Mc Mahon Pairing / Bar Theory

It is quite often that new tournament organisers will ask At what rank should the top McMahon bar be? This page is designed to give some guidelines to new tournament directors

There are three basic criteria for deciding the top bar:

• Every player that is deemed to have a reasonable chance of winning the tournament should be in the top group.
• The top group should not be so large that the number of rounds in the tournament is insufficient to find a unique winner.
• The top group should not be so small that the tournament is decided early, with the top scorer running out of viable opponents.

Winning Chances

The basic theory of McMahon Tournaments says that all players with a reasonable chance of winning the tournament should start above the bar. To measure reasonable chance is somewhat speculative. We should start by measuring the distance (in terms of rank) from the highest ranked player. A bar 3 or 4 ranks wide would not be undesirable, since players 3-4 ranks weaker can sometimes still defeat the stronger player in a game.

Unique Winner

It is generally desirable to have a unique tournament winner. Thus the bar should not be set in such a way that it is possible for two players in the top group to win all their games without facing each other. This puts an upper limit on the number of players above the bar at two to the power N, where N is the number of rounds in the tournament.

Sufficient Opponents

It is also desirable not to have a tournament winner too early. If a single player takes an unsurmountable lead before the last round, the remaining rounds will probably lack excitement. For this reason, the top group should be large enough that there are sufficient opponents for the strongest player to be challenged until the last round. An easy lower bound for this is to have more players in the top group than there are rounds in the tournament.

Magic Numbers

Some tournament directors like to make a top group consisting of a magic number of players. Magic numbers are 2 to the power of N e.g.: 2, 4, 8, 16, 32. Such a number has the advantage that, since the field halves after each round, the number of players in the top group will remain even until there is only one left. This has the advantage of preventing pairing players to a lower group, which could give such players an unfair SOS-disadvantage.

If the magic number is equal to the two-power of the number of rounds (eg: 3 rounds, 2^3=8 players above the bar), then the top group will function as a single elimination knockout tournament. Although this has the advantage that it unambiguously decides the tournament winner, it has the drawback that it is very bad at ranking the other players. If you want your tournament to reliably find the top three places, for example, then this number is too high. This number also grows quickly with the number of rounds, and is usually not viable with more than 3-4 rounds.

Criteria at odds

At times these criteria can clash. For instance a tournament may have a 6 dan, a 4 dan, three 3 dans, and three 1 dans. Magic number theory exponents would like to set the bar at 1 dan, whilst probability exponents might opt for 3 dan.

Rule of thumb

Several organisations, such as the EGF, BGA and AGA, have recommendations of the number of players in the top group. Although these recommendations vary, they mostly agree on the general range of numbers you should be looking at. As a good rule of thumb, try to have a number of players in the top group that is around twice the number of rounds in the tournament.

Collected Suggestions

• BGA and EGF recommendation
• AGA recommendation (PDF) See page 49 (page 55 of the PDF).
• Recommendation in C. Gerlach's Doctoral Thesis (PDF, German) See page 8 (page 12 of the PDF).

Handling players who skip rounds

Big events will often specify that you must promise to play in all rounds to start above the bar. The European Go Congress is a fair example. You can consider making exceptions to this rule. For instance, if a player asks in advance for something like a travelling bye, then he can be allocated a half point score for that single round. Generally though, you want those competing above the bar to play in all rounds to avoid tiebreaker oddities.