Chapter 9 The McMahon System

Chapter 9
The McMahon System

9.1 Overview

Most British Go tournaments use the McMahon system. The McMahon system is designed to ensure that games in a tournament are most likely to be even. Each player in the tournament starts off with a McMahon score (or MMS) that is equivalent to his or her grade. For example, a 4-kyu player would start at -4, and a 1-dan player would start at 0. However, players above a certain grade near the top of the tournament are demoted to this grade, in order to produce a pool of players with the highest McMahon score represented. This grade is known as the McMahon bar (see section 9.2). Each win for a player increases his or her MMS by one.

(Note: An alternative method, commonly used in Europe, sets the score of all players at 20 kyu or below to zero. Like the UK scale, each grade step corresponds to an increment of 1 McMahon point, so shodan has the value 20. A disadvantage of this scale, however, is that it is likely to make players below 20 kyu feel that they are not yet strong enough to enter a tournament.)

When pairing players against each other, the draw program (or human organiser) attempts to pair players with the same MMS against each other. This has the effect that, if a player enters at the wrong grade, their MMS will gradually come closer to that of players of their own strength. For example, if someone declares too high a grade, they are less likely to win, and so their MMS will stay the same while other players' scores rise - until finally the player meets those of roughly the same strength.

9.2 The Bar

Because a player's starting score is determined by their grade, a player who was 7 dan would have a massive advantage and the best chance to win the tournament, as such a player would start at a very good MMS. To counteract this, and to give as many people as possible an equal chance of winning the tournament, players at or above a certain rank all begin at the same MMS. This rank is called the bar. For example, if the bar is set at 3 dan (which is an MMS of 2) then no player can start at an MMS of more than 2, no matter what their grade. This gives a pool of players starting with the highest McMahon score. The optimum size for this pool depends on the number of rounds to be played. Rough guidelines are as follows:

3 rounds
4-8 players
4 rounds
5-10 players
5 rounds
6-12 players
6 rounds
7-15 players
7 rounds
8-18 players
8 rounds
9-22 players
9 rounds
10-26 players
10 rounds
11-30 players

The constraints on which these figures are based are

  • There must be a unique winner. This sets an upper limit for short tournaments.
  • If there are too few on the bar, these receive an unfair (and unnecessary) disadvantage.
  • If there are too many on the bar, the tournament will end without all of the top players having played each other.
  • Higher graded players should not run out of even game opponents.

If you are using Geoff Kaniuk's GoDraw to create the draw, it will set an appropriate bar automatically.

In effect the top players play a knock-out to determine the winner, so there should not be more than 8 players above the bar for a 3 round or 16 players for a 4 round tournament.

(Note: In some larger European tournaments, there is a supergroup, which is one point above the McMahon bar. This is used where there would otherwise be too many players above the McMahon bar.)

9.3 Handicaps

Although the McMahon system decreases the chances of uneven games, they still occur, especially where there is a large range of entry grades. The handicap in the McMahon system is normally one less than the current difference in the players’ McMahon scores, with a handicap of 1 meaning a no-komi game. If there is no handicap, colours are selected (more or less) at random. Therefore, a player may end up taking White even against someone on a McMahon score one better than them. It is normal to try to organise the draw so that, as far as possible, players play an equal number of games as each colour.

If either player declared a grade at or above the bar, then the game is even.

(Note: It is not a requirement that the handicap be one less than the McMahon difference; it is normal in Europe to use difference-2, and some tournaments use the difference directly.)

9.4 Sleeping Players

A player who misses a round (with the tournament director’s consent) sleeps for that round. For the purpose of producing the pairing, a sleeping player is deemed to have achieved an average score for each round missed. In order to prevent biasing the draw by matching players who sleep, an extra McMahon point is awarded after every two rounds missed instead of a half point for every round missed. This score increase does not, however, count as a win.

In more important tournaments, players in the top McMahon group (or Supergroup if there is one) are not allowed to sleep for any round. Sleeping players who would be in the top group are removed by reducing their initial McMahon score by one or possibly two points. This prevents the lower group players from interfering with top group players in later rounds.

9.5 Forfeits

If a player wins by default (usually because their opponent fails to show up), their MMS is increased by one. This counts as a win for the purposes of the tournament, but not for the purposes of the EGF rating system. It is not counted as a missed round for either player. (Note: In practice, this depends on how you choose to record the result; for simplicity, a win by default may well be entered as a normal win, in which case none of this applies.)

9.6 Overall winner

At the end of the tournament, the winner is the player with the best McMahon score.

There may be a tie for first place, and it is very likely that there will be ties for other positions. The tie breaker used varies from tournament to tournament; commonly used tie breaks include:

(Sum of Opponents Scores) SOS is the sum over all rounds of the final McMahon scores of the player's opponents. For each round that the player sleeps add the player's own initial McMahon score.

This is the most commonly used tie break in the UK.

(Cumulative Sum of Scores) CUSS is the sum over all rounds of the player’s individual McMahon score at the end of each round. This rewards the player for winning early in the tournament.
(Cumulative Sum of Points) CUSP is the sum over all rounds of the player's total winning points at the end of each round. Note that winning points at any round is the total sum of wins and jigos and does not include free wins or points gained by sleepers.

If two players start off on the same McMahon score, and end up with the same CUSS (having played all games) then their CUSP will also be the same. This follows since at any round McMahon score is equal to initial McMahon score plus winning points up to that round. So one cannot use CUSS and CUSP as first and second level tie-breakers. CUSP is easy to calculate by hand and is therefore recommended for use as the tie-breaker in small tournaments.

(Sum of Sum of Opponents Scores) SOSOS is the sum over all rounds of the SOS scores of the player’s opponents. For each round that the player sleeps add the players own SOS.
(Sum of Defeated Opponents' Scores; strongly deprecated) SODOS is the sum of the final McMahon scores of the player's opponents in those rounds that the player won. For each free win add the player's own initial McMahon score. For each jigo add one half the McMahon score of the opponent.

There is, however, a serious flaw associated with the concept of SODOS. This tie break depends on the scale chosen, and it is possible that the final ordering can change depending on the origin of the McMahon scale (i.e., whether is is 0 for shodan, as in the UK, or 0 for 20 kyu, as commonly the case in Europe). This tie break is therefore strongly not recommended.

It is a good idea to decide in advance what to do in the event of a tie, as even these tie breakers are not necessarily enough to separate players. For most tournaments, it is reasonable for first place to be shared. For more important tournaments, such as the Challenger's League, a play-off game is used.

Last updated Tue Jul 28 2009. If you have any comments, please email the webmaster on web-master AT britgo DOT org.