[EM] Juho--Why Largest-Remainder/Hamilton is unbiased
Juho
juho4880 at yahoo.co.uk
Sat Dec 9 15:36:30 PST 2006
Sometimes small portion of randomness is nice. People play cards and
are happy with he idea that random cards are dealt to the players. In
a two-party system the time each one of the parties stays in power
roughly corresponds to the value the parties are able to bring to the
people. In PR methods a small random component doesn't bother me
much. As you said, none of the discussed methods are really terrible.
But of course in general I feel most comfortable with methods that
respect the voice of the voters as exactly as possible.
When trying to reach maximum PR it may be good to count the votes of
the whole country and agree the number of seats based on those
numbers. If one arranges a separate counting process for each
district one probably introduces more randomness (rounding errors) to
the PR than what the selection between e.g. LR and SL does. "Regional
PR" is however quite ok, just like (the normal) "ideological PR". One
can provide both of these very accurately at the same time. But when
doing so one probably has to push the rounding errors somewhere else,
e.g. so that although each party gets the number of seats it is
entitled to based on national votes and each region gets its fair
share of seats, the candidates that will be elected at each region do
not exactly correspond to the opinions expressed by the voters in
that region. Let's say that a small party got 10% of the quota in all
of the 10 districts. In principle it is entitled to one seat (in a
quota based method). Now we must pick one of the regions where this
party gets its seat even if there would be unelected candidates that
got better results than the candidate of this small party in all
regions.
Sorry about the long paragraph. My only intention was to demonstrate
that there will be some rounding errors in any case and we just need
to decide where we want as exact results as possible and where some
more rounding errors can be allowed. There may also be other desired
properties like ability to order the candidates as discussed later.
Now back to the individual quota and divisor based PR methods. Each
one of them has some nice features and some less nice features. Many
of them have already been mentioned. I'll list some more of the to
point out some interesting features of them. (I hope I don't make too
many mistakes with the tricky details.)
LR and d'Hondt can guarantee that if I get 1/n of the votes, then
I'll get one of the n seats. SL and d'Hondt can order the candidates
for me, and I can then pick m best ones from the queue. The missing
support of the second feature makes the LR a bit "jumpy". There is no
fixed ranking of the candidates but the set of winners / order
depends on the number of seats. In many elections the number of seats
is however fixed and as a result the behaviour of LR is quite stable.
LR would maybe not be the best method when electing a team that will
get a new member every day since it would no be easy to decide in
which order the candidates should be chosen. Calling one candidate
back in order to replace her with two new ones doesn't sound like a
nice procedure. But for something more regular like parliamentary
elections where the number of seats has been decided before the
election I find LR very good (fair/unbiased and not jumpy).
There could be some "what ifs" with LR like "what if there would have
been one seat less in this region and my party would have gotten one
extra seat" but these problems exist also in other methods. For
example "if we had had LR instead of the current method my party
would have gotten the remaining seat with its larger number of extra
votes than the other partes had after the 1/n quotas were filled".
Other methods have their own verbal justifications. I however tend to
think that LR is a quite natural measure of justification (of fair
seat allocation). Its surprising features are just surprising
features of mathematics and not really a fault of the method. For me
SL is a "smoothened version of LR". It is ok when smoothness (or
ability to order the candidates) is what we want. And it is ok in
general as a PR method. But I just can't help seeing LR as the basic
method and the others as methods with additional tricks to tune the
method in some direction (e.g. to order the candidates or to favour
big parties a bit).
Sorry about the long and philosophical mail. I hope I managed to
describe why I find LR quite natural despite of its peculiarities.
And as you already noted, all the discussed methods are very
proportional. Some favour big parties a bit but otherwise I'd mostly
talk about unavoidable small rounding errors.
Juho Laatu
On Dec 9, 2006, at 22:35 , MIKE OSSIPOFF wrote:
>
> I'll speak of it in terms of PR, because that's where LR is used. A
> party's
> remainder can be from 0 to almost 1, and, on average, it's .5 That
> means .5
> quotas of remainder for each party. That means .5 remainder seats
> per party.
> So, since remainders, in the long run, are randomly ordered, a party's
> expectation is half of a remainder seat So, for instance, the
> party with
> 4.5 quotas has an expectation of 4.5 seats. So, on average, seats
> per quota
> is the same for all the parties.
>
> No doubt that isn't really a solid demonstration, but it's plausible.
>
> Because I'm used to single-winner methods, where merit differences are
> drastic, maybe I'm a bit over-dramatic in my criticism of
> apportionment and
> PR methods, none of which are really bad. In particular, LR, being
> unbiased,
> would be fine, in spite of its game-of-chance component. I've enjoyed
> betting small amounts at the craps table. But never bet what you can't
> afford to lose. So, Juho, say your favorite party is proportionally
> qualified for one seat. LR might give it one, two, or zero seats.
> Do you
> really want to play double-or-nothing with your representation?
>
> Mike Ossipoff
>
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