# [EM] Allocation bias summary

Michael Ossipoff email9648742 at gmail.com
Thu Jul 12 06:22:04 PDT 2012

```I was pointing out some things in a previous posting:

1. Systematically, predictably, favoring or disfavoring some parties,
districts or states is obviously the worst thing that an allocation
method can do. Unbias is the most important consideration in
allocation.

2. That means that the best allocation method would one that finds, by
trial-and-error, the allocation that minimizes the Pearson correlation
between q and s/q.I'll call that "Empirical Unbias", or "Minimum
Correlation".

3. But if you knew that a Weighted-Webster (WW) version used a
completely accurate probability distribution, then you'd know that
there is no systematic favoring of any particular parties, districts
or states, even though, in any particular allocation, the correlation
between q and s/q might be far from minimum.There'd be no bias. But
the "if" condition at the beginning of this paragraph is unattainable.
Hence my conclusion in paragraph #2.

I just want to add that it's a trade-off.

Using allocations found by Minimum Correlation will give unbias more
reliably than Webster or Weighted-Webster, whose unbias depends on an
assumption or guess.But Minimum Correlation (MC) will often give
allocations in which some parties', districts' or states' s/q differ
unnecessarily much from their ideal equal value. More deviation for
s/q, though that deviation is unbiasedly random.

Using allocations found by Webster will minimize, for every party,
district or state, the deviation of its s/q from the ideal equal s/q
value.  Weighted-Webster, because it differs so little from Webster,
will nearly achieve that too.
Both of those 2 methods are unbiased if you assume that their
probability distributions are correct. But, because you know that
those are just guesses or assumptions, you know that those two methods
can't have reliable true unbias. Least deviation, but that deviation
might might (very, very slightly) consistently favor some parties,
districts or states.

In principle, it could be that, in future apportionment discussions,
the states wont accept _any_ bias, especially if the direction of that
bias is known in advance. If so, then Webster would never be accepted
by the small states, because it is known to be very slightly
large-biased. (by up to 1/3 of one percent, according to something
that was quoted from Warren Smith. For comparison, Huntington-Hill,
the apportionment method currently in use here, is biased by up to 5.7
percent).

In principle, it could be that strategically-inclined voters in
Sainte-Lague elections would react even to the smallest amount of
bias, and would vote for larger compromise parties instead of for
their smaller favorites, in order to maximize the effect of their
vote, for forming coalitions in parliament.

If those things were so, then even Webster/Sainte-Lague's 1/3 of one
percent of bias would be too much, and something more unbiased would
be needed.

Maybe WW. With WW, you'd get rid of the predictable, known in advance
(though quite small) bias of Webster, while still keeping, very
nearly, Webster's minimization of the deviation of each state's,
district's or party's s/q from the ideal equal s/q value.

Two disadvantages of WW:

1. Someone might claim to know that its distribution approximation is
off in a certain way, causing bias in a certain direction. There could
always be someone saying that. Of course WW's bias should be
considerably less than the tiny bias of Webster. But, nevertheless, a
claim that there is bias in a known direction could turn against WW
the states that would be  (negligibly) disfavored by that alleged
bias.

2. WW is new, and doesn't have Webster's simplicity.

For those reasons, if Webster were unacceptable, for the reasons
postulated above (in the paragraphs beginning with the words "In
principle..."), and if complete unbias were demanded, then MC would
seem the best choice.

Of course, the starting point for MC's trial-and-error could be an
allocation by Webster or WW.

That's if Webster were unacceptable. But we don't know that it would
be. Hill, which is about 17 times as biased as Webster (based on a WW
formula recently quoted from Warren) is currently in use.By the way,
Hill was adopted, to replace Webster, because it was known that Hill
would take a seat from a large Republican state, and give it to a
small Democrat state. In the Congressional vote to replace Webster
with Hill, all of the Democrats voted in favor of that replacement,
and all of the Republicans voted against it. Actual bias has nothing
to do with it.

Sainte-Lague PR has been in use for a long time, and I haven't heard
of anyone voting for larger parties in order to increase the weight of
their vote, due to Websiter's (miniscule) large-bias.

So it seems to me that Sainte-Lague/Webster isn't having any
problems,due to its tiny large-bias, and isn't going to. So I suggest
that Webster/Sainte-Lague is the best choice for PR and apportionment.

Mike Ossipoff

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