[EM] Request comments on MMP?

[Olli Salmi]

At 00:11 +0100 6.8.2003, James Gilmour wrote:
>  > At 09:26 +0100 5.8.2003, James Gilmour wrote:
>>  >Different rules can produce different results:
>>  >d'Hondt favours the larger parties, Sante-Lague favours the smaller
>>  >parties, while modified
>>  >Sante-Lague tries to be neutral
>
>Olli replied:
>>  D'Hondt favours large parties, Sainte-Laguë is neutral, modified
>>  Sainte-Laguë makes the first seat more difficult, which favours
>>  larger parties.
>
>Not according to my calculations, using divisors 1, 2, 3, etc for 
>d'Hondt; 0.5, 1.5, 2.5, etc for
>Sainte-Laguë; and 0.7, 1.5, 2.5, etc for modified Sainte-Laguë. 
>Look at these two elections for
>five places with three parties R, S and T.

Sainte-Laguë favours small parties more than d'Hondt, but that 
doesn't mean it's biassed.

>Election 1: R 520 votes, S 340 votes, T 140 votes.
>Seats: d'Hondt R 3, S 2, T 0;  S-L R 2, S 2, T 1;  mod S-L R3, S 2, T 0.

Adams' method is biassed towards small parties. With nine seats 
there's a difference between Sainte-Laguë and Adams, if my 
calculations are right.
Seats: d'Hondt R 5, S 3, T 1;  S-L R 5, S 3, T 1;  Adams R 4, S 3, T 2.

>Election 2: R 520 votes, S 330 votes, T 150 votes.
>Seats: d'Hondt R 3, S 2, T 0;  S-L R 2, S 2, T 1;  mod S-L R 2, S 2, T 1.
>
>Unless I've made a mistake in my arithmetic, I don't think you can 
>say Sainte-Laguë is neutral.
>
>
>>  If we regard both methods as algorithms to find a suitable quota,
>
>I don't find this approach at all helpful.  Arithmetically the 
>result is the same, but the problem
>with the quota approach to the allocation of seats to parties is "so 
>many quotas each, but what do
>we do with the remainders to fill the last seat?"  The advantage of 
>the formula approach is that the
>arithmetic is integer (whole seats) and consistent all the way through.

The arithmetic works because it's the same phenomenon viewed in two 
ways. D'Hondt is easily explained as a highest average method: seats 
are successively given to the party that has the highest average 
number of votes. I haven't understood Sainte-Laguë that way.

On the other hand, if you say that both are methods to find a 
suitable quota, d'Hondt rounding down, Sainte-Laguë rounding off, you 
can compare the methods. If you use a series of divisors, the 
smallest quotient that entitles to election is the quota.

Let's suppose that we have an election where the quota is 100. 
There's a party that polls 150 votes and another one that polls 1050. 
With d'Hondt, the first party gets 1 seat and the second one 10. 
However, the bigger party has needed only 105 votes for one seat 
while the smaller party has needed 150. A smaller party always needs 
more votes per seat.

If we always round up, like in Adams, a party with 1050 votes would 
get 11 seats and and a party with 150 would get 2 seats. This favours 
small parties.

With Sainte-Laguë you use conventional rounding, so it favours larger 
parties if the fraction is rounded down and smaller parties if it is 
rounded up. It doesn't depend on the size of the party what the first 
decimal is, so the effect gets cancelled out in the long run. I have 
always presumed that the advantage in rounding up and rounding down 
are about equal.

This is how I've understood it since I read "Fair Representation: 
Meeting the Ideal of One Man, One Vote" by Michel L. Balinski and H. 
Peyton Young. Unfortunately I didn't understand their mathematics. It 
was somewhat more sophisticated than the explanation above. And 
probably more correct.

Here's one of the more sensible links about this.
http://www.janda.org/c24/Readings/Lijphart/Lijphart.html

Olli Salmi