The dilemma, as I see it, is this: a multiple transferable vote that doesn’t transfer approval until all current approvals win increases the unrepresented vote by diluting its voting power with already-represented votes that would otherwise have been exhausted. My solution: an adaptive approval threshold, which would reduce unrepresented and prematurely exhausted votes alike. It could be applied to any cardinal system.
A score ballot is used. The voter’s initial approval threshold (IAT) is MAX (or, alternatively, his original score for his favorite current hopeful) unless he opts for something else. Additionally, he may indicate a final approval threshold (FAT) that can be a score or a function of his original scores for current hopefuls; if not, a default may be used (if a score, I recommend MAX; if a function of scores, I recommend a power mean, specifically p=2, i.e. root mean square). The voter’s current threshold is given by: IAT-(IAT-FAT)/s, where s is the number of remaining seats. All votes max all hopefuls (originally scored) above their current threshold, min all hopefuls below it, and give the middle score to all hopefuls on it. If IAT or FAT are constant, this may result in some votes approving all hopefuls or no hopefuls, which votes’ thresholds should be moved just enough to alter them (or, more simply, all votes should approve their favorite hopefuls and disapprove their least favorite hopefuls).
- As threshold movement begins slow and rapidly quickens as the final seat approaches, much of the good of it would be got by simply saving FAT for the end.
- The voter should perhaps have the option to have his scores instead be normalized after each candidate is removed from the set of hopefuls as in Cardinal Baldwin. Normalization may also be desirable if IAT or FAT are constant, to avoid all-approval or no-approval votes (and thus the special rule for avoiding them) or even to better adapt the votes.
- An alternative to FAT as a fixed function of scores is to ignore (or not have in the first place) FAT and IAT and instead just make the current threshold itself a variable function of the scores. For example, it could be the p=s (or, more precisely, p=sv/(v-s), v being the number of voters) power mean score. This would (appropriately) set the threshold just below the best hopeful in the early rounds and just above the average hopeful by the final round.