[EM] Sainte-Lague vs d'Hondt for party list PR

Michael Ossipoff email9648742 at gmail.com
Sun Jul 1 15:51:28 PDT 2012

I always advocatred SL (Sainte-Lague) over dH (d'Hondt) for party list PR,
because, if you're using PR it's because you want proportionality,  and if
you want proportionality, then you want SL.

Then, more recently, I said that, since I don't think that we need PR
anyway (though I have nothing against PR), d'Hondt would be fine,
especially since it guarantees a seat majority to a vote majority.

But now I feel that I was mistaken to say that. For two reasons:

1. For fair inclusion, there should be no threshold. d'Hondt will
disproportionately exclude small parties. That matters, because PR can only
be justified, in comparison to a good single-winner method, if there's no
significant split-vote problem. You shouldn't have to worry that you need
to vote for a compromise because your favorite party might not have enough
votes to win a seat. That problem is, of course, worse in d'Hondt than in

2. In PR, the idea is that you don't have to compromise in your voting,
because you can just elect representatives (members of parliament
congress-members, etc.) of your party to parliament or congress, and _they_
can do the compromising, when necessary--but only when necessary.

But how true is that in d'Hondt? Not very.

Say your favorite party is significantly smaller than the ones that are in
major contention in parliamentary voting. If you vote for your party, and
it's a small party, d'Hondt will give it _significantly_ less
representation per person, as compared to the larger parties. So your
favorite party won;t have many seats with which to support coalitions. If
compromising, coalition-support, are necessary, then you'd do better, in
the d'Hondt PR election, to give your vote to a big party, so that you can
thereby add more seats to the coalition that you want to support.

That's no good. d'Hondt fails FBC, if FBC were extended to PR methods.

No, to fulfill the purpose of PR, Sainte-Lague, virtually free of
size-bias, is the method to choose. SL also puts every party as close as
possible to its correct proportional share.

Yes, SL isn't _entirely_ unbiased. Its unbias depends on the assumption (at
least in the regions of interest) of a uniform probability distribution for
parties, with respect to the scale of party vote totals. That assumption
isn't really correct. There are more likely to be more small parties than
large. The distribution-curve is most likely a decreasing function. From
that, one would expect Sainte-Lague/Webster to be _very slightly_
large-biased. That's probably true of Largest-Remainder too. Both of those
methods are strictly unbiased only if that probability-distribution is
flat, within the areas of interest.

But that bias isn't enough to matter, and doesn't bother me at all.

For example, even when that probability distribution is assumed flat, for
the vote or population range between each pair of consecutive whole numbers
of quotas (whole number values of the quotient of votes or population by
whatever divisor is being tried) Hill's method is significantly
small-biased, where, under that assumption, Webster (Sainte-Lague) is
completely unbiased. But, even so, Hill is only as biased as follows:

Consider a small party whose quotient, by the final divisor, is equally
likely to be anywhere between 1 and 2. And consider a large party whose
quotient is equally likely to be anywhere between 53 and 54.

The small party's expected s/v is only 1.057 times greater than the large
party's expected s/v.

And that's with Hill's blatant small-bias. So Websters particularly slight
large-bias wouldn't matter at all.

There are seat allocation method that have been proposed, by Warren Smith
and me, that seek more perfect unbias, without the assumption stated
above.. But they're more complicated, &/or without precedent. I cl;aim that
SL is quite good enough, and should be the only method used for allocation
of seats to fixed districts; or for allocation of seats to parties in

Well, maybe later, at some point, people might be interested in considering
those more perfectly unbaised methods.

Anyway, I emphasize that, in the United States, the voting system is
incomparably more important than the apportionment method.

However, it's also true that apportionment has been very fiercely fought,
and demonstrating that Hill is biased, even in the amount that I stated
above, and showing where a large state loses a seat because of Hill, might
show the people in that large state that Hill is unfair. And that will show
people that our current  ways of doing things can be wrong. And that will
make people more willing to look at the wrongness of Plurality. So I feel
that Hill's small bias should be publicized, especially when it can be
shown to have recently taken a seat away that a large state should have.

Hill's method, currently used for U.S. House of Representatives
apportionment, is usually known by the (incorrect) name of "Equal

Mike Ossipoff
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