Candidate Monotonicity and Proportionality for Lotteries and Non-Resolute Rules

We study the problem of designing multiwinner voting rules that are candidate monotone and proportional. We show that the set of committees satisfying the proportionality axiom of proportionality for solid coalitions is candidate monotone. We further show that Phragmén’s Ordered Rule can be turned into a candidate monotone probabilistic rule which randomizes over committees satisfying proportionality for solid coalitions.