# [EM] Simple dfn of proportionality & bias leaves no doubt: Webster wins

MIKE OSSIPOFF nkklrp at hotmail.com
Fri Dec 8 11:25:32 PST 2006

```As Joe pointed out, the transfer property leaves some room for
matter-of-opinion disagreement, at leasts for Hill vs Webster.

But let's go back to the simple definitions of proportionality and bias.

Two variables are proportional if they're related by a linear function.
Unbias means that seats per citizen stays as equal as possible for parties
of all sizes.

Let's graph the allocation methods. Seats as a function of
population-quotas. S(P).

Label the horizontal X axis "population-quotas". It represents the number of
quotas, usings the final quota, the one that gives the desired number of
total seats. Label the vertical Y axis "seats". Mark both axes with number
ticks spaced the same.

Draw a line from the origin, going up at 45 degrees. That's the Y = X line,
representing one seat per quota.

Webster is a step function. In fact all 5 standard methods are step
functions. Of course, having first awarded the 50 auitomatic first seats,
we're allocating the remaining 385 seats. When you graph Webster's step
function, you'll see that it's centered about the Y=X line, as close to it
as is possible with a step function. From zero to 1/2, Webster is level at
zero. Then, at 1/2, it goes up to 1. Likewise for each step.

Webster gives seats as a linear function of the number of population quotas,
as nearly as possible with integer numbers of seats.

Now, graph Hill. Hill rounds up sooner than Webster does. For instance,
rounding at the geometric mean instead of the halfway point. For example
where does Hill round up between 1 and 2? Their geometric mean is around 1.4
It's above the 1 seat per quota line. Hill gives small states more seats
per quota. That's unbias.

That early rounding is only pronounced for small states. The graph of Hill's
allocation isn't as linear, since it starts out more above the Y=X line, and
then approaches it closer for large states.

Likewise, Jefferson, Dean, and Adams will show their bias and
unproportionalilty if you graph them.

It isn't difficult to explain how Congress was snowed by Hill's jargon and
long impressive-looking proofs. The question is: How did Huntington & Hill
make their big error? Maybe they were looking for something complicated and
missed the obvious. Maybe they were partial toward something original that
they could call their own.

Mike Ossipoff

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