Round Robin scenarios for 3 players or 4 players (or teams)

This interesting discussion came up at our backgammon club last night.  Normally, when we have a tournament, we award the winner with club points equal to the number of players who attend.  The next runner(s) up get half those points, the next gets half again, etc.  Any player with a win gets some points and players with no wins get 0 points.  This works well when we use a double elimination bracket, which we do when we have 5 or more players.

Its common though towards the end of the year that we have less players, so lately we have had a few nights of 3 or 4 players.  On those nights, we do a round robin style tournament where each person plays each person once.  The end result can have several ties and so its less clear how to award points.  So finally, I documented each scenario and how many points we will award.  Of course this can be applied to round robins in any sport with 3 or 4 teams, or other games.

3 PLAYER RESULTS

--------------------

2-0, 1-1, 0-2 : 3 points, 1.5 points, 0 points (4.5 total)

1-1, 1-1, 1-1 : 1.5 points each (4.5 total)

4 PLAYER RESULTS

--------------------

3-0, 2-1, 1-2, 0-3: 4 points, 2 points, 1 point, 0 points (7 total)

3-0, 1-2, 1-2, 1-2: No head-to-head tiebreakers, 4 points winner, 1 point for each 1-2 player (7 total)

2-1, 2-1, 2-1, 0-3: No head-to-head tiebreakers, 2.5 points for each 2-1 player (7.5 total)

2-1, 2-1, 1-2, 1-2: For each of the ties, there will be a head-to-head tiebreaker, so clear 1st through 4th placings can be determined. 4 points, 2 points, 1 point, 0.5 points (since 4th place had a win) (7.5 total)

The goal is to try to award close to the same number of total points in each case that has the same number of players.  In a 3 player round robin with no ties, the points are 3 for winner, 1.5 for second, 0 for third because that person has no wins.  That's 4.5 total points. So when 3 players have a 3-way tie, each gets 1.5 points so that the total is again 4.5.  That works out perfectly but 4 players is a little trickier.

In a 4 player round robin with no ties, winner gets 4, second gets 2, third gets 1, fourth gets 0 (no wins).  That's 7 points total.  If we were to use a double elimination bracket for 4 players, which we do sometimes if we want to finish faster, then we end with a tie for 2nd.  We don't make the winner of the consolation bracket play the loser of the main bracket, for the sake of time.  In that case, we award 4 for winner, 2 each to the two players with 1 loss each, and 0 to the person with no wins.  That's a total of 8 points.  So the goal for the 4 player round robins where there are ties is to award either 7 points total or 8 points total, or somewhere in between.