The EMP becomes the IMP.

International Match Points are used in team matches and occasionally in club match point duplicate (Butler Imp Scoring). With standard scoring, one board with a swing potential of 2000 or more can overwhelm a x24 team of 4 match if all the other boards are part score comparisons or flat.  

The objective of converting point difference to IMPs is to have all boards contribute to a match result. The imp conversion constrains the effect of large swing boards and raise the significance of part score comparisons. IMP scoring is positioned between the extremes of total point scoring and point-a-board avoiding the inherent defects of either method but then introducing some deficiencies of its own. The IMP scale is a type of compression: The greater the points difference, the more points are squeezed into each successive IMP step.

The origin of the IMP is the EMP. The European Championships just prior to WW-II used a scale with a 12 EMPs maximum for points differences of 2000 and above. In the 50s a revised scale specified a 15 EMP maximum for points differences of 4000 and above. In the 60s the WBF set up the sub committee that created a 25 IMP scale for international competition which has matured to become the 24 IMP table of the present day.

The deficiencies of total point scoring for teams of 4 competition is exemplified by the result of the 1957  Spingold Teams .  ALAN TRUSCOTT, bridge columnist NEW YORK TIMES, tells the story  in a short  piece  he crafted in 2003. The Spingold event  is the end competition of the American Nationals.  Failing by 2,000 total points at the start of the final quarter, the contenders  had needed a big score  and were fortunate enough to be presented with a grand slam opportunity where it was possible to play, against the odds, and secure an overall victory  by a very slender margin. The sensitivity of the win to a clever, but nevertheless contrived play, led the way for the Americans to adopt the European method.

The conversion table printed on most personal score/system cards does, of course, carry all the characteristics of ‘design by committee’ but the detail does not seem to be readily available. However the basic principles can, I think, be uncovered.

 THE CURRENT TABLE

Point Difference	IMPS	Accumulation Rate 10pts
0	0	0.5	1/2
20	1	0.333333	1/3
50	2	0.25	1/4
90	3	0.25
130	4	0.25
170	5	0.2	1/5
220	6	0.2
270	7	0.2
320	8	0.2
370	9	0.166667	1/6
430	10	0.142857	1/7
500	11	0.1	1/10
600	12	0.066667	1/15
750	13	0.066667
900	14	0.05	1/20
1100	15	0.05
1300	16	0.05
1500	17	0.04	1/25
1750	18	0.04
2000	19	0.04
2250	20	0.04
2500	21	0.02	1/50
3000	22	0.02
3500	23	0.02
4000	24	0.00

Shown as a staircase graph, the scaling reveals an initial high RATE of accumulation of IMPs that diminishes as the point differences increases. Indeed the basic defining relationship is not the absolute imp value per se, but the imp accumulation rate or IMP increment and this relationship is part heuristic (seems right) and part mathematical. 

The rate at which imps accumulate through each table step is the step change in IMPs (always 1) divided by the width of the step measured in points. This value is the fractional imp change per point but it is more convenient to express the accumulation rate as a fractional imp change for 10pts (just multiply by 10). The 10pt imp rates are revealed to be common fractions of an IMP (1/2, 1/3, 1/4..1/10 etc). 

The ‘imping’ of point difference scores means allocating them within the 24 steps of the current table in such a way that that the desired bias appears in a consistent regular manner at least before any tailoring is applied.

With ‘y’ imps and ‘p’ points difference, the imp accumulation rate is (δy / δp) where δy is the change in imps corresponding to a change δp in points difference at a particular position along the conversion curve.

The imp accumulation rate is simply made inversely proportional to points difference ‘p’, and a good fit to the current table can be achieved.

The point conversion rate is the accumulation rate represented as an imp fraction for 10 pts.   The graphic of the current table data shows the underlying inverse proportional relationship.

The curve is imagined to be  continuous, (not stepped) so we can apply  elementary calculus: (δy / δp)->(dy/dp)

The adjustment ‘a’ is necessary to set the rate at 0 points of difference.

On integration this yields the conversion function…

” Ln” is the natural logarithm. The constants ‘R’ and ‘a’ are determined by the initial step and end values. If you are scared of natural logs use 

where the Log is base 10. 

To solve, set y to 24 and p to 4000 (a = 100 approx) R emerges as 6.5.

The initial imp step is defined as 20pts.  The ‘a’ parameter emerges as 120

The relation  has almost defined itself!.

It  is similar to the logarithmic chromatic frequency scale used on guitar with set fret positions.

The logarithmic conversion ensures the de-emphasis and reduced differentiation of big swing scores of 2000 plus and ensures an opposite emphasis for small scores less than 100.  However adjustment has obviously been made ‘on top of’ the base relationship. It would be desirable to separate common small swing types of scoring event (game/part score partscore/partscore game/sacrifice game/one down etc) where the tables might otherwise lump them together. So more imp steps than would be ‘natural’, would be applied by design, to a range where enhanced discrimination is considered beneficial. There might also be issues of equity between variants of the common scoring events that require imp boundaries to be shifted.   

A prototype table is probably generated for 10pt  boundaries, whole imps,  and common fractional rates.   This would then be tailored  by the “makes sense” boundary adjustments. Those tailored adjustments result in the flat regions and overvalues in steep sections of the conversion rate curve.

But the BIG question is why 24.  It is reasonable though….

Split the piece in two at the 2000 point difference mark, normal territory below, abnormal above. At a flat rate of an imp per 100 there are 20 imps below the mark.  In the abnormal region make the step 500. So a crude imp scale by heuristic rules has 24 steps assuming 4000 maximum.

The rest is fiddling having already done the strumming.