Re: More PR investigations: Log-penalty bugfix, house-monotone methods weaker than global methods, maxmin methods much worse

 Some interesting stuff Kristofer. As I've probably said before, it's no surprise to me that harmonic outperforms psi. I think psi is a naive way of dealing with scores that someone might think its a good idea before looking deeply into what actually happens.
House monotonicity makes sense in an apportionment setting or where people are voting for a single party, but not in a more complex PR environment. E.g. It's perfectly reasonable that the winner in a single-winner election need not be elected in an scenario where two are to be elected.
As for specifying proportionality - I think Phragmen is better than Monroe. I think Monroe has more failures of Independence of Irrelevant Ballots. Similar to Sainte-Laguë v Hamilton.
Toby
    On Thursday, 23 July 2026 at 16:07:37 BST, Kristofer Munsterhjelm via Election-Methods <[email protected]> wrote:  
 
 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.
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