Alternative Vote

Satyadev Nandakumar
Fri Aug 14 21:52:33 2015

In a recent article on the remarkable upsurge of Jeremy Corbyn in the Labour Party in the UK, the following claim appears.

The vote uses something called the Alternative Vote, where voters are asked to rank the four candidates in order of preference. If no one gets an outright majority on the first ballot, then the person with the least votes is eliminated, and everyone who voted for that candidate gets their votes shifted to the next candidate. If there's still no majority, the second-to-last candidate gets eliminated, and his or her votes get distributed to the remaining two candidates, where someone has to have a majority.

The alternative vote system is the following: suppose there are 4 candidates, A, B, C and D. Suppose no one gets 50 % of the votes, and D gets the fewest votes. Then D is expelled from the race, and the votes of D's supporters will go to their second preferences. So now, we are left with A, B, and C. Suppose there is still no one with 50 % of the votes, and C has the fewest.

C's voters who had C as their first choice will now have their votes added to A and B, according to whom among these they prefer more. (This may be their second or third choice, depending on whether they prefer D to A and B.) Some of C's voters may have had C as their second choice, since they voted for D as the first choice. Such voters now contribute to A or B, according to who their third choice is.

The article seems to claim that this process always produces a person with a clear majority. I think this is false.

A 3 candidate scenario.

We need to produce the following tie:

A  B

5  5

Let's say C contributed 1 each to A and B when it left the race. So the original scenario might have been,

A  B  C
4  4  2

We now need to produce a set of ballots which will result in the above scenario. Here's one:

   A    B     C
1  1
2  1
3  1
4  1
5       1
6       1 
7       1
8       1
9  2    3     1
10 3    2     1

When C disappears, 9 contributes to A and 10 contributes to B. Thus ties can result when there are only 3 candidates.

A 4 candidate case

It is natural to try to construct a 4 candidate case such that the previous scenario ensues when the 4th candidate exits the fray.

A B C D
3 4 2 1

becomes

A B C
4 4 2

which must become

A B
5 5

This is achieved in the following ballot.

   A   B   C   D
1  1
2  1
3  1
4      1
5      1
6      1
7      1
8  2   3   1   4
9  3   2   1   4
10 2           1

I think this can be generalized further to any number of candidates.

Random Ballots

Such tailor-made ballot scenarios may be unlikely. Will a random ballot necessarily produce a winner?

Political Preferences

Realistic ballots may behave in a very correlated manner. For example, in an election closely contested among two candidates, their supporters might rather give their second preference to a third "compromise" candidate than to the main rival. How does this affect the chances of an eventual winner?