# [EM] SL vs LR--Rounding is unavoidalbe because allocations are integer

Juho juho4880 at yahoo.co.uk
Sun Dec 10 12:47:28 PST 2006

```On Dec 10, 2006, at 20:50 , MIKE OSSIPOFF wrote:
>
> Juho--
>
> I tried to post this a few days ago, by forwarding the "failed
> delivery" message. It didn't post, and so now I'm resending it on a
> different computer:
>
> You spoke of Webster/Sainte-Lague having a rounding error, with the
> implication that Largest-Remainder/Hamilton doesn't have one. But
> rounding is quite unavoidable, since fractional seats can't be
> given (or at least are against the rules).

I agree. All methods lead to rounding errors (unless we cut the
candidates in fractions or give them unequal voting power :-).

> So rounding does _not_ count as a SL disadvantage in comparison to
> LR. SL's rounding. As I described earlier today, rounding makes SL
> a S(p) function that is as close as possible to a linear function.
> That's the definition of proportionality.

I guess this depends on what method you think is the ideal way of
allocating the seats and the unavoidable rounding errors to the
parties. (I didn't buy yet the idea that SL would be the ideal or
closest to the ideal.)

> LR isn't a function, due to its caprice. Well ok, I've heard of
> random functions.
>
> To answer an earlier question: LR _does_ have an additional
> problem, in addition to its nonmonotonicity: Its unnecessary random
> deviations from proportionality.

Could you give me a concrete example where LR performs worse than SL
(or unsatisfactorily in general).

> But LR is unbiased, and so I'd rather have it than Jefferson/
> d'Hondt. Some advocate d'Hondt on majority rule grounds: A party
> with a majority of the votes can't fail to win a majority of the
> seats. But that comes at the cost of bias in favor of large
> parties. And that brings back the old Lesser-of-2-evils problem:
> "If you vote Progresssive,  then the Progressives and the Democrats
> will get fewer votes than the Democrats would get if all the
> Progressives voted Democrat, because d'Hondt favors large parties.
> So you've got to anandon your idealism and pragmatically vote for
> the Democrats."

Yes. d'Hondt can give majority to a party with less than half of the
votes. Now we are however talking about how to allocate the
"fractional seats", which means that in most settings voters should
probably vote Progressives if that is their first choice. The
probability of getting a better result if Progressives vote Democrats
is probably typically smaller than the probability of getting a
result that is worse (replacing Progressive candidates with Democrat
candidates would make the result worse). This recommendation maybe
holds even if giving >50% of the seats to Republicans would be
considered a catastrophe and giving all Progressive seats to
Democrats would be considered an acceptable disappointment.

> That sounds too familiar. I don't want an electoral reform that
> will retain that problem.
>
> By the way, to reply to something more recent, if your party has a
> Hare quota, then it's difficult to imagine how SL could fail to
> give it a seat. In order to deny it a seat, SL would have to double
> the Hare quota to get its final quota. That would be unheard of.
> Adjsutments in quota size are quite small, to adjust the total
> allocation by one, or maybe two, seats. If your part has a Hare
> quota, it will get a seat. (A Hare quota is Total Votes/Total Seats).

I believe SL performs fine here.

The only deviation I found with your thoughts was the question if SL
or LR is closer to the ideal. I think different reasonings can be
presented. They are so similar that in practice this doesn't mean
much in most real life election settings (at least in typical
parliament elections) but for academic reasons defining also the
ideal targets is a good thing to do.

Juho Laatu

> Mike Ossipoff
>
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