The magic number: What’s with the fixation on 40% for a majority?

A common trope from the Canadian commentariat is that 40% national support should result in a majority government (see here from Nanos Research).  I’ve always been skeptical whenever a journalist trots out "rules" like this since they often ignore the complexities and subtleties of our system.

But 40% does seem roughly right: the last two majority governments received just above 39% of the vote. 

To test this, I've taken all the elections since 1945 to see the relationship between the popular vote received by each party and the proportion of seats they won in the House of Commons:

Each dot in this plot represents a single party's outcome in an election (e.g., in 2015, the Conservative Party won 32% of the national vote and got 29% of the seats in the HoC).  This plot shows that the relationship between the national vote and seats in the HoC is convex: smaller parties are under-represented, while larger ones are over-represented.  This attribute makes it easier to form majority governments since a party can win 50% of the seats with fewer than 50% of the votes. 

Looking at the data and fitting a quadratic line** shows that when a party gets 40% of the national vote, they should expect to get roughly 50% of the seats in the House of Commons.  The "40% for a majority" rule-of-thumb is pretty good! 

But there is another interesting pattern in the data: the relationship is quite clearly convex. That is, if you get below 30% of the national vote, you should expect to be "under-represented" in the House of Commons.  If you score over 30% of the national vote, then you should expect to get a disproportionate number of the seats in the House of Commons.  (This is shown on the chart as being above or below the 45 degree line - if you are on the 45 degree line, your national vote percent and representation in the HoC are the same)

This demonstrates that larger parties have an advantage in the FPTP system: two smaller parties that each get 20% of the national vote would expect to elect fewer MPs than a united party that gets 40% of the vote.  This provides a strong incentive for parties to unite and create broad coalitions (instead of smaller fringe parties).

Another implication is that relatively small differences in vote can result in large differences in the House of Commons.  Consider the current situation in Canada: the Liberals got just shy of 40% of the vote and received 54% of the seats in the House.  The opposition Conservatives got 32% of the vote and 29% of the seats.  An eight percentage point difference in vote results in a 25 percentage point gap in representation in the House of Commons. 

Put another way, changes in the House of Commons should be much more volatile than changes to the national vote.  This is very clear in the data, but that will be covered in a future post.

** An older version of this analysis on this site used a Loess regression.  I couldn't be bothered running the code again so just did this analysis in Excel using a polynomial regression.  The result is very similar and the conclusion doesn't change.