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It's not quite clear whether the correct spelling would be MacMahon or
McMahon, apparently both are used, but people seem to tend to the latter.

Please make very sure you have read the section about Swiss
system carefully; much of what follows just builds on that foundation.

The problem

It is clear that the Swiss system runs into problems when the number of
players grows very large. Worse, there is no simple solution if one is to
keep to the idea of treating everyone equally. A halfhearted attempt is the
accelerated Swiss, but it is no more and no less
than that. More drastic methods are needed.

One good way to solve the problem is to hold qualifying tournaments. This
adds another layer of selection, in effect it adds many more rounds to the
tournament, and therefore very much improves the system's capacity to handle
players. This is a very practical solution.

However, sometimes the organizers just refuse to give way to the pressure,
they want everybody to have a chance to participate, and there we have a far
larger tournament.

The solution

In Go, the kyu/dan rating system lends itself very nicely to help. The
estimates of how much just one stone stronger player wins varies from 66%
all the way to 85%, which is a lot. Accordingly, the winnage of a player two
ranks lower is a very rare occurrence.

Basically what McMahon does, is awarding initial points for rank. In effect,
what happens is that the system pretends that everyone has played a large
number of rounds, and this has set up the order of players. Giving points
for ranks may sound radical, and it is, but is easily justified by the large
winnage of just one rank difference.

These points are called McMahon points.

The tournament

What one does next is start the tournament as if it were a Swiss system, but
considering the initial MM points equal to wins. Also, for later rounds, the
MM points remain, and wins are just added to that score.

However, the primary criteria for prizes and good success should never be
the current MM score, but the number of wins, just as in Swiss system. In
effect, MM points are just a tool for being able to do a meaningful pairing.

Doing the top

It's not all that easy, however. The organizer needs to cut the increasing
points somewhere. It is entirely possible that a 4 dan in good form can beat
a strong 6 dan, and go on to win the tournament. To counter losing this,
one sets up so called top bar, that limits the top group.

The criteria for the top bar is twofold, and very simple after all:

everyone who has realistic chance of winning tournament should be in
top group

maximum size of top goup becomes surprisingly small

For example, in a tournament with 5 rounds (2^5=32) I'd set up top group of
at most 12 players. Why? Because the players in the next group interfere
with the top group people in any case. If there are more participants who
are considered winning candidates, then one needs to increase the number of
rounds. Further, usually one wants also the second place resolved.

And then the middle

Even given the large winning probability for one stone difference, setting
up the so-called McMahon ladder to 1 rank for 1 group can be dire
mistaek. Typically the 'thickness' of the ladder should be at the very least
half the number of rounds in the tournament, preferably at least equal to
it. This avoids having overly deterministic pairings: I for one would never
want to see someone seeing all his/her pairings decided before even the
first game is started.

A good solution to the thinness of MM ladder is to combine ranks into score
groups.

Finally the bottom

A bottom bar should be set so that the bottom pairing doesn't get too
deterministic either. The same numerical guidelines should be observed as
when setting up the top group. However, since there is special pressure for
including people in the top group (the potential winners) but not in the
bottom one, it need not be any larger than your average MM group.

In case the tournament has an odd number of players, the normal thing to do
is give a free round to the lowest-placed player who hasn't yet had one, and
counting the free round as a win -- after all, it wasn't the player's fault.