Differences in electoral rules and/or legislative, executive or legal institutions across countries induce different mappings from election outcomes to distributions of power. We explore how these different mappings affect voters’ participation in a democracy. Assuming heterogeneity in the cost of voting, the effect of such institutional differences on turnout depends on the distribution of voters’ preferences for the parties: when the two parties have similar support, turnout is higher in a winner-take-all system than in a power sharing system; the result is reversed when one side has a larger base. Moreover, the winner-take-all system has higher welfare if and only if the support is uneven. We compare the ‘size effect’ and the ‘underdog compensation effect’ under different systems. All systems induce an underdog compensation which is partial. Namely, unlike other costly voting models, the side with the larger support almost surely wins the majority of the votes. The results obtained in the rational voter model, characterized by the voter free-riding problem, continue to hold in other models of turnout such as ethical voter models and voter mobilization models.