## Bridgeditalia.it

**The New WBF IMP to VP Scales**
**Technical Report of WBF Scoring Panel**
*Technical Panel*: Max Bavin, Henry Bethe, Bart Bramley, Peter

**Introduction**
This documents presents the theory and algorithms for producing the newWBF conversion tables from:
The continuous scale gives a unique Victory Point (VP) to two decimalplaces for each integer IMP margin. The discrete scale, similar to existingWBF scales, gives a range of IMPs for each integer VP score. VP scores forboth scales range from 0 VPs for a maximum loss, to 20 VPs for a maximumwin. A drawn match results in a VP score of 10–10 to each team. If teamA wins

*V *Victory Points, then its opponent, team B, wins (20

*− V *) VictoryPoints.

These scales diﬀer in two signiﬁcant ways when compared to the old scales.

1. The continuous scale removes the phenomenon of ‘cusping’, where one
extra IMP gained could have led to a full extra VP. In the continuousscale, each extra IMP leads to a non-increasing fractional gain in VPs.

2. The new discrete scale removes anomalies in the old scales. In partic-
ular, the IMP range is non-decreasing for each subsequent integer VPscore. The old WBF scales violated this concavity rule, particularlyfor small IMP margins.

Both new scales (continuous and discrete) ensure that IMPs earned in a closematch are worth more than subsequent IMPs earned in a run-away match.

The new continuous scale is identical, except for some minor corrections, tothe USBF scale devised by Henry Bethe, that has been successfully used inthe round robin phases of North American team trials for some years.

**Theoretical Background**
Both continuous and discrete scales are based on an exponential mappingof IMPs to VPs described by the following formula
1.

*I *is the IMP margin (a non-negative integer);

*V*0 = 10 is the VP
0 when

*I *= 0);

*X *is the blitz
(maximum win) IMP margin determined by the formula:
where

*N *is the number of boards played in the current match. Notethat

*X *will in general not be an integer. The two parameters (

*V*0

*, X*)allow for possible variations in the choice of VP range and blitz pointrespectively.

2. Formula (3) comes from a study of some 200,000 individual board
records from BBO matches. These data indicated (when assumingroughly equally matched teams) a symmetric distribution with zeromean and standard deviation equal to 7.51 IMPs per board.

distribution is highly non-normal with secondary peaks at

*±*6 and

*±*11 IMPs, due to non-vulnerable game and vulnerable game and slamswings. If results for multiple boards are independent we can safelyuse the Central Limit Theorem to infer that the standard deviation
for

*N *boards will be 7

*.*51

*N *IMPs. Thus the blitz point

*X *= 15

*N*is seen to be very close to two standard deviations from the mean.

3. Henry Bethe also inferred from World Championship and other round
robin teams matches that the median IMP margin is approximately
5

*N *IMPs. If we assign this IMP margin to a VP score of 15 VPs(assuming a 0–20 VP scale) then about half the winning results willlie in the range 10–15 VPs and half in the range 15–20 VPs. Commen-surately, if

*V*0 is the VP score for a drawn match, 2

*V*0 is the maximumVP score and the median condition above becomes
Bethe also found from the same data, that between 5% and 8% of
all such encounters had margins that exceeded 15

*N *IMPs, a resultwhich is consistent with the roughly two standard deviations blitzpoint noted in item 2. above.

4. The parameter

*τ *is the so-called

*golden mean *and

*R *is its cube. Thus

*τ *and

*R *are just numerical factors and to six decimal places are givenrespectively by

*τ *= 1 ( 5

*− *1) = 0

*.*618034;
How these factors come into the scale is a direct consequence of theequations described in Item 6. below.

5. The mapping described by Eq(1) is a monotonic increasing and con-
cave exponential function of the IMP margin

*I*. That is, its ﬁrst andsecond derivatives (with respect to

*I*) are respectively positive and
negative. The concavity condition

*d*2

*V < *0 is regarded as an impor-
tant property of the IMP to VP conversion and ensures that VPs aremore sensitive to smaller IMP margins than to larger ones. Note thatlinear conversion scales, to which the old WBF scales approximate,have constant sensitivity at all IMP margins below the blitz point. Ofcourse, above the blitz point, the sensitivity is zero as each new IMPgained has no eﬀect on the maximum VP score (2

*V*0 = 20) alreadyattained.

6. The actual motivation for the exponential scale is discussed here and
in the next two items. Non-mathematicians can safely skip these tech-nical issues.

Let

*x *be a hypothetical IMP margin (not necessarily an integer) and

*V *(

*x*) be its corresponding VP score. We deﬁne the sensitivity

*S*(

*x*) ofthe scale by the ﬁrst derivative and the concavity

*C*(

*x*) by the negativeof the second derivative (which means

*C*(

*x*)

*> *0). That is

*C*(

*x*) =

*− d*2

*V*
Then the exponential scale deﬁned by Eq(1) is the unique solution ofthe following sensitivity equation and constraints (here

*A *and

*B *are
The sensitivity is modelled as a linear decreasing function of the VPscore. The three constraints respectively refer to the draw point, theblitz point and the median point – all mentioned previously.

7. Other models for the sensitivity are of course possible. For example:
The quadratic and square-root scales tend to give extreme IMP to VPconversions, sensitivities and concavities while the exponential andlogarithmic scales are more moderate. The WBF Scoring Panel con-sidered all four models before adopting the exponential scale.

8. As the IMP margin is always an integer and because we round the VPs
to two decimal points, the new continuous scale is really only a pseudo-continuous scale. We are content to refer to this scale as a continuousscale mainly to distinguish it from the discrete scale considered laterin this report.

The continuous WBF scale requires only the function deﬁned by Eq(1).

However, the discrete WBF scale requires its inverse. That is, given thewinner’s VP score

*V *, the IMP mapping

*I*
log

*{*1

*− *(1

*− R*)(

*V /V*0

*− *1)

*}*
provided

*V*0

*≤ V ≤ *2

*V*0. Note that both Eqs(1) and (8) apply only to thewinner’s VPs and IMPs respectively. Naturally, if

*I *is the IMP margin,then the winner’s and loser’s actual IMP scores must have been +

*I *and

*−I*
The next two sections describe in detail the algorithms that produce thecontinuous scales (to two decimal points) and the discrete scales for anyprescribed board numbers in a given match.

**The Continuous VP Scale**
The IMP margin is any non-negative integer, whereas the continuous VPscores are presented to two decimal places. Each IMP margin leads to aunique VP score. The VP range is from zero VPs (maximum loss) to 20VPs (maximum win). A draw corresponding to zero IMP margin resultsin a 10–10 VP score. The algorithm below determines the winners score of

*Vwin *(VPs) say. Then the loser will receive

*Vlose *= (20

*− Vwin*) (VPs).

The pseudo-codes below should permit programmers to convert the codesto their preferred programming language.

**The IMP-to-Continuous-VP Algorithm**
1. For IMP margin

*i *= 0 to ceil(

*X*) in steps of 1–IMP compute the VP

*i.e. *for

*i *= 0

*, *1

*, *2

*, · · · , *ceil(

*X*) and

*X *=

*N *with

*N *= no. of boards.

*Remarks*:ceil(

*X*) is the ceiling-function of

*X *deﬁned to be the smallest integerexceeding

*X *(

*i.e. *round towards inﬁnity). The function round(

*Y *)takes any real number

*Y *and rounds it to the nearest integer value.

A value ending in exactly point-ﬁve is rounded up to the next integer.

The function

*V*
0

*, X *)) is given by Eq(1). This formula essentially
gives the VP’s to two decimal places, while cutting oﬀ the maximumVP value at 2

*V*0 = 20 VPs.

2. The rounding in step–1 of the algorithm sometimes produces a viola-
tion of the Concavity Rule. Step–2 is designed to correct for all suchviolations. Generally the corrections to

*V *amount to only 0.01 of aVP. The concavity rule can be represented as follows. If (

*Vi−*1

*, Vi, Vi*+1)
is a sequence of three consecutive VPs corresponding to IMP margins

*I *= (

*i − *1

*, i, i *+ 1), then the Concavity Rule states:
(

*Vi*+1

*− Vi*)

*≤ *(

*Vi − Vi−*1)
That is, consecutive changes in the VP scale must never increase.

Computer code to correct for concavity violations depends on the ﬁrstdiﬀerence operator Diff deﬁned below.

Let

*V *= [

*V*1

*, V*2

*, . . . , Vn*] be a vector of length

*n*. Then the ﬁrst diﬀer-ence of

*V *, is deﬁned by the vector (of length (

*n − *1))
Diﬀ(

*V *) = [(

*V*2

*− V*1)

*, *(

*V*3

*− V*2)

*, . . . , *(

*Vn − Vn−*1)]
The pseudo-code to correct concavity violations is presented next.

*◦ *Recursion to correct violations
m := minimum i such that ddV(i) > 0if m > 1
V(m-1) := V(m-1) + 0.01Recompute all ddV(i) as above
end (if loop)TestViol := (m is not empty)
(a) Note the expression (ddV > 0) is a vector of zeros and ones
(a boolean vector). Thus TestViol := sum(ddV>0) > 0, is aboolean variable (0 = false, 1 = true) which determines if a con-cavity violation has occurred. If there are no such violations,then all the second diﬀerences ddV(i) will be either zero or neg-ative and hence each term ddV(i)>0 will be zero (

*i.e. *false). Aconcavity violation will therefore show up as a positive sum(ddV)and then TestViol will equal 1 (

*i.e. *true). The two zeros atthe start of the vector ddV are necessary in order to avoid inﬁniterecursions.

(b) The above routine is a recursion which will correct all such de-
tected concavity violations. For example for the case of 8 boards,only one correction is required. However for a 60 board match 24correction are required.

(c) The algorithm above is not unique in correcting for concavity vi-
olations. Other correction algorithms are possible, but we havefound the above to commonly require fewer iterations to removeall violations, particularly for matches with larger numbers ofboards.

These other correction algorithms, which include the
USBF scales incidentally, will diﬀer from the recommended ver-sion above by generally 0.01 of a VP only. So while these diﬀer-ences are tiny and will unlikely change the result of a tournament(let alone a match), consistency across the bridge world will bemaintained by everyone sticking to the same algorithm.

A spread-sheet accompanying this report gives the calculated continuousVP scales for several popular board numbers. Programmers can test theircodes by comparing with the tables in this spread-sheet.

**The Discrete VP Scale**
Generally, in major national teams events, the organizing body will want touse the new WBF continuous scales. However it has also been anticipatedthat in some lower level events (

*e.g. *congresses and sectionals) the organiz-ing body may prefer to stick to a discrete VP scale, perhaps to avoid theissue of decimal point scales.

This section of the report describes the algorithm to compute discrete VPscales which are commensurate with the new continuous scales.

We shall again describe the winners VP score which will now be an integervalue in the range [0

*, *20]. The mapping from IMPs to VPs while still depen-dent on Eq(1), or more accurately on Eq(8), will no longer be one-to-one.

Several integer IMP scores will lead to the same integer VP score. Thus foreach VP score

*V *, the mapping will generate a range of integer IMPs

*Ia *to

*Ib*. The end-points of each IMP range,

*Ia *and

*Ib*, are generally referred toas the ‘cusp points’.

**The IMP-to-Discrete-VP Algorithm**
1. Deﬁne the mid-score VPs,

*Vm *in the range (

*V*0 + 0

*.*5) to (2

*V*0

*− *0

*.*5) in

*Vm *= [10

*.*5

*, *11

*.*5

*, *12

*.*5

*, . . . , *19

*.*5]
2. Next use Eq(8) to compute the corresponding IMP scores

*m *= (1

*, *2

*, . . . , *10)
The integer IMPs

*Im *= [

*I*1

*, I*2

*, . . . , I*10] determine a ﬁrst-pass scale:
The ﬂoor-function of

*X *is deﬁned to be the largest integer not exceed-ing

*X *(

*i.e. *round towards minus inﬁnity). Note that the ﬁrst or drawrange of IMPs is [

*−I*1

*, I*1] rather than [0

*, I*1].

3. The corresponding concavity condition for the discrete scale is:

*I*2

*− I*1

*≥ *2

*I*1 + 1
(

*Ik − Ik−*1)

*≥ *(

*Ik−*1

*− Ik−*2);
That is, consecutive changes in the IMP range must never decrease.

The pseudo-code to identify and correct for concavity violations ispresented below.

*◦ *Define vector

*J *= [

*−*(

*I*1 + 1)

*, I*1

*, I*2

*, . . . In*];

*◦ *Recursion to correct violations
m := minimum i such that ddJ(i) < 0if m > 0
I(m) := I(m) - 1Recompute all ddJ(i) as above
end (if loop)TestViol := (m is not empty)
The above algorithm to correct for discrete concavity violations is very simi-lar to the corresponding continuous algorithm of Section 3 and can be inter-preted in the same way. We have tested this algorithm on all board numbersfrom 4 to 100 and no concavity violations were detected after the correctionrecursion was employed.

**Discrete Range Violations**
The design of the discrete scale can generate another problem, which fortu-nately is only a minor one. We call this problem a

*range violation *which canbe described as follows. Using Eq(8) we can determine an IMP score ˆ

*V *can be any of [10

*, *11

*, *12

*, . . . , *20].

*I *values will not in general be integers.

*I *values to be bracketed by the integer IMP ranges
computed by the discrete algorithm described above. That is, we wouldlike, for each

*i*
If this inequality is not satisﬁed for some

*i *in the range 2

*≤ i ≤ n*, then wehave a range violation. In words, this means that the discrete IMP rangedoes not bracket the continuous scale from which it was derived.

We have tested all board numbers from 4 to 100 and found only threeinstances of such range violations.

*N *= (5

*, *6

*, *7) and generally the sizes of the violations were quite small. Wesuggest, for the present, that we can live with these minor variations untilsuch time as a smarter joint concavity-range violation correction algorithmbecomes available.

The accompanying spread-sheet contains a page of discrete scales for thesame board numbers displaying the continuous scales. All these scales (bothcontinuous and discrete) have been computed using the algorithms describedin this report.

Please direct all technical enquiries to:

**Peter Buchen**

*email*:

Source: http://bridgeditalia.it/wp-content/uploads/2013/06/WBFAlgorithms.pdf

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