Weighted FPTP
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First, a caveat: I love coming up with new voting systems, but I'm very unconvinced that (in practice) this is at all a good idea.
Long story short, I tend to believe that the theoretical properties of a voting system are conditional on their legitimacy with the population, such that the best voting systems in practice are ones that are 1) familiar, 2) very simple to explain, 3) don't require maths. For real life, I largely approve of approval voting.
I also think that without a very rigorous plan and a willingness to spend your life implementing it, talking about voting systems is a lot like doing a crossword – it's a fun puzzle that vaguely mentions current events, but it has ~no actual connection to politics in the real world. This is plausibly true for most public policy discussions, but might be especially true for voting reform, because voting reform is almost-always against the interests of the current government who were voted in by the current system. There are all kinds of easy improvements that don't get implemented (cough anti-gerrymandering cough cough) because the people who decide about them are exactly the beneficiaries of the current messes. So I don't believe that writing blogposts like this has any chance of changing the world as it actually is.
But for the rest of this piece, I'm going to pretend I don't believe those things, so I can have fun thinking about voting systems.
In short: Weighted FPTP would be a parliamentary voting system where each representative gets votes in proportion to their vote-share in their district. If Anna wins her district with 60% of the vote, she gets 60 votes in parliament; if Billy wins his district with only 42% vote share, he gets 42 votes.
This has some properties:
1) it mildly counterweights gerrymandering. Gerrymandering usually works by putting Our Voters into districts with small majorities, and Their Voters into one district with a huge majority and several others with small minorities, therefore "wasting" a bunch of their votes. Under Weighted FPTP, this effect is lessened (but not removed!) because the huge-majority district delivers more votes to parliament than a small-majority district.
2) at the same time, because it's still FPTP, this system gives disproportionate reward to the winners: a candidate who gets 51% of the vote gets 51 more votes in parliament than the runner-up who got 49%. But this is both good and bad, because truly proportional systems can become dysfunctional when no faction has enough votes to get anything done.
3) it can encourage politicians in safe seats to keep working for their constituents despite knowing they're going to win anyway – even more so if the vote share used is "% of voting-eligible population" rather than "% of people who actually voted."