Photo: USA Today Sports

Gambling on the NFL is big business, especially after a 2018 Supreme Court decision striking down a federal ban on sports betting. Recent estimates suggest that as many as 46.6 million people will place a bet on the NFL this year, representing nearly one out of every five Americans of legal gambling age. As a result, there's been an explosion in sports betting content, most of which promises to make you a more profitable bettor. Given that backdrop, it can be hard to know who to trust.

Fortunately, you can trust me when I promise that I'm not going to make you a more profitable sports bettor. And neither will any of those other columns. It's essentially impossible for any written column to do so for a number of reasons I'll detail over the year. (I'm not saying it's impossible to be profitable in the long-term betting on the NFL, just that it's impossible to get there thanks to a weekly picks column.)

This column's animating philosophy is not to make betting more profitable but to make betting more entertaining. And maybe along the way, we can make it a bit less unprofitable in the process, discussing how to find bets where the house's edge is smaller, how to manage your bankroll, and how to dramatically increase your return on investment in any family or office pick pools (because Dave in HR and Sarah in accounting are much softer marks than Caesar's and MGM).

If that sounds interesting to you, feel free to join me as we discuss the weekly Odds and Ends.

Checking In On the Unders

In Week 7, I noted that unders had been hugely profitable so far this season and discussed structural reasons why unders tend to outperform overs (though usually not by enough to beat the vig). I also said I'd track the performance of the unders going forward to see if we could be profitable merely by mass-betting them every week. (My hypothesis was that unders would win somewhere from 50-52% of the time going forward.)

Unders started slowly but finished strong, going 9-5 in Week 11 for a 22.7% return on our wagers. Overall, unders have gone 40-29-2 since we started tracking; if you bet an equal amount on every game (and you saw all the same lines I saw, and all action was at -110), you would have turned a 10.4% profit. At $10 per bet, your total profit to date would be $73.64. If you had instead started with $160 and invested an equal percentage in every bet every week (rolling over your winnings or losses), you'd be up $83.16, or 52% of your starting bankroll.

Big Fortunes Are Built On Bad Bets

Back in Week 8, I laid out the math behind bet sizing, which is the process of deciding how much to bet based on the size of your bankroll and the size of your edge, known as "Kelly bet-sizing". That math advocates for much smaller bets than people would naively think: if you had a guaranteed 10% edge over the house, you should bet just 1/12th of your bankroll, and the long-run expectation is a 0.42% increase in your bankroll per bet. If you started with $5000 and wanted to become a millionaire, you should expect to need 1,265 bets. (And that's before considering taxes.)

Remember that this isn't a conservative prescription; this is the value that maximizes your long-run expected growth. Betting any other value would make the path to your first million dollars take even longer in expectation. (If you deviate sufficiently far from optimal bet sizing, you'll actually lose money over time, even when you're betting with a guaranteed 10% edge.)

If this seems bleak, well... so far, this is an extremely rosy picture of reality. Because this is the math for when we know the size of our edge, which will never happen. We might think we have a 10% edge, but there's a good chance we're wrong. And the damage to our bankroll of being wrong is not symmetric-- the gains if we're underestimating our edge are not as significant as the losses if we're overestimating our edge (which is more likely, anyway). Because of this, no serious gamblers "bet Kelly". Instead, they bet "fractional Kelly"-- meaning whatever the math says, they bet a fraction of that. 1/4th is a popular target, though depending on the nature of the bet, you might see even smaller fractions like 1/8th or 1/10th. (Generally, even 1/2 Kelly is considered overconfident and reckless.)

Again, I can't reinforce this enough: the reason for betting fractional Kelly isn't because you're risk-averse; it's because this is the number that maximizes your bankroll growth in the long run. At 1/4th Kelly, you're looking at 5,000 bets at an estimated 10% edge before you add that coveted second comma to your bankroll. But at anything other than 1/4th Kelly, you're looking at even longer. (Potentially infinitely long if you go broke in the pursuit.)

How do you make a living gambling? A massive amount of very risky, very patient work. How do you make a fortune gambling? You ignore optimal bet sizing, wager way too much, and hope you get absurdly lucky and get rich before you go bankrupt. You can go from $5000 to $1,000,000 in just eight bets if you go double-or-nothing on each and win them all. (Of course, even one loss leaves you broke. And the odds of at least one loss are 99.6%.)

The people who made the most money gambling are usually the ones who made the worst bets. (Most people who made millions gambling in America won it in the lottery despite the lottery being one of the worst bets available-- as little as 50% of money wagered is returned to the players, which would be like if sportsbooks let you bet against the spread, kept your wager if you lost, and returned your wager with no extra winnings if you won.)

There's a belief that if you want to get rich, you should study rich people and do what they did. This is spectacularly dumb because of something called survivorship bias: by only looking at trials that were successful you miss all the trials that were unsuccessful. If a thousand people tried a strategy and one wound up rich while the other 999 wound up destitute, that was a dumb strategy and you probably shouldn't be giving it a try.

We had a rare opportunity to see this in action last year. In 2020, Sam Bankman-Fried, a 31-year-old wunderkind with an estimated 11-figure net worth, posted a Twitter thread where he walked through the math behind Kelly bet-sizing and then just... disagreed with it. He essentially said that he was a special unicorn who was going to repeatedly bet five times Kelly because his value function was non-linear and so he was willing to accept more risk to maximize his return. But again, Kelly isn't conservative; that's the value that maximizes expected return!

(This blog post runs through a bunch of simulations to see just how lucky you'd have to be to wind up with more money betting 5x Kelly. The value was so extreme the author's laptop broke before he could find it.)

Bankman-Fried got rich by making a bunch of absurdly overleveraged bets based on a misunderstanding of the underlying math and continually getting lucky enough to not be punished for his error. And once he was rich, he continued making absurdly overleveraged bets until his luck ran out; his net worth fell by $9.1 billion on November 8th, 2022-- the largest single-day drop for anyone in history-- after a reckless bet resulted in a bank run on his company. His net worth continued to fall to zero, Bankman-Fried was convicted of seven counts of conspiracy and fraud, and he'll likely spend decades in prison.

This is a useful lesson. The numbers behind exponential growth through wagering are dismal, but the math is the math. And anytime you see anyone who has been extraordinarily successful but isn't preaching bankroll discipline, assume you're witnessing survivorship bias firsthand. The only sustainable path to profit is slow and steady; with any other approach, "up big" is merely a precursor to "down bad".

Or else there's always the second option: bet for fun, wager only what you can afford to lose, and count your losses as the cost of entertainment.

Since readers have asked for more AI-driven gambling advice, I've had ChatGPT summarize this in a series of zen koans:

1. A novice asked the wise gambler, "In the game of wagers, how do I attain riches swiftly?" The wise one replied, "Imagine the river's flow; to reach the ocean, one must navigate it drop by drop."

2. In the realm of chance, a seeker inquired, "Should I place my fortune on a single throw, doubling it each time?" The master replied, "Beware the seduction of haste; the road to wealth is paved with measured steps, not reckless leaps."

3. Along the path of wagering, a ponderer questioned, "Is it not wiser to wager large and court fortune boldly?" The sage responded, "The oak grows from a tiny acorn; so too does wealth from modest bets. Patience is the seed of abundance."

4. Seeking counsel, a gambler wondered aloud, "Why not embrace the full measure of Kelly's wisdom?" The elder replied, "The river whispers its depth, and the wise sail cautiously. Fractional Kelly is the compass that guides through turbulent waters."

5. In the realm of chance and uncertainty, a dreamer sought the secret to boundless wealth. The sage shared, "Consider the lotus; it blooms gradually, petal by petal. Likewise, your fortune unfolds in the careful cultivation of fractional dreams."

6. A gambler, eyes fixed on the stars, questioned, "Can one not amass wealth swiftly with bold bets?" The sage cautioned, "Like a shooting star, quick fortunes fade. Sustainable riches emerge from the quiet constancy of measured choices."

7. A skeptic challenged the wisdom of the method, saying, "Why not emulate those who bet recklessly and won?" The sage responded, "In the garden of outcomes, weeds of misfortune often outnumber flowers of luck. Beware the survivor's tale, for it is a narrow view of a treacherous landscape."

8. A prodigy, boasting of unparalleled insight, declared, "I am the exception, and I shall defy Kelly's counsel." The sage warned, "The path of arrogance is paved with broken fortunes. Even the unicorn, untamed by humility, may find itself in the abyss of overconfidence."

9. As the wise gambler observed the ebb and flow of fortune, a lesson echoed: "In the grand theater of chance, the protagonist is patience, and the climax is crafted with measured bets. Beware the allure of rapid ascension, for the summit built on haste crumbles swiftly."

10. In the dance of chance, consider the second path. Bet for joy, offer only what the wind may freely take, and let the echo of losses be the toll of amusement.

Lines I'm Seeing