¹ See Rae, Douglas and Taylor, Michael, ‘Decision Rules and Policy Outcomes’, British Journal of Political Science, I (1971), 71–90CrossRefGoogle Scholar, for proofs of these theorems and for a discussion of the assumption that individual preferences are of the City Block form.

² These necessary and sufficient conditions were established by Plott, Charles R.,‘A Notion of Equilibrium and its Possibility under Majority Rule’, American Economic Review, LVII (1967) 797–806.Google Scholar

³ This theorem is proved in Taylor, Michael, ‘The Problem of Salience in the Theory of Collective Decision Making’, Behavioural Science, XVI (1970), 415–30.CrossRefGoogle Scholar For a discussion of this result and of the results referred to in footnotes 1 and 2, and for their place in the theory of collective decision making, see Taylor, Michael, ‘Mathematical Political Theory’, British Journal of Political Science, 1(1971), 345–77CrossRefGoogle Scholar, and ‘The Theory of Collective Choice’, in Greenstein, Fred and Polsby, Nelson, eds., The Handbook of Political Science, Vol. 5 (Reading, Mass.: Addison-Wesley, forthcoming).Google Scholar

⁴ A utility function corresponding to the first (desired-ratio) part of this criterion is mentioned by Radner, Roy, ‘Mathematical Specification of Goals for Decision Problems’, Chap. II in Shelly, M. W. and Bryan, C. L., eds., Human Judgements and Optimality (New York: John Wiley, 1964), pp. 178–216.Google Scholar He calls it the ‘desired proportions utility function’.

⁵ In the pilot experimental games, in an attempt to add more interest and realism to the bare experimental situation, the Ss were told that they were a committee faced with building a new office block and that they were to decide on both the cost of the building and the distance it should be from London, given that cost and distance were not related. From comments made by Ss during these pilot games it became apparent that they were, in effect, introducing an unspecified number of additional dimensions, e.g. whether to build in the Green Belt, or the availability of commuter trains. To overcome this complication it was necessary to ask the Ss to make decisions on the abstract dimensions X and Y.

⁶ The value of k (simple majority, two-thirds majority, or unanimity) was specified appropriately in the different games.

⁷ This is how the requirement that the outcome be ‘stable’ was operationalized.

⁸ To evaluate these outcomes fully, it would be necessary to have a priori probability distributions of outcomes under the experimental conditions. These have not yet been determined.

⁹ The experimental games where the first five proposals were rejected so that the games terminated with the public policy equivalent to the starting proposal of five Xs and five Ys have been excluded from this Table on the grounds that these games terminated before a stable outcome to sequential voting was achieved.

¹⁰ A number of previous experimental studies of pairwise individual choice have found support for the hypothesis of additive utility, of which the City Block function is a special case. They have found that individuals treat each dimension separately and combine the differences along the dimensions into a single judgement by merely adding them. It was, in part, the results of these experiments which led to the incorporation of the City Block Hypothesis in a model of collective decision-making. It should be noted that these earlier experiments were with single individuals in isolation; whereas in the present study we wished to examine the choice behaviour of individuals who are engaged in making decisions collectively. In other words, we wished to examine individual behaviour in a political situation. Typical of the earlier studies is Attneave, F., ‘Dimensions of Similarity’, American Journal of Psychology, LXIII (1950), 516–56.CrossRefGoogle ScholarSs made judgements of similarity between pairs of visual objects (parallelograms, squares and triangles in three different experiments) varying on two or more of the dimensions: area, angularity, colour. Judgements made between all possible pairs were scaled to obtain ‘distances’ between the alternatives and although these distances were only approximately equal to the sum of the distances along each dimension of variability (as in the City Block model), they were definitely incompatible with an Euclidean model. Further experimental evidence for the additivity hypothesis is to be found in Adams, E. W. and Fagot, R., ‘A Model of Riskless Choice’, Behavioral Science, IV (1959), 1–10Google Scholar; Anderson, N. H., ‘Application of an Additive Model to Impression Formation’, Science, CXXXVIII (1962), 817–18CrossRefGoogle Scholar; Gulliksen, H., ‘Measurement of Subjective Values’, Psychomelrika, XXI (1956), 229–44CrossRefGoogle Scholar; Hoffman, P. J., ‘The Paramorphic Representation of Clinical Judgment’, Psychological Bulletin, LVII(1960), 116–31CrossRefGoogle Scholar; Rimoldi, H. J. A., ‘Prediction of Scale Values for Combined Stimuli’, British Journal of Statistical Pyschology, IX (1956), 29–40CrossRefGoogle Scholar; Yntema, D. B. and Torgerson, W. S., ‘Man-Computer Co-operation in Decisions Requiring Common Sense’, IRE Transactions on Human Factors in Electronics, HFE-2 (1961), 20–6CrossRefGoogle Scholar; and Shepard, R. N., ‘On Subjectively Optimum Selections among Multi-attribute Alternatives’, Chap. 14 in Shelly, and Bryan, , Human Judgments and Optimality, pp. 263–4.Google Scholar

¹¹ To evaluate these compatibility rank orders fully it would be necessary to have the a priori probability distributions of compatible votes, for the following reason. If there were only two logically possible utility functions, then a random distribution of discriminating ipcs would be 50 per cent compatible with the first decision criterion and 50 per cent with the second, although under experimental conditions the a priori distribution may not be equivalent to the random one. However, immediately the possibility of a third utility function is introduced, the random distribution of compatible votes in a table comparing the first pair of decision criteria may deviate from 50–50 if, e.g., the Ss’ choices are compatible with the third decision criterion, and votes compatible with that criterion are compatible with thefirstbut not with the second criterion. Because the a priori distributions are not available, no further analysis or discussion of the lower ranked decision criteria has been attempted in this paper.