In districts where only one seat is contested, the electoral formula (plurality or majority) should be a major determinant of the number of parties that receive votes. Specifically, plurality rule should generate two-party competition while other institutional arrangements should generate electoral fragmentation. Yet tests of these propositions using district-level data have focused on a limited number of cases; they rarely contrast different electoral systems and have reached mixed conclusions. This study analyses district-level data from 6,745 single-member district election contests from 53 democratic countries to test the evidence for Duverger's Law and Hypothesis. Double-ballot majoritarian systems have large numbers of candidates, as predicted, but while the average outcome under plurality rule is generally consistent with two-party competition, it is not perfectly so. The two largest parties typically dominate the districts (generally receiving more than 90 per cent of the vote), and there is very little support for parties finishing fourth or worse. Yet third-place parties do not completely disappear, and ethnic divisions shape party fragmentation levels, even under plurality rule. Finally, institutional rules that generate multiparty systems elsewhere in the country increase electoral fragmentation in single-member plurality districts. [ABSTRACT FROM PUBLISHER]