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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 detailed here. (I'm not saying it's impossible to be profitable 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 Those Unders...

Last week I noted that the "Under" bets were having a phenomenal season but cautioned that Vegas updates to new information, so even if there was an edge, it was probably gone already. So how did the unders do in Week 7? They... finished 7-7 and all the sports books pocketed in the vig.

Vegas is very, very good at this.

Bankroll Management? We Talkin' About Bankroll Management?

You might have noticed by now, but this column isn't super heavy on practical advice on how to maximize your return when betting on football (beyond the old standby to just limit what you bet to what you can afford to lose). I wanted to go against type today and take a serious look at a very important topic for professional sports bettors: bankroll management.

Your "bankroll" is "all the money you have available to wager", so "bankroll management" is simply deciding what percentage of that total to allocate to each individual bet. And if you want to be profitable in the long run, knowing how much to bet is every bit as important as knowing who to bet it on (if not more so).

Consider: for most people, picking games is a 50/50 affair. Let's say you had the best football model known to man. It's not perfectly accurate because there are always things like injuries or blown calls or other unknown variables that can impact the final outcome, but it's shockingly close; let's say that it predicts the winner against the spread a whopping 90% of the time.

If you're betting at standard -110 odds, you get a 91% return on investment for every correct pick, so the "maximally EV" move (or the move that results in you making the most money in expectation) is to bet 100% of your bankroll with every bet. If you have $1,000 to wager, then wagering $1,000 results in an expected profit of $718. (That's a $909 return 90% of the time plus a $1000 loss 10% of the time.) Wagering half of that results in an expected profit only half as large.

But if you max-bet your entire bankroll every time, the odds are better than 50% that you'll be broke within seven bets. Even a 90% bet misses 10% of the time, and if your entire bankroll is at stake, one wrong pick leaves you with no money left to bet. Even the best bet on the planet will leave you broke if you keep betting too much on it.

Let's look at a more practical example. You're never going to get a model that's 90% accurate, but let's say that I had a magic "odds wand" that I could wave and instead of the odds being biased in Vegas' favor they were biased in yours. Let's say instead of the standard bet having odds of -110 (meaning you need to bet $110 to win $100), I could change them so they had odds of +120 (a bet of $100 returns $120). Picking games remains a 50/50 affair, but all of a sudden Vegas is paying you a vigorish on every wager. And as Vegas demonstrates, making money's easy when you've got the vigorish on your side. (Or so you'd think.)

If you bet $100 on every game, your expected return would be $10 per game (or 10% of the amount bet). That'd be $120-$160 every weekend, or $2720 every football season. Not too shabby! But betting a constant amount results in "linear gains" and that's not the path to true wealth. Top gamblers instead want "exponential gains", where their bankroll gets bigger and bigger and bigger and their returns get larger every week. Why settle for $2720 every football season when you can grow that to $10,000 every season, or $100,000 every season, or $1,000,000 every season?

But let's say you learned the lesson from above and don't want to wager 100% of your bankroll on any pick. Let's say that you decide to bet 50% of your bankroll on every pick; this way, even a bad pick will never wipe you out and you should be expected to grow that bankroll week after week after week, right?

Ummmmm... wrong. Let's do the math. The law of large numbers says that over a large enough sample of bets, you're going to have as many hits as misses. So we can look at what happens to our bets in pairs of hits and misses. (The order of the hits and misses isn't important, but I'll walk through it both ways to demonstrate.) Say you start with $200, you bet $100 because it's 50%, and you win your first bet. You have the $100 you never bet, you get the $100 you bet back, and you also get $120 for winning; now your bankroll is $320. Now let's say you bet half of that again ($160 this time), and this time you lose it; now you're left with $160, or $40 less than you started with.

Let's say the wins and losses go in the other order. You have $200, you bet $100, and you lose. You're left with $100, you bet $50 of it, it wins and returns an extra $60. Now you're left with... $160 again, demonstrating the order of wins and losses doesn't matter. (For those who remember algebra well enough, this is because we're multiplying and the order of terms when multiplying is irrelevant to the final result.)

Every time you win a bet under this system, your bankroll increases by 60% (multiply by 1.6). Every time you lose a bet, your bankroll decreases by 50% (multiply by 0.5). And again, the law of large numbers guarantees that the more you make a bet, the more your proportion of wins and losses will equal the underlying odds (in this case, 50/50; the chances of winning the bet are the same, we only changed the payouts). So if your starting bankroll is S, your bankroll after N bets will be S * (1.6)^(N/2) * (0.5)^(N/2) (which is just your starting bankroll times the number of bets, half of which are wins and half of which are losses).

You can plug some values into the formula yourself and see that the more bets you make, the smaller your bankroll becomes. After 16 bets, your bankroll is down to 16.7% of its starting total (in expectation; it could be higher or lower simply because 16 trials aren't enough to be too confident you're actually at 50/50 wins and losses, but the more bets you make the more confident you can be that your actual bankroll will match your expected bankroll). After 100 bets, your bankroll is down to 0.001% of its initial value; if you started with $1,000,000, you should have a cool $14.27 left.

Again, I want to stress this because it's such a counterintuitive finding. If you had a fantastic bet, a bet that pays 10% profit in expectation, and you wager half of your bankroll on it every time, the net result is that you'll lose 10% of your bankroll for every bet you make. You can walk through the example as many times as you need, there's no trick there, that's actually how the math works out. If this sounds like it can't possibly be true, take it as just another sign from the universe that you aren't cut out to be a professional gambler.

So does this mean it's impossible to profit off of profitable bets? No, of course not. As I mentioned, there's always the "linear gains" route where you just bet a fixed amount on each contest. And if you really wanted exponential growth, there's a lot of complex math you can do to determine what percentage of your bankroll to bet each time to actually turn a profit over the long run. Conner Evans runs through the math here (and yes, there's a lot of math involved). The upshot is the optimal amount to bet is "S - F/R", where S is your chance of success, F is your chance of failure, and R is your rate of return. In the case of the hypothetical above, where Vegas gives you +120 odds on everything, you should bet 1/12th of your bankroll every time, and the expected growth is 0.42%.

That's not a typo, I didn't mean it's 4.2%. Your bankroll should grow by less than half a percent for every bet. Which sounds like a tiny amount, but remember that we're going for exponential growth. You can increase your bankroll by 0.42% every week of the season. (You could bet more than one game a week, but for exponential growth, you need your last bet to pay out before you make your next one, so let's say you're just picking one game a week.) That works out to 8.29% growth over the regular season. Run it through the playoffs and you're returning 10.1% of your initial investment every year. And that value compounds over time, too. After thirty years, you're not merely up 30 times 10%; instead, you'll have nearly 18 times as much money as you started with!

Which seems like a great deal. But this depends on us having an edge over Vegas in the first place (super hard!) and being able to accurately assess the size of that edge (also super hard!) and being able to retain faith in the process through the inevitable slumps (perhaps the hardest part of all!)

And then there's the fact that through the end of 2021, the S&P 500 had an annualized return of 11.88%. (The value can fluctuate depending on what timescale you're looking at but has usually trended toward around 11%.)

So even with my magic odds wand and a bunch of calculus to help us with optimal bankroll management, over the long haul, we'd still be better off just dumping that money into index funds.

Lines I'm Seeing