How do the Liberals manage to win elections but lose the popular vote?

Canada's "election that changed nothing" is over, and the new government is almost identical to the old government. Failed cynical power grab? Maybe. Waste of $600 million. Sure. Overwhelming mandate for the Liberals? Not really.

The Liberals ended up with a strongish minority of 159 seats in a 338-seat parliament, or 47% of the available seats, just shy of the 50% majority position. But they got to that with just 32.6% of the popular vote. The second place Conservatives gained 119 seats (35%) with 33.7% of the popular vote. The NDP fared even worse, winning just 7% of the seats with 18% of the popular vote.

Sure, this is just a function of the first-past-the-post electoral system we are stuck with here. Plus, the share of the popular vote won by the winning party - whether Liberal or Conservative - has been on a downward trajectory since about 1970, as has voter turnout, giving the 2021 Liberals the weakest mandate in Canadian electoral history. There are now SIX major parties in Canadian politics, not just the two of a hundred years ago.

But why is it that the Liberals seem to be able to form a government with only a third of the national vote, while the Conservatives perennially win the popular vote but are excluded from power? 

The Tories, of course, are convinced that it's all due to the nefarious dealings of the Liberals, and that  they have been unfairly treated by the system. The NDP and the Greens (and even the ultra-right PPC), arguably, HAVE been unfairly treated by the system, so it's no surprise that they are strongly pushing a proportional representation electoral system (an idea that the Liberals, who once professed to be in favour of it, have mysteriously dropped since they have been in power).

But the Liberals do not come by their seats by nefarious means. This is not a case of barefaced gerrymandering, such as plagues American elections. The fact is that most of the Conservatives' support is in Western Canada, specifically Alberta and Saskatchewan, and they win some Western ridings with over 80% of the vote. So, much of their Western support is "wasted", being far in excess of what they need to win the seats. The Liberals, on the other hand, have a much higher "vote efficiency", and so are able to win more seats with fewer votes, mainly because of their geographical distribution.

So, like them or like them not, the Liberals come by their election victories fairly. If, that is, you consider the first-past-the-post system to be fair...