I want to introduce a technique of combining two multiwinner approval methods using what I call vertical composition. The basic idea is that there are two "phases", both of which use approval ballots as their inputs, and the first phase filters out some candidates from entering the second phase. Voters, however, do not cast simple approval ballots, but ballots with trivalent values for each candidate - it is important to note that this is distinct from 0-1-2 score voting. The values are as follows:
- Approve in both phases (AA)
- Approve in first phase, disapprove in second phase (AD)
- Disapprove in both phases (DD)
The hypothetical fourth value, Disapprove in the first phase and Approve in the second phase, is excluded because it doesn't make sense.
My basic use-case for this technique is when the first phase is simple bloc approval voting that filters out anyone with more than, say, 75% disapproval (this threshold is a parameter), and the second phase is some proportional approval voting method.
This specific method would be useful for an organization that wants proportionality in its internal democracy but also wants to set up a barrier to prevent bad actors ("entryists") from joining the organization and winning a few seats proportionally while antagonizing everybody else in the organization. Voters would use DD votes to filter out the bad actors, and AD votes for candidates from different factions that are participating in good faith. A relatively high first-phase disapproval threshold, such as 75%, would make the method resistant to people overusing DD votes.