D'Hondt method: Difference between revisions
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The reason D'Hondt guarantees every party at least as many seats as they have more voters than that number of HB quotas is because there is always 1 HB quota more than the number of seats, so when a party has more voters than k HB quotas, it can divide its votes to do more than a quota per seat for k seats, whereas all other parties combined can at most do just under (((total number of seats + 1)-k)/(number of seats - (k - 1))) i.e. the most votes the other parties can have divided by the number of seats they're trying to take. For example, if there are 10 seats to be filled, and one party has over 3 HB quotas, then they can take at least 3 seats because they can do over 1 quota per seat for 3,and for all other parties to take at least (10-(3-1))=8 seats (the minimum required for them to deny the other party 3 seats, since you can't get 3 seats if 8 out of 10 seats are already allotted), the other parties can do at most just under ((10+1)-3)/8)=1 quota per seat for 8 seats.
The divisor in D'Hondt will always be equal to or smaller than a [[Hare quota]], because that is the largest divisor possible such that there are only as many winners as seats to be filled.
One easy way to do D'Hondt is to compare all parties except the party with the fewest votes (the last-place party) with the last-place party; if all other parties can split more votes per seat than the last-place party has for even one seat such that all other parties would be able to win a combined number of seats equal to or greater than the number of seats to be filled, then the last-place party can be eliminated, and this procedure repeated, to find a minimum number of seats each party must win in D'Hondt. Example: Suppose there are 8 seats to be filled, and 4 parties, A through D, with the votes being (in descending order) A: 10, B: 8, C: 3, and D: 2. Start by dividing every party's votes by just over the last-place party's (D's) vote total (just over 2). Each party can put more than 2 votes per seat for this number of seats: A: 4 (10/2 = 5, which moved down to the next-closest integer is 4), B: 3 (8/2 = 4, shifted down = 3), and C: 1 (3/2 = 1.5 shifted down = 1). In total, these parties have 4 + 3 + 1 seats; this is the number of seats desired, therefore, this is the final result.<br />
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