# Bucklin election

The Bucklin election is a voting procedure from the family of preferential elections , with which a single winner is determined.

The voters rank the candidates according to their preferences. If a candidate has an absolute majority of the first preferences, he is elected. Otherwise, the second preferences of all voters are added to the first preferences. If a candidate has now achieved an absolute majority, he is the winner. Otherwise, a further ranking of preferences is evaluated until a candidate reaches an absolute majority.

## Description of the voting method

The Bucklin election can be described as the iterative application of the following two steps to each rank (starting with the 1st and then ascending) of the order of preference:

1. Counting : The number of voters who placed him on the current rank is added to each candidate's previous votes.
2. Victory condition : If a candidate has a vote from the absolute majority of voters, the process is terminated. The winner is the candidate with the most votes.

It should be noted that several candidates can achieve an absolute majority at the same time, since the absolute majority basically relates to the number of voters, whereas the number of votes counted increases with each further iteration. If there are several candidates with an absolute majority, the candidate with the higher number of votes wins.

### extension

In the Bucklin election it is firmly specified that exactly one candidate per “round” is included in the count from each ballot paper. The voter must therefore state a strict total order of the candidates on his ballot paper.

The Bucklin election can be generalized by leaving it up to the voter to decide which candidates are included in the count in which round, i.e. allowing the voter to assign the same rank to several candidates and also to omit ranks completely.

This generalization of the Bucklin election is called Majority Judgment .

## example

Consider an election with four candidates A, B, C and D and the following preferences of the 14 voters:

number 2 3 4th 5
preferences 1. A 1. B 1. C 1. D
2. D 2. C 2 B 2. A.
3. C 3. D 3. A. 3. B.
4. B. 4. A. 4. D 4. C

With the first votes the following result results:

1 round
A. 2 2
B. 3 3
C. 4th 4th
D. 5 5

Since no candidate has achieved an absolute majority (8 votes), the voters' secondary preferences are added:

2nd round
candidate Rank 1 Rank 2 Overall votes
A. 2 5 7th
B. 3 4th 7th
C. 4th 3 7th
D. 5 2 7th

Still no candidate has the votes of an absolute majority of voters (8 votes), so the third-party preferences of the voters are evaluated:

3rd round
candidate Rank 1 Rank 2 Rank 3 Overall votes
A. 2 5 4th 11
B. 3 4th 5 12
C. 4th 3 2 9
D. 5 2 3 10

Now a candidate has an absolute majority of votes and the process breaks off. In fact, all four candidates passed the absolute majority at the same time. The winner is candidate B because he has the highest voting value at this point.

## properties

In social choice theory, there are a few criteria for determining the quality of an electoral system, among which the Bucklin election ranks as follows:

The Bucklin election fulfills the majority criterion, the mutual majority criterion and the monotony criterion.

The Bucklin choice violates the Condorcet criterion, the independence of clone alternatives, the later-no-harm criterion, the participation criterion, the consistency criterion, the reversal symmetry criterion, the Condorcet loser criterion and the independence from irrelevant alternatives .