The Voting System Problem

12 Mar

This prevents me from comitting suicide because…….You realise that you don’t have to settle for what we have now.

[Guest writer: DBoi]

Most people put a lot of time and thought into deciding how they will vote in an important election, but do you ever give any thought to the method used to determine the winner once your vote has been cast? Well you should! With a voting referendum coming up soon in the UK, voting systems have become a highly debated topic in recent weeks. But why should you care and what difference will changing the system even make? To answer those questions I thought I would draw your attention to a clever example that demonstrates just how crucial the voting system is. In fact, this example shows that the voting system can be just as important, if not more than, the actual votes themselves…

Suppose 5 candidates, conveniently named Alex, Bartholomew, Chester, Doris and Edwina are up for a position on a committee.  The 9 current members of the committee have to vote on the 5 candidates to choose 1 for the role. Everyone ranks the candidates from their most preferred to least preferred and the table below shows their rankings. So four members ranked the candidates in exactly the same way shown by group 1. Similarly a second group of three members ranked the candidates in the same way and the final two members ranked the candidates in the same way shown by group 3.

  Group 1 Group 2 Group 3
Ranking 4 members 3 members 2 members
1st A B C
2nd E C D
3rd D E E
4th C D B
5th B A A

I will now show, using 5 different voting systems that are in common use around the world, that we can produce 5 different winners based on the same vote.

Method 1 – Simple plurality procedure

This is sometimes known as first-past-the-post and it is simplest and most common method of voting. It is the current system used in the UK general election and is also used in Canada, India and some US election.

Each voter gets one vote and the candidate with the most votes is the winner. In our example, A receives 4 votes, B receives 3 and C receives 2, so Alex is the winner.

Method 2 – The plurality runoff procedure

Also known as the Alternative Vote (AV). This is the method that is proposed the UK changes to in the May referendum. It is currently used to elect: the Australian and Fijian House of Representatives, the leader of the Labour party and Liberal Democrats and most importantly the Oscar for Best Picture.

Each voter gets one vote. If a candidate has more than half the votes they are the winner. Otherwise, the candidate with the least number of votes is removed and another vote is taken between the remaining candidates. This continues until a candidate receives more than half of the votes. In this situation the first three rounds provide the same result and there is no winner. Candidates D and E are removed after the first two round and C is removed after the third. In round 4 everyone in group 1 and 2 will again vote for A and B respectively. The voters in group 3 will vote for B as it is higher than A in their ranking. So Bartholomew now has 5 votes and is the winner.

Method 3 – Pairwise Comparison

This is probably a fairer voting system than the previous two, in my opinion, but it is not widely used by any national government. The main reason for this is that it doesn’t always produce a winner or can even produce everyone as the winner! However, it is used by several major private organizations.

In this method we consider pairs of candidates being placed in a vote against each other. Every candidate goes in a vote against every other candidate. The winner is the candidate that beats all other candidates in the pairwise votes. For example E will beat A because A will have 4 votes from group 1 but E will have 5 votes from groups 2 and 3. We can see that Chester is the winner in this example. He beats A by 5 votes to 4, B by 6 votes to 3, D by 5 votes to 4 and E by 5 votes to 4.

Method 4 – Borda Count

The Borda count is named after the 18th-century French mathematician and political scientist Jean-Charles de Borda, who devised the system in 1770. It is currently used for the elections in Kiribati and  Nauru, which are apparently countries, but is more widely used throughout the world by various private organizations and competitions. This is the method used to determine the MVP in baseball, the winner of the Heisman trophy in college football and the Eurovision song contest winner.

This method gives points to candidates based how they are ranked by the voters. In our example with 5 candidates, each voter’s 1st choice is given 4 points, their second 3 points and so on. And so their last place candidate receives ‘nil poi’. Tallying up the points we can see that A gets 16 points, B gets 14, C gets 21, D gets 17 and E gets 22. So Edwina is the winner.

Method 5 – Approval Voting

This is a relatively new method of devising a voting system. It is not known to be used by government but it was used by the UN to elect Ban Ki-Moon as their Secretary General so it must be pretty good!

Each voter can vote for as many candidates as they want. You can give each candidate either 1 or 0 votes. The candidate with the largest number of votes is the winner. We would need to know more about each individual to know how they would vote using this system but if we assume that everyone in group 1 votes for their top 3 candidates and everyone in groups 2 and 3 votes for their top 2 candidates this wouldn’t be too unreasonable an assumption. Counting up the votes we see that Doris is the winner with 6 votes.

So we’ve got 5 different systems and 5 different winners! Which is the best, it could be argued, is entirely subjective, and often depends on the situation. You can have a look at the votes again and see who you think should get elected and which system achieves this (personally I’d plump for Chester). A lot of media coverage in the UK has been devoted to the political consequences of the referendum with very little explanation of the mathematics behind them. You have seen now, how several legitimate voting systems can provide completely different results for the same vote and I hope I have highlighted just how important the voting system is as well as shedding some light onto the workings of the current system and proposed new one.

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Posted by on March 12, 2011 in DBoi, High Brow


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6 responses to “The Voting System Problem

  1. Pep

    March 13, 2011 at 12:50 PM

    daaayum! maths boy can write! surprisingly interesting and I like the witty comments. great post DBoi!

    • Pep

      March 13, 2011 at 12:52 PM

      also you can delete this comment but there might be word missing in line 4 of “approval voting”.

      “We would need to (KNOW) more…” ????

      sort dat out boi!

  2. Clay Shentrup

    March 14, 2011 at 10:48 PM

    An even better method is Score Voting (aka Range Voting). Approval Voting is mathematically just Score Voting on a 0 to 1 “range” of scores.

    The most extensive Bayesian Regret calculations show Score Voting to be robustly superior to all politically viable alternatives from Condorcet to Borda to Instant Runoff Voting. Approval Voting is second best, and extremely simple and politically viable.

  3. Clay Shentrup

    March 14, 2011 at 10:48 PM

    Woops, here’s a link to those Bayesian Regret figures:


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