Understanding the First Past the Post “Seat Bonus”

Why is it that in some countries, the popular vote seems so out of whack with seats in parliament?

In a first past the post parliamentary election, there is usually a discrepancy between the popular vote and total seats won. Some have suggested that the high gap between these two in Singapore (in 2011, 60% of the popular vote translated to 93% of the seats) suggests some kind of manipulation or wrongdoing.

It doesn’t. It is purely a function of two factors:

• uniformity of voters and
• total average votes for the winning party.

Uniformity of voters means that there is little difference between the way people vote in one constituency to another. The range of results is narrow. So if in one place you see a 60% vote for the Cat Party, in another you would likely to see a range of 50-70% voting for the Cat Party and not numbers <20% or higher than 80%.

With a more diverse electorate, you may find 95% of the people vote for the Dog Party in one place and 80% voting for the Cat Party in another.

In an extreme case of voter uniformity (every single constituency votes the same), it would be possible for the ruling party to get 100% of the seats with just 51% of the popular vote. IE perfect uniformity = WINNER TAKES ALL.

Some Examples
Here are 4 hypothetical scenarios

1. High voter uniformity (8% std dev) & 55% popular vote
2. High voter uniformity (8% std dev) & 65% popular vote
3. Low voter uniformity (15% std dev) & 55% popular vote
4. Low  voter uniformity (15% std dev) & 65% popular vote

Based on normally distributed numbers, here is roughly the kind of outcomes you would see with these scenarios:

You can see from that table how a high standard deviation results in a wider min-max range than the low standard deviation.

The “bump” is the difference between the popular vote and the seats won.

Doing a cross of average votes and standard deviation, we can see how this advantage is affected.

Impact:

• The more uniform the electorate (ie low std dev), the higher the bump.
• The higher the average votes, the higher the bump.

Another way to look at it is that when there is high voter uniformity, more people are disappointed by the election outcome because their candidate did not get in.

Illustration

The graph below shows the distribution of the popular vote for countries 1 & 3. They both had an average of 55% carrying the winning party however one had a higher variation of results from one constituency to another.

This is illustrated as a wider min-max margin which visually looks like a steeper slope (as opposed to a flatter line for low variation).

The purple area represents the difference in voters “disappointed” between low std deviation and high std deviation countries. There are less people disappointed when there is a steeper slope ie higher standard deviation.

Singapore & The World

Singapore is a small country only 50km wide. It also does not have significant pockets of poverty or cultural ghettos. Voting across the country is fairly uniform. The standard deviation of votes for the ruling party from one constituency to another was 8% in 2011 and 9% in 2015.

By contrast, in 2011, Canada had an election and the Progressive Conservatives won with 40% of the popular vote. Per constituency, their vote ranged from 3.5% – 84% with a standard deviation of 19%. And in May this year, the UK went to the polls and the Conservatives won with 50.8% of the popular vote. Their vote ranged from 4.4% – 62.3% per constituency with a standard deviation of 14.5%.

These countries have both higher diversity and lower average votes carrying the ruling party. For this reason, there is a much lower discrepancy between the popular vote and representation of opposition in parliament.