Hi Max,
As I understand it, the paradoxical result
is the same as failing to produce a winner. I'm assuming that the electoral method being used produces a ranking consistent with majority choices, and that this ranking is the outcome of the election: it decides who is first, second, third place etc. In the problematic case (assuming three candidates) this method cannot produce a ranking because:
most people prefer A over B, and
most people prefer B over C, and
most people prefer C over A,
So, the ranking consistent with majority choices is paradoxical, or alternatively, it is impossible to rank the candidates, even though each individual voter has produced a logical ranking on their ballot sheet. Consequently, you do not have a winner.
A cannot be the winner, because more people prefer C to A.
B cannot be the winner because more people prefer A to B; and
C cannot be the winner because more people prefer B to C.
The method fails to rank the candidates and hence produces a non-result, and hence, no winner.
"A Condorcet method is an election method that elects the candidate that would win a majority of the vote in all of the head-to-head elections against each of the other candidates, whenever there is such a candidate. A candidate with this property is called the Condorcet winner. [...]
A Condorcet winner does not always exist in every election because the preference of a group of voters selecting from more than two options can be cyclic—that is, for each candidate it might be possible to select an opponent where the opponent would win a majority of the votes."
from here, my emphasis
Not producing a winner is a problem when one of the requirements of the method is that it does produce a winner, e.g. when electing a president.
This outcome: not producing a winner, is most definitely much more likely with this particular ranking method than with a single-vote, first-past-the-post method. With the latter method, the odds of a tie for first place is vanishingly small in a nationwide vote.
This, if nothing else, is a practical concern when choosing a method.
So I disagree that not finding a winner is just as likely with a first-past-the-post system. And I disagree that producing a cyclic/paradoxical ranking is just as likely with a one-vote-per-person system: as far as I can see, it's impossible to produce a cyclic/paradoxical ranking in this way -- the worst case scenario is that you get a tie.
If, on the other hand, we're using a different kind of ranking or PR method, then the outcomes will be different.
best,
Matt