¹ The election tabulations were collected from state and national yearbooks. The U.S. congressional returns have been collected together in Stokes, Donald and Iversen, Gudmund, “National Totals of Votes Cast for Democratic and Republican Candidates for the U.S. House of Representatives, 1866–1960,” July, 1962Google Scholar, mimeo, Survey Research Center, University of Michigan. Congressional Directories (Washington, D.C.: United States Government Printing Office) were used to update the Stokes-Iversen compilation and also as the source for tabulations requiring election returns in individual congressional districts. All percentages of the vote were computed from the votes received by the two major parties only.

² Rae, Douglas, The Political Consequences of Electoral Laws (New Haven: Yale University Press, 1967), p. 73.Google Scholar

³ Dahl, Robert A. in A Preface to Democratic Theory (Chicago: University of Chicago Press, 1956), pp. 147–149Google Scholar, estimates the linear model for U.S. House and Senate elections.

⁴ Somewhat similar notions of bias have been discussed by Butler, and MacRae, : Butler, D. E., The Electoral System in Britain Since 1918 (Oxford: Clarendon Press, Second ed., 1963), p. 196Google Scholar; and MacRae, Duncan, “Models of Legislative Representation,” mimeo, University of Chicago, 1969.Google Scholar This sort of bias can be estimated in four different ways: (1) as described in the text, where the seats-votes line is regarded as a “law”; (2) by regressing votes on seats and solving accordingly, (3) by the logit model described below, and (4) by manipulating the distribution of the district vote described below. A brief comparison of the four methods revealed small differences in most estimates when the bias was less than 5 per cent and the correlation between seats and votes was fairly high (usually the case); otherwise the estimates diverged. Confidence intervals for the estimated bias, if one believes they are appropriate for these data, can be constructed for the first three methods. For the complex case of method 1, see Bowker, Albert H. and Lieberman, Gerald J., Engineering Statistics (Engle-wood Cliffs, New Jersey: Prentice-Hall, 1959), p. 253Google Scholar; for methods 2 and 3, the usual confidence intervals in regression apply. Still another interesting bias is suggested by the question: What proportion of the seats does a party receive when it wins 50% of the vote? For the fitted line, this bias is, by the geometry of the situation, the product of the slope and the bias described in Figure 2.

⁵ Dixon, Robert G. Jr., “The Court, The People, and ‘One Man, One Vote,’” in Reapportionment in the 1970s, ed. Polsby, Nelson W. (Berkeley: University of California Press, 1971), p. 13.Google Scholar

⁶ Kendall, M. G. and Stuart, A., “The Law of Cubic Proportions in Electoral Results,” British Journal of Sociology, 1 (September, 1950), 183–197CrossRefGoogle Scholar; Kendall, and Stuart, , “La Loi du Cube dans les Elections Britanniques,” Revue Française de Science Politique, 2 (April-June, 1952), 270–276.CrossRefGoogle Scholar

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⁸ Thiel, Henri, “The Cube Law Revisited,” Journal of the American Statistical Association, 65 (September, 1970), 1213–1219.CrossRefGoogle Scholar

⁹ Butler, David and Stokes, Donald discuss the issue in detail in Political Change in Britain (London: Macmillan, 1969), pp. 303–312.Google Scholar

¹⁰ Sankoff, David and Mellos, Koula, “The Swing Ratio and Game Theory,” American Political Science Review, 66 (June, 1972), 551–554.CrossRefGoogle Scholar For a formal model taking into account some of the findings of the present paper, see Quandt, Richard E., “A Stochastic Model of Elections in Two-Party Systems,” unpublished manuscript, Princeton, 1972.Google Scholar

¹¹ Logit analysis is described in Thiel, Henri, Principles of Econometrics (New York: John Wiley, 1971), pp. 632–636.Google Scholar

¹² Stokes, Donald E., “Parties and the Nationalization of Electoral Forces,” in The American Party Systems, ed. Chambers, William N. and Burnham, Walter Dean (New York: Oxford University Press, 1968), 182–202.Google Scholar

¹³ See: Stokes; Polsby, Nelson W., “The Institutionalization of the U.S. House of Representatives,” American Political Science Review, 62 (March, 1968), 144–168CrossRefGoogle Scholar; and, for a detailed discussion of congressional competition which was particularly helpful in the present analysis, Mayhew, David R., “Congressional Representation: Theory and Practice in Drawing the Districts,” in Reapportionment in the 1970s, ed. Polsby, Nelson W., pp. 249–290.Google Scholar