geoff: McMahon Bar Summary (2015-07-10 14:22) [#10556]

Extensive Monte Carlo simulation has established the existence of a unique equilibrium grade Gc for a given entry in a tournament paired by Random or Swiss rules. I chose to start with Random pairing (play anyone, but no repeat games) because it seemed the simplest, and I wanted to see how often we get failure to pair everyone. Game results were simulated using a best-fit [3] to the EGD Winning statistics data.

The grade Gc has the property that the mean grade of opponents of group Gc is approximately equal to Gc (within 0.5 grade). In every Random or Swiss tournament, the average tournament score is half the number of rounds; call this the mid-score. The grade group Gc also has the property that the average score for the group is approximately equal to the mid-score.

For McMahon tournaments, the mid-score for a group of players of the same grade is the group's initial McMahon score plus rounds/2. By varying the bar we find that there is a specific value - again I call it Gc, which has both of the above properties when applied to the players at the bottom of the bar.

This means that in McMahon tournaments, if the bar is set to Gc, players at the bottom of the bar should see a fair mix of opponent grades and expect to win half their games. A further result is that for groups with grade below Gc, the probability of scoring above the mid-score is infinitesimal. For the group Gc the probability is greater than zero, and for the groups above Gc it is significant.

The technical details are presented in the reference [4] given in the parent page, and I look forward to comments and discussions here.