Should District 9 have resigned?

In the finals of the GNT, District 9 (D9) was down to District 21 (D21) by 77 imps after 45 boards, with 15 boards to play. They chose to forfeit the match. There are certainly many reasons to do so, "fatigue", "team fragmentation", "hopelessness", "hey, we have won this too many times already", "time for dinner", "sponsor's whim", etc. It is in their discretion to do so and I do not want to question their decision.

But what I would like to explore is a very simplified model of their likelihood of winning. I am going to use a  basic statistical model, looking at the results of the first 45 boards to determine a standard deviation, then periodizing it for the remaining 15 boards to determine their likelihood of recovering 78 imps using the easily criticized assumptions that the distribution of imps can be represented by a normal distribution around a mean and that the volatility of the first 45 boards is representative of the volatility for the remaining 15. It isn't difficult to challenge these assumptions in the abstract, and given that Meckwell with what may be the greatest record for comebacks in the history of the game plays for D9, they have additional problems of credibility in their application in this case.

But you need to start somewhere and the results may prove to be instructional. I have used the board scores in the BBO archives. (There is a one imp discrepancy between the totals and the deal by deal scores. I used the deal scores.) Over 45 boards, D21 averaged +1.73 imps per board. To compute the standard deviation, a measure of volatility, one sums the squares of the difference between the board score and the mean, divides by the number of boards, and takes the square root of the result. I computed a standard deviation of 7.86. (That is 2783/45=61.84, the square root of which is 7.86.)

Traditional statistical analysis would say that 68% of the occurrences should be within one standard deviation of the mean, 95% within two standard deviations, 99% within three. For example, the model would predict that 68% of the time, the score on a board will be between +9.59 (1.73+7.86) and -6.13 (1.73-7.86) imps in favor of D21.

To determine what that means over the 15 remaining boards, one periodizes the result by multiplying the standard deviation by the square root of the number of periods. 7.86 times the square root of 15 equals 30.4. D9 had to win back 78 imps; two standard deviations (95% of the time) is 60.8, three standard deviations (99%) is 91.2. But this is the probability of both tails and we want to look at just one, so the result is halved. In short, the model predicts that approximately 99% of the time, had they played on, D21 would have prevailed.

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