Adaptive Boosting

Machine learning studies the design of automatic methods for making predictions about the future based on past experiences. In the context of classification problems, machine-learning methods attempt to learn to predict the correct grouping of unseen examples. In the mid-1990s, Freund and Schapire introduced the meta-heuristic, Adaptive Boosting (AdaBoost), “…an approach to machine learning based on the idea of creating a highly accurate prediction rule by combining many relatively weak and inaccurate rules… Indeed, at its core, boosting solves hard machine-learning problems by forming a very smart committee of grossly incompetent but carefully selected members…” (Boosting: Foundations And Algorithms). As context for how boosting might work, the authors introduce the following toy problem in the opening paragraph to A Short Introduction To Boosting: “A horse-racing gambler, hoping to maximize his winnings, decides to create a computer program that will accurately predict the winner of a horse race based on the usual information...” As discovered by the early pioneers of expert systems in the 70s and 80s and as acknowledged by the authors, the biggest stumbling block to using experts is that many of them are unable to detail their decision process or even to rank order the importance of key variables. In light of this issue, Freund and Schapire point out that the beauty of boosting is that it builds on what experts can do not on what they cannot do, namely, given a specific scenario they are usually able to make a judgment in favor or against a particular outcome. For instance, we can ask an expert handicapper if a specific scenario - course and distance winner in last outing a week ago - would lead him to believe that it was more likely to win again or to finish out of the money. Note the phrase “more likely to” – this is a key strength of boosting - it asks for the balance of probabilities and not for the highly probable. The boosting phase combines many such simple, scenario-based rules into an overall weighted decision for an upcoming event. In its original specification, the defining quality of boosting was that it aggregated an incomplete set of “if-then” rules (decision stumps) that recursively address unexplored regions (areas for which previously chosen rules would give incorrect predictions) of existing data sets. The inherent strength of this approach is that it automatically leverages the key dimensions of Wisdom Of Crowds, namely, diversity, independence, decentralization, and aggregation For a worked example applied to NFL prediction, see James McCaffrey’s Classification And Prediction Using Adaptive Boosting. For the underlying theory of why wisdom of crowds works, see Scott Page’s excellent The Difference. And finally, Malacaria And Smeraldi explore the relationship between the AdaBoost weight update procedure and Kelly’s theory of betting and also establish a connection between AdaBoost and Information theory in On AdaBoost And Optimal Betting Strategies.

Biased Coin (Haghani & Dewey, 2016)

A recent paper by Haghani & Dewey (2016) sheds an unflattering light on subjects formally trained in finance as to their lack of basic knowledge with respect to probability and uncertainty – “If a high fraction of quantitatively sophisticated, financially trained individuals have so much difficulty in playing a simple game with a biased coin, what should we expect when it comes to the more complex and long-term task of investing one’s savings?” Though an otherwise interesting study, there are a couple of key points which do not receive adequate attention in the paper:

• Financial: Though the median final bankroll of $10,504 is derived in the footnotes, there is not sufficient attention drawn to it in the paper itself. Time-Value automatically generates this value whereas Expected-Value generates the wholly unrealistic $3,220,637.
• Psychological: The fallacy of “Playing With House Money” – “…you are offered a stake of $25 to take out your laptop to bet on the flip of a coin for thirty minutes.” What would have happened if the subjects had to pay $25 to play instead of being given it for free? 

No less a luminary in both the financial and gambling worlds than Ed Thorp says: “This is a great experiment for many reasons. It ought to become part of the basic education of anyone interested in finance or gambling.”

Evens-Equivalent Trades

Risk of Ruin (RoR) (Epstein 2009) provides an easily understood metric (probability of bankroll depletion before doubling it) with which to compare strategies. By way of illustration, let us assume that both you and your brother are recreational handicappers. You trade baseball, home-underdogs and he trades horse-racing, second-favorites, as follows:

Sport	Bank	Trades	Avg. Odds	Avg. Win Rate	Avg. Stake	Evens Odds	Evens Win Rate	Evens Stake	Expect. Value	Std. Dev.	EV.SD	Edge	Rsk Of Ruin
MLB	5,000	500	2.10	51.00%	250.00	2.00	53.37%	263.05	17.75	262.45	219	7.10%	7.11%
H-R	2,500	1,000	3.50	31.00%	100.00	2.00	52.62%	162.10	8.50	161.87	363	8.50%	16.53%

In the classic treatment of ruin, there is a working assumption of even-money trades to make the calculations tractable. To that end, we must first transform our real-world trades into their even-money equivalents with the same edge and volatility, see Krigman (1999). Despite having a smaller edge and a larger stake, you have a lower probability of depletion than your brother principally because you are risking a lower percentage of your bankroll per trade. Ideally, your RoR should be below 5% and to achieve this you both would have to either increase your bankroll or decrease your stake, as follows. [(MLB: 5%, 218.20 or 5730); (H-R: 5%, 54.97 or 4,546)].



Note that Edge's impact only equates with that of Volatility after 219 trades for you and 363 trades for your brother. And it takes a minimum of 806 trades for you and 1336 trades for your brother before you can be at least 95% confident that the combined effects of positive edge and mixed-bag volatility work in your favor to guarantee positive bankroll growth. In other words, despite having potentially successful trading strategies, you both will be well into your second season of handicapping before you can be sure of beginning to reap the benefits!

Juvenile Finish Position Ratings

In horse-racing, 2yos are by definition the least exposed runners at the racetrack. In the context of handicapping 2yo races, we can create a simple model to generate race-specific ratings based solely on finishing positions and number of runners per race in each horse's past performances record. These ratings will guide our elimination process. Remember that, in handicapping, elimination of probable non-contenders is always preferred to the selection of possible contenders.

• First, calculate "horses beaten" (n-f) and "horses beaten by" (f-1) from finishing positions (f) and number of runners (n) for each race in a horse's past performances. 
• Then, sum across all races for wins (w=Σ(n-f)) and losses (l=Σ(f-1)) respectively.
• Next, calculate a horse's posterior probability m=(w+α)/((w+α)+(l+β)). Prior wins (α) and losses (β) are derived from two full seasons of 2yo races and are equivalent to a horse finishing fourth of seven runners in a virtual race. Note that a first-time starter would automatically have a posterior probability of 0.50=(0+3)/((0+3)+(0+3)). 
• Finally, convert probabilities to performance ratings (min:112=8-00, max:126=9-00) using the following formula: r=((a+(b-a))*((m-x)/(y-x))), where a=out.min, b=out.max, x=in.min, and y=in.max of all runners in the current race.

The final rank ordering of horses is important, not the absolute performance ratings.
In summary, this finishing position rating system (fpr) does not take into account the strength of opposition, beaten lengths, weight carried, or finishing times; however, when it is based on a whole season of results, the fpr ranks correlate approximately 0.87 with the equivalent ranks from an Elo rating system.

As luck would have it, Sunday's Prix Marcel Boussac (Fillies' Group 1) - France's top 2yo fillies race - had a 0.92 correlation between fpr ranks and finishing positions, with the 10/1 winner (Wuheida) top-rated! Not scientific, nevertheless I get to keep the winnings.

Speed-Stamina Fingerprints

In 1981, Peter Riegel formulated an equation for the relationship between distance and time of athletics world-records. In 1982, Steve Roman adapted Riegel's equation to try and resolve the Secretariat Preakness controversy.
In a similar vein, we can generate unique speed-stamina, power-law fingerprints for racecourses (and horses) based on the best times for various distances. The simplest interpretation of these racecourse fingerprints is to confirm our expectations of the demands imposed by similar distances for different course configurations (Epsom is faster than either Ascot or Newmarket (lower y-intercept); Ascot and Newmarket have similar stamina profiles (same slopes)). Another possible insight might be how these course fingerprints reflect the potential impact on horses with different pace profiles (early speed at Epsom). A further analysis might be on how to better baseline and equate speed figures at different racecourses (use standard course with own speed-stamina equation). Finally, more controversially, using speed-stamina fingerprints for classic-generation (3yos) horses to match with course fingerprints in the lead-up to Group 1 contests (Epsom Derby) or for comparing performances from different classic generations (Frankel vs Sea The Stars).

Note the graph only shows best times and power-law equations for five, six, and seven-furlong races at Ascot, Epsom, and Newmarket and are for illustrative purposes only.

Wisdom Of Crowd Market Index (WCMI)

In sports markets, the probabilities implied by the prices on offer are a proxy for the wisdom of the crowd for that particular event. By adapting Shannon's Entropy formula, we can generate our own "Wisdom Of Crowd Market Index" (WCMI) to represent this information on a scale from 0 - 1.

      First, calculate implied probabilities of prices: x = 1.00 / d 
      (where d = decimal odds).
      Next, calculate log probabilities: y = log(x, n)
      (where n = number of runners).
      Then, multiply probabilities by log probabilities: z = x * y.
      Finally, sum products and subtract from one for final index:
      wcmi = 1 - (-sum(z)).      
    

Note that the index is at a minimum (0.00) when the market is completely uninformed about the outcome (all prices are the same) and at a maximum (1.00) when the market has closed (no price for winner and maximum price for all others). In the realistic exchange market above (snapshot of prices taken one minute before going in-play), the crowd is relatively uninformed (0.03) about the likely outcome and presents an excellent opportunity for the informed sports trader. Personally, I do not trade in any market with an index above 0.13 (approx).

It is very gratifying to note that FlatStats are now (29-Dec-2017) using our WCMI as a guide to those markets in which the crowd is less well-informed!

Fano Plane Transylvania Lottery And Trifectas

Jordan Ellenberg, in his excellent book "How Not To Be Wrong: The Hidden Maths Of Everyday Life", relates how he derived a mathematical model probably used by an MIT team ("Random Strategies") to successfully master the Massachusetts State Lottery ("Cash Winfall") from 2005-2012 winning an estimated $3.5 million. He illustrates the approach using a Fano Plane to master a variant of the Transylvania Lottery. Using the following seven triads (123, 145, 167, 247, 256, 346, 357), he outlines how it is possible to always guarantee a winning outcome from each draw.
From a handicapping perspective, it would be interesting to check for a subset of seven runner races for which these particular (or equivalent) bet combinations would prove to be a successful trifecta strategy. At the very least, you would have bragging rights for always getting at least two selections correct on one or more tickets! More generally, the field of Combinatorial Design may hold additional insights for other exotic bets.

Proebstings Paradox - Price Is Right If Marked-To-Market

Proebsting's Paradox refers to a situation in which a sports trader makes successively increasing bets on the same selection in a single market, ostensibly using the Kelly Criterion to calculate the stakes while taking advantage of better and better odds, only to ultimately face ruin.                 

 

The difficulty arises because the sequence of bets appears to cost more in total bankroll percentage than the Kelly Criterion would recommend as a standalone bet at the highest odds in the sequence.

In the Todd Proebsting example, the sports trader initially bets 25% of his bankroll on a 2/1 selection with a 50% win probability. Some time later, he is offered 5/1 on the same selection and calculates his Kelly stake at 22.5% leading to a total wager of (250 + 225) = 475. The problem with this result is that the Kelly stake for a single bet at 5/1 (assuming 50% win probability) is only 40% of bankroll (400) - leading to the theoretical possibility of ruin from betting at successively more attractive odds on the same selection in a single market.
To resolve this paradox, both Ed Thorp and Aaron Brown recommend that the sports trader should "mark to market" his bankroll after the initial 2/1 bet (reducing it by 12.5% from 1000 to 875) and use this updated position to calculate the 5/1 stake. In fact, to stay within the upper limit (400) defined by the standalone 5/1 bet, the trader also needs to reduce his estimated win probability to 42.5%! Using the above example, this would lead a sports trader to bet at most 146.25 = (16.71% * 875) at 5/1 giving a total wager of (250 + 146.25) = 396.25, which is roughly equivalent to the Kelly standalone stake (400) at 5/1 but with a lower upside!

Thinking Fast And Slow

It is opening day of your favorite meeting on your local circuit and, having downloaded the past performances, the program automatically applies the decision rules you so painstakingly put in place after months of careful analysis. It automatically highlights the recommended selections, prices, and stakes. You are in total control.
As luck would have it, just before placing the wager, you decide to look at the connections of your selections for confirmation of the wisdom of your program choices. To your horror, you notice that a relatively unexposed colt trained by a promising, young handler with a 7/1 offering is excluded. You make an "executive decision" to overrule the program selections and immediately place a double-unit wager on the unexposed colt only to watch the program top selection romp home at 7/2.
Recognize the scenario or some variant thereof? Daniel Kahneman, in his excellent book Thinking Fast And Slow, would tell you that System 2 (careful analysis) has just been trumped by System 1 (good story)!
The key lesson here is not to berate yourself for being human but to set in place measures to protect yourself from this kind of ambush. Focus your expert handicapping skills into determining the key factors not fully reflected in the starting prices and then use some level of automation to rate and rank your selections. Avoid looking for a good, coherent story and no fine-tuning!

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.

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).