Mathematics help predict each nation's chance of Arc success

France's Treve wins the 2013 Arc with Japan's Orfevre and Germany's Intello following behind. Photo:

Sound statistical basis for believing Japanese-trained horses are set for an overdue breakthrough in the Prix de l'Arc de Triomphe.

Prix de l'Arc de Triomphe 1988-2013

The Prix de l’Arc de Triomphe – Europe’s greatest race – may be at a tipping point. This could be the direct result of this year’s boost in prize-money; more likely, it is a symptom of the increasing pan-global magnetism of richly endowed horse races. Either way, the Arc is arguably on the cusp of a revolution.  

Three leading horses from Japan – headed by Just A Way – bid to underline the growing influence of Thoroughbreds from the East. Success for either the brilliant Dubai winner, or highly talented rivals Harp Star and Gold Ship, would electrify their nation and change the face of the Arc forever. Make no mistake, Japanese success has been coming and it now seems a matter of when, not if.

The threat from Japan is as obvious as it is potent. Orfevre’s runner-up finishes the last two years only added to the agony of Nakayama Festa’s narrow defeat by Workforce in 2010. Before that pair, Deep Impact came to Paris on the back of brilliant form back home, but could finish only third to Rail Link in 2006 before being disqualified for a banned substance. And before him was the gallant El Condor Pasa, caught only by Montjeu in the muddy renewal of 1999.

John Gilmore’s article underlined the awesome dimensions of Arc day that has made Longchamp the focus of attention out East. As my study in March showed, this isn’t small-sample theatre played out in one race. If you look back to the multi-coloured group bar chart that illustrated the relevant trends, the reach of Japanese horses is obviously expanding by the year.

In this article, I am going to use another quantitative framework to assess the fluctuating force of Thoroughbreds from various countries in the Arc. I will detail the mathematical method separately, but for those who find such things an understandable turn-off, the focus will be on explaining in plain English how the graph featured in this article is made, and, more importantly, what it implies.

The motivation is this: Imagine if you took a random walk back through Arc history all the way to 1988. At any point, you could stop and peer ahead of you into the past, but also turn around and remind yourself of what had already happened as you strolled back through the years. At the moment in time you chose to pause, you would see things both in the recent past and immediate future with the most clarity, but you could also peer a few extra years either way. You would be ideally placed to evaluate the instantaneous prospects of each nation at that precise moment, and to gauge the gradient of ascent or descent from past to future that each country was experiencing right there and then.

First using a fancy-sounding but widely used framework from financial forecasting, the method chosen distills all available information provided by the Arc results. It constructs a running forecast of the chance of each nation’s Thoroughbreds (according to the location of their pre-Arc training base) winning the race billed cleverly by the locals as: "Ce n'est pas une course, c'est un monument" – "It's not (just) a race, it's a monument".

The gory mathematical details can wait until the end (click here to read about the methods used). Instead, let’s get straight to what the numbers say about each country’s record in the Arc.


In the period under study (1988-2013), French-trained Thoroughbreds won 16 of the 26 renewals of the Arc, or roughly 62 percent. If you calculate the area under the blue line on the graph by a technique called integration you might not have worried about since calculus at school, it adds up to a total of only 15.1 wins.

Moreover, that success has been achieved pretty steadily and is buttressed by plenty of placed efforts. So why is the blue line consistently a hair below the observed rate of success? And why is it heading downward?

The answer is that each point on the line is equally influenced by both backward and forward projection. So, in effect, the rising success of Germany and Japan forces the projection for French-trained Thoroughbreds downward at its right tail, so that it balances the projected success for all future Arcs with what has already happened.

In short, the numbers are saying that where we are now in 2014 is a potential tipping point, or point of inflection, in the modern history of the Arc. Ignoring the precise national make-up of Sunday’s race, France is still by far the most likely country to win the Arc, but the times they are a-changin’. The projected rate of French success in future is less than 50 percent.


You could attribute the Arc wins of Marienbard, Sakhee, and Lammtarra to Dubai, as they were achieved by Godolphin’s trainer Saeed bin Suroor. But, according to the definition of the study, they go down as British because the pre-Arc base of the horses was Newmarket.

In total, British-trained Thoroughbreds have won five Arcs since 1988. So the fact that the forecast line averages a 22 percent success-rate when the actual rate is 19 percent seems to represent systematic over-prediction. The reason for the discrepancy, of course, is that placed horses also predict future wins and British-trained horses have been placed 14 times, with plenty of others finishing highly ranked, a rate that suggests that they could have expected to pick up another win somewhere with a better roll of the dice.

More significantly, as with France, the expected rate of success for British-trained horses is falling. This is a factor of both a general slide in quality (as suggested by my previously mentioned World Thoroughbred Rankings study) and the growing competitive threat of other nations. For any given year in the graph, the sum of vertical percentages must equal 100, so, if others get better, you get worse by standing still.

However, the model’s gloomy projection for British Arc success during the next few years is irrespective of the investment there of the Qatar cousins Sheikh Fahad and Joaan al-Thani -- now just acorn empires compared with the mighty oaks that they might become. (The latter’s Al Shaqab operation has already boosted France’s Arc record via last year’s heroine Treve, trained by Criquette Head-Maarek in Chantilly.)  


The gold line not surprisingly peaks in 2008, given that Irish-trained Thoroughbreds won in 2007 (Dylan Thomas) and 2009 (Sea The Stars). But the expected success rate was rising even before that, given Sinndar’s win in 2000. In total, then, three wins for Ireland in 26 renewals is a win strike-rate of 11.5 percent, but the expected success-rate averages only 9 percent and predicts only two wins during the period.

The reason should now be clearer: Oscar Schindler (1996), High Chaparral (2002, 2003), and Soldier of Fortune (2008) are the only placed horses. The intuition is that with fewer representatives, there is more volatility in the projection and therefore the percentage call is to expect fewer rare events – in this case wins – than has been the case.

Of course, if you were projecting Ireland’s likely success rate using all the evidence in the public domain, rather than just statistical inference, you would be more optimistic - the mood of my articles on Aidan O’Brien and Coolmore earlier in the year.


The highly optimistic projection for Japan’s first Arc success in the next few years has a sound statistical basis, as well as fitting the wider sweep of global racing trends.

Of course, we have been here before. The area under the dashed red line adds up to 0.84 Arc wins. In other words, the model thinks it 84 percent likely that Japan should have already won at least one Arc. If you backed Orfevre two years ago like I did, that is a frustrating though highly understandable conclusion, but it isn’t the nature of Japan’s near-misses but their frequency and recency that cause the bullish projection.

Remember, however, that if you extend the Japanese line to this year, you might expect roughly a 14 percent likelihood of victory, with greater than that in subsequent years. But, according to the latest ante-post book held by Ladbrokes, for instance, Just A Way is 7-to-1, Harp Star 8-to-1, and Gold Ship 11-to-1. The equivalent percentages add up to 32 percent, and, even if we exclude the layer’s vigorish (or over round, as we say in Britain and Ireland) it seems like 14 percent is a bearish expectation indeed.

There are two things that account for the difference: 1) The model doesn’t know the horses taking part this year, and 2) The projections, in effect, extend to infinity, with each future year seeing the gradient of victory regress back towards long-term rates.

So, for Japan, the model for the years beyond 2014 is something like this: 16 percent, 16 percent, 15 percent, 15 percent, 15 percent, 14 percent, 14 percent, 13 percent, 13 percent, 13 percent, 12 percent, etc. In other words, the recent spike is likely to continue for a while then flatten out somewhat, as the further ahead you project, the more difficult it is to make outlandish projections because the chaos of future events dampens the signal; so, the more you should assume the longitudinal signal of success (the entire period 1988-2013) is a better prediction than any small section of it, however recent. (Or, as a physicist might say, momentum dissipates as energy is absorbed by the environment.)

Right now, we should lean appropriately on what has already happened in order to predict what might. But, as the numerous financial crises have proved, mathematical models tend to go wrong when the stability of a system is impacted by new forces with nonlinear effects. Japan might be on the cusp of global horseracing dominance, but it probably isn’t. And the projection represented by the gradient of the red dashed line is intended to balance the likelihood of those different outcomes.


More subtly than with Japan, German-trained Thoroughbreds are surfing a rising tide of success. More dramatically than Japan, they have already sealed the deal in the Arc, via Danedream’s success three years ago (the second victory for the country, after Star Appeal in 1975).

It’s my view that the German racing brain may be the most rational in the world. Their restrictions on breeding from stock relatively free of chemical supplements to hay and water, coupled with a bias toward staying stock, is likely to instil their Thoroughbreds with the vital components of soundness and stamina that are in retreat elsewhere in the world of racing.

This is no overnight success either: Lando, Borgia, and Tiger Hill figured prominently in Arcs toward the end of the 20th century, an effect that can be seen on the graph from the upward surge of the black line in the equivalent period. Under the line, the area sums to 1.42 wins, meaning that German-trained Thoroughbreds might very well have struck again recently.    


It is a while now since Italian racing benefitted from the man who is the greatest thinker in the history of the turf, in my view. Federico Tesio’s book Breeding The Racehorse (first published posthumously in 1958) contains more insight into the sport than the total of everything else I have ever read or heard in my 25 years in the sport. To my mind, he was clearly fully deserving of the overworked term “genius.”

Tesio didn’t survive to see the greatest result of his brilliant breeding knowledge, Ribot, win the 1955 and 1956 Arcs. But Italian Thoroughbreds have a wider and richer history still. However, since Tony Bin’s victory at the start of this survey in 1988, there hasn’t been much sign of an Italian revival, and that is why the green line is beginning to merge with the x-axis.

Therefore, as previously explained, the model cannot be bullish because it must contain all past events and a reasonable prediction of all future wins. The area under the graph sums to only 0.78 wins - in effect assuming that victory for Italy in 1988 looks increasingly like an outlier.


The backward-and-forward looking aspect of this model is one of its most desirable aspects. As I said, it doesn’t know that Just A Way and his two travelling companions are lined up for this year’s contest, only that the gradient of Japanese success, described first by El Condor Pasa and most recently by Orfevre, is increasing rapidly.

After Sunday’s race, the situation will look different because a year of projection has now been realised. If a Japanese-trained horses wins, the model will become even more bullish about Japanese prospects in future, as it attempts to catch-up or track what has happened and put it all into a probabilistic framework with the attempt to project what might happen.

You can therefore understand the nonlinearity concerned – and possibly also how stock-market crashes happen. “All models are wrong, but some are useful,” wrote the statistician George Box who was the joint-father of the family of techniques that underpin many so-called time-series methods.

He’s so right, and the most dangerous of all models – just like the most ignorant of all humans – is the one that assumes it knows all the parameters and how they will continue to behave.

At the same time, it is much better to try to evaluate complex systems constantly, and to allow them to drift as new evidence comes in, rather than to rely on human assumptions or received wisdom. If any hypothesis has validity, it should be difficult to falsify.

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