I did some more investigation mainly into "quality-based" cardinal PR methods, where a function Q(outcome, ballots) outputs a score and maximum score wins. The results below are with the 1D Droop fitting concept of PR, with voters rating their favorite max, least favorite min, and everybody in between linearly scaled by distance. First, I found and fixed some bugs. Earlier I said that LPV0+ had truly bad goodness-of-fit VSE values, e.g. 3 seats, 10 candidates: Cardinal: LPV0+ VSE: -1.4730 Cardinal: Log-penalty (K = 1) VSE: -1.4669 Cardinal: Log-penalty (K = 10) VSE: -1.3790 Cardinal: Log-penalty (K = 1000) VSE: -1.3486 The implementation was buggy. After fixing the bug, I got these results: Cardinal: LPV0+ VSE: 0.4151 Cardinal: Log-penalty (K = 1) VSE: 0.4293 Cardinal: Log-penalty (K = 10) VSE: 0.4566 Cardinal: Log-penalty (K = 1000) VSE: 0.4611 It's still not all that great, but it's not sub-zero anymore either. It's 2-seat 10-candidate quantile fitting shows the best fit to be close to majoritarian/most center-favoring: Cardinal: LPV0+ q = 0.5 VSE: 0.7410 Cardinal: Log-penalty (K = 1) q = 0.5 VSE: 0.7949 Cardinal: Log-penalty (K = 10) q = 0.5 VSE: 0.8802 Cardinal: Log-penalty (K = 1000) q = 0.5 VSE: 0.9057 this is the same direction that Psi goes, and there seems to be a pattern: methods that relatively speaking count a large number of mediocre ratings the same way as a few favorites and a few loathed candidates do not do well by this proportionality measure. Psi is the typical example since the quality is digamma(delta + sum of the voter's rating of the proposed winners), so it judges a large number of meh just the same as a few favorites and the rest being actively disliked. On the other hand, Harmonic weights the best ratings higher because the voter's favorite rating counts at full weight, then the next at 1/2 or 1/3 weight, etc. Hence it also gives a few favorites plus a few disliked candidates a higher quality score than everybody being considered mediocre. I think I'll get back to this in a future post. Second, I implemented house monotone methods: a runner for sequential cardinal quality-based methods like harmonic, psi, etc.; and some glue code/an interface for the Schulze proportional ordering. Here are some results: 2 seats, 10 candidates, optimal quantile: Name q prop. VSE at q Schulze STV 0.3349 0.9998 Schulze prop. ordering (house monotone) 0.388 0.8945 Cardinal: Harmonic (Chamberlin-C) 0.2652 0.9196 Cardinal: Harmonic (Sainte-Laguë) 0.3487 0.9322 Cardinal: Harmonic (d'Hondt) 0.3816 0.9245 Seq Cardinal: Harmonic (Chamberlin-C) 0.2729 0.5911 Seq Cardinal: Harmonic (Sainte-Laguë) 0.3791 0.8401 Seq Cardinal: Harmonic (d'Hondt) 0.4066 0.8824 3 seats, 10 candiadtes, at Droop: Name prop. VSE at Droop Schulze STV 0.9992 Schulze proportional ordering 0.8915 Cardinal: Harmonic (Chamberlin-Courant) 0.6400 Cardinal: Harmonic (Sainte-Laguë) 0.8925 Cardinal: Harmonic (d'Hondt) 0.8006 Seq. Cardinal: Harmonic (Chamberlin-C) 0.6959 Seq. Cardinal: Harmonic (Sainte-Laguë) 0.8614 Seq. Cardinal: Harmonic (d'Hondt) 0.7613 (Chamberlin-Courant corresponds to delta->0 in the limit, because you only care about voters' favorites in that case; Sainte-Laguë and d'Hondt are delta=1/2 and delta=1 respectively.[1]) The results suggest that you pay a penalty for house monotonicity, and that this penalty is worse if the base method is factional-favoring rather than center-favoring - at least for these methods whose single-winner result is close by or at the median. (Presumably a house-monotone method based on IRV would have less of a penalty, but in exchange, it would face a single-winner penalty due to center squeeze.) That makes sense to me. I'd probably still say that the more serious problem with house-monotone methods whose single-winner variant picks canidates near the center is the dynamics. The first candidate can secure their position by moving closer to the center, because the first seat is chosen by a majoritarian/center-focused method. Then the second candidate gets a lot more room and can move outwards. However, there is then a potential instability. The outcome may flop between the second candidate being to the left and to the right of the center candidate, and the two-seat assembly is only balanced in expectation (with a lot of variance). Finally, I checked maxmin methods, also called "egalitarian" ones in some papers. These are quality-based methods (e.g. like Harmonic), but instead of the objective being to maximize the sum of each voter's satisfaction with the result, the objective is instead to maximize the satisfaction of the least satisfied voter. Ties are broken by considering the second-least satisfied voter and so on. Some results: 2 seats, 10 candidates, optimal quantile: Name q VSE at q Maxmin Cardinal: Harmonic (C-C) 0.2418 0.3583 Maxmin Cardinal: Harmonic (Sainte-Laguë) 0.2711 0.3824 Maxmin Cardinal: Harmonic (d'Hondt) 0.2877 0.3765 Maxmin Cardinal: LPV0+ 0.3582 0.4263 Maxmin Cardinal: Log-penalty (K = 1) 0.3586 0.4321 Maxmin Cardinal: Log-penalty (K = 10) 0.3619 0.4357 Maxmin Cardinal: Log-penalty (K = 1000) 0.3626 0.4365 Maxmin Cardinal: Psi (Sainte-Laguë) 0.3735 0.3570 3 seats, 10 candidates, at Droop: Name VSE Maxmin Cardinal: Harmonic (C-C) 0.1224 Maxmin Cardinal: Harmonic (S-L) 0.1492 Maxmin Cardinal: Harmonic (d'Hondt) 0.1751 Maxmin Cardinal: LPV0+ 0.3555 Maxmin Cardinal: Log-penalty (K = 1) 0.3593 Maxmin Cardinal: Log-penalty (K = 10) 0.3671 Maxmin Cardinal: Log-penalty (K = 1000) 0.3680 Maxmin Cardinal: Psi (Sainte-Laguë) 0.2512 5 seats, 10 candidates, at Droop: Name VSE Maxmin Cardinal: Harmonic (C-C) 0.0874 Maxmin Cardinal: Harmonic (S-L) 0.0745 Maxmin Cardinal: Harmonic (d'Hondt) 0.0712 Maxmin Cardinal: LPV0+ 0.2554 Maxmin Cardinal: Log-penalty (K = 1) 0.2611 Maxmin Cardinal: Log-penalty (K = 10) 0.2702 Maxmin Cardinal: Log-penalty (K = 1000) 0.2652 Maxmin Cardinal: Psi (Sainte-Laguë) 0.086 None are particularly good by this proportionality measure, and it gets worse the more seats there are. On the one hand, it may not be surprising after the fact - you could reason that maxmin only cares about the worst-off voter, which makes it too insensitive to disproportionality elsewhere: that it would sacrifice proportionality everywhere just to make a single voter slightly better off. On the other, you could also reason that if someone else's proportionality was affected, then *their* score would drop and they would be the new worst-off voter. I think proportionality needs to be specified explicitly (like Monroe) for it to work. Consider the single-winner case. It's hard to think of a function that would pick the candidate closest to the median voter if all you have to work with is each voter's satisfaction in isolation, and you're maximizing the minimum satisfaction, while, if you were to implement a Droop constraint, it could maximize the minimum satisfaction of a majority, hence that voter at that minimum would more likely be the median voter. But even if it were to be enforced by a constraint - like Monroe - it feels like the method would by its nature be pulling in two directions. So I suspect maxmin methods would still have trouble on this measure, even if they're constrained to be at least somewhat proportional. (One could then say: perhaps maxmin isn't meant to be proportional, but instead give minority factions more representation. And the maxmin version of a method does have a lower optimal quantile for 2-of-10. But the problem then is strategy: majorities can strategically represent themselves as minorities, but minorities can't do the converse.) -km [1] Strictly speaking "Chamberlin-Courant" here is *unweighted* C-C - with delta=0 Harmonic, you still give reps equal-weight votes in the assembly. You don't let somone who's the favorite of twice as many voters get twice the voting weight -- but if you were, it would reduce the disproportionality.