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Scoring a Duel |
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A duel is a series of deals played by the same set of
two pairs.
The result of each deal is scored in the fashion described
in „Scoring a Deal” – one of the pairs gains and the other one loses some amount
of points – say imps. For example, on five deals of the duel the scores may
look like: +7:–7
0:0 –2:+2 +11:–11 –1:+1.
Neither the sum of scores on all the deals (in our example
+15:–15) nor the average score on a deal (+3) are the proper
valuation of the result of a duel, since they do not take into account the
number of deals. The same overall result (or average result) is worth more
for the smaller number of deals, compared with the bigger number.
There is little doubt how the proper valuation should look like:
The probability distribution of the result of a
duel is the sum of distributions of a single deal, and as such it is approximately
a normal distribution with standard deviation P times greater (P = square root of the number of deals) than the deviation
on a single deal (approximately 5 imps).
Thus, results of a duel should be valued inversely proportionally to the
probabilities of their occurring. For example, if we want to value the result
using the scale 0, 1, 2,... 100, the positive part of the x-axis of the distribution
should be divided into 101 sections with equal integrals.
Such calculations were probably performed and the
result is a widely used method of scoring based on distributing 30 special
points between the two sides. These points are called Victory Points (VP),
hereafter referred to as Victors.
The tables converting imps to Victors use the close
approximation of the following formula:
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Win in Victors = |
where: R = Square root of the number of deals B = Balance for the winners – in imps with the exception
that: The winners are granted
at most 25 Victors |
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Examples:
1)
16 deals; over them one
pair won 37 imps, the second one 11 imps; thus, the balance is 37–11 = 26;
the win = 15 + 6.10 = 21 Victors; the distribution of Victors = 21:9
2)
16 deals; the result in
imps 100:10;
the win = 36 Victors; the distribution of Victors = 25:0
Using the Caracalla
flattening, 100 in the numerator has to be substituted by 30
and 100 in
the denominator has to be substituted by 1000.
In the past only 6 Victors were at stake, then this number was gradually
growing... reaching 30 nowadays. If 100 Victors were distributed we would
have the better perceived Percentage Victors:
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Percent win = |
where: R = Square root of the number of deals B =
Balance for the winners – in imps with the exception that: The winners are granted at most 90 percent |
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Using the Caracalla
flattening, 333 in the numerator has to be substituted by 100
and 100 in the
denominator has to be substituted by 1000.
Questions:
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1) |
To calculate the distribution of results
on a single deal. Due to oddities in the
present Bridge Scoring, this distribution will likely be odd as well. |
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2) |
To reproduce the construction principle
of distributing 30 Victors between the two sides. It could not be found !
(are the bridge circles idiosyncratic to mathematics?) |
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3) |
To
construct Victors for matchpointed duels (see Matchpointing
in „Scoring a Deal” ). |
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