Odds and Ends: Week 5

A good sports betting column should be backed by a profitable gambler with a proven track record. It should offer picks generated by a sophisticated and conceptually sound model. Most importantly, it should treat the subject with the seriousness it warrants.

This is not that column.

Instead, this will be an off-beat look at the sports betting industry-- why Vegas keeps winning, why gambling advice is almost certainly not worth the money, and the structural reasons why even if a bettor were profitable, anything they wrote would be unlikely to make their readers net profitable, too.

While we're at it, we'll discuss ways to minimize Vegas' edge and make recreational betting more fun, explain how to gain an advantage in your office pick pools, preview games through an offbeat lens (with picks guaranteed to be no worse than chance), and tackle various other Odds and Ends along the way.

Tracking the Unders

In 2022 and 2023, mass-betting the unders was incredibly profitable over the first 6 weeks and essentially just broke even after that. I hypothesized this year that maybe all we needed to do to make a killing was to start our "mass-bet the unders" strategy earlier in the season. Unders finished 6-10 last week, but are still 27-21 and net profitable since we started tracking.

If you put $10 on the under in every game (and you saw the exact same lines I saw and got all action at -110), you'd have lost $45.45 last week, but would still be up $35.45 for the year. (Briefly consider quitting while you're ahead.)

By contrast, if you'd taken the "snowball" approach-- starting with $160, rolling over all wins and losses, and devoting an equal percentage of your remaining bankroll to each bet-- your current bankroll would stand at $179.38, a $19.38 profit.

Why is the snowball strategy performing worse than the equal-bet strategy despite making the exact same picks? I'm so glad you asked because this leads us straight to today's topic:

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.

One popular betting 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 serious about making money with gambling, that's not how you bet. Instead, you use a variant of the snowball strategy, setting aside a fixed pool of money to gamble with and thinking of individual bets 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 a 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. But the real advantage of this strategy from a professional gambler's perspective is the fact that wins can snowball (which is why I refer to it as the "snowball strategy"-- ideally as in "rolling down a hill and getting larger as it goes" and not "sitting in the sun and getting smaller over time").

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, as our "betting the unders" experiment will hopefully demonstrate. 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 then 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.)

Kelly Criterion

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). You're still winning bets at a 50% rate, but all of a sudden Vegas is paying you a vigorish on every wager. 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.

Because every individual bet is +EV (or "positive expected value"), the most rational thing would seemingly be to max out every individual bet. But if you bet 100% of your remaining money on ten coin flips, there's a 99.9% chance you lose one of those bets and go to zero. Hardly a smart approach.

Let's say you internalize that lesson 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?

The Math

Wrong. Let's walk through 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 50% of that (or $100), 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 you bet half of that again ($160) and lose it this time; 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.

I want to stress this again because it's such a counterintuitive finding. You were betting a guaranteed +EV bet. The odds were in your favor on every bet you made. And yet you still lost 99.999% of your money betting it. Because of the way you are sizing your bets, you are offered a bet where the expected outcome is a 10% gain, and you are losing 10% of your bankroll every time you make it.

(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. With a fixed-bet system, that +10% opportunity actually pays out +10% per wager. But you're not going to make much money this way in the long run.

Snowball Strategy

If you really want to benefit from that sweet exponential growth action, you need to be using the snowball strategy. And you're going to have to find the optimal bet size using the Power of Math. Conner Evans runs through the numbers 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-- that's the "Kelly bet". If you are disciplined and stick to Kelly bet sizing, the expected growth of your snowball is 0.42%.

I have not misplaced a decimal, I didn't mean the expected return is 4.2%. Your bankroll should grow by less than half a percent for every bet. And again, this isn't a case of us being conservative-- the Kelly bet size is the mathematically optimal bet that maximizes the amount of money you make. It's the aggressive bet!

Calculating the Kelly criteria like this requires knowing unknowable things, like "What are my true chances of succeeding on this bet?" Because they can only guess at the actual underlying odds, real professional bettors instead bet something called "fractional Kelly". They might only bet half of what the Kelly criteria says they should based on their estimate of the size of their edge. I've seen compelling arguments that the ideal ratio is actually closer to one-quarter Kelly.

At one-quarter Kelly, you're making a whopping 0.18% expected profit per bet. At that rate, it will only take an average of 381 sequential bets (where one bet pays out before the next is made) to double your money! If you're treating every week of NFL action as a sequential bet and betting during the playoffs, it'll only take you a bit more than 17 years to turn a $10,000 bankroll into $20,000. What an opportunity!

Even with perfect Kelly bet sizing and its scorching 0.42% returns, you still need to make your bets sequentially to grow that snowball. At one sequential bet a week, that works out to 7.8% growth over the regular season. Run it through the playoffs and you're returning 9.7% 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 your initial stake); instead, you'll have nearly 16 times as much money as you started with!

Which seems like a great return. 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 10.7%. 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 have been 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 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.

Lines I'm Seeing

Last week, I predicted that the Dolphins had been losing, so they'd be Hungry For a Win. That was my mistake-- the Dolphins weren't at all Hungry, and that's why they'd been losing. (A system that can explain every possible outcome is infinitely useful because it's never wrong.) Our Motivational Differential picks might be 4-4 on the season, but that's not because Motivation Differential is a 50/50 system (like virtually every betting system), it's because I've only correctly estimated the Motivation Differential 50% of the time.

Which is a different thing. Somehow.

Minnesota (-2.5) vs NY Jets

Last week I noted that Brian Flores and the Minnesota defense Wanted It Very Badly. They're not just Hungry, they're downright Ravenous, and opposing quarterbacks are getting Eaten Alive. Jordan Love did throw four touchdowns against them, but only after falling behind by 28 points, and the Vikings also turned him over three times in the process.

That was last week against one of the greatest quarterbacks in Green Bay Packers' history; this week the team is merely facing Aaron Rodgers. This defense has Gotta Have It, and Rodgers is the one who's Gonna Give It to them.

Denver (-2.5) vs. Las Vegas

Denver's defense has been absolutely throttling teams so far; this is a unit that Very Much Wants You To Shut Up About the Offense, Please. What do the Raiders want? If Davante Adams' trade request is anything to go by, they're a collection of players that Want To Be Anywhere But Here. A game against a defense like the Broncos' will certainly make them want to be Not Here even more than ever.

The Other Side of the Pillow Lock of the Week

We started out naming our Lock of the Week after something that was sometimes cold and sometimes warm-- a stone. When that didn't work, we pivoted to something that was extremely cold-- the void of space. That went even worse, so now let's try something that's really not that cold, all things considered.

Kansas City (-5.5) vs. New Orleans

I kind of thought we were in on New Orleans this year, but the pseudorandom number generator is fickle and apparently it has decided it's out. It tells me it wants the Chiefs. I told it about the Rashee Rice injury, but it didn't care-- pseudorandom number generators are notoriously uncaring of context.

That's fine, we'll give it the Chiefs and it'll lay the 5.5 points. And if the pick fails for a fourth straight week, there's a whole universe of things-associated-with-coldness for us to explore in next week's Lock of the Week.