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The Relationship between Seats and Votes in Two-Party Systems*

  • Edward R. Tufte (a1)
Abstract

An enduring fact of life in democratic electoral systems is that the party winning the largest share of the votes almost always receives a still larger share of the seats. This paper tests three models describing the inflation of the legislative power of the victorious party and then develops explanations of the observed differences in the swing ratio and the partisan bias of an electoral system. The “cube law” is rejected as a description, since it assumes uniformity (which is not observed in the data) across electoral systems. Explanations for differences in swing ratio and bias are found in variations in turnout over districts, the extent of the “nationalization” of politics, and, most importantly, in who does the districting or reapportionment. The measures of swing ratio and partisan bias appear useful for the judicial evaluation of redistricting schemes and may contribute to the reduction of partisan and incumbent gerrymandering.

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*

I wish to thank Joseph G. Verbalis, Harold George, Nita Rome, Judith Rowe, and Richard A. Sun for their help in the collection and the analysis of the data. Orley Ashenfelter, Stanley Kelley, Jr., John L. McCarthy, Frederick Mosteller, Walter F. Murphy, Richard E. Quandt, and Dennis F. Thompson provided advice and criticism during the course of the work. Edward Franklin Cox, Gudmund Iversen, and David Mayhew generously gave me access to unpublished materials on congressional elections. The Center for International Studies and the Woodrow Wilson School of Public and International Affairs, Princeton University, provided financial support for the work. Computations were performed at the Princeton University Computer Center, operating under NSF grants GJ-34 and GU-3157. None of these individuals or institutions bears responsibility for the analysis or interpretations of the paper.

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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, 1962, 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.

3 Dahl, Robert A. in A Preface to Democratic Theory (Chicago: University of Chicago Press, 1956), pp. 147149, 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. 196; and MacRae, Duncan, “Models of Legislative Representation,” mimeo, University of Chicago, 1969. 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. 253; 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.

6 Kendall, M. G. and Stuart, A., “The Law of Cubic Proportions in Electoral Results,” British Journal of Sociology, 1 (September, 1950), 183197; Kendall, and Stuart, , “La Loi du Cube dans les Elections Britanniques,” Revue Française de Science Politique, 2 (April-June, 1952), 270276.

7 Butler, D. E., The British General Election of 1951 (London: Macmillan, 1952), pp. 275276 and the other Nuffield College election studies; March, James G., “Party Legislative Representation as a Function of Election Results,” Public Opinion Quarterly, 11 (Winter, 19571958), 521–542; “Electoral Facts,” The Economist, January 7, 1950, 5–7; Qualter, Terrence H., “Seats and Votes: An Application of the Cube Law to the Canadian Electoral System,” Canadian Journal of Political Science, 1 (September, 1968), 336344. Doubts about the cube law are expressed by Eldersveld, Samuel J., “Polling Results and Prediction Techniques in the British General Election of 1950,” in Pollock, James K., British Election Studies (Ann Arbor: George Wahr Publishing Co., 1951), pp. 7578; and in Brookes, Ralph H., “Legislative Representation and Party Vote in New Zealand: Reflections on the March Analysis,” Public Opinion Quarterly, 23 (Summer, 1954), 288291.

8 Thiel, Henri, “The Cube Law Revisited,” Journal of the American Statistical Association, 65 (September, 1970), 12131219.

9 Butler, David and Stokes, Donald discuss the issue in detail in Political Change in Britain (London: Macmillan, 1969), pp. 303312.

10 Sankoff, David and Mellos, Koula, “The Swing Ratio and Game Theory,” American Political Science Review, 66 (June, 1972), 551554. 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.

11 Logit analysis is described in Thiel, Henri, Principles of Econometrics (New York: John Wiley, 1971), pp. 632636.

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), 182202.

13 See: Stokes; Polsby, Nelson W., “The Institutionalization of the U.S. House of Representatives,” American Political Science Review, 62 (March, 1968), 144168; 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. 249290.

* I wish to thank Joseph G. Verbalis, Harold George, Nita Rome, Judith Rowe, and Richard A. Sun for their help in the collection and the analysis of the data. Orley Ashenfelter, Stanley Kelley, Jr., John L. McCarthy, Frederick Mosteller, Walter F. Murphy, Richard E. Quandt, and Dennis F. Thompson provided advice and criticism during the course of the work. Edward Franklin Cox, Gudmund Iversen, and David Mayhew generously gave me access to unpublished materials on congressional elections. The Center for International Studies and the Woodrow Wilson School of Public and International Affairs, Princeton University, provided financial support for the work. Computations were performed at the Princeton University Computer Center, operating under NSF grants GJ-34 and GU-3157. None of these individuals or institutions bears responsibility for the analysis or interpretations of the paper.

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American Political Science Review
  • ISSN: 0003-0554
  • EISSN: 1537-5943
  • URL: /core/journals/american-political-science-review
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