The expanding approvals rule: improving proportional representation and monotonicity

Abstract

Proportional representation (PR) is often discussed in voting settings as a major desideratum. For the past century or so, it is common both in practice and in the academic literature to jump to single transferable vote (STV) as the solution for achieving PR. Some of the most prominent electoral reform movements around the globe are pushing for the adoption of STV. It has been termed a major open problem to design a voting rule that satisfies the same PR properties as STV and better monotonicity properties. In this paper, we first present a taxonomy of proportional representation axioms for general weak order preferences, some of which generalise and strengthen previously introduced concepts. We then present a rule called Expanding Approvals Rule (EAR) that satisfies properties stronger than the central PR axiom satisfied by STV, can handle indifferences in a convenient and computationally efficient manner, and also satisfies better candidate monotonicity properties. In view of this, our proposed rule seems to be a compelling solution for achieving proportional representation in voting settings.

Fairness in Algorithmic Decision-Making: Applications in Multi-Winner Voting, Machine Learning, and Recommender Systems

“…q-Proportionality for solid coalition (q-PSC) [16]. For a rational number, q, a k-committee w ⊆ C satisfies q-PSC if for every positive integer and for every solid coalition U ⊆ V supporting some C ⊆ C such that |U| ≥ q, it holds that |w ∩ C | ≥ min{ , |C |}.…”

“…In addition, q-PSC is not guaranteed to exist if q ≤ n/(k + 1). Moreover, if q > n/k, any q-PSC committee must provide some counter-intuitive properties (see the work by the authors of [16] for the details).…”

“…Proportionality for solid coalition seems to be first considered by Dummett [15]. Many of its variants have been studied very recently [16,17]. For example, the weak q-PSC puts constraints only on solid coalitions supporting a limited sized committee, and asks a committee w to contain all candidates who are supported by these solid coalitions.…”

“…Weak q-PSC. [16]. A committee w ⊆ C satisfies weak q-PSC if the following holds, for every positive integer , every C ⊆ C such that |C | ≤ , and every C -solid coalition U of size at least q, it holds that C ⊆ w.…”

Measuring majority power and veto power of voting rules

“…For a voting rule, the majority loser criterion is satisfied if and only if the absolute majority loser paradox never occurs (Felsenthal and Nurmi, 2018;Diss et al, 2018). 15 (q, k, m)-majority criterion is even more general than the concept q-PSC formalized by Aziz and Lee (2017) if the latter is applied to single-winner elections. The weak mutual majority criterion defined by Kondratev (2018) is a particular case of q = k/(k + 1).…”