¹ The second ballot eligibility percentage was 5 percent in the 1958 and 1962 elections.

² Party positions are viewed as fixed over time. It should be recognized, however, that a shift of voters is, to a large extent, observationally equivalent to a shift in party positions.

³ Candidates eliminated from the second ballot may, and usually do, counsel their first ballot supporters about which candidates to support on the second ballot. A plurality wins the second ballot.

⁴ That is, voters must consider not only their preference regarding a candidate but also that candidate's chances of winning the election. For discussions on viability effects, see McKelvey, Richard D. and Ordeshook, Peter C., “A General Theory of the Calculus of Voting,” in Mathematical Applications in Political Science, VI, ed. Hernden, James F. and Bernd, Joseph C. (Charlottesville, Va.: University of Virginia Press, 1972), pp. 32–78 Google Scholar, and the debate between Kramer, Gerald and Casstevens, Thomas, in “Communications,” American Political Science Review, 62 (09 1968), 955–56CrossRefGoogle Scholar, and “Communications,” American Political Science Review, 65 (03 1971), 187–89CrossRefGoogle Scholar, concerning Casstevens' article, “A Theorem about Voting,” American Political Science Review, 62 (03 1968), 205–07CrossRefGoogle Scholar.

⁵ Viability will not affect choice in two-candidate races if voter utility is associated solely with “winlose” outcomes. In other contexts, where the margin of victory is important to the voter, viability effects may also be present in two-candidate races. See Riker, William H. and Ordeshook, Peter C., “A Theory of the Calculus of Voting,” American Political Science Review, 62 (03 1968), 25–42 CrossRefGoogle Scholar. Much more direct findings concerning strategic voting are reported in Castanias, Richard P. II, McGuire, Timothy W., and Rosenthal, Howard, “The French Voter as Rational Man” (Paper prepared for the Third World Congress of the Econometric Society, Toronto, Canada, 08 20–25, 1975)Google Scholar.

⁶ Rosenthal, Howard and Sen, Subrata, “Electoral Participation in the French Fifth Republic ,” American Political Science Review, 67 (03 1973), 29–54 CrossRefGoogle Scholar.

⁷ Our analysis applies to all metropolitan districts with second ballots, except for Corsica where vote fraud has been common.

⁸ Since voting is a function of both preference and candidate viability, some strategic voting undoubtedly occurs on the first ballot. However, casual empiricism suggests that strategic voting occurs much less often on the first ballot than on the second, decisive ballot. In any event, our models assume that voters always vote for their most preferred candidate on the first ballot.

Our claim here that voters reveal information about their true preferences on the first ballot is to some extent inconsistent with our view, expressed later, that spatially oriented voters vote probabilistically on the second ballot. A more accurate, if perhaps less intuitively appealing, statement of our overall approach is that we assume voters vote probabilistically on the second ballot, the probabilities being functions of the spatial position of the expressed first ballot choice of the voter and the spatial positions of the second ballot candidates.

⁹ That is, we deliberately ignore political socialization, campaigns, etc., as determinants of voting behavior.

¹⁰ See Rosenthal and Sen, pp. 35–36.

¹¹ Barnes, Samuel H. and Pierce, Roy, “Public Opinion and Political Preferences in France and Italy,” Midwest Journal of Political Science, 15 (11 1971), 643–60CrossRefGoogle Scholar.

¹² Wood, David M., “Majority vs. Opposition in the French National Assembly, 1956–1965: A Guttman Scale Analysis,” American Political Science Review, 62 (03 1968), 88–109 Google Scholar. In Table 1, the incumbent deputies have been classified by their party affiliation used in the election (as against their parliamentary group in the legislature). Of those seeking reelection in 1958, 75 percent had the median position; in 1962,56 percent. The only parties with over half the deputies not at the median were, in 1958: the PSU, the UNR, and the Right; in 1962: the Radicals and the Right. Similar results are obtained for the other Wood scales for each legislature.

¹³ Rosenthal and Sen, pp. 35–37. The approximation is tolerable because there are typically far fewer candidates on the second ballot than on the first. This implies that the differences of the distances from a voter to the various candidates will be large relative to the measurement error.

¹⁴ We also prefer models for which an analytical solution leads to global minimization of the loss function, in this case squared error. This leads us to regressions with a proportion as the dependent variable. As Theil points out, such regressions can have undesirable econometric properties. However, the fact that the vote proportion is modeled as a sum of proportions aggregated across spatial locations implies that in this case we cannot apply the attractive alternative models Theil discusses ( Theil, Henri, “A Multinomial Extension of the Linear Logit Model,” International Economic Review, 10 [10 1969], 251–59)CrossRefGoogle Scholar. We have to fall back on an assumption of approximate linearity over the range of the proportion under investigation and a reliance on the robustness of the estimates given our large number of observations.

¹⁵ Probabilistic models provide a means of coping with error introduced both in our approximation of the Left-Right distribution and in our use of a single dimension. Discussion of why aggregation problems force the direct expression of probabilities rather than deriving them from an individual utility maximization perspective is contained in Castanias, McGuire, and Rosenthal.

¹⁶ Rosenthal and Sen, pp. 40–52.

¹⁷ We eliminated, using newspaper sources, candidates who had withdrawn from the race but whose names were still on the ballot. The actual dependent variable then becomes the proportion of the vote cast for the “true” candidates. In our earlier paper, we used the heuristic decision rule of eliminating all candidates with less than 1,000 votes on the second ballot. This decision rule eliminated practically all the appropriate candidates for 1958, 1967, and 1968. For 1962, the approximation was somewhat worse. With the 1973 replication, we were forced to rely on the 1,000 vote rule.

¹⁸ Huff, David L., Determination of Intraurban Retail Trade Areas (Los Angeles: University of California, Real Estate Research Program, 1962)Google Scholar. The relationship to gravity models was suggested by Richard Staelin.

¹⁹ Rosenthal and Sen, pp. 44–46, 49–52. The nonlogarithmic models are used here.

²⁰ This restriction does not apply to the estimation techniques used for the all-candidates model (described below) in which all second-ballot districts are pooled to estimate simultaneously the parameters of the model which applies to all parties.

²¹ The “party” groupings are indicated in Table 1.

²² The constraint also implies that it is incorrect to assume uncorrected errors for observations within a given district. The ordinary least squares (OLS) estimates are thus inefficient. An estimation problem similar to ours has been addressed in McGuire, T. W., Farley, J. U., Lucas, R. E. Jr., and Ring, L. W., “Estimation and Inference for Linear Models in Which Subsets of the Dependent Variable Are Constrained,” Journal of the American Statistical Association, 63 (12 1968), 1201–13CrossRefGoogle Scholar. They obtain efficient estimates by deleting one observation per district and by applying a generalized least squares procedure. Their procedure cannot be applied directly because it would require that the same set of parties contest every district on the second ballot. A modified procedure we developed yielded estimated coefficients quite similar to those found by OLS. On the other hand, the results suggest that the standard error estimates produced by OLS are overly conservative.

²³ There are six, not nine, comparisons because of the equivalence of models I and III for two-candidate races.

²⁴ Rosenthal and Sen, p. 45.

²⁵ Use of the reciprocal is motivated in Rosenthal and Sen, p. 42.

²⁶ These variables are defined in Rosenthal and Sen, p. 43.

²⁷ Sen, Subrata K., “The Calculus of Voting: Empirical Studies for France,” (Ph.D. dissertation, Carnegie-Mellon University, 1971), Ch. 3, pp. 41–42 Google Scholar.

²⁸ For French applications see: Mendès-France, B. and Laumonier, L., “Une Application des Methodes de l'Analyse Statistique à l'Estimation des Déplacements de Voix entre les Deux Tours des Elections Présidentielles de 1965,” Revue Française de Science Politique, 17 (02 1967), 110–14Google Scholar, and Lancelot, Alain and Weill, Pierre, “Les Transferts de Voix du Premier au Second Tour des Elections de Mars 1967; Une Analyse de Regression,” in Les Elections Legislatives de Mars 1967, Centre d'Etude de la Vie Politique Francaise (Paris: A. Colin, 1970), pp. 373–88Google Scholar.