Divisor Choices and District Magnitude: Unintended Consequences in Modified Sainte-Laguë Systems

The choice of the first divisor in proportional representation formulas can have significant consequences for seat allocation, especially in multimember districts with large magnitudes (Gallagher 1992). This paper examines the modified Sainte-Laguë system, which Scandinavian countries have used for first-tier seat distribution (Elklit 1999). Unlike the original Sainte-Laguë sequence starting at 1.0, the modified version employs a higher first divisor—typically 1.2 or 1.4—to make it harder for very small parties to win representation. While this adjustment reduces fragmentation, it also lowers proportionality, and its effects become more pronounced as district magnitude increases. Denmark provides an instructive example. Before 2007, 135 first-tier seats were distributed across 17 districts with an average magnitude of 7.9. A reform reduced this to 10 districts averaging 13.5 seats. Retaining a first divisor of 1.4 under these conditions would have made it easier for small parties to secure district seats, triggering compensatory allocations. To prevent this, Denmark replaced Sainte-Laguë with D’Hondt—a formula that systematically favors larger parties. In the 2022 election, the largest party gained an extra seat beyond what 40 compensatory seats could offset, an outcome that is anomalous yet predictable given high party fragmentation. The paper investigates whether the intended effect of limiting small-party gains could have been achieved by raising the first divisor further within Sainte-Laguë (say to 1.7) rather than switching to D’Hondt, and how such adjustments would affect proportionality and overrepresentation. Methodologically, the study combines mathematical modeling of divisor sequences with seat–vote simulations across Denmark’s ten multimember districts. The paper includes a general model for tresholds of representation and exclusion depending on the size of the first divisor (developing Lijphart and Gibberd 1977). Simulations vary the first divisor from 1.4 upward to assess its impact on proportionality indices (Gallagher, Loosemore–Hanby) and on the distribution of compensatory seats. This approach allows us to identify thresholds where Sainte-Laguë retains its proportional advantages while reducing incentives for strategic entry by very small parties. The findings highlight how the choice of first divisor interacts with district magnitude, thresholds, number of parties and compensatory tiers to produce irregular outcomes that challenge assumptions about fairness and inclusiveness. By focusing on these trade-offs, the paper contributes to understanding the boundary conditions of proportionality and the unintended consequences of electoral engineering. It also offers actionable insights for reformers seeking to balance proportionality with governability in fragmented party systems. Specifically, it may improve the Danish electoral system if it comes close to achieve the politically stated purposes of the 2007 electoral system reform. Literature: Elklit, J. (1999) ‘What was the Problem? If a Divisor of 1.4 was the Solution?’, in E. Beukel et al. Elites, Parties and Democracy, Odense University Press, 75-101. Gallagher, M. (1992) ‘Proportionality, Disproportionality, and Electoral Systems’, Electoral Studies, 10(1), 33-51. Lijphart, A. & R. W. Gibberd (1977) ‘Tresholds and Payoffs in List Systems of Proportional Representation’, European Journal of Political Research, 5(3), 219-244.