Candidate Resignation Monotonicity in Approval-Based Committee Elections

Abstract: Approval-based committee (ABC) elections select a fixed-size committee of candidates based on voters' approval preferences. We study a setting where elected members may resign and formalize resignation monotonicity: when we re-run the voting rule after resignations, a resignation monotone rule must still elect all remaining previously winning candidates. We show that many prominent ABC rules fail this property, and by connecting our notion to prior work on ABC elections with dynamic candidate sets, we show that the justified representation (JR) axiom is incompatible with resignation monotonicity. It turns out that fractional committee voting does not suffer from this issue: rules based on using maximum flows in the network representation of an election instance can satisfy both representation axioms and resignation monotonicity. We design an integral version of maximum flow, the Maximum Payment Rule (MPR), and show that it satisfies a relaxed form of resignation monotonicity that only requires there to be some way of replacing the resigning candidates in a way that guarantees the PJR+ axiom. MPR is NP-hard to compute in general, but becomes tractable in structured domains. Finally, we study a strategic setting where a losing candidate may introduce new weak candidates to the election to try to become a winner. We show that all resignation monotone rules as well as many sequential rules are immune, but that the PAV rule can be manipulated in this way.