For many years, there has been a debate about how to figure out the true odds of an event simply by examining the market price. This article looks at why the favourite-longshot bias is not a bias. Read on to find out more.
To do this accurately, you must first determine which odds can be trusted to be an accurate reflection of the real odds, and then you have to remove the bookmaker’s margin from them. In many markets, Pinnacle is an excellent resource for getting an accurate view of the real odds, since they spend their time and money collecting the opinions of winning bettors, rather than sniffing them out and restricting them.
- Read: The truth about variance
But, the process gets tricky when you have to remove the bookmaker’s margin, so let’s take a closer look at how best to remove it from the lines of a given market. To do that, we have to try to get inside the head of the bookmaker and figure out how they apply their margin in the first place.
The conventional wisdom is that it’s in the best interest of the bookmaker to apply a margin to their lines equally. In other words, if they reduce the odds to 91% of the fair odds on one line, then they should reduce the other line (or multiple lines, in multiway markets) by the same proportion. For example, for NFL spreads and Totals, most bookmakers apply a margin of about 4.8%. Since the spreads are designed to make the probability of the two sides 50/50, their odds are determined by adding the margin to 100% to form an overround, and then multiplying this number by the true win percentage to yield a break-even win percentage for the bettors:
50% * 104.8% = 52.4%
This results in a line of -110/-110 on the two sides, since:
100 * [(52.4%/(52.4% - 100%)] ≈ -110
By doing it this way, the bookmaker gains the same edge or EV over the bettors regardless of which line they bet on, or at what odds. The bookmaker then becomes indifferent to whichever side they may have excess risk on because they have the same theoretical edge on the unbalanced money no matter what. Right?
The problem with this traditional theory is that in many, many instances, it has proven to be false. By examining the outcomes of real sporting events, and comparing them to the sharpest closing lines available, several highly credentialed authors have shown that there is a “favourite-longshot bias” or FLB. By this, they mean that there’s a bias in how the bookmakers add in the margin to their lines, with a greater amount on the longshot and less on the favourite than they should. This begs two questions: how do they apply the margin, and more fundamentally, why would they accept a smaller edge on the favourites but insist on a larger edge on the longshots?
There’s a subtle difference between balancing the EV in a market and balancing the MEG.
There are just about as many theories for why this happens as there are theories about the method the bookmakers use to add the margin in this way. But, I’m going to add another one to the mix that attempts to answer both questions at the same time.
My theory is this: The best interests of a traditional sportsbook (i.e., a market maker that may carry the risk on their money if there is unbalanced action) are not served when it creates the same edge on both sides of a market. Instead, they’re best served when it is indifferent to which side of a market carries excess risk, and this will happen when it creates the same maximum expected growth (MEG) on both sides of the market. The MEG for a line is the expected growth one would get when betting at the full Kelly fraction, determined by the Kelly Criterion.
There’s a subtle difference between balancing the EV in a market and balancing the MEG, and it will take a lot of math to work out a formula for how to remove the margin if the bookmaker applies it this way in practice. We’ll have to use some logarithms and a lot of algebra to answer the questions with my Theoretical Kelly Optimization (TKO) method. But if I’m right, then we’ll have an accurate method for removing the margin from a set of lines and better determining how large an edge we may have on our bets. And, we will have the reason why it’s in the best interest for the bookmakers to do it that way.
For my theory to be true, we have to prove that the expected growth of capital for both sides of a two-way market must be equal when the optimal fraction of the bookmaker’s capital is at risk on the respective sides. This fraction is, of course, determined via the Kelly criterion. Incidentally, this method is similar in concept to the theory put forth by Jonathon Brycki in his article ‘Who is responsible for the favourite-longshot bias’, written for Pinnacle in March 2019, but it seems he never gets to a final answer of what the optimal margin apportionment is. The correct balance of MEG will occur when the following equation for the expected values of the logarithm of wealth on both sides of the market is true:
E = p * log(1 + f1b1) + q * log(1 – f1) = q * log(1 + f2b2) + p * log(1 – f2)
p, q = true probability of the favourite and longshot winning, respectively.
f1, f2 = optimal fraction of capital at risk for the favourite and longshot, respectively
b1, b2 = posted odds for the favourite and longshot, respectively (i.e., decimal odds – 1)
And where we can solve for the true odds for both sides of the market (as a function of the implied probabilities of the posted odds), such that:
p1, p2 = implied probability of the favourite and longshot, respectively
b0 = true net fractional odds of the longshot
1/b0 = true net fractional odds of the favourite
To find the true odds, we will assume that the fractions of capital the bookmaker has at risk are right at the optimal amount, so we can use the simple Kelly Criterion formula to substitute in odds and probabilities for the fractions f1, f2, like so:
E = p * log(1 + f1b1) + q * log(1 – f1) = q * log(1 + f2b2) + p * log(1 – f2)
p * log(1 + f1b1) - p * log(1 – f2) = q * log(1 + f2b2) - q * log(1 – f1)
p [log(1 + f1b1) - log(1 – f2)] = q [log(1 + f2b2) - log(1 – f1)]
f1* = p – q/b1 and f2* = p – q/b2
We can substitute for f1, f2 and simplify:
p [log(1 + pb1 - q) - log(1 – q + p/b2)] = q [log(1 + qb2 - p) - log(1 – p + q/b1)]
p log[(1 + pb1 - q)/(1-p+q/b1)]
= q log[(1 + qb2 - p) - log(1 – p + q/b1)]
p log[(p + pb1)/(p + p/b2)] = q log[(q + qb2)/(q + q/b1)]
p log[(p (1 + b1))/(p + p/b2)] = q log[(q (1 + b2))/(q + q/b1)]
p log[(p (1 + b1) * b2)/((p (1 + b2))] = q log[(q (1 + b2) * b1)/((q (1 + b1))]
p log[((1 + b1) * b2)/(1 + b2)] = q log[((1 + b2) * b1)/(1 + b1)]
At this point, we can convert to decimal odds (O1, O2) for convenience, and then convert to implied probabilities (because implied probabilities are just the inverse of decimal odds):
p log[O1(O2 - 1)/O2] = q log[O2(O1 - 1)/O1]
p log[p2(1/p2 - 1)/p1] = q log[p1(1/p1 - 1)/p2]
p log[(p2/p1) * ((1 - p2)/p2)] = q log[(p1/p2) * ((1 - p1)/p1)]
p log[(1 - p2)/p1] = q log[(1 - p1)/p2]
b0 = p/q = log[(1 - p1)/p2] / log[(1 - p2)/p1]
b0 = log[p2/(1 - p1)] / log[p1/(1 - p2)]
b0 = log[p2/q1]
Now, this is the answer if the bookmaker has the optimal Kelly fraction at risk, which is a lot to risk on a single market. Does the answer hold up for smaller Kelly fractions on either side? Well, after plugging this very simple formula into Excel and examining the expected growth (EG) for smaller but equal fractions of full Kelly, I determined that the EG for each side matches very closely regardless of whether the bookmaker has money at risk on the favourite or the longshot. So, even when the bookmaker has much less than the optimal amount at risk on an individual market, it’s still indifferent to which side of the market the risk is on because it gets the same EG either way. And the tiny amounts of EG from hundreds (or thousands) of different simultaneous markets combine to give them the optimal balance between edge and risk.
Ok, so now we have a formula from which we can make testable predictions. That’s how we know a theory is most likely true if the actual data back us up. I’m no data wizard, but I know a couple of people who are. One of them, Joseph Buchdahl, has compiled many years’ worth of data on the results of soccer matches and matched them up with different proposed methods of finding the margin-free odds in his paper, ‘The Wisdom of the Crowd.’
Another one is Matt Buchalter, aka @PlusEVAnalytics on Twitter, who several years ago investigated different vig removal methods and found that the method of using a probit scale best matched the data. I have no idea what a probit scale is, but he was generous enough to post an Excel spreadsheet with a formula for it, so I compared his method to my TKO method of equal MEG, as well as to the margin proportional to odds, logarithmic function, and odds ratio methods that Buchdahl investigated. I crunched the numbers for a wide range of implied probabilities and, assuming a total two-way margin of 1.8% (like you might find in a very efficient Pinnacle market), plotted the percentage of implied margin added to each of the runners. The results are below:
The black line for equal margin represents what happens if there is no FLB. It’s very different than the rest. The curve for the logarithmic function is also somewhat different, but at least it slopes in the same general direction as the other FLB-based methods, with more margin added to the longshots’ implied probability and less on the favourites. I didn’t plot Buchdahl’s method of margin proportional to odds, as it would simply trace a horizontal line at an implied margin for each runner of 0.9%, since it works out that applying the margin this way adds a fixed percentage to the implied probability of each runner. For probabilities between 10-90%, the difference between this answer and mine is pretty trivial, so I didn’t want an extra line to obscure the small differences between the other methods - and those differences are indeed very small.
For this kind of two-way market, the odds ratio and probit scale methods almost exactly track the TKO method. This implies that either they’re all very accurate, or all mostly wrong. In fact, since the probit scale method is based on z-scores, there is evidence in Buchdahl’s new book ‘Monte Carlo or Bust’ that suggests it may be mathematically identical to the TKO method.
Now, in Buchdahl’s paper, his analysis of the data shows that the logarithmic function model much more closely tracks his method, and thus produces roughly the same superior results. Why, then, does itlook so different in my graph? Because the logarithmic function is particularly suited to modelling three-way markets (like the 1X2 markets for English soccer that Joseph analysed), but the odds ratio method is much better suited to two-way markets.
So, since the TKO method very closely matches the three best approximations of margin distribution according to the FLB, I think we have found the answer to both how, and why, the bookmakers do it that way. And, once you look at the advantage the bookmaker has over the bettor from an EG perspective (which factors in their variance) rather than from an EV-based one, you find out that the so-called FLB isn’t a “bias” at all. It’s exactly how the bookmaker should be applying its margin to profit equally from the bettor, whichever side of the market they choose to bet on.
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