I begin with a re-imagination of Approval (AV) and Score (SV) as the linear and logarithmic single-winner applications of the Nash bargaining solution (geomean-maximization). I then generalize geomean-maximization to multi-winner elections.
First, in order to make geomean useful for comparing sets including 0s, we must define it as the p → 0 power mean (M), by which: M(X) < M(Y) if X has more 0s than Y; if they have the same number of 0s, M(X) < M(Y) if X’s geomean positive number is smaller than Y’s. As the proportion of profiles giving a tie in 0s approaches zero as the number of voters increases toward infinity, so does the probability of the result changing if all positive numbers are replaced with 1s. Thus, if utilities are proportional to scores, the geomean utility-maximizing candidate is simply the one with the least 0s, making an approval ballot sufficiently expressive and AV a geomean utility-maximizing system.
But suppose a candidate’s score is instead proportional to a logarithm of his utility. In that case, the geomean utility of a candidate is proportional to the base of that logarithm to the power of his average score, making SV the geomean-maximizing system.
To generalize the Nash interpretation of AV to multi-winner, we consider the voter to approve any set of candidates he approves at least 1 member of (because, again, only 0s matter; if there is a tie here, it’s broken in favor of the set with the highest geomean positive number of approved candidates). If there are too many candidates to check every k-combination (k being the number of seats), we fill the seats sequentially, repeatedly electing the candidate approved by the most voters who do not yet approve an elected candidate (if there’s a tie, it’s broken in favor of the candidate whose election gives the set of elected candidates the highest geomean positive number of approvals).
To generalize the Nash interpretation of SV to multi-winner, we must identify the base (b) of the logarithm, let p=b, and let a voter’s score for a set of candidates equal the power mean of his scores for its members; the geomean utility-maximizing set is the set with the highest average score. For example, if b=2, the geomean-maximizing set is the set that maximizes the average voter’s root mean square score for its members. In the limit b → 1, it’s the set that maximizes the average voter’s average score. In the limit b → ∞, it’s the set that maximizes the average voter’s maximum score. If there are too many candidates to check every k-combination, we fill the seats sequentially, repeatedly electing the candidate whose election maximizes the average voter’s p=b power mean score for elected candidates.