# Neuberg formula

In duplicate bridge pairs tournaments, the Neuberg formula is a method of fairly adjusting match point scores achieved on boards which have been played fewer times than other boards. The objective is to estimate the number of match points that would have been earned if they had been played the same number of times as the other boards, while also attempting to give the board equal weight to the others.

Originally developed by Gérard Neuberg of France, the Neuberg formula is in widespread international use.

A board might have been played fewer times than others because:

• the movement was not completed, or
• there was a phantom pair, or
• one or more plays had to be cancelled because of irregularities, entailing explicit percentage assignments for those plays.

## Details

The method is:

• Add 1 to the number of match points scored. (If the North American match point system is in use, where each comparison is worth one point rather than two, add a half-point instead.)
• Multiply by the number of times the board should have been played (this should be the same number for all the boards in the tournament) and divide by the number of times it was actually played.
• Then subtract 1 (or ½, whichever was added above).

## Example

• Board played 6 times.
• Most other boards played 7 times.
• Pair X scored 4 match points (out of 10).
• Then (4+1) x (7/6) - 1 = 4.8333 (out of 12).
• Pair Y scored 9 match points (out of 10).
• Then (9+1) x (7/6) - 1 = 10.6667 (out of 12).
• The scores are usually then rounded to the nearest 0.1, so 4.8 and 10.7 respectively.

## Criticisms

• Wrong to give equal weight to boards played fewer times. A result achieved by a pair on a board played fewer times is a less reliable, higher variance, estimate of the pair's performance than a result achieved on a board played more times. It is misguided to aim to give equal weight to e.g. a 100% win achieved on the less-played board; it unfairly penalises a pair who have achieved a 100% win on a more-played board, since the latter's 100% win was less likely to have occurred by chance.
• Failure to account for diverse partnership strengths. If you play a board, and a different play of the same board was cancelled involving a weak pair that you would have beaten, the Neuberg formula does not compensate you for your cancelled (presumed) victory.

## Gérard Neuberg

The formula was developed by Gérard Neuberg, a French mathematician. He died at the end of 2016: there is a brief obituary in the French Bridge Federation magazine (January 2017) .

## Other uses

The formula can also 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.[citation needed]