Horse

Rating

W

L

%

Anchovy

90

16307

33698

32.6

Butler

89

12536

37469

25.1

Coconut

88

9553

40452

19.1

Donkey

87

6740

43265

13.5

Einstein

86

4869

45136

9.7

The relationship between a horse's rating and its winning percentage in this framework can be seen as nonlinear. Start at the bottom with Einstein: if his trainer could improve him by one point, his chance of winning would go from 9.7% to 13.5%, the same as Donkey. We say there is a marginal improvement of 3.8% at a marginal cost of 1 point.
The next marginal change of 1 point takes Einstein from Donkey to Coconut, as it were. Reading from the table this is 19.1% minus 13.5% or 5.6%  a bigger marginal improvement than from the step from Einstein to Donkey.
And so, in this example, there is an exponential growth in the marginal benefit to a horse's chance for every marginal point that it improves. But there is obviously an upper limit, a bound, to this imposed by the fact that no horse can have a greater than 100% chance of winning.
So, let's run the simulation again, holding all things equal expect that the difference between horses is now 2 points rather than 1 point. Coconut is still the same 88 horse but his rivals are spread out more in terms of ability; there is more variance in their exposed merit, in other words:
Horse

Rating

W

L

%

Anchovy

92

19317

30688

38.6

Butler

90

13265

36740

26.5

Coconut

88

8642

41363

17.3

Donkey

86

5439

44566

10.9

Einstein

84

3342

46663

6.7

Horse

Rating

W

L

%

Anchovy

98

27652

22353

55.3

Butler

93

13533

36472

27.1

Coconut

88

5876

44129

11.8

Donkey

83

2232

47773

4.5

Einstein

78

712

49293

1.4

Now, Anchovy is 54 on favourite and Einstein is just about a 1001 poke. This example perhaps resembles a Group race whereas our first simulation could serve as a proxy for a handicap. Let's reexamine the marginal benefit of improving from Coconut (rating 88 again) to Anchovy (now rated 98).
The marginal improvement is 43.5% at a marginal cost of 10 points. So, the marginal benefit of one point is now only 4.35%. It is obvious that the more spread out the ratings become, the less benefit there is to a horse's chance from a marginal improvement of one point. Hopefully, this is exactly what most punters would intuitively understand.
However, it is important to remember we are dealing with five horses who have exactly the same potential for improvement. If you remember the last blog, I noted that, in some situations where the subpopulations of horses are not connected  such as threeyearolds and older horses on the Flat, Dubaian form and European form, or novice hurdlers in Ireland and British handicap hurdlers  a horse's exposed ability is negatively correlated with its potential for improvement. (This was rephrased in several of the superb Twitter responses I received subsequently in the more familiar terms of horses being exposed/unexposed.)
So, now let's give one of our horses, Butler, greater upside than its rivals. To do this mathematically from my original sample of the Beyer speed figures of older horses in the US, I simply divided the population into subpopulations of horses who had a difference in average starts of five and recalculated the parameters of the distribution from which the Monte Carlo simulation makes a random draw. (For the technically minded, this second population had a fatter righttail and a lower peak, representing a greater chance of ratings distant from the mean and a higher standard deviation).
Now, Butler belongs to this lighterraced group and Anchovy, Coconut, Donkey and Einstein remain as they were. Let's go back to the first example, the handicapstyle encounter between closely matched rivals in which their median ratings are separated by only a point. Here is another reminder of that initial out turn of 50,000 simulations:
Horse

Rating

W

L

%

Anchovy

90

16307

33698

32.6

Butler

89

12536

37469

25.1

Coconut

88

9553

40452

19.1

Donkey

87

6740

43265

13.5

Einstein

86

4869

45136

9.7

Now, look what happens when Butler belongs to a cohort of horses who have raced five less times on average:
Horse

Rating

W

L

%

Anchovy

90

14832

35173

29.7

Butler

89

15797

34208

31.6

Coconut

88

8628

41377

17.3

Donkey

87

6255

43750

12.5

Einstein

86

4493

45512

9.0

...with the following placings:
1

2

3

4

5

14832

11507

8438

7967

7261

15797

9278

8187

8482

8261

8628

10990

10646

9925

9816

6255

9839

11306

11248

11357

4493

8391

11428

12383

13310

AVG

STD

MAX

MIN

MEDIAN

87.4

7.90

108

48

90

87.3

8.45

115

40

89

85.4

7.98

106

37

88

84.4

7.83

107

39

87

83.4

7.89

104

40

86

Anchovy has a higher rating than Butler going into the race (90 to 89) and the difference in their median figures is this same 1 point in 50,000 races. Anchovy's average rating (87.4) is also still higher than Butler's (87.3) but because the latter has more upside (represented by a higher standard deviation of his performance level (8.45 to 7.90) and a higher peak effort (115 to 108) in rare situations, he is actually more likely to win a race between them (making all the same assumptions I laid out in the last blog).
Remember, the change to Butler's distribution of expected ratings was not dramatic in and of itself. Here are some situations in British racing which would represent much bigger likely swings between horses that have had different numbers of starts or opportunities to achieve their best ratings:
 Two horses in a maiden when one has run three times and the other once
 Nursery handicaps in which the top weight has beaten horses with low ratings while its opponents have been beaten by rivals with bigger ones
 Irish novice hurdlers against British ones that have run in the Betfair Hurdle against exposed handicappers
 Dubaitrained horses who have thrashed locals and are now facing fancy outsiders with big figures in Group 1 races round the world
I'm certain you can list many other situations in which the opportunity cost of achieving a rating has not been discounted before projecting a horse's chance.
If you were confused by what I was saying about My Tent Or Yours, no implication was being made about his Betfair Hurdle form or the strength of the Supreme Novices' form or whether you would go skint laying horses who were clear toprated at the Festival (whether you go skint has far less to do with your ability than your temperament, in any case.)
Instead, the point is that if you were trying to create a computerised SP and one of your inputs was (hopefully) a horse's form, there must be a term in the equation representing opportunity cost, so that the likely distribution of all its future ratings reflects the importance of prior opportunity. It is not an overcomplication, it is an extremely important correction to an oversimplification of some who make bogus inferences from figures, especially on the television.
What we are working towards here is a mathematical understanding of the importance of upside compared with exposed ability. Many of you whom I have met over the years have an extremely good intuitive grasp of this nonlinear dynamic, one ability which makes them a superb punter. My brain does not work like yours, unfortunately, in that I have to quantify my edge before having a bet.
I do not trust my instincts, partly because the single most dominant influence over my cognitive development was my maternal Grandfather who impressed on me the beauty of numbers and that everything in life  even those things assumed to be the preserve of the aesthete  were underpinned by quantitative processes. In some ways, he has made my life hell but mostly I feel lucky to have been inspired in this way. And, my mantra is that most things which come from your superior intuition are amenable to a test for cognitive bias. Or, as in the case of my friend Tom Segal, just evidence of a brilliant, intuitive mind.