First I want to bring an enlightening podcast episode to your attention.
đïžHow a Professional Sports Bettor Really Makes Money (Bloomberg Odd Lots)
Joe and Tracy interview pro gambler
. In under an hour you can learn a ton about the sports gambling industry. With sports leagues shoving it down our throats, young and old alike, I feel itâs important to understand whatâs going on. And itâs not pretty.
Iâve talked about how this industry is rigged in âFreeâ Markets Wet Dream. This podcast echoes the problems but it encompasses so much more â opportunities, cultural impact, and basic mechanics. There are lots of misconceptions about sports gambling as well.
This is a list of some of the less-obvious ideas found in the interview:
This exchange is the betting parallel of fundamental vs arbitrage investing:
Tracy (07:49):
So just so we can better understand this dynamic, walk us through your sort of day-to-day as a professional sports gambler. What kind of opportunities are you trying to identify and then how do you decide how much money, for instance, to apply to each individual bet?
Isaac (08:05):
Yeah, so itâs a great question. So it really depends on what sports are in season. So right now, you know, end of the NBA, but thereâs a lot of MLB things like tennis are year round. And so it does depend on the sports.
There are in general two ways of identifying profitable sports bets. The first is you can take sort of a market-based approach. And by a market-based approach, what we mean is there are tons of sports books all out there and as Joe mentioned, theyâre all offering all of these different kinds of bets. And if you constantly scroll through all of the odds, youâre going to find slight mispricings. So letâs say everybody has the Yankees as two-to-one underdogs and one sportsbook has them as a three-to-one underdog. You can identify that as off market and you can place that.
Tracy (08:45):
Oh, I see. So youâre not saying that the platforms have the odds wrong, youâre trying to identify outliers among the platforms.
Isaac (08:51):
Thatâs exactly right. yeah. So thatâs probably the main that the majority of professional or winning sports betters make money is by identifying markets which are simply mispriced. And for that you donât need any special sports knowledge, you just need to have a screen with all the odds and constantly be scrolling through them and looking for price changes and looking for books that are slow to update.
The other way to do it, which is a lot harder and a lot [more] rare, is to basically create your own numbers. So you say, okay, I know exactly how much each player is worth. I know what the weather is going to be today, I know these matchups and so Iâm going to generate kind of from scratch from my own, model the odds and then apply that to the market. And when it comes to major liquid markets like the NFL or the NBA or the MLB, thatâs really, really hard to do. And there are very, very few people who can do that, but those are the people who make the most money.
The house edge on typical bets in sports is in the ballpark of 5% if you have have to risk $110 to win $100 on a coin flip. The house edge is ensures a healthy long term profit for the sportsbooks if they can avoid smart bettors. But the edge is small enough to keep bad bettors coming back. They win enough to think they have a chance.
On my flight back from NJ, I read applied mathematician David Sumpterâs The 10 Equations That Rule The World (he was the author of the sports analytics book Soccermatics as well).
Itâs a good book for introduction to the below topics especially since it provides lots of real-world applications about problems we encounter on a regular basis. For example, he uses Bayesâ Theorem to show why forgiveness is not just a gracious thing to do but a statistically sound choice. He also bodies Jordan Peterson if youâre into that.
The book is sequential â the equations build on preceding ones to build rich models that underpin profitable business and life decisions.
Taking the baton from Odd Lots, we can use chapter 3âs Confidence Equation and chapter 6âs Market Equation to establish a basis for determining if we actually have an edge.
Letâs get into the details.
Sumpter defines the equation:
h * n ± 1.96 * Ï * ân
where:
h = edge or signal per trial
n = trials
Ï = standard deviation
The 1.96 gives away equationâs identity â itâs the 95% confidence interval.
The best way to understand it is by demonstration.
If you have 3% edge on a bet with a standard deviation of 71% and make 100 bets your realized edge will be:
.03 ± (1.96 * .71)/ â100
.03 ± .14
Your realized edge will have a 95% chance of falling between -11% and +17%
While the confidence interval contains zero, you cannot be particularly sure that the signal is positive and that the gambling strategy works.
The value of this equation is often best seen in reverse. We can invert the expression to ask:
âHow many trials do I need to be 95% sure that my edge is positive?â
h/Ï > 2/ân
*Note: The 2 comes from rounding 1.96 up. Sumpter doesnât mind sacrificing precision to make the formula memorable.
đĄMoontower readers will observe that h/Ï is a measure of risk to reward and can be interpreted as a Sharpe ratio.
The bet described above is ascribed to a hypothetical gambler named Lisa. Notice that Lisa not out of the woods even if she gets to make this bet 2,300x.
Sumpter explains the problem:
During those six years, other gamblers might have picked up on her edge and started to back it. The bookmakers may adjust their odds and the edge disappears. The risk for Lisa is that she doesnât realize that her edge has gone. It takes over one thousand matches to be confident that an edge exists. It can take just as many expensive losses to realize that it has disappeared. The profits that grew exponentially fast now crash down and decay exponentially fast.
Notwithstanding the ever-present problem of âdid the world change while I was deploying my strategyâ the blunt math is instructive:
Most amateur investors are vaguely aware that they need to separate the signal from the noise, but very few of them understand the importance of the square root of n rule that arises from the confidence equation. For example, detecting a signal half as strong requires four times as many observations, and increasing the number of observations from 400 to 1,600 allows you to detect edges that are half as large. It is easy to underestimate the amount of data needed to find the tiny edges in the markets.
These ideas were fundamental in options training. You can see them applied in:
Understanding Edge (10 min read)
If You Make Money Every Day, Youâre Not Maximizing (23 min read)
During the 1700s, mathematician de Moivre pioneered combinatorics (i.e., how many ways can you be dealt a full house). The combination formula relies on factorials which become computationally impossible when numbers get large, especially in the 18th century. Scottish academic James Stirling showed how, at large n, the binomial distribution can be approximated by the normal bell-curve.
In 1810, Laplace developed the idea of moment-generating functions to describe features of distribution. This allowed him to study how the shape of the distribution changes as random outcomes are added together. Laplace demonstrated something truly remarkable: irrespective of what is being summed, as the number of outcomes we sum increases, the moments always become closer and closer to those of the normal curve.
While there were tricky exceptions to be grappled with:
the result that Jarl Lindeberg finally proved in 1920, it is known today as the central limit theorem, or CLT. It says that whenever we add up lots of independent random measurements, each with mean and standard deviation Ï, then the sum of those measurements has a bell-shaped normal distribution with a mean ÎŒ and a standard deviation of Ï.
To take in the vast scope of this result, consider just a few examples. If we sum the results of 100 dice throws, they are normally distributed. If we sum the outcomes of repeated games of dice, cards, roulette wheels, or online casinos, they are normally distributed. The total scores in NBA basketball games are normally distributed (illustrated in the bottom panel of figure 3). Crop yields are normally distributed. Speed of traffic on the highway is normally distributed. Our heights, our IQs, and the outcome of personality tests are normally distributed.
Whenever random factors are added up to come, or whenever the same type of observation is repeated over and over again, the normal distribution can be found. De Moivre, Laplace, and, later, Lindeberg created the theoretical bounds within which the confidence equation can be applied. What they couldnât know, and what scientists have since found, is just how many phenomena can fit under that same curve.
We now jump to chapter 6 to see how this concept applies to investing.
Sumpter lays it out:
dX = h * dt + f(X) dt + Ï * Δ
This looks very similar to the stock diffusion equation known as Brownian motion where h is the drift and Ï * Δ is the random component. The signal and the noise respectively.
But thereâs a wrinkle in this version.
We acknowledge:
The key assumption for the central limit theorem is that events are independent. In roulette, one spin of the wheel doesnât depend on the last; the central limit theorem applies.
But not all financial mathematicians understood that the central limit theorem didnât apply to markets.
âThis brings us to the f(X) term in the market equation. I havenât seen that before.
It is a feedback function.
We turn back to Sumpter:
When I met J. Doyne Farmer in 2009, he told me about a colleague at one trading firmâwhich, unlike Farmerâs own company, had lost a lot of money during the 2007/08 crisisâwho referred to the Lehman Brothers investment bank crash as a âtwelve sigma event.â As we saw in chapter 3, 1-sigma events occur 1 time in 3, 2-sigma events occur about 1 time in 20, and a 5-sigma event about 1 time in 3.5 million. A 12-sigma event occurs 1 time in, well, Iâm not sure, actually, because my calculator fails when I try to find anything larger than a 9-sigma.
The simple signal and noise market model assumes independence in price changes. Under the model, future values should thus follow the ân rule and the normal curve. In reality, they donât.
On the stock market, one trader who sells causes another to lose confidence and sell too. This invalidates de Moivreâs central limit theorem. Fluctuations in share prices are no longer small and predictable. Stockholders are herd animals, following each other into one boom and bust after another. Introducing the f(X) term into the equation means that traders donât act independently from each other, but it does assume they have short memories. It again invokes the Markov assumption, this time to say that tradersâ feelings about the near future change as a function of their feelings now. Seen this way, the Market equation can be thought of as combining the Confidence equation, for separating signal and noise, with the Influencer equation, for measuring social interactions, in a single model.
Instead, as the theoretical physicists in Santa Fe showed, the variation in future share prices can become proportional to higher powers of n, such as nÂČ/Âł or even proportional to n itself.
This makes markets scarily volatile.
While I havenât seen this market equation before, the topic is not foreign. In Thinking In N not T you learn how the presence of auto-correlation underestimates an assetâs volatility!
In a world where even laypeople know Talebâs favorite gym lift, itâs banal to point out that we donât inhabit Mediocristan. And yet, experience suggests itâs not so banal that weâve internalized the implications of non-gaussian distributions.
Maja finds that non-mathematicians seldom take the time to reflect on the assumptions that underlie the models she uses. They see what she does as predicting the future, rather than describing future uncertainty. Last time we met for lunch, together with her colleague Peyman, she told me, âThe biggest problem I see is when people take the results of models literally.â Peyman agreed. âYou show them a confidence interval for some time in the future, and they take that as true. Very few of them understand that our model is based on some very weak assumptions.â
Whatâs possibly more upsetting is what the equation means for people that spout âreasonsâ.
The core message of the market equation is to be careful, because almost anything could happen in the future. At best, we can insure ourselves against fluctuations without needing to know why they have occurred. When the markets temporarily melted down and bounced again at the start of 2018, Manoj Narang, CEO of quantitative trading firm MANA Partners, told the business news organization Quartz that âunderstanding why something happened in the market is only slightly easier than understanding the meaning of life. A lot of people have educated guesses, but they donât know.â
If traders, bankers, mathematicians, and economists donât understand the reasons markets move, then what makes you think that you do? What makes you think that Amazon shares have reached their peak or Facebook shares will continue to fall? What makes you so confident when you talk about getting into the housing market at the right time?
Sumpter leaves us with what Iâd describe as irreducibly vague advice:
The most important lesson from the market equation, a lesson that applies not only to our economic investments but also to investments in friendships, in relationships, in work, and in our free time. Donât believe that you can reliably predict what will happen in life. Instead, make decisions that make sense to youâdecisions you truly believe in. (Here you should use the judgment ie Bayes equation, of course.) Then use the three terms in the market equation to prepare yourself mentally for an uncertain future:
Remember the noise term: there will be many ups and downs that lie outside your control.
Remember the social term: donât get caught up in the hype or discouraged when the herd doesnât share your beliefs.
And remember the signal term: that the true value of your investment is there, even though you canât always see it.
Finally, hereâs a fun excerpt that I want to point out because everyone knows this person â Mr. My Way:
I am sitting in a cafĂ© in the late afternoon and watch him come in. He shakes a waiterâs hand and then does the same thing with the barista, exchanges smiles and a few words. He doesnât see me at first, and as I stand up to go over to him, he spots someone else he knows. A round of hugging ensues. I sit back down again, waiting for him to finish.
His celebrity here partly derives from his former life as a professional soccer player, and because his face is often on TV, but he is also popular because of how he holds himself: his confidence, his friendliness, the way he takes the time to talk to people, sharing a few words with everyone.
Within a few minutes of sitting down with me, he is into his spiel. âI think I make a difference because I show them my way of doing things. I think thatâs lost sometimes,â he says. âI just do my thing, I tell it as it is, and I am honest, because thatâs what is needed in this game. âIâve got a lot of contacts. A lot of meetings like this one, you know, keeping connected. You see, people want to talk to me because I have unique way of seeing it. Because of my background, you know, a way that no one else has quite got, and thatâs what Iâm aiming to de liver when I sit down with you.â These observations are interspersed with anecdotes of his playing days, a bit of name-dropping, and rehearsed stories, complete with well-timed jokes.
He smiles, looks me straight in the eyes, and, at times, makes me feel like Iâve asked for all this information. But I havenât asked for it. I wanted to talk about using data, both as it is employed in the media and within the game of soccer. Unfortunately, Iâm not getting anything useful. I call this type of man âMr. My Way,â after the song Frank Sinatra made famous. The careful steps, the standing tall, and the seeing it through provide the basis for each of his stories. It can make a beautiful melody, and for the two or three minutes during which my current Mr. My Way is hugging and greeting his way into the cafĂ©, it entertains those he meets. But it only works provided he moves from one person to the next. Now, here am I, stuck in this position, with nowhere to go.
Iâve enjoyed hearing behind-the-scenes stories about players and big matches, and finding out about life at the training ground. Moving from being a fan to being someone who is confided in by those close to the action was, to use the biggest clichĂ© possible, a dream come true. I still love hearing those stories and seeing the real world of my favorite sport for myself. But more often than not, the interesting bits are accompanied by âheroicâ tales of Mr. My Wayâs âvision,â followed by accounts of how their progress has been foiled by a cheating adversary or how they could do things better than anyone else if they had been given half a chance. Because of my background in math, these guys often feel they have to explain their thinking process to me. They start by telling me that I have a different way of looking at things than they do, without actually asking me how I look at things. âI think stats are great for thinking about the past,â he will tell me, âbut what I bring is insight into the future.â
After that, he will explain how he has a unique ability to spot a competitive advantage. Or how it is his self-confidence and strong character that help him to make good decisions. Or how he has cracked a way of picking out patterns in data that I have (he assumes) missed.
His tales tend to include a digression to times that didnât go quite as well for him. âIt was only when I lost concentration that I started to make mistakes,â he tells me. But he always returns to emphasizing his strengths: âWhen I stay clear and focused, I get it right.â
What I hadnât understood when I started working with sports statistics was just how much time I would have to sit listening to men telling me why they believed they were the special one. I should have known better because this doesnât just happen in sports. I have experienced the same thing in industry and business: investment bankers telling me about their unique skill sets. They donât need math because they have a feeling for their work that their quantitative traders (known as quants) can never have. Or tech leaders explaining to me that their start-up succeeded because of their unique insights and talents. Even academics do it. Failed researchers describe how their ideas were stolen by others or, when they succeed, they tell me how they stuck to their principles. Each of them did it their way. Here is a difficult question to answer: How do I know whether someone is telling me something useful or not? The guy Iâm sitting with now is obviously full of it. He has talked about himself nonstop for the last hour and a half. But many other people do have something useful to say, including, on occasion, Mr. My Way. The question is how to separate the useful stuff from the self-indulgent stuff. The difference between a Mr. My Way and a mathematician can be summed up in one word: assumptions. Mr. My Way barrels through the world confident everything he assumes is true really is true.
In case you start feeling too smug about the bullshitters in your work sphere, stop to consider the second order effect of knowing that some signals are more verifiable than others. See the 1-min read The Paradox Of Provable Alpha.
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