[tournament-org] WhereIsTheBar

[Geoff Kaniuk]

There is overwhelming support for a bar at 1k with a spread from 2k to 
1d, and a few suggestions allowing handicap above the bar.

It seems that we expect the 5d to win no matter where we set the bar, 
but in order to run a McMahon tournament we are going to set the bar to 
1k anyway.

One guide to setting the bar is that all players above the bar should 
have a reasonable chance of winning the tournament. As stated we assume 
that all players are correctly rated. This means we know the win 
probability for each pairing.

Let us suppose the bar is at 1k in our entry of:
5d 1d 1d 1k 1k 2k 2k 3k 3k 4k 4k 5k .....

Then given a plausible tournament, the top players have these chances of 
winning all three games:

PLAYER	OPPONENTS	PER ROUND PWIN	  PROB WINNER
1k	1k 1d 5d 	0.50*0.40*0.02 =  0.40%
1d	1k 1d 5d	0.60*0.50*0.06 =  1.8%
5d	1k 1d 1d 	0.98*0.94*0.94 = 86.6%

I think this makes it clear that the 1k and 1d have effectively no 
chance to win this tournament. Not much changes if you set the bar to 1d.

Going back to basics, the McMahon system is designed to provide a fair 
pairing at every round. It does this by assigning an initial MMS 
determined by your grade. The fairness comes about because we assume 
grades are realistic and always pair players on the same MMS where possible.

The winner is the player who ends on the maximum MMS. If there is only 
one strongest then he or she will have a massive advantage. Hence we 
have a bar - and in the old days there were plenty of 4d and 5d around 
in the top group so no problems.

If an an even pairing is not possible, then it is common for players 
below the bar to play with handicap - often MMS difference-1 but this 
can of course be varied.


We have agreed so far that setting the bar at 1k or 1d produces an 
unfair tournament. Therefore in order to continue play in the spirit of 
McMahon, we should consider raising the bar to 5d. For the sake of 
clarity this is equivalent to setting the bar at 6d.

In the example given, setting the bar at 1k produces an odd number in 
the bar, so one player chosen at random will have to play down.  The 5d 
therefore might play a 2k in round 1.

Setting the bar at 6d avoids this kind of problem. We pair for the 
smallest MM difference so the 5d will play the 1d in round 1 with a 
handicap of 3 stones by usual rules.

This levels the playing field somewhat but calculations show that this 
still favours the 5d and better would be: handicap is the straight MMS 
difference.

My conclusion is that in these anomalous cases setting the bar anywhere 
without handicap goes against the basic principles of McMahon pairing.

In looking at variants of setting the bar with handicap say in a 
super-group just leads to complications (e.g. when top players who miss 
rounds are supposed to be below the bar).

The simplest and most elegant solution to this problem is just to set 
the bar to 1 higher than the highest grade. Assign handicaps by straight 
MMS difference where needed. Then let McMahon do its job without any 
further interference.

Geoff


33 Ashbury Close, Cambridge CB1 3RW 01223 710582

On 02/10/2018 00:21, Richard Wheeldon via tournament-org wrote: