But … I suspect 99.99% of the “real elections” referenced in that had no more than 3 viable candidates. We have a combination of a low number of voters with a high number of reasonable choices.
As a sanity check, I ran my own simulations using 100 voters each picking a full (no duplicate ranks) ranking of 7 candidates, picked uniformly at random from the 7! candidate permutations.
Slightly less than half(!) turned out to have a Condorcet winner. Worse, we won’t have 100 voters, and the fewer there are the worse the chances. Worse again, allowing rank duplicates also boosts the odds of not having a Condorcet winner.
Against that, rankings across voters in real life will doubtless be much more correlated than permutations picked at random, so “less than half” should be viewed as worst-case.
I don’t know how to model that, though. Cutting the number of candidates to 3 and voters to 21, about 92% had a Condorcet winner, far closer to the 98.5% cited in the quote. It’s easy to imagine that real-life ranking correlation closes that gap.
Heh - and going back to 7 candidates, changing the number of voters from 100 to either 99 or 101 boosts the Condorcet-winner rate to about 63%. When there are an odd number of voters, and equal rankings aren’t allowed, no one-on-one contest can tie. When there are an even number of voters, they can tie, and then there’s a substantial chance that “the best” candidate doesn’t lose any one-on-one contest, but does tie on at least one. Not a Condorcet winner then.
Another reason I like voting systems better when they’re not too clever for their own good .
Anyway, bottom line: best guess is that we’ve grossly overestimated the odds of getting a Condorcet winner on the first try, due to our unusual combination of relatively few voters faced with an unusually high number of viable alternatives. Still think we probably will, but it will no longer be a real surprise to me if we don’t.