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Proportionality and the Limits of Welfarism

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Document pages: 33 pages

Abstract: We study two influential voting rules proposed in the 1890s by Phragmén andThiele, which elect a committee or parliament of k candidates whichproportionally represents the voters. Voters provide their preferences byapproving an arbitrary number of candidates. Previous work has proposedproportionality axioms satisfied by Thiele s rule (now known as ProportionalApproval Voting, PAV) but not by Phragmén s rule. By proposing two newproportionality axioms (laminar proportionality and priceability) satisfied byPhragmén but not Thiele, we show that the two rules achieve two distinctforms of proportional representation. Phragmén s rule ensures that all votershave a similar amount of influence on the committee, and Thiele s rule ensuresa fair utility distribution.Thiele s rule is a welfarist voting rule (one that maximizes a function ofvoter 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 welfaristfairness properties cannot be guaranteed by Phragmén-type rules. Thisformalizes the difference between the two types of proportionality. We thenintroduce an attractive committee rule which satisfies a property intermediatebetween the core and extended justified representation (EJR). It satisfieslaminar proportionality, priceability, and is computable in polynomial time. Weshow that our new rule provides a logarithmic approximation to the core. On theother hand, PAV provides a factor-2 approximation to the core, and this factoris optimal for rules that are fair in the sense of the Pigou--Dalton principle.

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