We study the design of voting rules for committees representing heterogeneous groups (countries, states, districts) when cooperation among groups is voluntary. While efficiency recommends weighting groups proportionally to their stakes, we show that accounting for participation constraints...

In the United States, the legitimacy of presidents who do not obtain a majority of the popular vote is often questioned. Debates on electoral legitimacy tend to revolve around the relative importance of the Electoral College and the popular vote. We develop a theory of electoral legitimacy...

We consider a model of optimal apportionment in which individuals derive utility from the probability that the collective decision matches their own will. We suggest an iterative algorithm for the numerical optimization of the utilitarian social welfare function and apply it to the case of the...

This paper provides a theoretical foundation that supports the degressive proportionality principle in apportionment problems, such as the allocation of seats in a federal parliament. The utility assigned by an individual to a constitutional rule is a function of the frequency with which each...