¹ This refers to the following works: Black, Duncan, The Theory of Committees and Elections (Cambridge University Press, 1958)Google Scholar; Davis, Otto A. and Hinich, Melvin J., “A Mathematical Model of Policy Formation in a Democratic Society,” in Mathematical Applications in Political Science, II, ed. Bernd, J. (Dallas: Southern Methodist University Press, 1966), 170–208Google Scholar; Davis, and Hinich, , “On the Power and Importance of the Mean Preference in a Mathematical Model of Democratic Choice,” Public Choice, 5 (Fall, 1968), 59–72CrossRefGoogle Scholar; Davis, and Hinich, , “Some Results Related to a Mathematical Model of Policy Formation in a Democratic Society,” Mathematical Applications in Political Science, III (Charlottesville: University of Virginia Press, 1967)Google Scholar; Dorfman, Robert, “General Equilibrium with Public Goods,” IP 95 (Berkeley: Institute of Business and Economic Research, Center for Research in Management Science, June 1966)Google Scholar; Downs, Anthony, An Economic Theory of Democracy (New York: Harper and Bros., 1957)Google Scholar; Garvey, Gerald, “The Theory of Party Equilibrium,” American Political Science Review, 60 (March 1966), 29–38CrossRefGoogle Scholar; Hinich, Melvin and Ordeshook, Peter, “Plurality Maximization vs. Vote Maximization: A Spatial Analysis with Variable Participation,” American Political Science Review, 64 (September, 1970); 772–791CrossRefGoogle Scholar; Hotelling, Harold, “Stability in Competition,” The Economic Journal, 39 (March, 1929), 41–57CrossRefGoogle Scholar; Jerome Rothenberg “A Model of Economic and Political Decision Making,” (Rothenberg has a number of models; in one the party is interested in platform, and thus similar to the alternative model, but it does not take into account counter moves by the other party), in The Public Economy of Urban Communities, ed. Margolis, J. (Baltimore: Johns Hopkins Press, 1965)Google Scholar; Schumpeter, Joseph A., Capitalism, Socialism, and Democracy (New York: Harper Bros., 1950)Google Scholar; Smithies, Arthur, “Optimum Location in Spatial Competition,” The Journal of Political Economy, 49 (June, 1941), 423–439CrossRefGoogle Scholar; Tullock, Gordon, “A Simple Algebraic Logrolling Model,” American Economic Review, 60 (June, 1970), 419–426.Google Scholar

A number of these articles assume that the party either maximizes the number of votes it receives or maximizes its plurality (these are not equivalent to maximizing the probability of winning the election). Consequently some of them are not meant to be a description of winner-take-all elections (as in the U.S.) but of an ideal type (such as a benevolent dictator) or of a coalition government.

² With single peakedness the incumbent need not present its platform first. While Downs and others have claimed that abstention will pull the parties apart, this is not true if the parties want to maximize the probability of being elected (although it is possible to have other globally stable equilibria with both parties having a 50 per cent chance of winning and the parties not being identical). See Wittman, Donald, “Theories of Optimal Political Party Decision Making,” unpublished Ph.D. thesis in Economics, University of California, Berkeley, 1970.Google Scholar

³ If the parties present lotteries it is possible for the extremes to prefer a risky lottery and the median voter to prefer a nonrisky (degenerate) lottery; but then there always exists another risky lottery which the median voter and one (and only one) of the two extremes prefer to the original risky lottery.

⁴ For a more detailed presentation see Arrow, Kenneth, Social Choice and Individual Values, Cowles Commission for Research in Economics, Monograph No. 12 (New York: Wiley, 1951).Google Scholar

⁵ Other ways of making the model more complicated do not change the essential point that the parties have a wide variety of platforms which will yield the same probability of winning, and thus the parties will have some other criterion for choosing among platforms than just winning the election. For example, if the voters are very hesitant to vote against the incumbent, then the incumbent can win on almost any platform. The interesting question becomes: which winning platform will the incumbent choose?

⁶ Actually single peakedness does not imply transitivity if lotteries are allowed (see Zeckhauser, Richard, “Majority Rule with Lotteries on Alternatives,” Quarterly Journal of Economics, 83 (November, 1969); 616–703.CrossRefGoogle Scholar Nor does lack of single peakedness imply a lack of transitivity; however, it is not clear when these cases such as (1) Latin-squarelessness or (2) closed compact sets would arise.

For theoretical arguments against single peakedness see Chapman, David, “Models of the Working of a Two-Party Electoral System, Part I,” Papers on Non-Market Decision Making, 3 (Fall, 1967), 19–38Google Scholar; Sen, Amartya, Collective Choice and Social Welfare (San Francisco: Holden-Day, Inc.)Google Scholar; Ward, Benjamin, “Majority Rule and Allocation,” Journal of Conflict Resolution, 5, No. 4 (December, 1961), 379–389CrossRefGoogle Scholar; for empirical arguments see Duverger, Maurice, Political Parties: Their Organization and Activity in the Modern State, trans. Barbara, and North, Robert (London: Methuen and Company, 1954)Google Scholar; Donald E. Stokes, “Spatial Models of Party Competition,” in Elections and the Political Order, Campbell, A., Converse, P., Miller, W., and Stokes, D. (New York: John Wiley and Sons, 1966), 161–179.Google Scholar

For articles on the other side see Arrow, Kenneth J., “Tullock and an Existence Theorem,” Public Choice, 6 (Spring, 1969), 105–112CrossRefGoogle Scholar; Black, Duncan, “On Arrow's Impossibility Theorem,” The Journal of Law and Economics, 12 (October, 1969), 227–248CrossRefGoogle Scholar; Niemi, Richard G., “Majority Decision Making with Partial Unidimensionality,” American Political Science Review, 63 (June, 1969), 488–497CrossRefGoogle Scholar; and Tullock, Gordon, “The General Irrelevance of the General Impossibility Theorem,” Quality Journal of Economics, 81 (May, 1967), 256–270.CrossRefGoogle Scholar

⁷ It should be noted that all the work concerned with a multidimensional issue space has had to put restrictions on the weighting of the issues or on the distribution of voter attitudes in order to avoid intransitivity.

⁸ See San Francisco Chronicle (Final edition, September 23, 1968), p. 1, col. 2. It is also very likely that a considerable number of the 60 per cent who still would have voted for Wallace had McCarthy as their second choice.

⁹ Gallup Opinion Index, No. 30, 4.

¹⁰ Gallup Opinion Index, No. 32, 8.

¹¹ Gallup Opinion Index, No. 32, 7.

¹² Nixon—36, Johnson—44, Wallace—12, No Opinion—8. (Gallup Opinion Index, No. 33, 11); Rockefeller—40, Johnson—37, Wallace—13, No Opinion—10. (Gallup Opinion Index, No. 33, 13); Nixon—48, Rockefeller—42, No Opinion—10. (Gallup Opinion Index, No. 34, 9). It should be noted that in earlier surveys, the relative ranking of the two major party candidates was not affected by the inclusion (or omission) of George Wallace as a third party candidate.

¹³ Prewitt, Kenneth, “Political Ambitions, Volunteerism, and Electoral Accountability,” American Political Science Review, 64 (March, 1970), 5–17.CrossRefGoogle Scholar On page 10 Prewitt says, “Among the nearly 500 men who govern the cities of the San Francisco Bay Area, about onefifth will be planning to retire voluntarily at the end of the present term and another 30 to 50 per cent will be planning to retire after only one more term.” Furthermore, he shows that the councilmen do not worry about going against the majority and thus they do not seem to formulate policies in order to win elections.

¹⁴ Pentagon Papers (New York: Bantam Books), 1971.

¹⁵ There are some other variants. For example, Riker, in The Theory of Political Coalitions (New Haven: Yale University Press, 1962)Google Scholar suggests that the parties are only interested in a minimal winning coalition. This is in response to the usual formulation of the traditional model which assumes that parties maximize the numbers of votes they receive. One reason for forming a minimal winning coalition is that the rewards are divided among fewer people. These rewards may be of office or of policy.

¹⁶ This example also shows why the concept of extremist may not be very useful. The notion of extremist is no longer well defined nor can friends and enemies be as easily distinguished as in the single peaked case. To have been in a minority in an election does not imply that had your most preferred position been offered as a choice to the voters it would not have been preferred by a majority. Thus the “extremist” position may not be extreme.

¹⁷ This probably characterizes the 1968 U.S. presidential election where each party would have had a greater probability of winning if it had moved “left” (see Table 3).

¹⁸ Given certain restrictions on the utility functions of the parties and on the probability of the voters making an incorrect choice, it can be shown that the platforms that the parties end up choosing will result in each party's having a 50 per cent chance of winning the election.

¹⁹ Unless there is more than one office, in which case the parties may agree to split the offices. There may be limited collusion with respect to campaign spending. See Curry, R. and Wade, L., A Theory of Political Exchange (Englewood Cliffs, New Jersey: Prentice-Hall), 1968.Google Scholar

²⁰ Whenever collusion occurs no assumption about the order of presentation of the party's platforms need be made.