Proportionality and the Limits of Welfarism

We study two influential voting rules proposed in the 1890s by Phragm\'en and Thiele, which elect a committee or parliament of k candidates which proportionally represents the voters. Voters provide their preferences by approving an arbitrary number of candidates. Previous work has proposed proportionality axioms satisfied by Thiele's rule (now known as Proportional Approval Voting, PAV) but not by Phragm\'en's rule. By proposing two new proportionality axioms (laminar proportionality and priceability) satisfied by Phragm\'en but not Thiele, we show that the two rules achieve two distinct forms of proportional representation. Phragm\'en's rule ensures that all voters have a similar amount of influence on the committee, and Thiele's rule ensures a fair utility distribution. Thiele's rule is a welfarist voting rule (one that maximizes a function of voter utilities). We show that no welfarist rule can satisfy our new axioms, and we prove that no such rule can satisfy the core. Conversely, some welfarist fairness properties cannot be guaranteed by Phragm\'en-type rules. This formalizes the difference between the two types of proportionality. We then introduce an attractive committee rule, the Method of Equal Shares, which satisfies a property intermediate between the core and extended justified representation (EJR). It satisfies laminar proportionality, priceability, and is computable in polynomial time. We show that our new rule provides a logarithmic approximation to the core. On the other hand, PAV provides a factor-2 approximation to the core, and this factor is optimal for rules that are fair in the sense of the Pigou--Dalton principle.