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.