Kelly And Mutually-Exclusive Outcomes (AvK)

In AvB events, such as baseball, basketball, or football, the general advice to only bet the overlay is technically correct. However, in an AvK event, such as horse-racing, with a number of mutually-exclusive outcomes this advice is not strictly correct. For example, in the following racecard (sorted in decreasing e.v order), even though the handicapper has rated Bravo's win probability (π) at 29%, it is an underlay and not included in the list of bet selections:

h	p	π	e.v	Σ(π)	a.k	b.k	%
Echo	21.00	7.00%	1.470	0.070	0.048	0.977	2.49%
Charlie	5.50	20.00%	1.100	0.270	0.229	0.947	2.78%
Bravo	3.25	29.00%	0.943	0.560	0.537	0.951	0.00%
Delta	6.00	13.00%	0.780	0.690	0.704	1.047	0.00%
Alpha	2.50	31.00%	0.775	1.000	1.104	0.000	0.00%

But, if Bravo's price was to drift to 3.35, then this underlay is now added to the list!

h	p	π	e.v	Σ(π)	a.k	b.k	%
Echo	21.00	7.00%	1.470	0.070	0.048	0.977	2.56%
Charlie	5.50	20.00%	1.100	0.270	0.229	0.947	3.05%
Bravo	3.35	29.00%	0.972	0.560	0.528	0.932	1.18%
Delta	6.00	13.00%	0.780	0.690	0.695	1.015	0.00%
Alpha	2.50	31.00%	0.775	1.000	1.095	0.000	0.00%

Kelly betting is predicated on maximising the logarithm of the handicapper's bankroll over the long-term. But, in the short-term, that goal is translated into not losing specific events when the price is right! The key role played by overlays in mutually-exclusive events is that there must be at least one such betting option available in any event on which we wish to bet. Beyond that, the specific choices will only be governed by maximising the logarithm of our bankroll!

Note: Blindly backing high probability combinations such as Alpha, Bravo, and Charlie (total win probability = 80%) will eventually lead to ruin. In order to calculate the correct stakes, make sure table is sorted by column 'e.v.' in descending order!

Vendire-Ludorum Excel Add-In

Treat yourself to an Excel Add-In (32-bit and 64-bit [Windows] that includes some of the standard functions on which we have come to rely in our daily sports trading. Among the functions available are the following:

• Edge,
• Expected Value,
• Time Value,
• Kelly Single Stake,
• Kelly Mutually-Exclusive Stakes,
• Kelly Simultaneous Independent Events (5),
• Mark-To-Market,
• Risk-Of-Ruin, and
• Wisdom-Of-Crowd Market Index (WCMI).

In addition, there is a sample spreadsheet highlighting the various functions through worked examples from Haigh, Paulos, and Yao.

Mark-To-Market And Hedging

Following careful analysis of the next race, your assessment of the odds is 4/1(5.00) - a 20% edge on the market price of 5/1 (6.00) - with respect to your selection. Flushed with confidence, you place a $250 (4%) win bet on InTheMoney at 5/1 (6.00). Time to sit back and wait for the profit to roll in? Maybe!
Consider for a moment -  what is the current market value of your investment?. When you place the initial trade, its market value is $250=[($250*6.00)*(1/6.00)]. Roll tape and the market turns in-play as InTheMoney sets the early pace with measured fractions. Turning into the stretch it looks an even-money chance at worst to win. Freeze frame and consider once again - what is the current market value of your investment?. Assuming the market is now efficient with respect to InTheMoney’s win probability, the updated market value is $750=[(($250*6.00)*(1/2.00)]. In other words, at this point in the race, you have already won $500=[$750-$250]. Fast forward to the finish-line and InTheMoney is beaten by a late closer, JustInTime. Now, the really interesting question is - how much have you lost? If you had no choice but to only back your selection before the race, then you have lost $250. But, if you also had the option in-play to hedge the win bet at even-money, then you lost $750! (see Weighing the Odds in Sports Betting (Ch.4)).
In summary, no trade is complete without both a back and a lay bet or, in other terminology, an opening and a closing position.

Using our knowledge of time averages, we can select a lay price and calculate a hedging stake to maximize our median bankroll over time.

Horse-Racing Overlays

Sports traders sometimes conflate AvB events with AvK (K = N-1) events (N = number of entrants). The prototypical AvB game is a football match and the equivalent AvK example is horse-racing. Some experts encourage traders to identify a single overlay in both events and bet accordingly. Whereas this advice is generally correct for AvB events, it is not correct for AvK events.
In horse-racing, you should only select those races with at least one overlay for further examination. But, as Ravi Phatarfod points out in his excellent 1996 paper Betting Strategies In Horse Races, a gambler is more likely to correctly assess that the winner will be one of three horses than being able to correctly assign win probabilities to each individual horse. And, as John Haigh illustrates in Taking Chances, Kelly betting on horse races also encourages us to spread our risk across multiple entrants including, on occasion, those from all three categories of bets, favorable (positive expected value), fair (zero expected value), and unfavorable (negative expected value) bets. This approach guarantees over time that you will minimize your risk of ruin (total loss of capital).

Note: You must enter horse details in descending e.v. order only.

Engineer, Physicist, And Statistician

Many years ago, as a young postgraduate student, two colleagues (an engineer (E) and a physicist (P)) and I would meet every Saturday for an early lunch to discuss the week’s events in our respective disciplines and to give our differing perspectives on world events. It was an innocent and idealistic exercise driven by youthful enthusiasm and naivete. Naturally, our discussions ranged from the sublime to the ridiculous and everything in between.

Time passed and as our careers evolved we drifted apart and lost contact. Then a few years ago, I unexpectedly ran into E at Royal Ascot. After we engaged in some good-natured banter about the humbling nature of the aging process and introduced our respective wives, our attention turned to the Group 2, 6f, Coventry Stakes (2yo). I asked E what he liked in the race and how he made his selections. He turned to me in disbelief and said, “For 2yo races, I use the method you recommended to me back in the day to identify a select band of unexposed horses to exploit throughout the season.” Completely bewildered I said, “Remind me”. He then proceeded to outline an adaptation of the Bayesian Bandit (Thompson Sampling) “explore-exploit” strategy as used in the multi-armed bandit problem. To which, I blurted out, “You mean, it works!”. I quickly pointed out that I must have thought at the time it was a strategy worth exploring but that all the kudos should go to him for exploiting it so successfully. Engineers rule by defeasible reasoning.

Later that evening, my wife teased me by asking if I had given mathematical advice to everyone I had ever met and when I looked surprised by the question she added wickedly, “My hero, so brave, so strong!”

True Talent Levels

It is easy to conflate Which team is number one? with Which team wins today? The former is decided over the course of a regular season and is primarily driven by skill (talent) but the latter changes on a daily basis and when there are two roughly equal opponents it is principally governed by luck!
With respect to being number one, the definitive review of the field is provided by Langville and Meyer (2012) in Who's #1?: The Science of Rating and Ranking. And, in terms of estimating true talent levels, Adam Dorhauer delivers two excellent articles worthy of publication, Elo vs. Regression to the Mean: A Theoretical Comparison and Regression with Changing Talent Levels: The Effects of Variance.

Cheltenham 2017: Supreme Novices Hurdle Handicapping

Once more unto the breach, dear friends, in our annual attempt to find live longshots to finish in the money in the Supreme Novices Hurdle (G1) at Cheltenham 2017. As ever, our approach is based on the following premises: 

• Supreme Novices Hurdle is similar to Kentucky Derby - young horses, many attempting graded stakes, championship race for first-time with little form in book.
• Eliminate non-contenders and whatever remains, no matter how improbable...
• Avoid horses with pedigree mismatch to former winners [Melon, Crack Mome].
• Avoid horses from small fields [Labaik], [Elgin].
• Late speed important [Magna Cartor, Glaring, Capitol Force].
• Poor Cheltenham Form [River Wylde].
• Poor FPR [Pingshou, High Bridge].
• Minimum price 10/1 [Ballyandy, Bunk Off Early].

This leaves Beyond Conceit 20/1 and Cilaos Emery 16/1 as our selections!
Note, given the limited exposure of all the runners, we are not saying that those we have eliminated are not going to win - simply that they did not meet our criteria for live longshots to run in the money. The key takeaway, as always, is using a process of elimination not selection for identifying contenders.
Footnote: In their respective next outings, Cilaos Emery won G1 (Punchestown) and Beyond Conceit finished second in G1 (Aintree).

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!