Discussion

The way percentage scores depend on the number of participants in the classical method is unsatisfactory. Equal performances are rewarded differently. Looking for a method that would give better results when comparing sessions with different tops we arrived at Neuberg as a matter of course. Also the Ascherman system, promoted by Herman de Wael, leads to the desired result.

Introduction

To illustrate the unsatisfactory properties of the classical method I only have to point to the example given in the introduction of the discussion about the Neuberg formula.

One may approach the problems arising from different tops in a number of ways.

1. ignore (method 1, factoring)
2. Choose a standard value for the number of scores per board and convert the result of other boards to this value using a well defined procedure. (the Neuberg formula and alternatives 4 and 5).
3. Choose a measure for the result that does not show this dependence on the number of results (the Ascherman method).

Which method should we use?

The point score, x, of a board is a discrete variable with distribution p(x_i). We assume that the probability of a any result, x_i, is independent on the size of the field. On a certain board say, the probability to bid and make slam is 10%, irrespective of the number of players. That implies that the corresponding k_i and m_i, on average, are proportional to the size of the field. Only the methods of Ascherman and Neuberg obey this requirement. See Ascherman, eq. 9 and further.

The Neuberg formula and the Ascherman method lead to exactly the same results regarding the order of the participants. The only difference is that the mutual scores are somewhat closer together in the Ascherman approach.

A misunderstanding about Neuberg

There is a general consensus that Neuberg should be used for boards within one session that have a deviating number of regular scores. The cause may be an artificial adjusted score, or a board that was not played at some table, or a fouled board, or it may just be that the chosen movement contains board groups that are not all played the same number of times.

But for the case of a contest of several sessions, where the number of pairs varies per session, the consensus is less general. Yet, it is a misunderstanding to think that Neuberg was not intended for this application. In fact, we derived the Neuberg formula starting from the requirement that the percentage scores should be the same for a small field and a large field.

Neuberg and the NBB

[ for this discussion, see the Dutch version ]

Implementation in scoring software

Most scoring programs already use Neuberg for cases where within onse session one has boards with a different top. To apply Neuberg also to competitions of several sessions probably only small modifications are required. The session and the resulting percentage score may be left as is. Only when the results are incorporated in the overall competition standing the percentages are converted, and one may use equation (6) for this. At first, the participants may be surprised by this, e.g. when the session result shows 60.0% and the competition standing 59.5. A convincing explanation might be "we normalize everything to 16 pairs per session; this time there were only 12 pairs, so we applied a small correction".
The program should offer a choice for the value of N. This value should be the same for all sessions and rounds of the competition.

Changing over to Ascherman would be a more radical break with the past. We won't go into the practical implementation here.

Others about Neuberg and combining sessions

Among experts the use of Neuberg, also for combining sessions, is self-evident.

• Max Bavin writes about the Neuberg formula:
  However, it is acknowledged that there are difficulties, not the least being:-
  (a) it is difficult (very difficult) to understand; and
  (b) it is even more difficult (though not impossible) to perform such calculations without the aid of a computer.
  For this reason, and for this reason alone, the formula has not been written into the Laws of Bridge as a "must do it this way".
  http://www.ebu.co.uk/documents/laws-and-ethics/articles/neuberg-formula.pdf

  Some further quotes:

• A similar method can be used for example in a club competition when it is desired to give equal weight to scores achieved over a number of sessions, but there were different numbers of tables at each session. http://en.wikipedia.org/wiki/Neuberg_formula
• All methods are legal.
  EBU regs and current software require #1 ( = Neuberg, P.S.), which Max Bavin and Herman De Wael recommend.
  (David Stevenson on "Combining sessions with different tops") http://blakjak.org/lwz_ste0.htm
• Indeed in a personal conversation with Gerard Neuberg in April 1992, he confirmed that it is his opinion too, that the formula that is generally known under his name, should be used whenever different boards have obtained different numbers of results, regardless of the reason why this occurred.
  (Herman De Wael on the same subject) http://www.hermandw.be/bridge/calcula/calcul27.html

Final remarks

If you want a top shared by 3 pairs in a field of 10 to yield the same result as a top shared by 30 pairs in a field of 100 you automatically arrive at Neuberg or Ascherman.

None of the methods discussed takes into account that the average strength of the subgroup may be different from the group as a whole.

The Neuberg method should not be too difficult to implement and to explain. This is the appropriate way for those who want to cling as much as possible to old and proven methods.

But evidently also the Ascherman method deserves more interest, as it is a much more elegant solution.

So far the NBB (Netherlands Bridge Union) has never taken a serious interest in the Ascherman method. Thanks to Herman De Wael the method has been brought to live and saved from oblivion. In Belgium it has been used for several decades.

Ascherman came to his ideas 65 years ago. The proposition that in the Netherlands everything happens 50 years later is somewhat optimistic.