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.
Each win for a player increases his or her MMS by one.
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.
At the end of the tournament, the winner is the player with the best McMahon score.
Often, there is a tie for first place. The tie breaker used varies from tournament to tournament, but one common method is Sum of Opponents’ Scores (SOS)—here the McMahon scores of each player’s opponents are summed, and the player whose opponents did best is the winner.
Because ones starting score is determined by ones grade, this implies that 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, 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.
Although the McMahon system decreases the chances of uneven games, they still occur. 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.
If either player declared a grade at or above the bar, then the game is even.
For every two rounds a player misses, his or her MMS is increased by 1 (but it is not increased by 0.5 for a single missed round). In any event, this MMS increase does not count as a win.
If a player wins by default (usually because their opponent fails to show up), their MMS is increased by one but they are not awarded a win. This is not counted as a missed round for either player.
Note: This really depends on how the tournament organiser chooses to record the result in the program; for simplicity, a win by default may well be entered as a normal win, in which case none of this applies.
More detail can be found in the document McMahon tournament pairing rules written by Geoff Kaniuk, the creator of godraw— the computer program used to run most UK Go tournaments.