A federal election is held on Monday, September 20, 2021. The outcome of this election is uncertain, as the polls indicate a tight race between Liberals and Conservatives. Canada's first-past-the-post electoral system has more intricate competitions than the United States because many of the races are between three parties rather than two. In this blog I illustrate some of the 3-party dynamics using so-called `ternary diagrams'.

The above diagram shows the vote shares of the Liberal Party and the Conservative Party along with the vote share of the leading third party (New Democratic Party, Bloc Quebecois, or Green Party) for each of Canada's 338 electoral districts (also known as "ridings") in the October 2019 election. The colours of the dots indicate the party (red for the Liberal Party, dark blue for the Conservative Party, light blue for Bloc Quebecois, orange for the NDP, and green for the Green Party). The vote shares increase from 0% to 100% along the three axes as indicated. There are essentially for segments in the diagram. In the left corner are all outcomes where the Liberals received more than 50% of the vote, and in the right corner are all outcomes where the Conservatives received more than 50% of the vote. These are each parties' "safe" ridings, and each of the two parties has their bastions of support—the Liberals more in urban areas, and the Conservatives more in rural areas. The corner at the top has outcomes where a third party received more than 50%: mostly the Bloc Quebecois and a few NDP seats. These three 50%-plus corners are shaded in yellow. The triangle in the middle, shaded in light green, is the competitive area where no party received more than 50% of the vote. This is where elections are competitive. At the bottom of this inner triangle are the two-way competitions between Conservatives and Liberals, while at the top-left corner are competitions between Liberals and third parties. There are fewer direct competitions between Conservatives and a third party (mostly the NDP in urban areas).

It is worthwhile looking at two "battleground" provinces in particular, British Columbia and Quebec. Below are diagrams for electoral districts in these two provinces.

In British Columbia, the majority of electoral districts are inside the light-green battleground zone. The Liberal Party has few safe seats, whereas the Conservative Party has a number of safe seats in rural and suburban districts. The NDP dominates in a number of districts on Vancouver Island and in Metro Vancouver, while the Green Party has one particular strong district (Saanich-Gulf Islands). The single white dot is for an independent candidate. There are a fair number of competitive races in BC, mostly between Liberals and Conservatives.

The diagram for Quebec looks rather different from BC. Liberals, Conservatives, and Bloc Quebecois have numerous "safe" ridings. However, in the top-left corner of the light-green competition zone, there are a fair number of ridings where Liberals and Bloc are close to each other. These are extremely competitive districts where elections are decided. These shifts are very important in determining the overall election outcome.

So how competitive are individual races in the forthcoming election? There are actually 47 electoral districts where the vote gap was less than 5% in 2019. The table shows these races in ascending order of the gap between winner and competitor. Fifteen of these "battleground" districts are in Ontario and Quebec each, eight in British Columbia, four in Nova Scotia, three in New Brunswick, and one each in Manitoba and the Yukon Territory. Small swings can decide these ridings, as well as strategic voting. If you live in any of these 47 districts, your vote may matter more than elsewhere!

Technical Note: There is one small caveat about ternary diagrams. There are a few instances where there are four-way races. However, in a ternary diagram the three shares need to add up to 100%. Thus the vote shares were normalized so that the three parties that are being compared add up to 100%, excluding votes for any fourth or further party in constituencies. This means that the vote shares are not with respect to the total votes in each electoral district, but with respect to the sum of the Liberal Party, Conservative Party, and leading Third Party.