# [EM] Apportionment (biased?) let me add some more confusion to the mix :)

Joseph Malkevitch malkevitch at york.cuny.edu
Sun Dec 10 15:05:50 PST 2006

```Dear Election List,

If you look at the technical papers on apportionment, especially
Balinski and Young's work, you will see that one of the most
difficult problems to address is the issue of "ties." Ties not only
occur when two states have identical population but also when you
apply the divisor interpretation of the the "divisor" methods.
Loosely speaking the reason why divisor methods violate not giving a
state the integer above or below its exact quota (when its exact
number of seats that should be given to it is not an integer) is that
states of about the same population (not to mention exactly equal
population) should be treated equally from the pairwise equity point
of view. What this means is that Webster, say, will give the same
number of seats to states with approximately the same population even
if this means being overly generous or ungenerous to other states.
Thus, loosely speaking, these methods try to treat equally situated
states in an equivalent manner even if this means that "quota" (as
defined above) gets violated. One way to try to get around this is to
use discrete optimization methods that optimize some global criterion
rather than look at pairwise equity.

Note that Balinski and Young have a "technical meaning" for the word
"bias."

Regards,

Joe

On Dec 10, 2006, at 5:43 PM, Warren Smith wrote:

>
> Actually, I claim EVERY apportionment method so far discussed is
> biased,
> in the sense it will, under the right circumstances, systematically
> always-down-round
> one class of states and always-up-round the other.  (Just make the
> small states
> all have exactly the right sizes and the large states all have the
> right
> sizes, and voila, this'll happen.  You can make pretty much all the
> methods prefer larger or prefer smaller states, at your whim, by
> setting up
> the populations right in your contrived scenario.)
>
> Is there a way to get around that?  Yes:  "randomized rounding."
>
> The idea would be you use
> a random number generator as part of the input into your decision
> to round
> a state up or down, and in such a way the expected net gain, was zero.
>
> Example: 5.3  -->  5 with probability 0.7 and   --> 6 with
> probability 0.3.
> (That is for absolute unbiasedness.  Also important is ratio-
> unbiasedness,
> which you can also assure by the same kind of method.)
>
> OK, so, here is a possible such method: do this kind of rounding.
> If the total number of congressmen comes out wrong,
> then try again, and keep trying until it comes out right.  The end.
>
> This method seems totally unbiased.
> (Incidentally, the same idea was suggested in the 1980s for rounding
> floating point numbers inside computers.
> Biases can build up and result in large errors, and randomized
> rounding prevents that.
> This is a good idea but no computer hardware I know of implements it.
> The "round to even" approach is often used, which tries to get
> unbiasedness
> but isn't perfect.)
>
> Only problem with it is, it is randomized!!
>
> Warren D Smith
> http://rangevoting.org
>
> ----
> election-methods mailing list - see http://electorama.com/em for
> list info

------------------------------------------------
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

Phone: 718-262-2551 (Voicemail available)

My new email is:

malkevitch at york.cuny.edu

web page:

http://www.york.cuny.edu/~malk

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