# D'Hondt method

## Contents

#### The method

The method for allocating seats in proportion to the list-vote share is like an auction, but one in which the bids are rigged. It's only a notional auction, of course: it all happens on computers, with no people involved, but an auction is a good way of describing it.

It works like this:

• The seats are bid for successively: all parties bid for the first seat, and one party wins that seat, then all parties bid for the second seat, and so on.
• For each seat, a party's bid is calculated by dividing its votes by the number of seats it now has plus one:
• On the first bid it bids all its votes. (It has no seats, and zero plus 1 = 1, and votes divided by 1 = votes.)
• When it wins its first seat, it can bid only half its votes for the next seat; and when it wins its second seat it can bid only one-third of its votes for the next seat, and so on.

#### Example 1

SeatNo Party VotesLeft CnstySeats ListSeats SeatsPlusOne Bid
1 LargeParty 6000 0 0 1 6000 WIN
1 SmallParty 4000 0 0 1 4000
2 LargeParty 6000 0 1 2 3000
2 SmallParty 4000 0 0 1 4000 WIN
3 LargeParty 6000 0 1 2 3000 WIN
3 SmallParty 4000 0 1 2 2000

The example above shows a region with three seats, two parties and 10,000 voters. The threshhold for the third seat is therefore 3,333 votes. The Small Party has more than that - it has 4000 votes. (The example leaves out constituency seats, which affect the result, but not the method.)

• At the first seat both parties bid all their votes, and the Large Party wins that seat.
• At the second seat, the Small Party's votes are more than one-half of the Large Party's votes, so it wins the seat.
• At the third seat, one-half of the Small Party's votes is less than one-half of the Large Party's votes, so the Large Party wins the seat.
• So the 60%-40% vote-shares produce 2:1 seat-shares, which is as closely proportional as can be achieved.

#### Example 2

SeatNo Party VotesLeft CnstySeats ListSeats SeatsPlusOne Bid
1 LargeParty 8000 0 0 1 8000 WIN
1 SmallParty 2000 0 0 1 2000
2 LargeParty 8000 0 1 2 4000 WIN
2 SmallParty 2000 0 0 1 2000
3 LargeParty 8000 0 2 3 2667 WIN
3 SmallParty 2000 0 0 1 2000

In this second example, the votes-shares are now 80%-20%, so the Small Party's vote is less than the 3,333-vote threshold, and the resulting seat-shares are 3:0. The vote-shares are 80%-20%, but the seat-shares are 100%-0%, because matching seat-shares to vote-shares is arithmetically impossible.

#### Constituency seats

Constituency seats are brought in at the first seat, so parties with larger numbers of constituency seats start at their proper disadvantage.