How Winners Win: A Framework for Breaking Down Expected Value
By Matthew Buchalter, PlusEV Analytics
Diversity is one of the most important keys to good scientific inquiry.
I’m not talking about diversity of race, gender, sexual orientation, etc. – those are fine but they’re superficial. I’m talking about intellectual diversity – different ways of thinking about things and different ways of doing things.
Gambling Twitter is a great source of intellectual diversity. Here is a recent example:
Once you take the LV out of CLV, you end up with mere correlation with EV.— truepokerjoe (@truepokerjoe) July 14, 2021
Your bet on Conor vs Floyd was not genius.
Spanky is a line grinder; for them, CLV is understandably the end all.
For handicappers, it's just another tool in the box,l.
EV w/o CLV is beautiful.
Here we have Spanky saying that beating the closing line is the path to long term success followed by Poker Joe’s response that “EV without [closing line value] is beautiful”. Ridiculous nicknames aside, both of these guys have plenty of credibility as winning bettors and sharp thinkers. Spanky has been the subject of several high-profile articles and runs an operation that is reminiscent of a Wall Street trading floor. Poker Joe is the author of “Sharper: A Guide to Modern Sports Betting“. They both obviously know their stuff. So what is going on here?
Positive expected value (EV) is at the heart of every winning bettor. In this article we’re going to deconstruct EV and explore the different (diverse!) skills that enable winning bettors to be successful.
Fundamentally, the definition of EV is straightforward and non-controversial. The EV of a bet is the sum over all possible outcomes of the (outcome probability x outcome win/loss). On a standard American roulette wheel, the EV of a $10 bet on red is calculated as follows:
|Red||18 / 38||+ 10|
|Black||18 / 38||– 10|
|Green||2 / 38||– 10|
EV = (18/38 x 10) + (18/38 x (-10)) + (2/38 x (-10)) = – 0.526, or a loss of 52.6 cents. The EV is commonly expressed as a percentage of the bet amount, so -0.526 / 10 = -5.26%.
So there’s your definition, but before we continue any further it might be helpful to talk about what this mathematical object called “expected value” really means. I’m going to have to get a little metaphysical here, so if you don’t care you can feel free to skip ahead and you won’t hurt my feelings. The one thing it is NOT is a forecast of any single observation; it is impossible to put $10 on red and walk away with a net loss of 52.6 cents, either you will win $10 and over-perform your EV due to good luck, or you will lose $10 and under-perform your EV due to bad luck. What you have to do to truly understand EV is to imagine an infinite number of parallel universes in which the outcome resolves. Your set of results across all of these universes would contain a bunch of +10s and a bunch of -10s, and they would average out to -0.526. Now, gamblers don’t play in an infinite number of parallel universes, BUT they do think about their results over the “long run” where there there is a large enough number of separate bets that the good luck and the bad luck tend to balance out in much the same way. Over a large enough time horizon, bettors with positive EV WILL win and bettors with negative EV WILL lose.
The tricky thing about calculating EV in a sports betting context is that the outcome probabilities are impossible to directly measure. The best we can do is use models to estimate them. A roulette wheel is designed to work with the laws of physics such that a properly spun ball has an equal chance of landing in all 38 slots, but there is no such design that governs the complex outcomes of a sporting event.
So calculating EV depends on estimation of probabilities, which is difficult. Fortunately, we don’t have to do it alone. Sportsbooks set and move odds according to market forces, and markets are notoriously good at estimating probabilities. How good are they? Well, that depends on your views on “market efficiency”; that is, the ability of markets to accurately reflect the entire universe of information.
If markets were perfectly efficient, what would that mean? It would mean that a game lined -110/-110 would have exactly 50% probability of each side winning. More generally, it would mean that every bet would have identical EV on both sides of it, and that EV would be a negative amount equal in magnitude to the vig charged by the sports book. For a -110/-110 bet, the EV on each side would be -4.55%. This is a tough one to swallow, because it would make positive EV impossible to achieve and as a consequence, no bettor would even win over the long run. While it’s true that most people who claim on social media to be long-run winners are lucky and/or liars, trust me that there are some legitimate winners out there.
Okay so let’s take our assumption of perfect market efficiency and soften it a little, to allow for the possibility that some bettors can have +EV at some points in time. It’s fair to assume that people who have +EV will bet and bet heavily, enough to move the market back in line with where it should be; meaning that +EV opportunities are self-effacing. Take this idea to its logical conclusion and you get the theory that the closing line is perfectly efficient.
If the closing line is perfectly efficient, then the EV = -vig relationship derived above is true for the closing line. From there, you can get the EV of any bet by using the difference between the line you bet and the closing line. This is the central doctrine of the church of Closing Line Value, or CLV. This is Spanky’s world.
Like any good church, the idea that the closing line is perfectly efficient requires an element of faith – in this case, faith that the sharpest money is betting large enough to move the markets more than anyone else. These markets can get pretty large, especially for sides and totals on major sports. If I find a method or angle that nobody else in the world has, I’m obviously going to bet it but my bankroll is modest and even if I grow it steadily with good Kelly bet sizing, it could take years before I’m making a dent in global markets. Until then, there’s no guarantee that the market is necessarily going to follow me and if it doesn’t, and if my angle is truly good, then I will have achieved positive EV without CLV. This is Poker Joe’s world.
So this is the dichotomy. Arbers, line grinders and steam chasers live in Spanky’s world where CLV is the be all and end all. In this world, the sports themselves are incidental – you don’t have to know anything about the teams or the players, all you need to know are the markets. Modelers and originators live in Poker Joe’s world where CLV is certainly nice to have but it’s not absolutely necessary. These two approaches may seem like opposites but they’re really just two different means to the same end: +EV. That doesn’t mean there isn’t animosity; the steam chasers get called parasites who sponge off others’ blood sweat and tears, and the originators get called idealistic try-hards who think they’re better than everyone. It’s similar to the rivalry in finance between technical analysis and fundamental analysis.
We’ve established that CLV is a component of, but not the only component of, EV. Time to explore what that means by building out a mathematical framework for EV.
EV = sum (outcome probability x outcome win/loss).
For a typical sports bet where there are only two possible outcomes, win and lose, we can simplify this to a formula that’s going to be much easier to work with:
EV = win probability x (bet odds, in decimal form) – 1.
Multiplying this by (closing odds / closing odds) and rearranging:
EV = (bet odds, in decimal form / closing odds, in decimal form) x (win probability x closing odds, in decimal form) – 1.
The first term, (bet odds in decimal form / closing odds in decimal form), measures the change between the odds at the time of bet and the odds at the close. This is Closing Line Value or CLV. Let’s re-center this by subtracting 1 so that the absence of closing line value translates to CLV = 0.
CLV = (bet odds, in decimal form) / (closing odds, in decimal form) – 1.
The second term, (win probability x closing odds in decimal form), looks almost like an EV itself. It’s just missing a -1 at the end, and it uses the closing odds instead of the bet odds. What we’re really looking at is what the EV of the bet would have been, if the bet had been placed at the closing odds. Let’s call this EVC, or EV at the Close.
EVC = (win probability x closing odds, in decimal form) – 1.
Substituting these newly defined objects into our original formula:
EV = (1 + CLV) x (1 + EVC) – 1.
A few observations:
- In the absence of closing line value, this simplifies to EV = EVC. This represents Poker Joe’s world where it’s possible to obtain EV without CLV.
- If the closing line is perfectly efficient, then EVC = -vig (not zero!) and EV = (1 + CLV) x (1 – vig) – 1. This represents Spanky’s world where CLV is the key to victory; more specifically, enough CLV to at least overcome the vig.
- There are multiple paths to +EV: through CLV, through EVC or through a combination of both.
Just like how Return on Investment is an observed estimate of the theoretical EV, we can create ROC, or Return on Close, as an observed estimate of the theoretical EVC:
ROI = (net win/loss) / (bet amount).
ROC = (net win/loss evaluated at closing odds) / (bet amount).
One thing I’ve skipped over until now…with multiple sports books offering odds on the same event, “odds” are not a singular value. It seems obvious that “bet odds” should be evaluated using the odds at whichever book you placed the bet, but what about “closing odds”? If we’re talking about market efficiency, then we really should be using an average of the closing odds across all sharp books in the market (defining “sharp books” as the ones that move on their own action, as opposed to the ones that copy other books’ lines). So let’s amend the definition of CLV and EVC to make them more specific:
CLV = (bet odds at your book, in decimal form) / (closing odds at market consensus, in decimal form) – 1.
EVC = (win probability x closing odds at market consensus, in decimal form) – 1.
ROC = (net win/loss evaluated at market consensus closing odds) / (bet amount).
With this change, it now becomes clear that there are in fact two distinct ways to generate CLV: by finding a book with better odds than the market consensus at the time you place the bet, and by having the market consensus odds shorten between the time you place the bet and the close. So let’s split CLV into two parts:
Line Shopping Value: You can think of this as “CLV across space”.
LSV = (bet odds at your book, in decimal form) / (market consensus odds at the time of your bet, in decimal form) – 1.
Line Improvement Value: You can think of this as “CLV across time”.
LIV = (market consensus odds at the time of your bet, in decimal form) / (closing odds at market consensus, in decimal form) – 1.
CLV = (1 + LSV) x (1 + LIV) – 1
EV = (1 + LSV) x (1 + LIV) x (1 + EVC) – 1.
ROI = (1 + Observed LSV) x (1 + Observed LIV) x (1 + Observed ROC) – 1.
The only trick is that the weighted averages have to be done sequentially to ensure that the ROI = (1 + LSV) x (1 + LIV) x (1 + ROC) – 1 result holds for the totals – check out the formulas in the weighted average row in the attached sheet to see what I mean.
This whole exercise may seem like pointless mathematical masturbation, but I assure you it’s much more satisfying than that. To see why, let’s look at three hypothetical bettors’ results over the same number of bets:
|Bettor||Observed LSV||Observed LIV||Observed ROC||Observed ROI|
In terms of straight profit/loss, the three bettors all have equal results. But, if I had to bank on one of them to be a long-term winner, without a doubt my first choice would be Alice, my second choice would be Bob and my last choice would be Charlie. Why? Because the information coming to us through observations of LSV, LIV and ROC contains different ratios of signal to noise. A bettor’s LSV can be evaluated immediately when the bet is placed, without any uncertainty at all; therefore, it does not take a large sample of data to establish a bettor’s ability to consistently generate +LSV. Moving on to LIV, the observations can have some variance in them; for example, lines may shift after your bet due to an unforeseeable injury or lineup change, and such movement can be either favourable or unfavourable in a random way. For this reason, it takes a larger sample to establish a bettor’s ability to consistently generate +LIV. Finally, observed results have more random variance in them than anything else – you can see this by the “Observed ROC” column in the example sheet above. It takes a much larger sample to establish a bettor’s ability to consistently generate +EVC than it does for either +LSV or +LIV.
|Metric||Calculation||Skills Indicated||Sample Size Required to Demonstrate +EV|
|Line Shopping Value (LSV)||LSV = (bet odds at your book, in decimal form) / (market consensus odds at the time of your bet, in decimal form) – 1||Line shopping, steam chasing, arbitrage, sourcing outs||Small|
|Line Improvement Value (LIV)||LIV = (market consensus odds at the time of your bet, in decimal form) / (closing odds at market consensus, in decimal form) – 1||Origination, betting early, beating openers, predicting market moves, front-running sharp money||Medium|
|Closing Line Value (CLV)|
CLV = (bet odds at your book, in decimal form) / (closing odds at market consensus, in decimal form) – 1.
CLV = (1 + LSV) x (1 + LIV) – 1
|All of the above||Medium|
|Return on Close (ROC)||ROC = (net win/loss evaluated at market consensus closing odds) / (bet amount)||Origination, with novel approaches that are better than what the market is doing||Large|
|Return on Investment (ROI)|
ROI = (net win/loss) / (bet amount) – 1
ROI = (1 + CLV) x (1 + ROC) – 1
ROI = (1 + LSV) x (1 + LIV) x (1 + ROC) – 1
|All of the above||X-Large|
Keep this framework in mind when evaluating a bettor’s results (either your own or someone else’s) – it will give you a much clearer picture than win/loss results alone! And while any of these metrics and skills alone is potentially good enough to make a winning bettor, the best case scenario would be to perform well in all of them – expanding your skill set is always a +EV move!
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