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Comments on McMahonPairing/BarTheory [#3192]

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geoff: Comments on McMahonPairing/BarTheory (2015-10-29 00:25) [#10602]


Following the simulation study reported in Tournament Simulation and McMahon Bar Summary?, there is further useful information to be had regarding McMahon Bar Theory. The following comments follow exactly the same headings as in the main page.

Winning Chances

When the bar is optimally set, players at the bottom of the bar have a very low chance (typically less than 5%) of winning even half their games, let alone winning the tournament. The probability increases dramatically as the player grade moves up from the bar value.

The bar-depth (distance from maximum grade to bar grade) never exceeds 3 in the simulations. Its most likely value is 1, but a bar depth of 0 is possible in 30% of cases (for a 3 round tournament).

Unique Winner

For a 6 round tournament, 64 players above the bar would swallow the whole tournament in many cases. Clearly the desire for a unique winner cannot always be satisfied. However, the probability of a unique winner in a 6 round tournament is about 60% at the optimal bar setting, so tie-breaks are not always necessary.

Sufficient Opponents

In some tournaments it can happen that the few strongest players have no meaningful opposition all the way to the end. For example consider an entry like 1x3d, 3x2d, 0x1d, 2x1k, 1x2k, 1x3k ... in a (real-life) 6 round event. To get 7 players in the top group means setting the bar at 2k, and this means the 3d might play the 2k in round 1 - a very unfair game!. If we raise the bar to 2d (so there is no gap in the bar group) then the 3d starts running out of viable opponents at after round 3, but at least then will be playing the best of the rest.

Magic Numbers

Eventually there will have to be games with unequal McMahon scores if the magic number is less than 2^R. One of the magic numbers often assumed is that the bar population should be even. However if this comes at the cost of possibly very unfair games, then a case can be made for an odd bar.

Criteria at odds

The magic number ideas are useful only when the higher grades are densely and evenly populated at the higher grades. In reality this is often not the case, and the bar setting is sensitive to the actual grade distribution. The methods developed in Tournament Simulation exploit the winning probability between players of any grade to produce an optimal bar setting.

Rule of thumb

There is actually quite a lot of disagreement in the tables, especially when the number of rounds is 6 or more. It is very difficult to provide a recommendation which depends solely on the number of rounds. Nevertheless the above rule of thumb lies within the range of bar populations obtained in the simulations for all rounds from 2 to 10.


  • The bar population should lie between R and 2^R where R is the number of rounds.
  • The bar-depth should preferably lie between 1 and 2 and should not exceed 3.
  • If there is a grade with zero players near the top grades, carefully consider the implication of reducing the bar to below the gap. This increases the bar population and may lead to excessive bar-depth.


It is common practice in many Go organisations to disallow handicaps for players above the bar. In the French system, handicaps above the bar are minimised rather than forbidden.

In the extreme case of a very fragmented entry it may be impossible to satisfy both of the conditions in the useful rules above. If the bar setting means that 3d players are paired against 2k players then some means should be found to level the playing field. If this is not possible then some other pairing method such as Swiss or Swiss Handicap, or even Random Handicap could be considered.

Another possibility is to stick to basic McMahon rules, but allow some handicap games above the bar. Indeed the EGF has recently removed the restriction of handicaps above the bar for class A tournaments.

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