Partial Round-Robin TournamentsA popular format for backgammon tournaments is the partial round-robin.
A full round-robin sees every player play every other player in the tournament. If there are n players in a tournament, a full round-robin needs n − 1 rounds to complete. This many rounds is not practical if you have more than about 10 players, so many tournaments opt to use some sort of "partial" round-robin.
A partial round-robin might have anywhere from 5 to 15 rounds, depending on the time available.
Termination MethodsThere are several methods you can use shorten a round-robin tournament. We will look at two:
- Truncation: In a truncated round-robin, all players play every round up to a predetermined number. For example, all players might play for six rounds.
- Elimination: In an elimination round-robin, players continue to play until they have lost a set number of matches. For example, if the limit is 3, every player continues to play until he has lost three times.
A truncated round-robin is good when you want to rank all participants from first to last. This method also maximizes the amount of play for everyone.
An elimination round-robin is good when the primary goal is to find an overall winner of the tournament, perhaps to award a trophy or cash prize. The advantage of this method is that only players who still have a chance of winning continue to play. (There are no "meaningless" matches.)
Matching Up PlayersWhen running a partial round-robin, an important question is how to match up players in each round — who plays who? There are two common pairing methods:
- Random Pairing: Each player plays an opponent chosen at random.
- Swiss Pairing: Each player plays an opponent who has the same record so far in the tournament. For example, suppose that after three rounds you have won two matches and lost one match. That means in the fourth round you will play someone else who has also won twice and lost once.
Tournament EffectivenessIs one pairing method better than the other?
To answer that question we first have to figure out what we mean by "better." One definition of better is how good the tournament is at separating the wheat from the chaff.
In my compter simulation each player has a different skill level. One player is the strongest and, on average, will win more often than any other player. There is a lot of luck in backgammon, and just because you are the strongest player doesn't mean you will win the tournament. But, over the long term, you should win tournaments more often than weaker players. And that's what we mean by tournament effectiveness.
If there are n players in the tournament, the average player wins about 1/n of the time. You would expect the strongest player to do much better than this. How much better depends partly on how the tournament is set up. If the strongest player wins more often under one format than he does under another format, we say that the first tournament format is more "effective."
A SimulationI wrote a computer simulation to find out which pairing method was more effective. The simulation works as follows.
There are 64 imaginary players, each with a different skill level. To implement the different skill levels, I ranked the players from 1 to 64. Let D be the difference in rank between two players. When two players play each other, the higher ranked player wins with a probability of 0.50 + (0.0025 × D) and the lower ranked player wins with probability of 0.50 − (0.0025 × D).
So, for example, suppose Player #1 (the best player in the tournament) plays Player #64 (the worst player). The difference in rank is D = 63. The favorite in this match-up will win 50 + (0.25 × D) percent of the time, or 65.75%. The underdog will win only 34.25% of the time.
I ran the tournaments using various setups. For each setup, I ran the simulation 1,000,000 times and tabulated how often each player won. In particular, I wanted to see how often Player #1 won to determine how effective that tournament setup was.
Swiss versus Random Pairing in a Truncated Round-RobinIn a truncated round-robin tournament, nobody gets eliminated; everyone continues to play for k rounds. The winner of the tournament is the player with the most wins. (If two or more players are tied, the winner is chosen at random among the tied players. At least that's how it worked in my simulation; in a real tournament you might have some kind of tie-breaker.)
I simulated a truncated round-robin tournament with k = 6, 9, 12, and 15 rounds. Each tournament was run twice — once with players paired at random, and once with players paired using Swiss pairing. The tournament was run 1,000,000 times and results were tabulated to see how often each player #1 through #64 ended up the tournament winner.
An average player should expect to win the tournament 1/64 of the time, or about 1.56%. Player #1 (the strongest player) should do substantially better than this. And Player #64 (the weakest player) should do substantially worse.
The tables below summarize the results. From the tables, you can see that in a 6-round tournament with random pairing, Player #1 wins about 3.047% of the time (almost twice what you'd expect for an average player). In a 6-round tournament with Swiss pairing, Player #1 does even better. He wins 3.286% of the time. This is a significant difference. Strong players are definitely favored by Swiss-style pairing.
top players do better when Swiss pairing is used.
The color coding in the tables shows which pairing style you'd prefer based on how high ranked you are as a player. Green is the higher probability of winning. In a 6-round tournament, the top 23 players do better with Swiss pairing than they do with random pairing. (The fact that the colors alternate back and forth between Players #23 and #26 is simply a reflection of the small random variations you'd expect in the simulation results. If I had done more simulations, these variations could have been reduced.)
Longer tournaments favor the better players too. With random pairing, the top player won 3.047% of the time in a 6-round tournament, 3.5% of the time in a 9-round tournament, 3.877% of the time in a 12-round touranament, and 4.235% of the time in a 15-round tournament. With Swiss pairing, the top player won 3.286% of the time in a 6-round tournament, 3.673% of the time in a 9-round tournament, 4.063% of the time in a 12-round tournament, and 4.424% of the time in a 15-round tournament.
As the top player wins more, it leaves less equity for the other players. In a 6-round tournament, the top 28 players showed positive equity (won more than 1/64 = .015625 of the time). In a 9-round tournament, only the top 27 players showed positive equity. In a 12-round tournament, only the top 26 players showed positive equity. And in a 15-round tournament, only the top 25 players showed positive equity.
Swiss versus Random Pairing in an Elimination Round-RobinIn an elimination round-robin tournament, players continue to play until they have lost a specified number of matches. For example, players might continue playing until they have lost three times.
I simulated an elimination round-robin tournament with 1, 2, 3, and 4 losses. Each tournament was run twice — once with players matched at random, and once with players matched using the Swiss pairing. The tournament was run 1,000,000 times and the results tabulated to see how often each player #1 through #64 was the last to survive. The tables below summarize the results.
A single knockout round-robin tournament is the same as straight elimination tournament and always takes 6 rounds. A double knockout round-robin takes about 9.862 rounds to find a winner. A triple knockout round-robin takes about 12.935 rounds. And a quadruple knockout round-robin takes about 15.844 rounds.
The surprising thing about these tables is that there is no advantage to tournament effectiveness of using Swiss pairing rather than random pairing. The strongest players do equally well with either system.
there is no difference in tournament effectiveness
between Swiss pairing and random pairing.
There is, however, one advantage to Swiss pairing. The average length of the tournament is slightly less with Swiss than with random pairing. For example, in a 3-loss tournament, random pairing takes 12.94 rounds on average to finish and Swiss pairing takes only 12.63 rounds on average to finish.
But as a practical matter this difference is not very significant when you consider the extra work required to run a Swiss tournament. In fact it's likely the time saved by doing a simple random draw outweighs the time savings for Swiss suggested by the simulation.
(6.000 rounds rand)
(9.862 rounds rand)
(12.935 rounds rand)
(15.844 rounds rand)
SummaryIn a truncated round-robin tournament, the strongest players do better when Swiss pairing is used.
In an elimination round-robin tournament, the strongest players do equally well regardless of whether random pairing or Swiss pairing is used.