[EM] Juho--Why Largest-Remainder/Hamilton is unbiased

[Juho]

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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