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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'll detail shortly. (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 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 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 have gone 7-5-1 since then; if you bet an equal amount on every game (and all action was at -110), you would have turned a 10.5% profit. At $10 per bet, your total profit to date would be $13.64.

## 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.

I mentioned last week that if you started with a $160 bankroll and simply bet the under in every single game, you'd have turned a $187 profit. Or alternately maybe you'd have returned a $166 profit. It depends on what betting strategy you used!

One popular strategy among more casual bettors is simply putting a flat amount on each wager. If $10 is high enough to be interesting and low enough to be affordable, then you could just put $10 on every bet you make to give yourself a rooting interest. Or you could put $1, or $100, or $50,000. The optimal value is going to vary from person to person, but again, the goals are "low enough to be affordable, but high enough to be interesting".

When you're trying to make serious money with gambling, that's not how you bet. Instead, you set aside a fixed pool of money that you're going to gamble with, and all of your bets are made as a percentage of that overall pool. This is known as your bankroll. As your bankroll (hopefully) grows over time, your bets grow, too. If you have a few bad weeks and your bankroll shrinks, your bets shrink, too.

Setting aside your bankroll in advance is another great way to cap your potential losses; make sure your starting bankroll is not more than you can afford to lose and resolve to yourself that if it's gone, you're not going to buy back in again.

One advantage of percentage-based bets is your losses are capped; you can never lose more than your initial bankroll. Another advantage is that your wins can snowball. If you start with a $160 bankroll, put $10 on every bet, and go 16-0 on picks every week, you'll make $145 every week. After ten weeks, that's $1450 in profit. That's nice.

If you start with that same bankroll and put 1/16th of it on every game, after ten weeks you'll have made $102,732 in profit, a 642x return on your initial investment. (The secret to becoming a wildly profitable bettor, it turns out, is just winning every single bet you ever make.)

But betting 1/16th of your bankroll on every game is a terrible idea. If the $10-on-every-game bettor has a catastrophic 0-16 showing in Week 11, he or she loses $160 and is still up $1290 on the year. If the 1/16th-on-every-game bettor has a catastrophic 0-16 showing, he or she loses $102,892 and has no money left to rebuild.

The key to long-term bankroll management, then, is sizing bets to maximize long-term growth while ensuring that a cold streak won't completely wipe you out (because cold streaks eventually come for everyone). One big rule must be "never have all of your bankroll in play at the same time". Had that manager only bet 1/20th of their bankroll on each game, they'd "only" have lost $100,000 and would still have more than $20k left to work with. (Of course, if they'd bet 1/20th on previous weeks, their bankroll wouldn't have been so large to begin with.)

One can use math to calculate exactly what percentage of one's bankroll to risk based on how confident one is in a given bet. The formula used is called the Kelly criterion, so using this method is often known as "Kelly bet sizing". And it returns values that are much, much, much smaller than you'd probably guess.

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.) You have an expected profit of 10% of every dollar wagered, you should be virtually printing money.

Let's say you've internalized 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 have a guaranteed +EV bet, the key to maximizing your long-term return is betting less money each time. If you're betting half your bankroll on a bet that pays 10% profit in expectation, the long-term 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. One could always stick to the "linear gains" route where you just bet a fixed amount on each contest. But if you really want to benefit from that sweet exponential growth action, you're going to have to do the math to determine what percentage of your bankroll to bet each time to 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. This likely sounds like a tiny amount, like we're being especially conservative, but this is not the conservative play, this is the aggressive one. This is the value that maximizes long-term exponential growth. Because they can never know the actual underlying odds of any given bet, real professional bettors bet something called "fractional Kelly" (which itself isn't so much the conservative play as it is the aggressive, optimal play after controlling for uncertainty.)

With a 10% edge and perfect Kelly bet sizing, 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. That value compounds over time, too. After thirty years, you're not merely up 10% a year times 30 (or 4x); 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. (The optimal Kelly bet when you don't have an edge is 0%.) It also depends on being able to accurately assess the size of that edge (which is functionally impossible) and being able to retain faith in the process through the inevitable slumps, which might be the hardest part of all. (It's very hard during a cold stretch to tell whether you're just getting unlucky or whether your edge was not nearly as big as you initially thought. More often than not, it's the latter.)

And then there's the fact that over the last 30 years, the S&P 500 had an annualized return of about 10.4%. 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.

Should you be betting Kelly, fractional Kelly, or some other percentage-based approach to bet-sizing? If you just love doing math and look for random excuses to do it in your day-to-day life, then sure. Personally, I'll just stick to tossing a few bucks on some action when it looks interesting.

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