[EM] Derivation of Identric-mean as round-up point for Bias-Free method

[Richard Lung]

Just a preliminary remark. Divisor methods and quotas some times 
distinguished. Thus there is the Droop quota and corresponding D'Hont 
divisor method. Divisor methods regarded as belonging to apportionment, 
as by Jefferson and by Webster, not to carve out party seats, which is 
too restrictive of personal choice.


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)
>
>
> ----
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