# [EM] Sainte-Lague vs d'Hondt for party list PR

Juho Laatu juho4880 at yahoo.co.uk
Tue Jul 10 00:14:57 PDT 2012

```On 2.7.2012, at 13.58, Raph Frank wrote:

> For example, 26 parties at 1.5% and one party at 61% for a 49 seat parliament would split the seats, 20 for the large party and 29 for split between the micro parties.  The micro parties get 59% of the seats for 39% of the vote.

I only now checked the numbers in this Webster / Sainte-Laguë example. It seems that even with first divisor 1 the largest party would get 23 seats. The example is still valid, just the numbers seem to be inaccurate.

Divisor methods are based on a fixed seat allocation order (based on the number of votes of each party) that does not depend on the number of seats. This means that if there are N small parties of equal size and one large party, there will be some fluctuation in the results when the number of seats grows. There is some number of seats (S1) that will all go to the large party, and next N seats will all go to the small parties (assuming no other ties than those between the equal size small parties). Size S1 thus favours the large party, and size S1+N favours the small parties. This is what I called fluctuation above.

In the given example S1 = 20 (all seats to the large party) and S1+N = 20+26 = 46 (one seat to every small party, still 20 to the large one). In the S1 seats case the large party gets 100% of the seats with 61% of the votes. In the S1+N seats case the large party gets 43.48% of the seats with 61% of the votes. Or in other words, all 20 seats with only 12.2 quotas (7.8 extra seats), or only 20 seats with 28.06 quotas (8.06 seats too little).

In real life party sizes usually vary more, and as a result the fluctuation is not as radical as in this kind of extreme examples. Modified first divisors can be used to eliminate strategic splitiing of parties. Use of the second divisor for strategic purposes is more difficult than using the first divisor, so there may be no need to modify the second divisor, although there is similar (but proportionally smaller) fluctuation also around the second (S2 = 87), third (S3 = 154) and later seats.

Juho

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