Favourite-Longshot Bias: A Bias or Not a Bias? - That is the Question
Posted 14th November 2023
In April 2022, former poker player and betting analyst Dan Abrams published an article on Pinnacle which, to my mind, is one of the most insightful pieces of work on the subject of sports betting that may ever have been produced. The trouble is that it's a difficult read on account of the heavy mathematics which features in it, and may have readers giving up before the end. Indeed, it has only been recently that I have revisited his work and fully appreciated what it was he was trying to say. It's something I have been skirting about over recent years myself, but Dan had joined up the dots. Only now have I seen what it was that he did. What he was dealing with was our old friend the favourite-longshot bias, the observation, in many sports betting markets, that prices on favourites tend to be less unfavourable (i.e., a smaller margin weight) than those for the longshots, as measured by actual level-stakes outcomes. In contrast to traditional academic theories that conisder this bias a reflection of the suboptimal underbetting/overbetting of favourites/longshots respectively by (demand-side) bettors as described by Prospect Theory, Dan concluded the favourite-longshot bias is in fact not a bias at all, but the logical outcome of rational (supply-side) bookmakers seeking to optimise their management of capital growth. In this article, I'm going to attempt to recast what Dan's was saying in a way that is hopefully easier to make sense of. Well, hopefully.
Imagine, just for fun, that you're a bookmaker offering bets on six for a die roll. What sort of odds are you going to offer? The fair odds for such a proposition are obviously 6 to land it and 1.2 to fail. Suppose you want a 3% margin on such a book, how will that change those numbers? This depends how you spread the 3%. If you apply 3% to both sides, your odds reduce to 5.83 and 1.165. If you do that, your customer has a 1.7% chance of showing a profit after 1,000 bets on the favourite 'no-six'. By contrast, however, they have a 33.6% chance of showing a profit after 1,000 bets on the longshot 'six'. The chart below shows the bettor's outcome probability distribution. The area under the orange curve to the right of the profit line is much larger for the longshot (33.6% for backing 'six') than the favourite (1.7% for backing 'no six').
It would thus seem rather daft to apply the margin equally to both sides like this. Yes, as the bookmaker, your expected value is the same on either side, but the longshot ('six') option is carrying the majority of the short-term risk.
If you're a clever bookmaker you will probably want to manage your risks. You might rightly start to wonder how exactly you would apply this 3% margin to balance the risk of losing to your customer after 1,000 bets such that, whatever the odds, that probability is the same for both favourite and longshot. In this example, you could use the binomial distribution (or the normal distribution if you were lazy because a sample of 1,000 is big enough for the two to be equivalent) to crunch the numbers. I've done it using my yield distribution calculator. If I reduce my odds to 5.5 and 1.179, in other words much more for the longshot than the favourite, there is now a 10% chance my customer will make money off me after 1,000 bets either on the favourite or the longshot. Although my risk has increased on the favourite, I've reduced it on the longshot. The profitable areas under the curves below are now exactly the same size (i.e., 10% of their respective total areas).
To achieve this risk-equalisation, I've had to reduce my margin weight on the favourite to just 1.78%. On the other hand, the margin weight on the longshot is now 9.09%. From the bettor's perspective, this also translates into dramatically shifted expected values, as marked by the high points on both curves.
Those who've followed my material over the years may notice that this symmetry of risk is very similar to my discussion of the symmetry of likelihood I published for Pinnacle in the context of the relationship between expected value and betting odds. To all intents and purposes, this is the same thing, just from the bookmaker's perspective rather than the sharp bettor.
Thus, to balance likelihood, the bookmaker has to impose an asymmetry to the margin weights applied to the odds, with smaller margin weights applied to the favourites and bigger margin weights applied to the longshots. These asymmetrical margin weights now look just like the favourite-longshot bias.
What has all this got to do with Dan Abrams article for Pinnacle? Dan's theory was as follows:
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 creates the same maximum expected growth on both sides of the market. The maximum expected growth for a line is the expected growth one would get when betting at the full Kelly fraction, determined by the Kelly Criterion.
To prove his theory was correct, Dan had to show 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. As he points out, this fraction is, of course, determined via the Kelly criterion. Now is not the time to diverge into a lengthy discussion on Kelly staking. Hopefully most of you are already familiar with it; for the rest I refer you to my latest book Monte Carlo or Bust or to my previous article on the pitfalls of Kelly staking in sports betting. Given this theory Dan managed to show, with some messy mathematics, that for both sides of a 2-way book to have the same maximum expected growth, the true (margin-free) odds of the longshot, ol, would be given by:
where p and q are the probabilities of the favourite and longshot implied by the bookmaker's odds respectively. Knowing ol it's then straightforward enough to calculate the true odds of the favourite, of, using:
Let's take my 'six' and 'no six' odds of 5.5 and 1.179 and retrospectively remove the margin using Dan's equation. With p = 1/1.179 and q = 1/5.5, plugging these into the equations above, we arrive at true odds of 1.2 and 6 respectively. Thus, Dan has shown that his method of equalising a bookmaker's maximum expected growth on both sides of a 2-way betting book is equivalent to my method of equalising the likelihood of outcomes after a specified number of bets. A bookmaker's maximum expected growth is balanced on each side of a 2-way market when the likelihood of loss after a specified number of bets is also balanced.
There is some material in Monte Carlo or Bust that shows this equivalence rather beautifully (pages 254 to 256). At the time I produced it, I was not aware of Dan's methodology. Dan has taken my material and added an additional robustness to it, for which he should commended.
Dan also showed that his method for removing the margin in 2-way market is almost identical to both the 'odds ratio' method and 'margin proportional to odds' methods I have published with my Wisdom of the (Pinnacle) Crowd betting methodology.
So much for the theory of balanced likelihood; do real world data suggest this is how bookmakers actually apply their margin? My tennis betting datasets offer the most logical way of testing the theory. Taking Pinnacle's closing odds for the last 10 seasons on the ATP and WTA tours (2014 to 2023) and cleaning the data for obvious errors, I compiled a set of 47,269 matches with a total of 94,538 betting odds. Along with actual unit stake profits and losses from these bets (based on the match results), I also calculated their expected losses, given that a bet's expected loss is given by [bookmaker's odds / true odds] – 1, where the true odds were calculated using Dan's equation. Then, sorting the bets in ascending order of expected loss, I plotted the actual and expected losses on a chart, which is reproduced below. It reveals clearly that the longer the odds (which have larger expected losses), the greater the losses, as would be expected given the larger margin weights.
The match between the trend in actual and expected losses given this data sorting is very close. I could have also included the expected bankroll evolution using the 'odds ratio' and 'margin proportional to odds' methods for true odds calculation but their curves would have almost precisely overlain the blue one above and would have been indistinguishable. A t-test comparing actual and expected profits for these 94,538 bets give a p-value of 0.50 implying there is no hint of a statistical difference between them at all, quite a result for such a large sample.
There is one final question to ask: why is this asymmetry in margin weights between favourites and longshots, more commonly called the favourite-longshot bias, arising? Typically, a bias in cognitive behaviour implies a deviation from a strategy that would be considered fully rational. If Dan is correct, however, the favourite-longshot bias may arguably not be a bias at all, but rather a rational application by bookmakers of classical probability theory via the balancing of outcome likelihoods and maximum expected bankroll growth. Or to put it more simply: “it's just variance, stupid.”
Dan's idea also lends further support to the suggestion that the favourite-longshot bias is largely a supply-side phenomenon, as argued in my 2015 article on the favourite-longshot bias, in that it is deliberately created by bookmakers to exploit square bettors, rather than a demand-side phenomenon arising as a result of bookmakers responding to behavioural preferences of their customers.
It should be noted that this analysis has considered 2-way betting markets only. The draw option in 1X2 football betting markets will presumably change the mathematics of Dan's analysis. I wouldn't have the slightest idea where to begin with that. Certainly, the equivalent 'odds ratio' method for margin removal is not as well suited to 3-way markets. Markets with more runners will be even more complicated. Nevertheless, I think Dan has made significant strides in showing the favourite-longshot bias may actually be largely a consequence of rational decision making seeking to optimise the risk management and likelihood of bankroll growth.