I am confused about how three-way ties are settled in round robin tournaments. That is to say, what happens when 3 teams have the same record, say 13-1. Who gets 1st place?

If it was a two-way tie, then the head-to-head matchup would tie-break. But if three teams end up with a single loss, the head-to-head matchup doesn't work, as each of the teams has beaten one of the other teams, while lost to the second team.

As an example consider this tournament https://www.warlight.net/MultiPlayer/Tournament?ID=23460

Teams N, O, and K each finished with a 13-1 record. Team N beat Team K, while Team K beat Team O, and Team O beat Team N. So how does the tie-breaker work?

Head-to-head, Team N beat Teak K, so N should place ahead of K right?

And team K beat Team O, so K should place ahead of O, right?

But Team O beat Team N, so O should place ahead of N. But that can't happen.

It seems to me that the three teams should be tied, since no one team beat both of the other teams. However, if you look at the tournament, Team N placed first. Does anyone know why Team N placed first here?

In particular, Team O beat Team N. Why would Team O place behind Team N?

When there is a tie for first place (or any placing), that can't be broken by tiebreakers, then the winner is determined randomly, same with second and third place, so the fact that you got third place simply means you got unlucky.

Thank you, I did not know that.