Chopped and Skewed: The Mathematics of Points Betting
By Matthew Buchalter, PlusEV Analytics
NOTE: I have no affiliation with any of the companies or products mentioned, and I do not endorse any specific provider or service.
An Australian bookmaker called PointsBet has been in the news quite often recently. They expanded into Illinois, and they signed marketing deals with NBC Sports and the PGA Tour. Normally this is not something I would find too interesting – this is not a betting industry news blog after all (if you want that I’d recommend Alfonso Straffon and/or Captain Jack). However, PointsBet is a little different. In addition to the standard betting options, they also offer something that’s unique, at least in the North American market: the less-than-creatively named “points bet”. Time will tell whether bettors love it or hate it, but as a mathematician I find it very interesting. So, let’s explore the world of points betting.
Most sports betting propositions can be thought of as taking the form of (quantity) (over or under) (value). Even point spread bets like Giants +3.5 can be restated as (Giants final score – Cowboys final score) (over) (-3.5). There’s a number, if you’re on the right side of it you win, if you’re on the wrong side of it you lose. It doesn’t matter whether you win/lose by 1 point or by 50, your result is the same.
With points betting, instead of winning/losing a fixed amount depending on whether you’re on the right or wrong side of the number, the amount you win or lose grows according to the margin by which you beat (or lose to) the number. So if you bet “$10 per point” on Giants +3.5 and they win by 10, you win $135. If they lose by 6, you lose $25, etc.
Before we start our analysis, a couple of important notes about this unique format:
- For this to be a viable product, you can’t have unlimited wins and losses…so PointsBet puts a cap on both the maximum win and the maximum loss. This cap is different for each bet and is shown in the bet details so make sure you look at it before you bet.
- Normally books make their money by charging a vig, like -110, on both sides of a bet. That doesn’t really work with this format, so what PointsBet does instead is put a spread between each side of the bet; for example, you can take the Giants +3.5 or the Cowboys -4. This will seem outrageously unfair at first glance, but it’s only because it puts the book’s margin more “in your face” than the traditional vig; objectively it’s not any better or worse.
Ready? OK, let’s go.
I’m going to explain the mathematics of points betting using three made-up example bets that might be found in an NBA game. If you’re mathematically inclined and you want to follow my calculations, I’ll provide the assumptions in parentheses – please don’t DM me about how bad the assumptions are because I don’t care, I’m just trying to illustrate the concepts as simply and accessibly as possible. If you have no interest in reproducing my calculations, I don’t blame you – just ignore the stuff in parentheses.
Bet #1: LeBron James points over 29.5 / under 27.5 (assume Poisson distribution with mean 28.68)
Bet #2: Time of first points scored over 29.5 / under 27.5 seconds (assume exponential distribution with parameter 0.0245, conditional on >=4 with 0 probability for <4)
Bet #3: LeBron James steals x blocks x 1st quarter points over 29.5 / under 27.5 (assume independent Poissons with means 1.2, 1.2, 10.1) – I made this particular one up, but PointsBet DOES offer weird “multiplier” props that look like this!
For all three bets, assume that both wins and losses are capped at a maximum of 100 units.
I’ve set each of these bets up so that if we were betting traditional over/unders, 28.5 would be the fair line for each of them. Each bet has an approximately 50% probability of over 28.5 and 50% probability of under 28.5; at the risk of giving you painful flashbacks to middle school math, each of these three bets has a median of 28.5. And if we’re operating a normal sports book with normal bet types, we’re done here. The median is all we care about.
With points betting, the median doesn’t matter much at all. Because our win or loss scales with the magnitude of the number, the mean is really what we care about. For sports bettors who are used to thinking in medians, this can be a jarring shift. We can describe the relationship between median and mean by looking at the “shape” of the probability distribution and how symmetrical or asymmetrical it is; the technical name for this is skewness.
Let’s explore mean, median and skewness using our three hypothetical bets:
Bet #1: LeBron James points over 29.5 / under 27.5
This distribution is almost completely symmetrical – it is barely skewed at all. The mean of this distribution is 28.68, very close to the median of 28.5.
For an “over” bettor, the payoff is (unit size) * (X – 29.5) where X = LeBron points.
This makes the EV of the bet = (unit size) * (E[X] – 29.5) = -0.82 units.
For an “under” bettor, the payoff is (unit size) * (27.5 – X).
This makes the EV of the bet = (unit size) * (27.5 – E[X]) = -1.18 units.
The cap of 100 units doesn’t have any impact because it’s impossible for these bets to win or lose more than 100 units with this distribution.
Bet #2: Time of first points scored over 29.5 / under 27.5 seconds
This distribution is much less symmetrical. The median is still 28.5, but the minimum possible value is 4 and it’s plausible to have values of 50, 100 or even 150 or more. The mean of this distribution is 44.60, significantly higher than the median. We call this distribution “right skewed” because there is a disproportionate amount of probability in the right, or upper, end.
For an “over” bettor, the payoff is (unit size) * (X – 29.5) where X = time of first score.
This makes the EV of the bet = (unit size) * (E[X] – 29.5) = +15.10 units. (+EV alert!)
For an “under” bettor, the payoff is (unit size) * (27.5 – X).
This makes the EV of the bet = (unit size) * (27.5 – E[X]) = -17.10 units.
But, we failed to consider the 100 unit win/loss cap which DOES come into play here.
For over 29.5 bettors, wins are capped when X>=129.5 and losses are not capped. So the real distribution we’re interested in is an adjusted version of X that gets “chopped” at 129.5.
The mean of this “chopped” distribution is 42.82, so the EV of the over including the cap is +13.32 units.
Bet #3: LeBron James steals x blocks x 1st quarter points over 29.5 / under 27.5
Now, I have no idea how much interest there is among the recreational betting population in these feats of mathematical gymnastics disguised as props. They do exist and they’re fairly common across major sports, and they are tough as shit to price properly. PointsBet does have an analytics team, and stuff like this seems almost like it was invented for the purpose of putting them to the test. The more difficult a line is to price, the more potential opportunity there could be for advantage bettors…so let’s dive in.
Because we’re multiplying things together, X = 0 if ANY of steals, blocks or 1q points is zero. This results in a huge probability mass at zero, and a mean of 6.45 (Remember, the median is still 28.5!) You guessed it, this one is “left skewed”.
EV of over 29.5 = -23.05 units
EV of under 27.5 = +21.05 units
|Bet||Skewness||EV of over 28.5 -110||EV of under 28.5 -110||EV of Points over 29.5||EV of Points under 27.5|
|LeBron points||None||-0.0455 x bet amount||-0.0455 x bet amount||-0.82 x unit size||-1.18 x unit size|
|Time of first points||Right||-0.0455 x bet amount||-0.0455 x bet amount||+13.32 x unit size||-15.32 x unit size|
|LeBron steals x blocks x 1Q points||Left||-0.0455 x bet amount||-0.0455 x bet amount||-23.05 x unit size||+21.05 x unit size|
Unfortunately for us, PointsBet oddsmakers understand the impacts of skewness and chopping just as well as you now do…so they won’t necessarily price their points betting offerings at the median as I stupidly did in these examples. That means that value will not be easy to find without some pretty sophisticated distributional analysis. For that reason, I don’t recommend these for amateurs who are looking for +EV propositions. However if you do choose to play them, a few words of advice:
- If a line looks too good to be true, be careful. Think about whether you’re evaluating it in your head against the median or the mean, and ask yourself how much skewness there is and in which direction.
- Look for bets where the chopping will cut your potential losses more than your potential gains. For example, “under 90” if it’s limited to +/- 100 units on something that can’t be negative but can be over 190, such as the the time of some event.
As I’ve said many times, the theory of “antifragility” says that anything new, whether it’s a new sport, new league, new book or new betting product, can be a potential benefit to advantage players.
Good luck if you play, may the skewness be favourable to you and may your losses get chopped more than your winnings!
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