The Relationship between Seats and Votes in Multiparty Systems

The relationship between a party's popular vote share and legislative seat share—its seats—votesswing ratio—is a key characteristic of democratic representation. This article introduces a general approach to estimating party-specific swing ratios in multiparty legislative elections, given results from only a single election. I estimate the joint density of party vote shares across districts using a finite mixture model for compositional data and then computationally evaluate this distribution to produce parties' expected change in legislative seats for plausible changes in their vote share. The method easily extends to systems with any number of parties, employing both majoritarian and proportional electoral rules. Applications to legislative elections in the United States, United Kingdom, Canada, and Botswana demonstrate how parties' swing ratios vary both within countries and over time, indicating that parties under majoritarian electoral rules are subject to unique and possibly divergent geographic—political constraints.