The Condorcet Method has a huge disadvantage that it is hard to understand.
Being simple to understand is crucial for voting systems and I would argue is the most important factor since an election without public buy-in is worse than worthless.
Understanding is one thing, counting is another. Plurality voting is great because it’s easy to count by hand and audit with volunteer observers. I’ve personally observed the counting of ballots as a volunteer and it gave me a lot of confidence that the election had been conducted fairly according to the law.
While I would really love to see what outcomes could be produced by a Condorcet method I can’t imagine actually implementing one without relying on computerized counting and software. This makes it impossible for volunteers to audit due to the need to audit the hardware and software of the computers doing the counting. If only security experts are capable of auditing such a system then our whole democracy is at their mercy. That’s essentially a technocracy.
Sweden has an extremely complex system for tallying votes and very very few people can explain the exact algorithm that turns a collection of ballots into a list of names of people in Parliament. Even people that follow politics closely only have a vague idea of how it works. Yet most people feel the system is fair and reasonable and there is no real push to change it or make it simpler to understand.
If your business relies on some big pile of spaghetti code, you might be very reluctant to change it, but you also would be unlikely to recommend that design to someone who was starting from scratch (unless they were a competitor).
I'm not recommending the Swedish system per se. Just pointing out that as long as people feel they understand how to vote to nudge the result in their desired direction, and that the outcome (ie. who ends up in parliament) feels reasonable and representative then people probably don't care too much about the details of the voting system
The Condorcet property (not method) is easy enough to understand.
"Imagine this candidate ran in a two person race against each of the other candidates in turn. If he'd win ALL those races individually, he should win when running against all of them at the same time too."
Now, explaining various methods which guarantee this property is not as easy - nor is it easy to understand why you might prefer one such method to another.
But this isn't actually a big deal. In the many, many countries that use proportional representation, it's the outcome ("parties share of seats in parliament should be roughly the same as parties share of votes in the election") which is agreed upon. The actual mathematics to achieve it are somewhat counterintuitive and not many people are aware of them, but that doesn't matter. We can all see it works, and so we would in an election with, say, Ranked Pairs.
Well, I'm not really seeing people complain about the math itself, but instead more about fraud in the votes themselves, or machines/people cheating when tallying.
Just to be clear, the difference would be something like having to say B won because he was the most common second choice and beat every other candidates in a one on one. Versus saying B won because it had the most first choices, yet did not have a majority of first choices.
I think it's easy enough to understand that while someone had more first choices, they didn't have a majority first choices and since someone else had more second choices they took the win.
Being simple to understand is crucial for voting systems and I would argue is the most important factor since an election without public buy-in is worse than worthless.