Different institutional systems determine different mappings from election outcomes to power shares. We study such institutions within a mobilization model as well as in a model with rational voters. Assuming heterogeneity in the cost of voting, the effect of all such institutional differences on turnout depends on the expected closeness of the election: when two parties are expected to have similar support, turnout is higher the closer the system is to a winner take all one; the result is the opposite when one party has a larger expected base. We compare competition effect, size effect and underdog effect under different systems and show the robustness of all comparisons to changes in the number of parties and across models. Turnout is also shown to increase with the number of parties in the proportional system.