Ensuring proportionality: a logical model for compensatory seats added to multi-member constituencies

Abstract

How many compensatory seats are needed to ensure (full) proportionality in two-tier electoral systems with multi-member constituencies? This question is answered by estimating the discrepancy between seats and votes rigorously, probabilistically and empirically. In 1919, Pólya showed that the Jefferson/D'Hondt method favours large parties. This seat surplus accumulates across constituencies. In some countries (e.g., Portugal, Poland, Spain and Turkey), this accumulation directly leads to quantifiable disproportionality in the final parliament. Elsewhere compensatory seats counteract this accumulated seat surplus. Using a logic-first approach, closed formulas for the required number of compensatory seats are derived given regional and national apportionment methods, the number of relevant parties $n$ and the size of the largest party $p$. If there are $c$ constituencies, it is derived logically that a two-tier system built on local D'Hondt apportionment in multi-member constituencies needs around $\frac{c}{2}(n-\frac{1}{p})$ compensatory seats to ensure proportionality. In words, that is half a compensatory seat per constituency times a number that increases with the number of parties and the size of the largest party. The validity of the derived formulas is analysed empirically using the extensive recent CLEA dataset. Many quantities do not depend much on the total parliament sizes, which means that the variance can be reduced by artificially varying the house sizes. Portugal, Poland, Spain and Turkey all employ a one-tier D'Hondt system and the analysis shows that if they were to add compensatory seats to the system, reasonable numbers would be 40, 75, 75 and 120, assuming that no other properties of the systems change.