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.

Betfair Pari-Mutuel Equivalence

For those handicappers fortunate enough to have access to both Betfair and Pari-Mutuel markets for the same events and who wish to arbitrage their positions for a "no loss" outcome, they should add the following formulae to their toolset:

• o = 1 - (d * 1/(x - 1)) and
• d = -((o - 1) * (x - 1))

where o = betfair decimal odds, d = pari-mutuel dollar payoff, and x = betfair tax (combination of commission and discount). These prices are equivalent in terms of expectation and volatility..

Betfair InPlay Hedge Stake

Trading an event in-play on Betfair is not for the feint of heart as, ultimately, no position is safe until it is successfully hedged. Psychologically, however, if you have carried out a fundamental analysis of the event then you want to be paid a premium for that analysis should your selection prove to be successful. On the other hand, Cumulative Prospect Theory confirms that we hate losing (loss aversion >= 2.25) more than we enjoy winning. In order to balance those conflicting forces, you could calculate a hedge stake to green-up your position, as follows:

  z = (s*(o+m-1))/(h+m-1)
  where  z = hedge stake
        s = original stake
        o = original price (back)
        m = win multiple (ratio of win payout to loss payout, if greened up)
        h = hedge price (lay)

For example, if I back a selection for $100 @ 6.00 and wish to green-up at 2.00 then the default option is to lay $300 @ 2.00 for a guaranteed $190 whatever the result of the event. By contrast, the above calculation (e.g., m = 2.25), gives a stake of $223.08 with a win payout of $264.74 and a loss payout of $117.66 giving you a win premium!

Trailing Low Threshold (Max Drawdown)

Given Loss Aversion, the painful effect of an unexpected loss is at least twice the joyful effect of an unexpected gain. One of the most frustrating scenarios in sports trading is moving into the black early in the day only to finish it in the red. Sound familiar? Psychologically, we reset the baseline to zero each day even though intellectually we may be focused on generating an annual income. Because we are loss averse it is not possible to simply ignore these daily downswings as it impacts our overall confidence level. I would recommend (even to those handicappers who do not accept session handicapping) setting a trailing low threshold. For example (specific numbers used for illustrative purposes only):

• Day Bankroll: $1000
• Max Drawdown: 20%
• Low Threshold: $800 = (80% * $1000)
• Day High: $1350
• Trailing Low Threshold: $1080 = (80% * $1350)

In other words, even though you set an initial low threshold of $800, should you go into profit on the day ($1350) the low threshold is increased to maintain the 20% drawdown from the day high. This allows sufficient flexibility to continue trading without suffering the negative emotional impact of losing all your profit on the day.

Betfair In-Play Trading (Minimax Regret)

Opportunity Loss (Regret) plays havoc with the emotions of In-Play Traders. One psychologically valid approach is to use Minimax Regret. For example, in a horse-race, assume your selection (AtTheWire) is on offer to back at 3.50 (Win Market) and to lay at 1.80 (TBP Market) and your calculation of edge dictates a stake of 100. In the Win Market, at what price and with what stake should you trade out In-Play to minimize regret? As the above table shows, trading out at less than or equal to 1.80 for 100 is the optimal choice! Note that backing your selection in the Win Market is equivalent to stating that, at a minimum, you expect your selection to contest the finish. Marked-to-Market (TBP Market), your selection is on offer pre-race at 1.80 to contest the finish and this price represents your best exit point In-Play (Win Market).

Discounted Harville v1.17 (VBA Functions - Excel 2007+)

Discounted Harville spreadsheet (Discounted_Harville v1.17) with VBA Functions for Excel 2007+ (DHExactaOdds, DHTrifectaOdds, and experimental DHSuperfectaOdds). Change the values of lambda, and rho to approximate the Henery (lambda = 0.76, rho = 0.62) and/or Stern models (See Donald B. Hausch, Victor S. Y. Lo, and William T. Ziemba, Efficiency Of Racetrack Betting Markets, London, Academic Press, 1994, pp. 478).

Chasing Losses and Abandoning Profits

With respect to trading, humans are better described as loss averse than as risk averse, Because the pain of a $100 loss far outweighs (between 200% and 500%) the pleasure of a $100 profit, when faced with a situation framed as a probable failure we gamble and when faced with a situation framed as a probable success we quickly take profits. In other words, we neither maximize profits nor minimize losses! This is the normal human condition.
As a result, whatever approach we take to reducing the impact of chased losses or abandoned profits must always allow for some of both. Trying to completely defy human nature is simply pointless and guaranteed to fail!
One approach that I would recommend is to use a form of session trading. Set aside a daily bankroll based on both the number of markets you wish to trade and the maximum number of units you are willing to risk for some minimum profit target. Critically, you must factor in a small percentage for the possibility of chasing losses.

Handicappers Edge and Traders Hedge

If you define yourself as a gambler/handicapper/punter then you are focused primarily on identifying a unique advantage in a sporting event that is not already accounted for by the public odds. By contrast, if you define yourself as an investor/speculator/trader then you are focused primarily on taking an initial position in a market and then protecting that position against a negative move.
The former approach is fundamental in nature and the latter technical. Both strategies have their respective Achilles heels! With fundamental approaches, you never know in advance if you have an edge in an upcoming sports event; with technical approaches you are uncertain when to exit a particular market by either “greening up” or “redding down” your current position.
In terms of money management, handicappers generally preach the Kelly Criterion and use Sportsbooks while traders affirm “Bold Play” and frequent betting exchanges. In general, the age profile of handicappers is higher than that of traders simply because betting exchanges are a relatively recent innovation. Handicappers try to automate their plays by using systems whereas traders use bots.

Absence of Proof is not Proof of Absence!

Francis Bacon (1620) “For a man always believes more readily that which he prefers.” If we leave aside the contentious issue of whether or not it is possible to be a profitable sports trader using a wholly fundamental approach to handicapping, we can readily identify three evidence-based views of ‘form’:

• Evidence of Presence (EoP),
• Evidence of Absence (EoA), and
• Absence of Evidence (AoE).

The natural tendency to confirm our initial opinion leads us to look for supporting evidence (EoP) in both the historical record and in statistical trends with the obvious danger of the logical error of affirming the consequent. By contrast, evidence that eliminates a contender (EoA) is the recommended approach with its natural link to the logically correct method of denying the consequent. Finally, lack of evidence (AoE) neither confirms nor denies the status quo but many neophytes seem to confuse it with EoA.

Even-Equivalent Probability of Ruin

Epstein (2009) provides a detailed, mathematical analysis of the risk of ruin within the broader context of the basic theorems of gambling. Without loss of generality, it is worth providing a layman’s interpretation of how the probability of ruin impacts the choices made by the average handicapper.
In the classic treatment of ruin, there is a working assumption of unit bets at even-money in order to make the calculation of the risk of ruin more tractable. To avail ourselves of this analytical solution, we need to transform our real-world bets into their even-money equivalents, see Krigman (1999).

For example, if you are in the fortunate position outlined above with a 42% strike rate at an average price of 2.50 then your advantage is the equivalent of having a win probability of 52.02% at 2.00 with both the same expected value and standard deviation (0.25, 6.169) as the original bet and with a probability of ruin equal to 24.07%. Obviously, it is also possible to estimate stake size given a specific price and preferred risk of ruin.

Gamblers Ruin at Casino Incroyable

A nice mixture of mathematics and humor - Gamblers Ruin at Casino Incroyable with passing reference to "Bold Play", "Minimum Gain", and "Gamblers Ruin".