National Electoral Thresholds and Disproportionality

We model the conditional distribution of seats given vote shares induced by national electoral systems using a stochastic threshold of representation and a disproportionality parameter that regulates allocation for parties above the threshold. We establish conditions for the parameters of this model to be identified from observed seats/votes data, and develop a Maximum a Posteriori Expectation-Maximization (MAP-EM) algorithm to estimate them. We apply the procedure to 116 electoral systems used in 417 elections to the lower house across 36 European countries since WWII. We reject a test of model fit in only 5 of those systems, while a simpler model without thresholds is rejected in favor of our estimated model in 49 electoral systems. We find that the two modal electoral system configurations involve higher thresholds with seat allocation for parties exceeding thresholds that does not statistically differ from perfectly proportional allocation (32.76% of all systems); and systems for which we cannot reject the absence of a national threshold but exhibit disproportional seat allocation for parties eligible for seats (38.79% of all systems). We also develop procedures to test for significant changes in electoral institutions and/or the distribution of seats.