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 offbeat 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 touch on various other Odds and Ends along the way.
How to Lose Money on Good Bets
Last week, I explained that there's no betting strategy that will return a guaranteed profit (in expectation) when you don't have an edge. Is there perhaps a betting strategy that will return a guaranteed loss (in expectation) when you do have an edge, though? Heck yeah, there is (and it's more common than you'd think)!
Let's prove it.
Imagine I'm offering to play a coin flip game with you. You start with $100; every round you bet half of your money, and if you win, I'll pay out 150% of your bet. In the first round, you bet $50, and if you win, you get $75 back. This is a massive 25% edge in your favor. (If I'm paying out an extra 50% on your wins, why is the edge only 25%? Because you only expect to win half the time.)
Because you're wagering a fraction of your bankroll, those gains would seemingly compound over time. Your expected bankroll after 1 flip is 12.5% bigger (a 25% expected increase applied to half your bankroll at a time), so after 2 flips it should be (1.125 x 1.125) times bigger, after three flips it should be (1.125 x 1.125 x 1.125) times bigger, and after 196 flips it should be (1.125)^196 times bigger, right? 1.125^196 is about 10,614,374,333 (let's just round it to 10 billion, shall we?), so multiply that by your starting bankroll and you should be the world's second trillionaire. That's definitely how the math works, right?
Well, not exactly. Let's play it out as a sequence of wins and losses. You start with $100, bet $50, and win, leaving you with $175. Then you bet $87.50 (half your bankroll), you lose, and you have $87.50 left. After one win and one loss, you've lost 12.5% of your starting stake.
(Note that the order of wins and losses doesn't matter. If you lost your first flip, your bankroll would drop to $50, and then if you won your second flip, you'd gain $37.50, leaving you right back at $87.50 after two flips.)
Assuming you finished with an equal number of wins and losses, instead of being a trillionaire, 196 rounds would leave you with... about 2 hundredths of a penny.
How the heck are we going to settle that tab up? I never should have let you start betting fractional cents.
How to Make a Trillion Dollars (Very, Very Slowly)
The problem isn't that the bet is secretly rigged against you; it's that you're betting too much of your bankroll each time. This is even easier to illustrate if we assume you have to bet 100% of your money each round—all it takes is one bad flip to bankrupt you, and the more flips you make, the more certain that catastrophic loss becomes.
So what if we don't bet a percentage of our bankroll? What if we just pick a flat number and bet it every time? If we start with $100, we wouldn't want that flat number to be $100, because there's a 50% chance we're out of the game before we have a chance to strike it rich (if we lose on the first flip). At $50, there's a 25% chance that we go broke before we hit escape velocity (if we lose our first two flips).
If we want to ensure a 99% chance that we don't go bust on a string of losses to start, we need to wager enough that we can survive six straight losses—so let's put down $15 on every round. Every winning flip pays out $22.50, every losing flip costs $15.00.
With these parameters, we're going to return a near-guaranteed 25% profit (assuming we don't fail seven straight flips to start and go bust). By the time we've bet $1000, we will have won $250. By the time we've bet $1,000,000, we're walking away with $250,000 in free money, and eventually we'll reach our goal of becoming the world's second trillionaire.
We'll just have to wager about $4 trillion to get there. At $15 a flip, that's about 267 billion flips. If we flip once every 15 seconds for 8 hours a day for 250 days a year (essentially turning the coin flip game into our full-time job), we'll get there in... a little bit over 500,000 years.
(It's not a bad gig, all things considered; it still pays out $900 an hour—decent work if you can get it.)
How to Make a Trillion Dollars (As Quickly As Possible)*
Okay, we need to start out betting a low value like $15 to ensure we don't go broke before the fluctuations in luck have evened out. But in the long run, linear gains are for suckers—once we already have a few million in the bank, there's no reason we need to keep betting such a low value. We could bet, say, $100,000 and still have a long enough runway where a string of 10 straight losses wouldn't wipe us out.
It seems like the optimal strategy would be to bet some percentage of our bankroll, even if not as much as 50%. If only there was some way to calculate the mathematically optimal percentage which would maximize our long-term growth.
If only...
Oh yeah, there totally is! The formula used is called the Kelly criterion, so using this method is often known as "Kelly bet sizing" or "betting Kelly". And it returns values that are probably a lot smaller than you'd guess.
Let's quote Wikipedia!
The Kelly bet is: f = (p / l) − (q / g) where:
- f is the fraction of the assets to apply to the security.
- p is the probability that the investment increases in value.
- q is the probability that the investment decreases in value (q = 1 − p).
- g is the fraction that is gained in a positive outcome. If the security price rises 10%, then g = 0.1.
- l is the fraction that is lost in a negative outcome. If the security price falls 10%, then l = 0.1. In gambling, typically the entire wager is lost, which corresponds to l = 1.0.
I know math is scary, but I think we can do it. In our coin-flip hypothetical, "p" and "q" (the probability of a win and loss, respectively) are both 50%. "l" is 1.0 (because we forfeit all of our stake with a loss), while "g" is 1.50 (because we get our entire stake back plus an extra 50% with a win). So that's (0.5) / (1) - (0.5) / (1.5), which works out to 0.1666666..., or 1/6th. Looks like our $15 bet was a pretty good place to start!
If we make full Kelly bets, what's our typical growth per flip? A win/loss pair at 1/6th of our bankroll takes us to 1.0417 times our starting value, so the profit per bet is about 2.06%. At that rate of return, we're only 1129 flips away from being a trillionaire*. We can get there in less than five hours! That's certainly much better.
The REAL Fastest Way to Make a Trillion Dollars
What's with all those asterisks I keep sneaking in? Well, when I said that last way was the fastest way to a trillion, I lied. (But wait, don't I often say that I'd never lie to you? Yeah, well, I lied.)
Kelly bets give you the absolute fastest possible rate of growth over the long run if you assume average luck. But the real fastest growth rate is just to assume perfect luck. If you bet your full bankroll on every flip, you can make a trillion dollars in just 26 flips (assuming you win every bet). That's less than 10 minutes. So sure, if you absolutely need to make a trillion dollars and you can't spare five hours to do it, just ignore the Kelly criterion and assume you'll get lucky every time. Surely that'll work out well.
Should You Be Betting Kelly?
Nah. Kelly only works if you both have an edge and can estimate the size of that edge. If both of those things were true, you wouldn't be reading me every week (unless you just read for the dazzling bon mots, I suppose). And the numbers I used above painted an overly rosy picture; even the best gambler in the world isn't getting a 25% return.
Let's say you convinced Vegas to pay you a vigorish instead—say they let you make all of your bets at +120 odds. That's a 10% return per bet, which seems like it should be nearly as good, right? At a 10% return, the optimal Kelly bet would be 1/12th of your bankroll, just half as much as at a 25% return. Which should cut our typical growth in half, right?
Wrong! Typical growth per bet at 1/12th of our bankroll is 0.42% per bet. At that rate, it would take more than 5,500 bets to make our first trillion dollars.
Let's say our model only identified one bet per week that was so tilted in our favor. There are 18 weeks in the regular season, plus four weeks in the playoffs. If our bankroll grew by about 0.42% per week (on average), that would be a 9.65% return on investment every year. That's pretty good!
Of course, the annualized return on the S&P 500 over the last 30 years is about 10.3%, so you'd have likely been better off just tossing that money in the market.
Personally, I say save the math for the people who know what they're doing. I'd find a value that's big enough to be interesting but small enough to not be painful and stick with that.