McMahon Pairing
McMahon Pairing is a method of pairing players in a tournament. Developed by Lee McMahon and Bob Ryder in the 1960's to address the problem of severely unbalanced pairings in the Swiss pairing method. McMahon pairing generalizes Swiss Pairing by assigning different initial scores to various players. McMahon pairing has become the default choice for amateur Go tournaments in Europe and America. (Chess tournaments still use the Swiss system.)
McMahon Pairing Method Details
- Before the First Round: Order the field of players by strength. Divide the field into bands of players based upon number of rounds and gaps in the distribution of players strengths (if any gaps). Players in the same band will be paired against each other in the first round. The system contains two bars, the top bar and the bottom bar. The top bar defines the strength above which all players are in the same band. This band contains the strongest players, who are said to be above the bar. This band above the bar should contain all players that are thought to have a chance to win the tournament. The bottom bar functions the same way, but defines a playing strength below which all players are in the same bar.
- The bands are assigned a starting McMahon Score. There are two main ways to do this. The first is to set the McMahon Score of the band above the bar to 0, with all other bands starting at a negative score. So the next weaker band is -1, down to the weakest band which is -(M-1) for M bands. The other is is to set the McMahon Score of the band below the bottom bar to 0, with all other bands receiving a positive initial score. In effect, the early slaughter rounds of the Swiss pairing are replaced by assigning a number of virtual won games to stronger players. The stronger the player the greater the number of won virtual games assigned to that player.
- First Round: Pair players within the bands, starting with the top band, according to one of the methods outlined in Tournament Pairing. Each player that wins their game has one added to their McMahon Score. The McMahon Score of players that lose does not change. Note that by assigning the top band an initial McMahon Score of 0, the score among the top players is the same as if the tournament was a Swiss pairing tournament.
- All Subsequent Rounds: Pairing players with the same McMahon Score against each other using one of the methods outlined in Tournament Pairing. The process is repeated till only one player has a perfect undefeated record provided there are a sufficient number of rounds.
Tournament Placings
First place in the top band is the tournament winner. Placings are ususally based upon number of wins and one or more tie breakers. Placings can be assigned for each band. By allotting prizes to each band, all players in the tournament have a goal for which to strive.
Problems with McMahon Pairing
- Dividing the field into bands requires the tournament director to make decisions that are based, at least in part, on discretion and experience rather than a fixed, known algorithm.
- In most cases, the number of players with the same score will be odd for one or more rounds. An odd number of players can not be paired. At least one player must be paired against someone with a different score. This may result in a different group of players with the same score having an odd number of unpaired players. The problem may thereby ripple across the score groups. This problem is shared with Swiss pairing.
- Due to limits on the number of rounds possible in a weekend tournament relative to the number of players, it is possible that more than one player will have a record of all wins. In that case, the final ranking is usually based upon the following criteria:
- McMahon Score (Number of wins plus initial MMS)
- additional tie breakers
- This problem is shared with Swiss pairing. One may argue that this problem is due to the tournament director assigning too many players to the top band, rather than an inherent problem with McMahon pairing tournaments.
Computer Programs for McMahon Pairing
A number of programs apply additional criteria when creating the pairings. The main programs are
Historical Information
The system is named after Lee McMahon of New York Go Club, but it was spirited out of NY into Britain where it (eventually) found its current form.
Tim Hunt writes...
"The remarkable thing about the history of the McMahon system is that such a successful tournament system arose as the result of a transatlantic misunderstanding. According to Francis Roads (revered BGA member) who was there at the time, this is what happened:
The first three British Go Congresses (1968, 1969 and 1970) were run as handicap, or open + handicap tournaments. Then for the 4th (1971) BGC they tried the McMahon system, because they wanted a system where most games were approximately even games, but where everyone was really playing in the same tournament, rather than being split up into different classes. The McMahon system they used was loosely based on and named after the McMahon system used by the New York Go Club. What the British did not realise was that the New York system was, in fact, a club ladder/grading system, not a tournament system. Anyway, the BGA used it as a tournament system and, apart from one small bug (later) the first time around, it seemed a success. So much so that eventually it spread to Europe and back to America.
The bug: the mistake they made the first time around was to say if you win you move up one, and if you lose you go down one (rather than staying the same). This meant that if your McMahon score was even in the first round then it was odd in the second round and vice-versa and similarly in later rounds. This meant that people who started with an even McMahon score (almost) never played people with an odd McMahon score and so effectively they had two separate tournaments going on side-by-side. Oops! They got it right the second time around."
It is often called MacMahon too, but nobody is quite sure why. The BGA Journal Feb 1978 refers only to MacMahon for example. Although a few years previous to that, McMahon was exclusively mentioned.
Personally I would be interested in finding more information on the original usage in New York Go Club.
See Also:
- Tournament Pairing
- Swiss Pairing
- Tournament Directing Software
- McMahon Example
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British Go Association McMahon Tournament Pairing Rules
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Christoph Gerlach's thesis (PDF file, in German)
-
Cambridge Go club page on McMahon
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Geoff Kaniuk on McMahon
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Olli Lounela's description of McMahon (
Wayback machine link (2002) to Olli Lounela's page )
- McMahon explained by Christoph Gerlach
- According to page 95 of the 1986 Ranka Yearbook (at least as quoted
here), the original McMahon system was invented by Lee McMahon and Bob Ryder of Bell Labs.
-
RGG: MacMahon and Swiss Tournaments
(WME ongoing since 7 September 2005)
Explanation
Where McMahon pairing differs from Swiss pairing is in the initial starting scores. The players begin with a Score determined by the rank at which they entered. So for instance, a 5kyu would begin at 2 points higher than a 7kyu. The important exceptions are introduced by the bar, upper or lower.
- Above the (upper) bar are placed all those deemed to have a reasonable shot at winning the tournament. So if the bar is at 5dan, then a 7dan a 6dan and a 5dan would start on the same score, all 1 point above a 4dan beneath the bar.
- A lower bar is not always included, but beneath this we treat all players to be the same rank. If the lower bar is 19kyu, then a 35kyu and a 20kyu would start at the same score.
Now the tournament director attempts to pair players with similiar scores together using one of the methods specified in tournament pairing or, perhaps, another method. After each round, each players score is increased by one, one-half or zero points. One point is awarded for a win, or the opponent fails to show. One-half point is awarded to each player for a draw or no-result game. Zero points are awarded for a loss, or when both players fail to show. We call the score, perhaps unsuprisingly, the McMahon Score. Byes are normally awarded a half point, but this is not always the case.
Variants
- Aguilar-McMahon
Migeru: The 2005 Iberoamerican Go Tournament is being run (on KGS) by dividing the participants into 4 groups initially:
7d- 1d -- 3 McMahon points -- 14 players 1k- 6k -- 2 McMahon points -- 44 players 7k-13k -- 1 McMahon point -- 43 players 14k-20k -- 0 McMahon points -- 37 players
This was actually on the suggestion of Fernando Aguilar who thought this addressed a number of concerns that players had been expressing over the years, and the results of trying out different variations of the Swiss/McMahon system. The tournament will last 8 rounds.
What was the general opinion as to the success of this format?
Phelan: I can't be sure now, but from what I remember it was favorable. The biggest problem was higher level players not playing some rounds. Some lower level players were ranked below them, despite playing all games in the tournament. Results at (link broken)
http://www.goprat.es/TIG2005.TXT