¹ ‘Policy space’ will be defined below. For the time being, it is the set of feasible and mutually exclusive alternatives for the resolution of a given issue.

² We shall see later how this result follows from our formal analysis.

³ A distance, or metric, of this kind is known as a city-block metric. The reasons for using it, rather than certain other kinds of utility function which have been used in similar studies, will be given later.

⁴ This one-dimensional result has already been shown to follow from a wider class of utility functions than those considered here. Roughly speaking, each individual's utility function need only decline monotonically from his optimum point. See Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958), chapter 4.Google Scholar

⁵ The overlap may consist of only a single point - as with when m is odd, or as in ‘degenerate’ cases when several optima all lie on a line parallel to one of the coordinate axes.

⁶ Theoretical discussions of additive utility can be found in Adams, E. W. and Fagot, R., ‘A Model of Riskless Choice’, Behavioral Science, IV (1959), 1–10Google Scholar; Debreu, G., Topological Methods in Cardinal Utility Theory’, chapter 2 in Arrow, K. J., Karlin, S. and Suppes, P., eds., Mathematical Methods in the Social Sciences, 1959 (Stanford: Stanford University Press, 1970)Google Scholar; Aumann, R. J., ‘Subjective Programming’, chapter 12 in Shelley, M. W. and Bryan, G. L., eds., Human Judgements and Optimality (New York: John Wiley, 1964)Google Scholar; Luce, R. D. and Tukey, J. W., ‘Simultaneous Conjoint Measurement: a New Type of Fundamental Measurement’, Journal of Mathematical Psychology, I (1964), 1–27CrossRefGoogle Scholar; and Luce, R. D. and Suppes, P., ‘Preference, Utility, and Subjective Probability’, chapter 19 in Luce, R. D., Bush, R. R. and Galanter, E., eds., Handbook of Mathematical Psychology (New York: John Wiley, 1965), vol. 3, pp. 267–72.Google Scholar

⁷ The equilibrium problem for circular indifference curves is discussed, but no general conclusions are reached, in Tullock, Gordon, Toward a Mathematics of Politics (Ann Arbor: University of Michigan Press, 1967), chapter 2.Google Scholar

⁸ Black, Duncan and Newing, R. A., Committee Decisions with Complementary Valuation (London: William Hodge and Co., 1951), section IIGoogle Scholar; Plott, Charles R., ‘A Notion of Equilibrium and its Possibility under Majority Rule’, American Economic Review, LVII (1967), 787–806.Google Scholar Similar utility functions are used in the recent mathematical work on Downsian theories of political party competition, where the ‘dominant campaign strategies’ are the same as the present equilibrium points; the most recent paper, with references to the earlier work, isDavis, Otto A., Hinich, Melvin J., and Ordeshook, Peter C., ‘An Expository Development of a Mathematical Model of Policy Formation in a Democratic Society’, American Political Science Review, LXIV (1970)Google Scholar.

⁹ A contract curve between two optima is the locus of points which are ‘Pareto optimal’ for just those two individuals. The second condition, says Plott, ‘can be modified to account for the point being a maximum for more than one individual’ (p. 790, n. 8).

¹⁰ Black, and Newing, , Committee Decisions, pp. 23–5.Google Scholar

¹¹ The ‘paradox of voting’ refers to the possibility that a set of transitive individual rankings of some alternatives can be aggregated in a pairwise fashion to yield intransitive social decisions. See Duncan Black, The Theory of Committees and Elections. A more general result is due to Arrow, Kenneth J., Social Choice and Individual Values (New York: John Wiley, 1951; 2nd edn, 1963).Google Scholar

¹² See Black, Duncan, ‘The Decisions of a Committee Using a Special Majority’, Econometrica, XVI (1948), 245–61CrossRefGoogle Scholar; and, for a correct proof of this result, see Arrow, Social Choice and Individual Values, 2nd edn, chapter 7.

¹³ Buchanan, James M., The Demand and Supply of Public Goods (Chicago: Rand McNally and Co., 1968), pp. 110–14.Google Scholar

¹⁴ Sen, Amartya K., ‘A Possibility Theorem on Majority Decision’, Econometrica, XXXIV (1966), 491–9.CrossRefGoogle Scholar The most comprehensive account of these conditions is Ken-Ichi Inada, ‘The Simple Majority Decision Rule’, Econometrica, XXXVII (1969), 490–506.Google Scholar

¹⁵ Dummett, Michael and Farquharson, Robin, ‘Stability in Voting’, Econometrica, XXIX (1961), 33–42;CrossRefGoogle Scholar the most recent work of Pattanaik and Sen, including a summary of their earlier results, is Sen, Amartya K. and Pattanaik, Prasanta K., ‘Necessary and Sufficient Conditions for Rational Choice under Majority Decision’, Journal of Economic Theory, 1 (1969), 178–202.CrossRefGoogle Scholar

¹⁶ Kramer, Gerald H., ‘On a Class of Equilibrium Conditions for Majority Rule’, Cowles Foundation Discussion Paper No. 284, Yale University (New Haven, 1969).Google Scholar

¹⁷ A spatial model of collective decision-making in which the individual preference orderings cannot be represented by utility functions of any kind (since they are based on a lexicographic ranking of the policy dimensions) is proposed in Taylor, Michael, ‘The Problem of Salience in the ‘Theory of Collective Decision-Making’, Behavioral Science, XV (1970).Google Scholar In this model, which clearly lies beyond Kramer's assumptions, the existence of equilibrium points is established.

¹⁸ Attneave, F., ‘Dimensions of Similarity’, American Journal of Psychology, LXIII (1950), 516–56.CrossRefGoogle Scholar

¹⁹ Householder, A. S. and Landahl, H. D., Mathematical Biophysics of the Central Nervous System (Bloomington, Indiana: The Principia Press, 1945)Google Scholar; and Landahl, H. D., ‘Neural Mechanisms for the Concepts of Similarity and Difference’, Bulletin of Mathematical Biophysics, VII (1945). 83–8.CrossRefGoogle Scholar

²⁰ See, for example, Gulliksen, H., ‘Measurement of Subjective Values’, Psychometrica, XXI (1956). 229–44CrossRefGoogle Scholar; Rimoldi, H. J. A., ‘Prediction of Scale Values for Combined Stimuli’, British Journal of Statistical Psychology, IX (1956), 29–40CrossRefGoogle Scholar; Adams, E. W. and Fagot, R., ‘A Model of Riskless Choice’, Behavioral Science, IV (1959), 1–10Google Scholar; Yntema, D. B. and Torgerson, W. S., ‘Man-Computer Cooperation in Decisions requiring Common Sense’, IRE Transactions on Human Factors in Electronics, HFE-2 (1961), 20–6CrossRefGoogle Scholar; Anderson, N. H., ‘Application of an Additive Model to Impression Formation’, Science, CXXXVIII (1962), 817–18CrossRefGoogle Scholar; Shephard, R. N., ‘On Subjectively Optimum Selections among Multi-attribute Alternatives’, chapter 14 in Shelley, M. W. and Bryan, G. L., eds., Human Judgements and Optimality (New York: John Wiley, 1964).Google Scholar Shephard (pp. 263–4) also notes the indirect evidence provided by certain studies in which learning and remembering classifications of a set of multi-attribute objects were found to be much more difficult when the classification was based on several interacting attributes than when it was based on only one attribute.

²¹ ‘Wilbur D. Mills: A Study of Congressional Influence’, American Political Science Review, LXII, (1969), 442–64, quote from p. 462.Google Scholar Manley goes on to discuss the relevance of ‘swing votes’ without elaborating a model to account for their impact.