balashiha on FIBS
Many competitions use head-to-head record as a tiebreaker. I
personally find this somewhat unsatisfactory in backgammon. In many
types of competition, the fact that A beats B means that overall A is
superior. In backgammon, it means to some extent just that A was
luckier in that match.
You can also of course have playoffs. Playoffs can be awkward if you
have an uncomfortable number of players.
Why do you need to break ties though? If you have ties, just split
the prize money. It is necessary to have clear winners?
Chess and checkers also have ties. I know very little about those
games, but one tiebreaker you could use is most wins (which means a
player going 4-2-2 would beat one going 3-4-1) or fewest losses.
Bridge sometimes uses fewest losses as a tiebreaker. Backgammon does
not have ties.
One warning about round-robins. Unless you have meaningful prizes for
each match, or there are social pressures, you may have situations
where one player is in contention and another just doesn't care. By
"social pressures" though I mean something as simple as that Thursday
is club night and everyone comes out to play. I've played in bridge
tournaments like this and even teams out of contention play for a kind
of moral redemtpion.