I'd say that the Droop Quota isn't very good when the quota is so small.

I agree, I'm not allocating seats in a real life election so this is just purely theoretical.

A more realistic quota for most elections would be in the thousands, so rounding up to the next integer would be a very small percentage difference. I think this is largely where the problem stems from.

So that's the core reason, makes sense to me.

But if we're going with 4 as the quota anyway, I think we're just minimising the amount over quota they go. So we have the following seats "owed" to each party:

That's pretty smart. This is the algorithm as I understand it:

If there are still seats remaining to elect, award seats to the party with the smallest difference between seats won and "party quota" (seats_won - votes / quota). Repeat until all seats filled

(For simplicity I'm breaking ties in favour of the first party)

So my example would start from the automatic seats [5, 6, 12]. Award seats based on the largest remainders method as usual, stopping when all parties has been awarded a seat: [6, 6, 12], [6, 6, 13], [6, 7, 13].

The remaining 4 seats would be filled like this

[7, 7, 13], [7, 7, 14], [7, 8, 14], [8, 8, 14]

Thank you very much for the help!

Hello,

I've found a weird edge case in using the Droop quota for the largest remainders methods in proportional representation. Consider the following election results for 100 voters and 30 seats:

The droop quota is 1 + (100 / (1 + 30)) = 4.23. There are various flooring strategies, but they all should yield 4.

A total of 23 seats are automatically allocated:

There are still 7 seats remaining, but there are only 3 parties. It doesn't matter what the remainders are, there are only 3 remainders.

The examples on Wikipedia assumes that the remaining number of seats are less than the number of parties. This election does works if there are 20 seats total:

The quota is 1 + (100 / (1 + 20)) = 5.76 ~= 5. 19 seats are automatically allocated, the largest remainder is party A with 0.6, hence the result is A: 5, B: 5, C : 10 seats.

My question is, what is supposed to happen in this situation? Will there just be unfilled seats in parliament? How can this be fixed or mitigated? Did I make any mistakes above? Thank you in advance

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

@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!

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:

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:

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:

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.