship between the greatest number and the greatest good, so that majoritarian democracy cannot be said to assure a joint maximization of utility even on the too-optimistic assumption that it leads directly to a maximum individual rate of satisfaction. This point is explored by Plamnatz, John, Man and Society (London: Longmans, Green & Co., 1963) pp. 12, 21–36 Google Scholar, and Dahl, Robert A., Preface to Democratic Theory (Chicago: University of Chicago Press, 1956), pp. 90–102 Google Scholar.

² For example, consider three people, A, B, and C, deciding policies by simple majority-rule. The core would then list four winning coalitions (ABC), (AB), (AC), and (BC); four blocking coalitions (ABC), (AB), (AC), and (BC); and four losing coalitions (A), (B), (C), and the empty set (ø), included for convenience. These categories are applied in a number of important contemporary works of which three are especially relevant: Riker, William H., The Theory of Political Coalitions (New Haven: Yale University Press, 1962)Google Scholar; James Coleman, “Control of Collectivities and the Power of Collectivities to Act,” RAND P-3902; and Shapley, Lloyd S., “Simple Games: An Outline of a Descriptive Theory,” Behavioral Science, 7 (1962), 59–66 CrossRefGoogle ScholarPubMed. The present use of “blocking coalitions” is broader than the use suggested by the game theorists, and my term “core” is different from its sophisticated cousin in that field.

³ See Murakami, Y., Logic and Social Choice (London: Routledge and Kegan Paul, 1968), p. 23 Google Scholar for an elaboration of the concept of decisiveness, and for a formal analysis abundant in suggestive insight See also Bachrach, P. and Baratz, M. S., “The Two Faces of Power,” this Review, 56 (1962), 947–952 Google Scholar, for an argument substantiating the utility of thinking about decisiveness in both senses (e.g., winning and blocking).

⁴ These limits (and the resulting analysis) are defined only for binary decisions. This is useful if the analysis is to be related to the commonest forms of decision found in nominally democratic institutions. For a useful analysis of such procedures see Farquarson, Robin, Theory of Voting (New Haven: Yale University Press, 1969)Google Scholar.

⁵ For example, suppose that an effective domain containing individuals A, B, … Z is governed dictatorially by A. Then the largest possible losing coalition would be (B, C, … Z) and L_max would take the value 25.

⁶ For example, if member A dictated policy for a domain of any size, D _min would have a value of 1, since a coalition as small as that could impose (or block) policies. Notice that we are dealing in potentialities here, and it is not asserted that this coalition will in fact form at any given moment.

⁷ For example, consider majority rule among three individuals, A, B, and C. The smallest possible decisive coalition contains 2 members (e.g. AB, AC, or BC) and the largest losing coalition contains 1 member, A, B, or C. The condition of pure democracy is satisfied since the former exceeds the latter by one.

⁸ Since the largest possible losing coalition contains all the members of the domain, its value is n. And since he is not a domain member, the dictator's potentially decisive coalition contains no members who count, and n − 0 = n.

⁹ Since n/2 + 1 is a slightly more restrictive requirement than (n + 1)/2, the limit is only n/n + 1 for n even. This is not an artifact of the measure, but a logical property of numbers as they operate in political institutions. Where n is anything but very small, the resulting difference is trivial. For another instance in which the seemingly trivial fact of an even number of members alters a theoretical value, see Black, Duncan, The Theory of Committees and Elections (Cambridge: The University Press, 1963), esp. pp. 14–25 Google Scholar.

¹⁰ It is possible to simplify the definition further by eliminating the term L _max. Since D _min + L _max must equal n (each member in the domain being included in one of the two), we can substitute (n − D _min) for L _max in equation 2, with the result that:

Because of its less obvious connection with the institutional structures and their political consequences, I have not used this simpler expression in the analysis of Q's structural determinants. It is, however, employed in the appendix.

¹¹ And this blocking coalition may turn itself into a winning coalition through bargaining. See Barry, Brian, “A Theory of the Offensive Veto,” in his Political Argument, (London: Routledge, Kegan Paul, 1965), pp. 245–249 Google Scholar. It is interesting to note that this fact did not escape the attention of a group which produced a number of very important opportunities for small blocking coalitions, the framers of the United States Constitution. See Notes of Debates in the Federal Convention 1787 Reported by James Madison, reprinted (Athens, Ohio: Ohio University Press, 1966), p. 62 Google Scholar.

¹² In a real sense, this requirement dissolves the government: gives no group the authority to bind any other, save through bargaining. Professors Buchanan and Tullock go on to draw the erroneous conclusion from this that the rule of unanimity is to be rejected only on the ground that it leads to excessive decision-making costs. This conclusion would follow only in a world devoid of bargaining and, in effect, devoid of dynamic disturbances in the social environment. See Buchanan, James and Tullock, Gordon, The Calculus of Consent, (Ann Arbor: University of Michigan Press, 1962), p. 88 CrossRefGoogle Scholar.

¹³ That is holding j or k constant, the other variable depresses Q identically be it j or k.

¹⁴ Under conditions of strong competition (zero-sum payoffs, with side payments) and perfect information, this should in fact occur. See Riker, op. cit., p. 47, for a theoretical development of the point. For another argument leading to a similar prediction, consult Gamson, William A., “A Theory of Coalition Formation,” American Sociological Review, 26 (1961), 373–382 CrossRefGoogle Scholar.

¹⁵ The truth of this assumption is, of course, a major question in contemporary debate on representative political democracy, and no verified theory to the contrary exists. See Pitkin, Hanna, The Concept of Representation (Berkeley: University of California Press, 1967)Google Scholar, for a most useful survey of the difficulties and ambiguities entailed by claims that elites do in fact represent publics. For an equally useful survey of theories about the role of parties in such representation, see Ranney, Austin, The Doctrine of Responsible Party Government (Urbana: University of Illinois Press, 1956)Google Scholar. Two especially interesting empirical works are Eulau, Heinz et al, “The Role of the Representative: Some Empirical Observations on the Theory of Edmund Burke,” this Review, 53 (1959), 742–756 Google Scholar; and Miller, Warren and Stokes, Donald, “Constituency Influence in Congress,” this Review, 57 (1963), 45–56 Google Scholar.

¹⁶ An interesting and useful application of a similar analysis appears in Power, Max S., “A Theoretical Analysis of the Electoral College and Proposed Reforms,” (unpublished Ph.D. thesis, Yale University, 1970), Ch. 2Google Scholar.

¹⁷ Again, a very strong assumption.

¹⁸ This does not, however, seem an essential distinction in light of strategic bargaining, and in view of the fact that policy proposals can be and are stated negatively as well as positively.

¹⁹ In other words, the analysis depends upon the existence of a regime core, and our structural variables are ways of inferring the core limits.

²⁰ Capitalism, Socialism & Democracy (New York: Harper & Row, 1947)Google Scholar.

²¹ Flathman, Richard E., The Public Interest (New York: John Wiley & Sons, 1966)Google Scholar; and Barry, Brian, Political Argument (London: RoutJedge & Kegan Paul, 1967)Google Scholar.

²² For example, the more democratic institution could be majority-rule among three individuals, A, B, and C, where the winning coalition AB was formed so that only two members were satisfied. Meanwhile, a dictatorship with A in charge of the same group could form the winning coalition ABC so that all three were satisfied. In general, high consensus with low democracy may produce higher frequencies of satisfaction than low degrees of consensus with high degrees of democracy.

²³ See the Appendix.