[EM] a question about apportionment (was: basic fairness question)

[robert bristow-johnson]

On Apr 15, 2011, at 8:59 PM, ⸘Ŭalabio‽ wrote:

> 	“Owen Dalby” <Owen.Dalby at Gmail.Com>:
>
>> 	I apologize if I am asking a dumb question, but would appreciate  
>> any honest and practical advice from this list. I am conducting an  
>> election among a group of colleagues who are all graduates of a  
>> fellowship program. 45 people will vote on perhaps 30 candidates  
>> for roughly 15 seats.
>
> 	If these people are not paid, thus it costs group nothing for their  
> services, just let everyone wanting to have a seat.  If money is an  
> issue, one should think about the ideal size of the legislature:
>
> 	If the legislature has 1 tier, then the size of it should be the  
> squareroot of the electorate.  The squreroot of 45 is:
>
> 	7


how well does this work for a large population?  if that rule applied  
to the U.S., we would have about 17,500 in the House of  
Representatives in Congress.  might be a little unwieldy.

would there be some threshold where this rule changes to one where the  
legislature is a smaller portion?  i s'pose, if i think about it, i  
can think up an asymptotic function that approximates sqrt(N) for  
small N and k*N for large N where k is the constant of proportionality.

____


this is interesting.  in the U.S., the number of Representatives is  
435, no matter what the population is, but they have to be allocated  
proportionally in some sense of the word.  this is called  
"apportionment" and is done to decide how many Representatives go to  
each state.  once that is done, states that are apportioned more than  
one representative have to go through a another slugfest (which is  
more local and more subjective) in how that state is Redistricted.

now that we have had a new census in the U.S., they're gonna do that  
again.  10 years ago (after the 2000 census), there was some dispute  
about the method because North Carolina felt they got screwed of  
receiving the last apportioned Representative, which barely went to  
Utah.  (there is a geographic shift of the center of mass of the  
population in the U.S toward the west. there was a sorta big deal or  
milestone when this population centroid moved from east of the  
Mississippi to west of it.)

anyway, the rule they currently use is this Huntington-Hill method.   
and i've been wondering exactly how the mathematics work in deriving  
it.  as anonymous IP 96.237.148.44, i posted on 14 February 2009 (UTC)  
a reasonably clear mathematical posing of the same question to:

     http://en.wikipedia.org/wiki/Talk:Huntington%E2%80%93Hill_method

using Wikipedia's LaTeX markup.  so it's clearer there than here.   
all's i can say is that the conceptually simplest method of  
proportional allocation, with a fixed number of Representatives and  
that guarantees each state a minimum of one Representative is:

    Number of Reps for State k:  N(k) = ceil( q*P(k) )

where

    SUM{ N(k) } = N
     k

and

    SUM{ P(k) } = P
     k

and where N is the total number of representatives (now 435 in the  
U.S.) and P is the total population of the electorate (or of those  
represented, in this case we're counting non-voters) and P(k) is the  
population of each state.

q starts out as arbitrarily low (so the SUM{N(k)} is less than N) and  
is monotonically increased until SUM{N(k)} is equal to N.

now, this just seems to me to be the most consistent rule that is  
applied to every state that makes it proportional (to an integer  
number of representatives for each state) and makes certain each state  
gets at least one representative.

can someone explain to me how the Huntington-Hill method is better.   
in this section:

   http://en.wikipedia.org/wiki/United_States_congressional_apportionment#The_Method_of_Equal_Proportions

they say it's because "the method ... minimizes the percentage  
differences in the size of the congressional districts."  can someone  
point me to a proof of that?  i asked the same question in that  
article's Talk page and have been unsatisfied with the responses.

can anyone here (like Warren) spell this out?  what so good about  
sqrt(n*(n+1)) instead of just n?

thanx,

--

r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."