# [EM] Possibly making Sainte-Lague even more STV-like

Kristofer Munsterhjelm km_elmet at t-online.de
Thu Sep 5 02:42:44 PDT 2013

```On 09/04/2013 11:19 AM, Vidar Wahlberg wrote:
> On Wed, Sep 04, 2013 at 12:14:36AM +0200, Kristofer Munsterhjelm wrote:
>> If you're electing just one seat, then C should win; anything else
>> would be unfair to a majority. But if you're picking two, then if
>> you give the first seat to C, giving the second to L will bias the
>> assembly to L and giving the second to R will bias the assembly to
>> R. So the right outcome for two seats would be {LR}. But {C} is not
>> a subset of {LR}, so house monotonicity is not desirable. You might
>> argue that {C1, C2} would fix the problem, but that would just push
>> the problem itself into the three-seat case.
>
> Very well explained. Some quick thoughts this early morning:
> I fully agree with you that it would make much more sense for {LR} to
> win in a 2-seat election.
> If you were to elect some of the seats using a quota, then resolve any
> remaining seats using a pairwise method, I do see some issues once you
> get more parties into the equation, let's assume a 4-seat election:
> 28: A>B>D>F>E>C
>  1: B>C>A>F>E>D
> 27: C>B>D>F>E>A
> 11: D>B>F>A>C>E
>  7: E>B>D>C>A>F
> 26: F>B>D>A>E>C
>
> B is despite receiving very few first preference vote, still a candidate
> that every voter rank high, but depending on how you'll weight down the
> votes after distributing the directly (quota) won seats, B may end up
> winning no seats at all. That's not quite fortunate either.

For candidate elections, the Droop proportionality criterion regards not
just single candidates, but also sets of these. So if you have an
extreme election of the form:

n: A>X>B>C>D
n: B>X>C>A>D
n: C>X>A>B>D

then the criterion doesn't disqualify X. Instead it says that the
council must come from {ABCX}. The problem is, it says nothing about
whether X should be elected, either; that's up to the method to decide.
Condorcet-based methods try to generalize the trade-off logic that makes
Condorcet elect compromise candidates in the single seat case to
multi-seat cases, subject to the constraints given by the
proportionality measures they're designed to obey (Droop proportionality
for CPO-STV and Schulze STV, party list Sainte-Laguë reduction for
CPO-SL and my unnamed method).

You can do slightly better than Plurality with respect to vote-splitting
by extending DAC or DSC (Descending Acquiescing Coalitions or Descending
just weight the number of voters voting for each acquiescing/solid
coalition by dividing by the Sainte-Laguë divisor for the number of
party members that have been elected and are in that coalition. (For
instance, if you've picked two H candidates and an FrP candidate so far,
the set {H FrP V}'s support would be divided by f(3) = 7.)

However, like IRV, DAC/DSC tends to elect from "the strongest wing"
rather than a true compromise. It's also house monotone, so the argument
against house monotonicity applies to it as well. Finally, it fails to
redistribute the votes in the example I used for the unnamed method,
where the voters can benefit from strategy.

So a Sainte-Laguë based on DAC or DSC is better than one based on
Plurality - and we know how to do the generalization - but there isn't
much else to be said about it. (It might also be weakly monotone, which
would be nice since STV isn't.)

> Not saying that the monotonocity criterion must be fulfilled. In your
> example I would agree that in a 2-seat election then L and R should win
> each their seat (and C should win in a 1-seat election), but in my
> example above I would expect B to win at least one seat, regardless of
> the amount of available seats (with the possible exception of 3 seats,
> where 3 parties stand out, but still not enough to reach the quota).

Just a note here. Monotonicity can refer to one of two things. First,
there's the per-seat monotonicity that I call "house monotonicity": that
the outcome for larger assemblies should contain the outcomes for
smaller ones. That's not generally desirable, as the explanation gives
(although some times it might be required by external concerns, e.g.
when determining an ordered party list).

Then there's voter monotonicity. This is the type of monotonicity that
IRV is known for failing. A voter monotonicity criterion states that if
the voter does something that intuitively one would think would help X
(respectively harm X), that shouldn't harm X (respectively help X).
"The" monotonicity criterion, in single-winner elections, is that moving
X closer to the top of one's ranking shouldn't make X lose, i.e. that X
shouldn't lose if some of the voters who ranked him decide they like him
better.

And to explain my reference to "weak monotonicity" above, I generalized
the monotonicity criterion to two types for multiwinner elections. "Weak
monotonicity" means that, if X is in the outcome (for party list PR, if
party X has n members), then if a voter ranks X higher, X shouldn't drop
out (in party list, lose one or more seats). Strong monotonicity is that
if someone ranks a subset of the candidates in the outcome higher (not
necessarily by the same amount), none of them should drop out. I suspect
strong monotonicity is too strong and that you can't have both it and
setwise proportionality. But then again, I once thought that of weak
monotonicity, too.

>> Second, a voter may gain undue power with additional preferences.
>> Say a voter's preference is H > FRP. Then when a H seat is chosen,
>> that will deweight his preference for H over AP (say), but it won't
>> deweight his preference for FRP over AP. Thus some of his pairwise
>> preferences get counted at full strength even though he got his
>> first choice.
>>
>> If you want to go down this sequential deweighting route, I think
>> you should instead deweight the ballots themselves. So say H gets a
>> seat. Then everybody who voted for H first should have his ballot
>> deweighted, including later preferences (e.g. FRP > AP). That method
>> isn't summable, but it's better[1]. You'd end up with something
>> somewhat similar to Forest Simmons and Olli Salmi's "D'Hondt without
>> lists", but with Sainte-Laguë instead of D'Hondt. See http://lists.electorama.com/pipermail/election-methods-electorama.com/2002-August/008561.html
>> .
>
> preferences remaining at full strength.
> Some months ago I believe I actually tried this approach (that is,
> reweighting the entire vote and not just the vote strength for parties
> who already won some seats), but I scrapped the idea and the code after
> receiving some peculiar results. I can't rule out that the
> implementation was flawed, I might give this another shot.

The "D'Hondt without lists" implementation weights according to rank
position too. E.g. someone who ranks H>FRP>V with an FRP member elected
still gets his full preference for H>FRP, but not for FRP>V. Perhaps
that makes it more well-behaved than your attempts at doing the same?

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