A picture, they say, is worth a thousand words. If they’re right, then the new book by Joseph Buchdahl, ‘Monte Carlo or Bust - Simple Simulations for Aspiring Sports Bettors’, is worth about a million. The author of ‘Squares & Sharps, Suckers & Sharks’ has followed it up with a comprehensive guide for how to make randomness work for you, rather than against you.
In his previous books and articles for Pinnacle’s Betting Resources blog, Buchdahl has presented many mathematical montages to evaluate everything from the trustworthiness of tipsters to your optimal stake. In this book, he shows us − quite graphically − how randomness and variance work, so that we know what to expect in many common sports betting scenarios and can learn how to use them to make valuable predictions.
There is a lot of math to slog through, but he uses it skillfully to provide pieces of evidence, layered one on top of the other, like a prosecutor building a case against a crooked tipster. The math and the Monte Carlo simulations first tell us what the world should look like, and then he shows us whether or not real-world occurrences and claims actually look like what we should expect them to. When Joseph shows us the pictures, even those who find the math difficult (or even impossible) will see his case come into focus.
Buchdahl starts off our journey gently, with some context for why he named the book after Monte Carlo, and why the type of randomised simulations he’s using bear that name too. The reason is that the grand casino there is both famous for its swank, as well as infamous for the streak of 26 consecutive black winners that occurred at the roulette wheel there over 100 years ago. It’s a great metaphor for the statistical analysis to follow, for which we must be properly prepared.
Principles and formulas needed
In the second chapter, we’re introduced to the basic principles and formulas we will need when working out the probabilities of many crucial aspects of sports betting. Not only do we have to concern ourselves with how often teams will win, but also how often will we beat the odds (decimal odds, if you please). Beyond that, we need to know not just the average outcomes, but the variance surrounding the average because 50% of us will be below average. To find the right answers, he outfits his readers with binomial, normal, and lognormal distributions and a visual representation of what each of them means.
As he dives deeper into the material, Joseph shows us the equations we’ll need to answer simpler questions, as well as step-by-step instructions on how to build an Excel simulation to answer the more complex ones. You’ll need to do some prep work if you want to replicate his results that show you what winning and losing look like in the choppy seas of sports betting.
But by giving us the tools to investigate on our own, Buchdahl provides something very few other betting authors do. He doesn’t just teach us how to fish – he gives us a blueprint for how to construct a rod and reel, and shows us how to interpret our own results. Only caught two fish today? Maybe that’s expected in 1 out of 31 days. Or maybe you’re just not a very good fisherman.
In defence of the Closing Line Value (CLV) hypothesis, Joseph presents ample evidence (from years of his own painstaking data analysis) that the average closing line in efficient betting markets is like a sonar device to map out when you’re fishing in fertile waters and when you’re not. In the chapter called ‘Winning’, he states: “I have already made the case that the accuracy of a price (or a line) is largely dependent on three things. Firstly, the quality of the bookmakers’ prediction model, and their willingness to reflect the ‘true’ probabilities in their odds and lines. Pinnacle, as I’ve argued, are more willing to do this than the recreational bookmakers. That’s the reason why you can find expected profit from the latter but not the former when comparing them.”
"I would need many thousands of wagers to demonstrate any meaningful statistical evidence of skill against bookmakers’ margins."
Yet, using Pinnacle’s closing lines to find value is just a starting point to reeling in the maximum profit. He follows up this point with Monte Carlo simulations showing how often you’ll win given your estimated edge, how often you may lose, and how to determine your staking so you have enough bait left in your tackle box when the inevitable drawdowns hit. These Monte Carlo simulations show that even a small sample of CLV can't just come about through luck.
In fact, to quote Buchdahl: "The more relevant point, however, is the speed at which we’ve identified a signal... against the background noise of randomness. Typically, for betting systems with a mean expected value of 1.72%, I would need many thousands of wagers to demonstrate any meaningful statistical evidence of skill against bookmakers’ margins. By using price movements instead, I could achieve this in as few as 65 bets."
The prospect of losing, and how best to avoid it
No one likes to think about losing, but in his chapter on it, Buchdahl makes us look at the prospect of it and how to best avoid it. A key element of his analysis (borrowed from Nassim Taleb) is that often when bettors calculate their long-term return on investment (ROI), they’re "conflating ensemble probability and time probability" since "there is no ‘next day’ after ruin." The lesson here is that if you go bust, or have a tiny fraction of your original bankroll left, due to the inevitable variance your theoretical expected returns for future bets won't matter.
In the chapter called 'Staking’, Joseph compares various legitimate methods and a couple of baseless ones. It boils down to this − once you find an edge and a reasonable estimate of how large it is, how do you best exploit it? Starting with fixed staking plans, he later does something novel by reinterpreting different staking plans like unit-loss and unit-win as Kelly-variants to compare them to Kelly bet sizing directly. His novel approach pays huge dividends in this chapter, when he uses the statistical concept of "z-score" to concoct a new staking method called "unit-z" that tracks his reams of real-life betting results much better than other typical staking methods.
"All Martingale has really achieved is a change in the distribution of risks."
I won't spoil the surprise in this section for you, but I will say that this is ground-breaking work. On the other end of the distribution of staking plans, we have the infamous Martingale. While most savvy bettors realize that it's foolish, they may not be able to explain exactly why. Buchdahl breaks it down for us, ultimately concluding that: "All Martingale has really achieved is a change in the distribution of risks. The trade-off for gaining one extra outcome with a positive expectation is another with a much greater negative expectation, relative to the equivalent outcome for unit staking. This risk asymmetry is the source of the inherent danger associated with the Martingale strategy."
In the chapter on tipping, we get a thorough analysis of how likely tipsters are to provide truly "winning" tips. Let's just say the news is not good, particularly for Fixedmatches.org, whose claimed results deviate so far from what Buchdahl's analysis paints as the likely reality that he concludes their results are probably just concocted. The final nail in their coffin is that they don't even show enough long winning streaks to jibe with their touted win rate. I guess the conclusion here is that not only are they not good tipsters, but they aren't very good cheaters on top of it.
The Bayesian inference
In the ‘Odds and Sods’ chapter, Joseph gives us a brief explanation of Bayesian inference, which is the mathematical method for updating the estimated probability of something based on recent results. He simulates how this method changes the posterior probability from different starting points, namely that there's a 1%, 10%, and 50% chance that he's a skilled bettor. After showing us the evidence, he concludes that: "You can see that the evolution of my belief in the probability I am skilled is quite sensitive to the initial prior probability; indeed, it’s one of the weaknesses of Bayesian inference. I will much more quickly conclude that I am wholly skilled if my initial starting point was a larger probability that I am."
Here, I take some exception that this result shows a weakness of the method. In fact, I think it's a strength that when starting with only a 1% chance of being skilled rather than just lucky (a figure that he shows strong evidence for later), it takes much more data to move the needle to "most likely" skilled than if we believed it was a 50/50 chance initially.
Another common gambling fallacy that he debunks here is data mining, writing: "Data mining effectively reverses the process of inference. Rather than test an a priori hypothesis that some variable, or set of variables, may cause a particular level of betting profitability, a pattern of profits is perceived to have a cause simply because it has happened. If you have no idea what is behind a pattern, you will have no idea why it disappears if it does."
It's another example of a case where bettors may try to take an easy shortcut to finding an edge, only to be sorely disappointed after all the results come in. The good news, though, is that when you do find a true and sustainable edge, it doesn't need to be very large in order to benefit from it over time, since he tells us: "You only need a tiny advantage for the influence of compounding to deliver the rewards in the long term."
In the last chapter, ‘A Cautionary Tale’, Joseph zooms back out to the big picture. He talks about the different models of bookmakers and how they can, and perhaps must, coexist. He gives us some insight into how this abundance of randomness in betting naturally causes the thrills associated with it and talks about how recognising the reality of our predisposition to gambling will do more to help problem gamblers than heavy-handed prohibition or regulations.
My favourite quote from his conclusion, though, is this: "One can never know whether one path through life is better or worse than another; you only get to live one path within the Monte Carlo model of life, where there are potentially an infinite number of paths, and where the shape of the distribution is unknown. As I would say to all gamblers, focus on the process, not the outcomes, or as Buddha once said: 'Happiness is a journey, not a destination.'"