[EM] Derivation of Identric-mean as round-up point for Bias-Free method
Richard Lung
voting at ukscientists.com
Thu Sep 7 10:17:40 PDT 2023
When I happened to remark that divisor methods and quota methods are
sometimes distinguished, this is so. Robert Newland did so, in his book
on Comparitive Electoral Systems.
Indeed, I didn't say why party lists restrict personal choice. (And you
didn't deny it.) But Enid Lakeman did say why, in How Democracies Vote.
In so many words, It is because with transferable voting, the voters
decide how their votes will count from first preferences onward, while
with party lists, it is the parties who more or less determine personal
representation.
Richard Lung.
On 07/09/2023 06:04, Michael Ossipoff wrote:
>
>
> On Tue, Sep 5, 2023 at 23:59 Richard Lung <voting at ukscientists.com> wrote:
>
>
> Just a preliminary remark. Divisor methods and quotas some times
> distinguished.
>
>
> I don’t know what you mean by that.
>
> I didn’t say that the divisor methods are quotas. I didn’t say that
> quotas are divisor methods.
>
> I defined “quota” for its use in what I was saying. That use of that
> word isn’t new or unusual. It’s found in divisor method discussion.
>
> Thus there is the Droop quota and corresponding D'Hont divisor
> method.
>
>
>
> I don’t know what you’ve heard of, but I’ve never heard of Droop quota
> in a definition or discussion of d’Hondt.
>
> But maybe someone has proposed a method that he calls “d’Hondt”, &
> maybe his method uses the Droop quota, which I’ve heard of being
> sometimes proposed, & sometimes used, in STV.
>
> Divisor methods regarded as belonging to apportionment
>
> That term was probably first applied to apportionment proposals, but
> nonetheless d’Hondt is the Jefferson divisor method, & Saints-Lague is
> the Webster divisor method.
>
> Those two list-PR methods are usually defined, & at least partly
> implemented by a systematic procedure, rather than the by the
> implementation often or usually specified by the definitions of the
> divisor methods proposed & used for apportionment. ..at least in
> earlier apportionment discussion.
>
> as by Jefferson and by Webster, not to carve out party seats,
> which is too restrictive of personal choice.
>
>
> Carve out?
>
> I’m not sure, but you seem to be saying that party-list PR restricts
> personal choice.
>
> But you didn’t say why you think so.
>
>
>
>
>
>
>
>
> On 06/09/2023 05:05, Michael Ossipoff wrote:
>>
>> Greetings list-members—
>>
>> …
>>
>> In 2006, I proposed an allocation divisor-method that I called
>> Bias-Free, which eliminates bias. I’d like, in this message, to
>> better explain my derivation of Bias-Free (BF).
>>
>> …
>>
>> Instead of defining “bias”, I’ll just let the derivation of BF
>> tell what it guarantees, and anyone can decide whether that’s unbias.
>>
>> …
>>
>> Outline of derivation of Bia-Free (BF):
>>
>> …
>>
>> First, to define the terms in the explanation, I should say what
>> a divisor-method is:
>>
>> …
>>
>> Divide total votes by total seats. That’s the Hare Quota.
>>
>> …
>>
>> Divide each party’s votes by the Hare Quota, & round off to one
>> of the two closest integers. (Each divisor-method uses a
>> different round-up point.)
>>
>> …
>>
>> Allocate seats according to those rounding-results.
>>
>> …
>>
>> If the number of seats thus allocated equals the legally-ordained
>> number of seats, then that’s the final allocation.
>>
>> …
>>
>> Otherwise, try the procedure using another number to replace the
>> Hare Quota, & call that new number the quota. Repeat the above
>> procedure, using that new quota instead of the Hare Quota.
>>
>> …
>>
>> Find (by trial-&-error, or by some systematic-procedure) a quota
>> such that the resulting number of seats allocated equals the
>> legally-ordained number of seats.
>>
>> …
>>
>> In the explanation below, “quota” means “quota” as defined above,
>> or a number of seats equal to the quota.The Hare Quote too is a
>> “quota” as the term is used below.
>>
>> …
>>
>> The object is for the average seats per quota to be unity,
>> averaged over an interval between two integer numbers of quotas.
>>
>> …
>>
>> q = quotas.s = seats.R = the round-up point between a & b.
>>
>> …
>>
>> Above the round-up point, s/q = b/q.
>>
>> …
>>
>> Below the round-up point, s/q = a/q.
>>
>> …
>>
>> …because, below the round-up point a party would have a seats, &
>> above the round-up point a party would have b seats.
>>
>> …
>>
>> Integrate b/q from R, to b.
>>
>> …
>>
>> Integrate a/q from a to R.
>>
>> …
>>
>> Add the two integrals together.
>>
>> …
>>
>> To average over the interval, divide by b – a, the total amount
>> of quota in the interval.
>>
>> …
>>
>> i.e. divide by 1.
>>
>> …
>>
>> Set that average s/q in the interval equal to 1, because it’s
>> desired for it to be 1.
>>
>> …
>>
>> Solve for R.
>>
>> …
>>
>> R = (1/e)((b^b)/(a^a)).
>>
>> …
>>
>> That quantity is called (a special case of) the identric-mean of
>> a & b.
>>
>> …
>>
>> Someone expressed concern that the unbias would be spoiled
>> because the size of parties has a nonuniform
>> probability-distribution. But he didn’t say why he thinks so, I
>> don’t know what that probability-distribution has to do with
>> anything said in the derivation.
>>
>> …
>>
>> The identric-mean has been much discussed by mathematicians.
>> But,from what was said in an academic paper (I’ll cite it below),
>> it wasn’t proposed as the round-up point for an unbiased
>> divisor-method before I proposed it here in 2006.There were two
>> academic journal-papers about that proposa, in versions starting
>> in 2008.
>>
>> …
>>
>> Here are the two academic-journal references:
>>
>> …
>>
>> “The Census and the Second Law: An Entropic Approach to Optimal
>> Apportionment for the U.S. House of Representatives”.
>>
>> …
>>
>> By Andrew E. Charman
>>
>> …
>>
>> It was in _Physics and Society__, or _Journal of Physics and
>> Society_, in 2017.
>>
>> (The latest version of the article was in 2017)
>>
>> …
>>
>> The citation said:
>>
>> …
>>
>> Cite as arXiv.1712.09440v3 [physics.soc.ph <http://physics.soc.ph>]
>>
>> …
>>
>> I don’t know the page or Journal-volume & the issue-numberr, or
>> if that information is encoded in the numbers above.
>>
>> …
>>
>> The other paper was:
>>
>> …
>>
>> “Optimal Congressional Apportionment”
>>
>> …
>>
>> By Robert A. Agnew.
>>
>> …
>>
>> …in The American Mathematical Monthly, for 2008, volume 115,
>> number 4 (April 2008).
>>
>> …
>>
>> Pp 297-303(7 pages)
>>
>>
>> ----
>> Election-Methods mailing list - seehttps://electorama.com/em for list info
>
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