Thresholds and Payoffs in List Systems of Proportional Representation

Abstract Building on to the earlier work by Stein Rokkan and by Douglas Rae, Victor Hanby, and John Loosemore on the relationship of parties' vote shares and seat shares at the district level in list proportional representation (P.R.) systems, we propose sets of formulas for (1) the thresholds of representation and exclusion that a party has to cross to win its first seat, and (2) the minimum and maximum proportions of the vote required to obtain a given number of seats. The electoral systems on which we focus our analysis are the main variants of list P.R. that have been used in practice: the d'Hondt, modified Saint-Laguë, and largest remainder systems. The thresholds of representation and the minimum vote shares tend to show greater differences among systems than the thresholds of exclusion and the maximum vote shares. These differences, however, may be outweighed by two other factors: the number of seats in a district, and the number of parties presenting lists.