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

[Warren Smith]

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