[EM] Joe: Bias

[MIKE OSSIPOFF]

I'd said:

If we consistently and systematically give more seats per person to smaller 
states,

You wrote:

What is your definition of a small state?

I reply:

For apportionment purposes, a low-population state.

I'd said:

that is biased.
That's what Hill does. Hill is biased.

You replied:

How do YOU measure bias?

I reply:

I defined bias as systematic disparity in seats per person. Or, more 
conveniently, seats per population quota.

But I found out that you were right: Defining bias, and applying the 
definition in a thorough, precise way is easier said than done. Trying to 
apply my definition precisely opens a can of worms.

So, for right now, let me just say that prioportionality's definition seems 
much easier to apply: The seat allocations are truly proportional if the 
allocations are a linear function of the states' population quotas, and if 
that function goes through the origin, so that F(0) = 0.

Obviously that's impossible when only whole seats can be given. But Webster, 
and only Webster, is a step function that is symmetrical about the 
1-seat-per-quota line, and stays as close to it as possible. So Webster is 
the most proportional allocation.

And, if unbias means a uniform seats per quota, then, by its symmetry about 
the 1-seat-per-quota line, and by staying as close to that line as possible, 
Webster is more unbiased than other methods.
I'm using that looser meaning of unbias instead of trying to stricly apply 
"bias is a systematic disparity in seats per quota". Anyway, fortunately the 
Constitution speaks of proportionalilty rather than unbias, doesn't it? That 
makes things easier, since it's obvious and undeniable that Webster is the 
most proportional allocation, due to its symmetry about, and closest 
possible proximity to, the 1-seat-per-quota line.

You continued:

Can you provide the data for bias based on your definition of small state 
and bias?

I reply:

Using actual allocation results? I haven't done that. But Balinsky & Young 
did. I haven't seen their book for about 18 years.

You continued:

Furthermore, apportionments can be used for other problems other than the US 
House of Representatives. Does this data show no bias too?

I reply:

PR? I haven't measured actual allocations. I know that B&L showed that 
Webster minimizes unproportionaly, as measured by a sum that they describe. 
But, for the reason I've mentioned, Webster's best-proportionality prize 
seems undeniable.

I tried to download Ernst's paper, but the computer had trouble doing so. I 
want to check it out. Thanks for the reference. I'm not bigoted, and I'm 
curious and interested to hear the defense of Hill's method.

Mike Ossipoff

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