Chew Pairings

A pairing system tailored for Scrabble tournaments.

Updated Sat Jul 2 14:02:08 EDT 2005 for tsh 2.960.

This section discusses the theory behind the Chew pairing system, which will be implemented in a forthcoming release of tsh. For general pairing theory, see the section on pairing theory and tsh. For information about how to implement pairings in tsh, please refer to the section on configuring tsh.

History

The Chew pairing system is a descendant of the Fontes-Boys modified Swiss system used at the 2000 CNSC, the two-victor pairing system used at the 2003 CNSC and the NSA “Basic Approach” used at the 2004 NSC and the 2005 CNSC. It refines the basic ideas of these pairing systems, which concentrate on fairly pairing players near the end of a high-stakes tournament, but is flexible enough to use in any tournament setting.

For the benefit of any Chess or Go tournament directors who may be reading this document, Scrabble pairings differ from these other games in the ready availability of a standard tie-breaking or secondary ranking criterion (the difference between the total number of points scored by a player, minus his opponents’ total), and in their comparably shorter timing (almost all games give players 25 minutes each, so rounds can start hourly).

Goals

The following are desirable properties of a pairing system. It is likely impossible to satisfy all of these properties simultaneously.

Fairness: the final ranking of players by the customary win-loss record and cumulative spread should be equivalent to a ranking according to their level of performance in the tournament. Three possible ways of measuring the latter are to average the final rankings of the opponents of each player (Average Opponent Ranking or AOR, to add the number of wins and spread of the opponents of each players (Sum of Opponent Scores, SOS or Buchholz), and uninitialised iterated performance ratings (IPR, the ratings that players would have if they had all arrived at the tournament unrated).

Implementability: a good human director can pair 50–100 players by hand if the pairing system is not too computationally demanding. Computers can currently do exhaustive searches for optimal pairings up to groups of about 16 (for which there are 2,027,025 possible pairings), but can easily apply complex rules to larger groups to find pairings which match well-designed criteria.

Inclusivity: a player should not needlessly be excluded from contention for a prize by being paired with someone who is out of contention. For example, if with one round left to play the top three players have N, N and N–1 wins and the top two have a 500-point spread advantage over #3, then #3 needs to play #1 or #2 in order to have any reasonable chance of finishing first.

“Monagony”: players should play each other as few times as possible. Repeat pairings can prevent other players from catching up to the repeaters, and do not accurately measure the repeating players’ ability to defeat a wider field.

“Exagony”: players travelling together to a tournament do not like to play each other, as many players travel to vary their opponents. At the World Championships, unnecessarily pairing together players from the same country may make it difficult for them to earn an additional place at the next WSC.

Incentivization: a player should not be placed in a position where tieing or losing is strategically preferably to winning. For example, a player who has clinched a place in a two-player final should not be paired with a possibly weak player whose victory would send him to the finals.

Monotony: a lower-ranked player should not be paired so as to make it more likely that he will become the overall winner than that a higher-ranked player will.

Theory

Consider the following scenario. Nine rounds have been played of a 10-round tournament, with no repeats so far. The top four players are 7–2 and the next four players are 6–3, and each group has players with spreads +700, +500, +300 and +100. If only victor (overall prizewinner, a term I use to avoid confusion with the winner of an individual game), how should the players be paired in the final round?

It would not be uncommon practice to pair the players king-of-the-hill, allowing repeats, that is, 1–2, 3–4, 5–6 and 7–8. While this violates inclusivity by shutting #5–8 out of contention, many would argue that #1–4 have earned the right to contend amongst themselves.

Suppose however that #1 has already played #2–4 and #3 has already played #4, but none of #1–4 have played any of #5–8. Pairing 1–5 and 4–6 would better satisfy both monagony and inclusivity. Or to put it negatively, pairing 1–2 instead of 1–5 would give #6 two reasons to complain: that he was unfairly shut out of contention, and that the winning player didn’t play a wide field.

This is the inspiration behind Chew pairings, and here is the basic idea:

  1. Calculate who the currently lowest-ranked possible victor L is.
  2. Calculate the smallest number of repeats R that can be permitted in a pairing of all the players from #1 down to L.
  3. Calculate the smallest N for which #1–N can be paired with at most R repeats.
  4. Pair #1–N.

Ranked Too High To Contend

Curiously, it is possible for a player to be eliminated from contention while one or more players ranked immediately below them are still in contention. Here’s an actual example from Albany NY, July 2005.

With one round left to play, the top six players in the second division had the following wins and spreads: 14/+868, 14/+256, 13/+1469, 13/+509, 13/+376, 12/+538. If we try pairing the top four, then player #4 cannot win. In a 1–2 3–4 pairing, the winner of the 1–2 game will win the tournament. In either of the other pairings, #1 and #2 only need to win their last game, and if neither does then #3 will win.

Now suppose that the top flight includes #5. Then because we have to include #6 to even out the flight, it's quite possible for #4, #5 and #6 to beat #1, #2 and #3 and for #5 to end up having the highest cumulative spread among the players who end up at 14 wins.

Here’s another example from the sample tournament. With one round left to play, the top eight players in the third division had the following records: 12/+465, 12/+332, 12/+200, 12/+194, 11/+301, 11/+279, 11/+113 and 11/+103. If we pair the top N, for varying values of N from 2 to 8, then the Nth ranked player has a chance at winning for N equal to 1, 2, 3, 4, 7 and 8 but not 5 or 6. The reasons why are left as an exercise for the reader.

Implementation

As of the current version of tsh, the “ChewPair” command performs the calculations necessary to determine the highest possible rank that each player can attain.

A future version will implement the rest of the algorithm.

Usage

The current implementation can be and has been used with diffiucluty to semiautomatically perform Chew pairings. It's not really recommended for production use yet, but a fully functional version should be ready no later than September 2005.