Reformulating the Cube Law for Proportional Representation Elections

REFORMULATING THE CUBE LAW FOR PROPORTIONAL REPRESENTATION ELECTIONS REIN T AAGEPERA University of California, Irvine Le cube law was proposed around 1910 to express the con- version of a party's vote shares into its seat share in two-party plurality elections with single-seat districts. This article develops predictive seat-vote equations for a much wider range of elections, including those involving many parties, single- and multi-seat districts, and diverse seat allocation rules such as plurality and list proportional repre- sentation (PR). Without any statistical curve fitting based on the seat and vote shares themselves, the basic features of the conversion are predicted using exogenous parameters: magnitude and number of districts, number of parties, and total size of the electorate and of the assembly. The link between the proposed equations and the original cube law is explicated. Using an existing data base, the fit of the predictive model is examined. On balance, this model accounts well for the conversion of votes to seats, and for the deviation from proportionality in PR systems. A fundamental issue. for democracies is the extent to which elections reflect the popular will. One important aspect of that concern is the degree to which voter preferences for different parties and candidates are reflected in electoral outcomes. The way in which this question is commonly studied is by looking at the relationship between the aggregate vote share for can- didates of a given party (or other group- ing) and the aggregate seat share received by candidates of that party (Rae, 1971). Some electoral systems expressly aim at proportional representation (PR), that is, seat shares equal to vote shares. However, the so-called PR systems differ widely among themselves in the degree to which they deviate from ideal PR, largely because different district magnitudes (i.e., the number of seats per district) are used, and different numbers of parties have evolved. Discussion of electoral reform in such countries often revolves not around the principle of PR, but around whether and how to make the system somewhat more or less proportional. However, such discussion has not beeh guided by any general quantitative rule predicting the specific degree of deviation from PR in the systems existing or proposed. For plurality elections, it is well known that proportionality between vote shares and seat shares cannot be expected. In this case, however, a votes-to-seats conver- sion rule has been proposed, the so-called cube law. Devising a comparable expres- sion for the PR elections would have con- siderable practical importance in guiding the discussions of electoral reform. A AMERICAN POLITICAL SCIENCE REVIEW VOL. 80 NO. 2 JUNE, 1986