# McMahon tournament pairing rules: functional specification

## See the current Organisers Handbook McMahon section which reproduces and updates the part of this document necessary for running a BGA tournament.

## 1. Purpose Of The Document

This document aims to provide a comprehensive statement of all the rules governing the administration of the pairing in McMahon tournaments. The intended audience includes tournament directors (preparing for a tournament); referees (having to answer detailed questions on the pairing), and software developers (producing software for pairing programs).

It provides a functional description suitable for software implementation, such as in GoDraw.

Much of the original material for these rules is found in the BGA Organisers Handbook [Ref. 1] and the EGF Guidelines [Ref. 2]. However neither document covers all the rules in sufficient detail to satisfy the requirements of the intended audience.

## 2. Overview Of Tournament Styles

The main styles for serious Go tournaments considered here are McMahon and Swiss.

In the McMahon tournament, players are given an initial McMahon score depending on their strength. After each round the McMahon score increases by either one, half or zero McMahon points depending on the result of the game. The player with the highest McMahon score at the end of the tournament is the winner.

A Swiss tournament is the same as a McMahon tournament, except that the initial McMahon score is the same for all players. This style is used in special circumstances, e.g. for qualifying tournaments.

In both styles of tournament, the pairing method attempts to pair players in the same McMahon group (players with the same McMahon score). The Swiss tournament can achieve this at every round if the number of players is 2^{N}, where `N` is the number of rounds.

No formal rules are specified for other tournament styles such as Round Robin, as these are usually used for side events distinct from the main tournament.

## 3. Game Score

The outcome of a game between two players can be one of the following results. The Score below is the number of McMahon points awarded for each kind of Result:

Result | Score | Explanation |
---|---|---|

Win | 1 | |

Lose | 0 | |

Jigo | ½ | |

Unresolved | ½ | Game ends in triple Ko or otherwise cannot be resolved. |

Free Win | 1 | Opponent fails to show. |

Default | 0 | Player fails to show. |

Void | 0 | Both players in a pairing fail to show. |

## 4. McMahon Score

The traditional grading system (Dan/Kyu) is mathematically awkward as there is no grade between 1 kyu and 1 dan. Various McMahon scales have been introduced to deal with this. One method is the ‘Zero Shodan’ scale used in the UK which sets shodan at zero McMahon points, and kyu players then have negative points along the scale. Another method is the ‘Zero 20 kyu’ scale commonly used in Europe which sets the scale of all players at 20 kyu or below to zero. In both scales, each grade step corresponds to an increment of 1 McMahon point, so in the latter scale, shodan has the value 20.

### 4.1 Initial McMahon Score

Each player in the tournament is allocated an initial McMahon score with value equal to the grade of the player in McMahon points. All the players on the same McMahon score form a McMahon group. The initial McMahon scores of the top groups of players are lowered to a common level called the McMahon bar.

### 4.2 McMahon Bar

The McMahon bar is chosen to ensure that:

- All the stronger players have an equal a-priori chance of winning the tournament.
- Higher graded players do not run out of even game opponents.
- There is a unique winner after the final round.

It then follows that in a tournament with `N` rounds there should be at most 2^{N} players in the top McMahon group. For long tournaments this number would be excessive, and in practice tables have been devised to specify the population of players above (greater than or equal to) the bar.

#### 4.2.1 McMahon Bar (BGA style)

The following table [Ref. 1] defines the population of players above the McMahon Bar in BGA tournaments:

Rounds | Population |
---|---|

3 | 4–8 |

4 | 5–10 |

5 | 6–12 |

6 | 7–15 |

7 | 8–18 |

8 | 9–22 |

9 | 10–26 |

10 | 11–30 |

11 | 12–35 |

12 | 13–40 |

Starting with the highest grade present, the bar is lowered to maximise the population above the bar, but within the range defined by the above table. If there are sufficiently many strong players, the bar may be higher than the traditional 4 dan setting.

#### 4.2.2 McMahon Bar (EGF style)

In most European tournaments the McMahon bar is set at 4 dan, and the population of players above the bar must conform to the following table:

Rounds | Population |
---|---|

5 | 10–16 |

6 | 12–24 |

7 | 15–30 |

8 | 20–40 |

9 | 25–50 |

10 | 35–60 |

If the number of 4 dans or stronger above the bar exceeds 6(`N` - 2), where `N` is the number of rounds, then a McMahon Supergroup [Ref. 2] is formed. The rules for the supergroup are defined in the next section.

Note 1: The formula is not consistent with the maxima defined in the above bar table.

#### 4.2.3 EGF Supergroup

When required, the Supergroup satisfies the following rules:

- The McMahon bar remains at 4 dan.
- The Super bar is one McMahon point above the McMahon bar.
- Exactly 4(
`N`- 2) players are promoted to the Supergroup by having their initial McMahon score increased to the Super bar.

If the number of 4 dan and stronger players is larger than the limits mentioned here then it is necessary to form a second supergroup. The details for this and the choice of which players go into the supergroup varies according to the tournament status (European; Toyota Tour and others). See separate documentation [Ref. 2, Ref. 3] for details.

Note 2: Toyota Tour tournaments [Ref. 3] specify McMahon bar populations of 10–12 players for a 5 round tournament, and 12–16 players for a 6 round tournament. If the population exceeds this, then the supergroup has 12 players for a 5 rounder, or 16 players for a 6 rounder.

### 4.3 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.

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.

## 5. Ranking and Tie-Breaks

At the end of the tournament the player with the highest McMahon score is the winner. If more than one player has the same final McMahon score then the tie is resolved using either one or two tie-breaks (whether one or two are used depends on the tournament status—see separate documentation [Ref. 2, Ref. 3]). Several kinds of tie-breaks are in use and the following sections detail the calculations of these.

Note 3: In Toyota tour tournaments, the first tie-break is SOS and no other tie-break is applied.

### 5.1 Cumulative Sum of McMahon Scores [CUSS]

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.

### 5.2 Cumulative Sum of Wins [CUSP]

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 recommended for use as the tie-breaker in small tournaments [Ref. 1].

### 5.3 Sum of Opponents’ McMahon Scores[SOS]

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.

### 5.4 Sum of Opponents’ SOS[SOSOS]

SOSOS is the sum over all rounds of the SOS scores (Section 5.3) of the player’s opponents. For each round that the player sleeps add the players own SOS.

### 5.5 Sum of Defeated Opponents’ McMahon Scores[SODOS]

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.

Note 4: There is a serious flaw associated with the concept of SODOS.

It has been mentioned in Section 4 that different organisations may choose different origins for the McMahon scale. Nevertheless we would expect that the pairing and ranking process should be independent of the (completely arbitrary McMahon origin).

For example a McMahon score in the Zero Shodan scale is converted to the Zero 20 Kyu scale simply by adding 20 to the score. So the ordering by McMahon score is indeed independent of the origin. For CUSS, one is summing McMahon scores over a fixed number of rounds `N`, so the conversion is:

`CUSS _{Zero 20 kyu}` =

`CUSS`+ 20

_{Zero Shodan}`N`

and again the ordering of players within a McMahon group is unaffected by the shift in origin. The same argument applies to SOS and SOSOS as we are adding the McMahon scores of all opponents shifted by the same amount.

However for SODOS we are not summing over all opponents, only over those that have been defeated. So the offset for different players will vary, depending on the number `D` of defeated players:

`SODOS _{Zero 20 kyu}` =

`SODOS`+ 20

_{Zero Shodan}`D`

In this case ones position in the rank list will vary depending on which particular McMahon origin has been selected—the ordering is not invariant!

This happens for example when two players are initially one point apart; the weaker wins one more game than the stronger; and ends up having the same McMahon score and same SOS, but a lower SODOS. When converting as above, the weaker gains 20 extra points through having an extra win and may then end up above the stronger.

For this reason we strongly recommend that SODOS is not used as a tie-break.

## 6. Uneven and Handicap Games

The pairing method tries to pair players in the same McMahon group, but this aim could be frustrated because either the group is odd or because the only available opponents for a player come from different McMahon groups.

### 6.1 Odd McMahon Groups

When a McMahon group is odd, it is necessary to find a player from a neighbouring group in order to complete the pairing. In this case the McMahon score difference is 1; the player with the higher score is playing down, while the player with the lower score is playing up. However the game is still treated as an even game as far as allocation of colours is concerned. Organisations have evolved different rules to define who gets drawn to play up or down in this circumstance.

#### 6.1.1 Rules for playing up/down (BGA Style)

If no one in the group has played up or down then we choose a player at random. Otherwise the probability that a player is drawn up or down depends on whether the player has been drawn up or down in a previous round, as defined in the following table

Past round | Future round | ||
---|---|---|---|

Direction | Result | Priority up | Priority down |

down | win | high | low |

up | lose | low | high |

down | lose | low | low |

up | win | low | low |

A player who plays up/down and achieves the expected result, may later play down/up to balance the strength of opponents. However if you play down and lose, you do not get unfairly punished by then playing up and probably losing again! Likewise if you play up and win, you do not get unfairly rewarded with an easy game you are likely to win.

#### 6.1.2 Rules for playing up/down (EGF Style)

In many European tournaments, players in neighbouring McMahon groups are seeded by SOS (Sum of Opponents’ McMahon Scores). Then the strongest in the group playing down is paired against the weakest in the group playing up.

### 6.2 Handicap Games

Handicap games might be awarded if suitable opponents cannot be found in the same McMahon group, and such games satisfy the following rules:

- All games above the McMahon bar are played without handicap stones.
- The number of handicap stones awarded depends on the difference in McMahon scores (
`MMS diff`) as in the following table.

Style | Handicap |
---|---|

European | MMS diff - 2 |

UK | MMS diff - 1 |

Leamington | MMS diff |

The original McMahon philosophy is to reduce the number of handicap games as far as possible, and certainly the reduced McMahon difference achieves this.

A game with an adjusted McMahon score difference of zero is treated exactly like an even game with regard to choice of colours as discussed in Section 7.6.

At the bottom end of the draw handicaps may have to be awarded in excess of 9 stones. An alternative is to award komi points instead of the remaining stones. The BGA suggests twice the value of even game komi; the author of this document suggests 20 points per stone based on club experience; and the EGF suggests 15 points per stone.

In order to prevent excessive handicaps at the bottom end of the draw, the entry grade has a cut-off point called the bottom bar. Players below the bottom bar are promoted for the purposes of the tournament. The bottom bar is 30 kyu in the UK and 20 kyu in most European countries.

## 7. Pairing Rules

Many different strategies have been used to produce acceptable pairings either by hand or by computer. A manual pairing which satisfies all the rules is very time consuming and likely to be feasible only for very small tournaments (fewer than twenty players over no more than 3 rounds). For most serious tournaments, computer pairings are recommended. The rules governing the pairing (whether manual or computer) are set out in the sections below.

### 7.1 There are no byes

Go players can travel many hundreds of miles to enter a tournament and would rightly get upset having to take a bye. The organisers must ensure that there is a ‘ghost’ player able to play to make up even numbers as required.

### 7.2 Players meet only once

Players should never be required to repeat a pairing, except in very special circumstances. Cases have occurred for unresolved results and void games—see Section 3.

### 7.3 Pair players from the same McMahon group

The draw aims to minimise the difference in McMahon scores between paired players. Various strategies have been devised [Ref. 4] for pairing players in the same McMahon group including: Random, Split & Fold, and Split & Match. The last two of these require seeding the players in the McMahon group and this is usually done using one or other of the tie-breaks defined in Section 5.

### 7.4 Minimise uneven games

When a McMahon group is odd the pairing rule used is discussed in Section 6.1 above. Where required, handicaps are allocated according to Section 6.2 above.

In either case the draw aims to minimise the number of uneven games for a player.

### 7.5 Pair players from different countries or clubs

Players below the bar should be paired from different countries or clubs where possible. Players above the bar are usually paired without regard to country or club. Note that this does sometimes cause complaints above the bar!

The bar referred to here is the Current McMahon Bar which starts off at the initial McMahon bar discussed in Section 4.2 and increases by 1 each round. This prevents an otherwise ever increasing group of players from possibly having to play someone from their own country or club.

### 7.6 Games alternate in colour

The draw aims to alternate a player’s colours, and in any case should attempt to minimise colour bias. Where the McMahon score difference is 1 or less, and the player’s colours are balanced, then colours are chosen at random

## 8. References

- BGA Organisers Handbook (Old)
- Fujitsu Guidelines (Old)
- Toyota Rules & Regulations
- EGF referee workshop

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