Proportional representation with droop quota and quota rule

akazukin5151

Hello,

I was running property-based tests and it seems to have found a situation where the Droop quota violated the quota rule. As a largest remainder method, I thought the Droop quota will always satisfy the quota rule. I don't think it is my code being buggy because I reached the same results when calculating by hand. You can also use this unrelated calculator and it gives the same results. In fact, its code uses my algorithm. I found another project which I didn't manage to compile, but the code should also give the same results.

I am aware of the mathematical proofs behind the method and the quota rule, so I hope all three of us made a mistake!

Consider this election with 4 parties contesting 360 seats, with the following vote counts

p1, p2, p3, p4 = 885292, 50089, 1536, 87859
house_size = 360

sum == 1024776

Calculate the droop quota

quota = 1 + floor(sum / (1 + house_size))
quota == 2839

Divide each party's votes by the quota

p1 / quota = 311.83233532934133
p2 / quota = 17.643184219795703
p3 / quota = 0.5410355759070095
p4 / quota = 30.94716449454033

Give each party their automatic seats based on the floor of the above:

party	seats
1	311
2	17
3	0
4	30
total	358

There are 2 seats remaining, so give one seat to the two parties with the largest remainder - they are party #3 (0.94716449454033) and party #1 (0.83233532934133)

The election result is thus:

party	seats
1	312
2	17
3	0
4	31
total	360

The problem with this result is that party #1's seats violated the quota rule. The lower and upper quotas are the vote percentages times the house size:

expr	result	lower quota	upper quota
p1 / sum * house_size	310.99978922223	310	311
p2 / sum * house_size	17.596079533478534	17	18
p3 / sum * house_size	0.5395910911262558	0	1
p4 / sum * house_size	30.86454015316518	30	31

Party #1 received 312 seats, which violates the quota rule. The additional seat could be awarded to party #2 or #3 without violating the quota rule. Notably, party #1 was automatically allocated its upper quota, and gained another one due to its large remainder.

I'm following the rules on the Wikipedia page for the largest remainder method -- as do the two other implementations. You can see my code here. Please do leave a comment if you have an insight. Thank you.

PS: I'd post this to the proportional representation category but there was no new topic button

Edit: here are some other parameters that appear to violate quota:

house_size = 72, p1 = 80183, p2 = 34027, p3 = 403586, p4 = 30472
house_size = 144, p1 = 80183, p2 = 34027, p3 = 803270, p4 = 7888
house_size = 216, p1 = 35570, p2 = 24675, p3 = 798357, p4 = 30291
house_size = 288, p1 = 80183, p2 = 34027, p3 = 705602, p4 = 23398

They all appear to share some common characteristics: the largest party has over 80% of the vote; their vote share has a large remainder, and the floor of their vote share is their upper quota. In other words, to construct this scenario, find p such that floor(p) == upper_quota(p) and remainder(p) is one of the largest compared to the other parties.

Andy Dienes

@akazukin5151 said in Proportional representation with droop quota and quota rule:

The lower and upper quotas are the vote percentages times the house size:

If you are using Droop quotas to compute the seats then you must also use the Droop quota to compute the upper/lower quotas. What you have done is use the Hare quota to define the upper/lower quotas but then use the Droop quota to fill the seats.

Toby Pereira

Also Droop Quota isn't a method. You can have e.g. largest remainder or highest averages methods that use the Droop quota.

akazukin5151

@andy-dienes Oh, that makes sense. How do I use the droop quota to calculate the lower and upper quota though? I can sort of see the similarities between p1 / sum and the hare quota (total_votes/total_seats). But if I substitute p1 into total_votes and sum (of all votes) into total_seats for droop, I get 1 + floor(p1 / (1 + sum)), which will always be greater than 1. Even if the 1+ and floor is dropped, I still get 311 for party #1's upper quota. Thank you so much for your help!

Andy Dienes

@akazukin5151

floor( vote / droop_quota ) is the lower quota, ceil( votes / droop_quota ) is the upper quota.