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

[Juho]

Yes, random allocations are yet another way to balance the situation.  
Since the population of a state changes quite slowly some fixes  
(random, time division, voting power) may be wanted to reduce the  
risk of continuously getting less seats than the population would  
give right to. Changes in the support of political parties are maybe  
faster and therefore do not need similar fixes as badly.

One more tool that can be useful in some situations is the  
hierarchical structure of the states/parties. To guarantee that  
certain set of states/parties will not be underrepresented they could  
form a team/alliance. When seats are allocated to that team they  
could lose (in typical allocation methods) only one seat to rounding  
errors instead on many of them losing a seat. Geographic alliances  
would maybe be more natural than e.g. an alliance of small states.

I already mentioned the different voting power. A simple method in  
that direction would be to elect one representative from every state  
and give her voting power in relation to the number of people she  
represents. Or maybe large states would be given n seats with 1/n of  
the voting power of the state etc. Maybe the building where these  
representatives will work has a fixed number of physical seats =>  
fill those seats and allocate voting power according to that.

Juho Laatu


On Dec 11, 2006, at 0:43 , 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
>
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