Dosage Late-Speed And Novice Hurdler Championship Race

As mentioned before, National Hunt racing is not my discipline. However, Cheltenham's four-day graded-stakes, championship meeting is a worthy challenge of one's handicapping skills. Effectively, this is the Breeders Cup of jumps racing. Lacking in-depth knowledge of this code, I seek to focus on novice races where historical, pedigree knowledge is as informative as current form. Turning to the Supreme Novices Hurdle (G1) and, given my working assumption that dosage is not factored into the starting prices, the following details are observed:

• Dosage Comparison With Former Winners;
• Past Performance Indicators Of Late Speed; and
• Live Longshot Prices.

I approach the task as follows using a process of elimination:

Note that I am not claiming any great insights and will be pleasantly surprised with a positive result.

Handicapping: Benford's Law, Shannon Entropy, And Twenty Questions

Imagine a horse-race (i.e. Benford Law Stakes) with the following distribution of win probabilities.

This toy example is not as unrealistic as you might expect at first glance. Look at the very good approximation by Benford's Law of Starting Price position for roughly 20,000 GB flat races 2004-2013 inclusive.

Then, in simplest terms, the inherent uncertainty of the Benford Law Stakes race outcome is best represented by Shannon's Entropy: H(x) = -SUM((x)*log(x)) = 2.87, which number also suggests (under optimal conditions) the minimum number of yes/no questions (i.e. 3) the handicapper should ask himself to identify a potential winner. Taking our lead from Shannon-Fano Coding, we should iteratively divide the entrants into two approximately equal groups of win probabilities (i.e. 50%) and use Pairwise Comparison to eliminate the non-contenders using at most five questions.
Once again, this restriction is not as unrealistic as it might first appear. Slovic And Corrigan (1973), in a study of expert handicappers, found that with only five items of information the handicappers' confidence was well calibrated with their accuracy but that they became overconfident as additional information was received. This finding was confirmed in a follow-up study by Tsai et al (2008).

Time-Value Vs Expected-Value Or Likely Poverty Vs Average Riches

In his excellent book, A Mathematician Plays The Stock Market, John Allen Paulos outlines a potentially disastrous, trading strategy that clearly illustrates the difference between expected value and time value summary statistics or, in his case, mean (expected value = +10% -> average riches) and median (time value = –16.43% -> likely poverty) performance.

This extreme example points to the obvious advantage of knowing the most likely outcome of an investment and raises the interesting question of how to summarize a trading portfolio in time value terms?

By way of illustration, assume a trading portfolio (illustration only) that includes just two sports, baseball and horse-racing, with the following profiles:

It is now immediately apparent that, even though both profiles have positive expected values, they are losing propositions as reflected by their evens-equivalent, negative time values.

In summary, expected value summarizes the average performance across all traders and is of critical importance to the bookmaker whereas time value best reflects the most likely, individual outcome and is of paramount value to the individual sports trader!

Ensemble Averages Vs Time Averages

Evaluating Gambles Using Dynamics outlines his idea of time averaging in contrast to ensemble averaging in a discussion of the St Petersburg Paradox. Note the parallels with volatility drag, median outcomes, and expected values!

Doubling Rate Entropy And Kullback-Leibler Divergence

Cover and Thomas (2006) show that, in a horse race, a handicapper has an expected wealth growth-rate equal to that of an investor who wins every race minus a measure of uncertainty of the race and minus the difference in the win probability distribution used by the handicapper and the distribution of true win probabilities. Intuitively, this makes sense. To reduce the race uncertainty, we should focus on open betting markets with as few runners as possible. To reduce the win estimates difference, we should meld our betting line estimates with that of the crowd (Benter, 2004). In terms of a simplistic equation:

   Long-Term Profit = Clairvoyance – Race Uncertainty – Market Divergence.

My experience is that most handicappers focus too much on trying to become clairvoyant and not enough on selecting open races and factoring in the “wisdom of crowds”.

Euro-Style Handicapping

As a long-distance, handicapper of Euro races, I look for events in England, France, and Ireland (Grade 1 Horse-Racing Countries) with a high degree of chaos (I have entropy scores for all race-types). From this initial list, I have selected seven race-types that I focus on exclusively. Because of the high level of uncertainty in these races, the market (wisdom of crowds) is a less successful predictor. With a Bayesian-based, Elo-Class algorithm, I generate my own performance figures. Using the performance figures for all entrants in a chosen race, I run a Monte-Carlo simulation of 1000 races that automatically generates a realistic odds-line as final output. Critically, as long as there is at least one overlay (almost always) in the chosen race, I finally run a Haigh-like, Kelly-variant algorithm that selects the final contenders. (As I have stated elsewhere, this list may include both overlays and underlays).

Cheltenham Supreme Novices Hurdle (G1)

Though not my discipline (National Hunt Racing), Cheltenham's four-day graded stakes meeting is an exception. Effectively, this is the Breeders Cup of jumps racing. Lacking the in-depth expertise required to handicap this form of thoroughbred racing, I seek to focus on novice races where historical knowledge is as informative as current form. To that end, the Supreme Novices Hurdle has many similarities to the Kentucky Derby - young horses, many attempting a graded stakes, championship race for the first-time with little form in the book. Using dosage analysis of the "in-the-money" finishers over the last ten years and ranking the current field against this metric to identify "live" outsiders, shortlisted two interesting contenders - Shaneshill (14/1) and Tell Us More (25/1). Though not necessarily the most likely winners, these two selections are the best matched to previous contenders on dosage and, therefore, worthy of punting in both the win and show markets!

Analytical (Kelly) And Numerical (Solver) Solutions

Many sports trading problems yield to both an analytical and a numerical solution.



In the above example, the numerical solution (using Solver in Excel) to minimizing the difference between expected value and volatility drag over a sequence of similar bets equals the analytical solution (using Kelly) for the same sequence!

Equivalent Single Bet

With multiple bets (illustration only) in a single win market, what is the equivalent single bet that best summarizes the overall position?

As the worst win-loss outcomes are to either win only the minimum profit or lose the total stake, then the most informative summary position is a combination of both scenarios.

Volatility Drag

Aaron Brown, author of The Poker Face Of Wall Street, makes a strong case for the negative impact of volatility drag on expected value with respect to the Kelly Criterion in the following posts:
  * Short-Term Variance
  * Risk Of Ruin And Kelly Betting
  * Bankroll Performance Simulator
  * Betting Strategy.

The above before and after illustrations show a worked example of setting stakes to match a zero difference between expected value and volatility drag.

Time Distance And Fatigue

Notwithstanding the strengths and weaknesses of the dosage approach to handicapping, it is worth reviewing the excellent article by Steven Roman on speed-stamina profiling. Looking forward to the Breeders Cup at Santa Anita, one could consider generating dirt and turf course profiles (based on track records at different distances) to provide insights into the challenges faced by European shippers. Going further, one could profile those shippers (based on best performances at equivalent distances in Europe) to identify live-longshots for exotic plays. Obviously, I am glossing over the difficulties of generating meaningful numbers using winner final times, beaten lengths, varying track configurations, and qualitative track conditions. Nevertheless, our goal should be to look for live contenders to fill tickets and not on demanding mathematical accuracy!

Beta-Binomial (Wins And Losses)

In trading sports events, it is necessary to continually update one's opinions conditioned on new information. In many sports, the only information available is in the form of wins and losses. In that context, two technical articles worth reviewing include the following excellent contributions - Regression To The Mean And Beta Distributions, and Market Efficiency and Bayesian Probability Estimation via the Beta Distribution.

Scripsi Exposui Feci

As the Latin triple asserts: Scripsi, Exposui, Feci ("I wrote, I explained, I did"), the trading posts below are not ivory tower musings but are the product of real-world experiences, though obviously not in that order.

Antifragile Trading (Small Losses, Big Gains)

Nassim Taleb’s latest book, Antifragile: Things That Gain from Disorder, defines a new concept, Antifragility. Operationalizing this concept in the world of sports trading would mean creating an approach that is explicitly designed to benefit from market volatility. In other words, an antifragile trading system would be characterized by a procession of small losses periodically punctuated by large gains - for example, live longshots. However, nobel laureate Daniel Kahnemann, Thinking, Fast And Slow, would point out that the pain endured by a succession of small losses will not be emotionally compensated by an iteration-ending large gain. Obviously, most humans are too fragile to handle antifragility!

Overlay Markets And Multiple Selections

When handicapping horse races, it is critical to focus on those markets in which there is at least one overlay. However, contrary to received wisdom, the professional sportstrader does not just trade the specific overlays but instead trades one or more additional horses (possibly including an underlay) with the goal of spreading his risk while maximizing his long-term median income. An excellent worked example of this approach is detailed in Appendix V of Taking Chances.

Graded Stakes: Dosage And Live Longshots

With respect to handicapping classic generation (i.e. 3yo colts and fillies) graded stakes races around the world, it is often difficult to get the relevant data (past performances, trainer statistics, and so on) to make an informed selection. However, pedigree details are usually available and using dosage it is certainly possible to identify some live longshots.

For example, in the recent French 2000 Guineas, using past renewals (DI=1.85, CD=0.43) as a guideline dosage would have identified Lucayan (DI=1.80, CD=0.50) as the number one ranked contender (Won, 33/1).

Food for thought?

Finding Good Bets In Lotteries

An MAA award-winning paper from 2011, Finding Good Bets In Lotteries, combining expected value and portfolio theory.

Edelman Sharpe Ratio

David Edelman, Quantitative Finance lecturer, handicapper, and author derives a sports trading version of the Sharpe Ratio on page 28 of "The Compleat Horseplayer".
SR = (ProbWin - (1 / DecimalOdds)) / Sqrt(ProbWin * (1 - ProbWin))
For example, with the following 'investments', A is judged to be slightly better than B in terms of expected return per unit of risk:
A: ProbWin = 45%, DecimalOdds = 2.60
SR = (0.45 - (1 / 2.6)) / Sqrt(0.45 * (1 - 0.45))
= 0.13142
B: ProbWin = 31%, DecimalOdds = 4.00
SR = (0.31 - (1 / 4.0)) / Sqrt(0.31 * (1 - 0.31))
= 0.12973

Benfords Law Favorites and Exotic Bets

In various horse-racing jurisdictions (e.g. Australia, UK, Ireland, France), there is a very strong correlation between the winning rates of favorites and Benford's Law. In other words, favorites win approximately 30% of races, second favorites approximately 18% and third favorites approximately 12%. One could conceivably use this information to generate some tickets for Daily Double, Pick 3, 4, 5, or 6 exotic pools by using a random number generator and a "Benford distribution" of win rates. Though unscientific in validation, this method proved invaluable to me over the Breeders Cup weekend (given many upsets to expected outcomes)!

Information Calibration And Confidence

In 1979 [Studies in Intelligence, Vol. 23, No. 1 (Spring 1979)], a study of expert handicappers demonstrated an interesting interaction between information and confidence. There were two key findings. First, as soon as an experienced handicapper has the minimum information (seven plus or minus two variables) necessary to make an informed judgment, obtaining additional information generally does not improve the accuracy of his selections. Second, additional information does, however, lead the handicapper to become more confident in his judgments, to the point of overconfidence. It appears that handicappers have an imperfect understanding of what information they actually use in making judgments. They are unaware of the extent to which their judgments are determined by a few dominant factors, rather than by the systematic integration of all available information.
As ever, if the handicapper cannot find variables that account for sufficient variance in outcomes over and above that provided by market prices then he will not have an edge and will lose his bankroll.