[EM] Fwd: Proportional election method needed for the Czech Green party - Council elections

[Peter Zbornik]

---------- Forwarded message ----------
From: Raph Frank <raphfrk at gmail.com>
Date: Tue, Apr 27, 2010 at 2:51 PM
Subject: Re: [EM] Proportional election method needed for the Czech Green
party - Council elections
To: Peter Zbornik <pzbornik at gmail.com>


On Tue, Apr 27, 2010 at 9:18 AM, Peter Zbornik <pzbornik at gmail.com> wrote:
> thanks for your information and the short explanation on STV.
> I was thinking about d'Hondt's method in general.

D'Hondt is equivalent to the Jefferson Method.  It is clearer why that
is proportional.

1) pick an initial divisor
2) divide each party's vote total by the divisor
3) For each party round down to the nearest whole number of seats
4) If the total number of seats is correct, then finish
5) Otherwise, update to a better divisor and repeat (go to 2)

Lots of divisors will give the correct number of seats, but they
always give the same number of seats per party.

So, you take each party's vote total, divide it by a number and then
round downward.  This means that the method is proportional, except
for rounding errors.  The divisor will work out to be around (votes
cast)/(seats).

Sainte-Lague rounds to the nearest whole number rather than rounding
downwards.  This is why Sainte Lague is fairer (though there can be
strategy issues for smaller parties).

Anyway, the process for d'Hondt is equivalent to:

The initial divisor is set equal to the number of votes received by
the largest party.

When you divide all the other parties' totals by this value, they all
give a fraction less than one, so none of the other parties receive
any seats.  The largest party gets 1 seat.  This is the same as
d'Hondt.

When updating the divisor, we reduce it by just enough so that 1
additional seat is assigned.

If party has N seats and V votes, then the divisor must drop below

divisor = V/(N+1)

before it will get the next seat.

So, according to the update rule, we reduce the divisor so that at
most one more party gets a seat.  Therefore, we need to find the party
who gets its next seat at the highest possible divisor.

So, we pick the party with the highest

V/(N+1)

and we set the divisor so that they get 1 more seat.  So, we set the
divisor to slightly below the above number.

This means that the party who has the highest V/(N+1) gets the next
seat in each step.

However, this is exactly what d'Hondt does.  It just doesn't calculate
the divisors at each step.
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