This blog is the result of reading this paper by William Benter on horse race handicapping, and from email and twitter exchanges with Sean Dickert. Writing this was fairly revelatory for us, but is just a re-hashing of Benter's paper.

Something that has troubled us for some time now is the idea that the public odds (that is, the odds you are taking to place your bets) should be included in the modelling process. My knee-jerk reaction to this claim is that it won't really matter whether you incorporate them or not: if your model indicates positive expected value on a given bet, then taking into account the market's odds will never change that positive sign to a negative, and vice versa. Therefore it seems that taking into account the public odds could have a similar effect to having a less aggressive staking strategy; or, if you are simply betting the same stake no matter what the percieved edge is, it would have no effect.

This logic is flawed, however. The conceptual mistake is the failure to remove the margin from the public odds; if "fair" public odds (i.e. odds with implied probabilities that add up to 100%) are used in one's modelling process, their inclusion can change your expected value from positive to negative. To arrive at this conclusion, we first need a better understanding of what is required for a bettor to be profitable against the public, or a bookmaker's, odds.

More specifically, we are interested in the conditions that a bettor's model must satisfy to beat the long-term profit implied by a bookmaker's margin (i.e. 1/(1+margin) - 1). Is it necessary that the bettor's model fits the data better (i.e. is better at estimating the true underlying probability of the events being bet on) than the bookmaker's odds? It turns out that this is not required. To better understand this point, let's focus on a betting market on a binary event that occurs with "true" probability p, and where the odds-setter (i.e. the bookmaker) is not adding any margin to their odds.

Suppose that both the bettor's model and the model used by the bookmaker to set odds are equal to the true probability of the event plus a random noise term. This means that both the models will fit the data identically. However, despite the models having identical fit, if the bettor place a bet whenever their model indicates a higher probability than the bookmaker's, this will generate positive value in the long-run. This may seem counterintuitive; it initially did to me. The logic is fairly simple: given the nature of the two models, on average the true probability of the event will lie in between the bettor's estimate and the bookmaker's estimate. Therefore, in situations where bets are placed (i.e. the bettor's probability estimate is greater than the bookmaker's) the true probability will on average be greater than the bookmaker's probability — which means I have positive expected value (recall that the expected value of a 1 unit bet is true_prob/implied_prob - 1). Conversely, when the bettor's estimate of the probability is lower than the bookmaker's estimate, no bets are placed. Therefore, despite the fact that the two models fit the data equally well, the party who has to set odds is at a disadvantage. This disadvantage arises because they are not able to incorporate information from the bettor's model into their odds-setting process. On the other hand, the bettor is able to incorporate the information from the bookmaker's model into their betting process (because the odds-setter reveals their estimate in the form of offered odds). And, because the combination of the bookmaker's model and the bettor's model together will fit the data better than either of the model's alone, this generates positive expected value for the bettor. (The models together will fit the data better due to wisdom of the crowds logic; when you combine more and more guesses that are equal to the true probability plus noise, you will converge on the truth - the classic example is many people guessing how many M&Ms are in a jar.)

Above I said that the bettor has the advantage because he can incorporate information from both his model and the bookmaker's model. However, in the process outlined above, where did this occur? It doesn't seem like the bettor incorporated the market odds into his process. In fact, because the bettor is using the rule "I will only place a bet whenever my estimate of the probability of an event is greater than that of the bookmaker", they are informally incorporating the information from the bookmaker's model. If the bookmaker's implied probability is below my probability estimate, I am inferring that the true probability of the event is above the bookmaker's (and consequently I have positive expected value), while the opposite inference is made if the bookmaker's estimate is above mine. If this description still feels a bit hand-wavey, another way to describe why the odds-setter is at a disadvantage is that they have to provide odds that cannot be improved upon by bettors in order to not lose money; conversely, the task of the odds-taker (i.e. the bettor) is much easier in that they simply need to improve upon the odds-setter's estimates in some way. The worst-case scenario for the odds-taker is that they make no money, while the best-case scenario for the odds-setter is that they make no money.

Of course, in practice, odds-setters don't provide odds to the public at prices that reflect their estimates of the underlying probability of the event. They increase their odds, as they should. Unless the odds-setter's probability estimates reflect all relevant information, they will lose money by offering odds equal to their probability estimates; otherwise (in the absolute best-case scenario that their odds cannot be improved upon) they will just break-even. The only way an odds-setter could make money offering odds equal to their estimates (with no margin added) is if the bettor, by incorporating their model's information, somehow actually make the odds worse (which is pretty hard to achieve - see example below). This is why it is pretty unreasonable for people to take our live model's odds on Twitter and treat them as prices we would offer to the public (as a bookmaker, or on an exchange). We would never offer those odds, for reasons outlined above.

What's written so far may seem like a convulted way of describing something trivial. Essentially, all of the above is simply to say that if the bettor's model can be used to improve upon the probability estimates implied by the bookmaker's odds, they will profit (if there was no margin being applied; otherwise they will make more than what's expected given the margin). This is critical to understand. It implies that the overall fit of your model (i.e. basically how "good" your model is) is not necessarily relevant to its betting success. That is, model A can be more profitable than model B even if model A fits the data significantly worse. For a specific example in golf, suppose that the bookmaker is setting odds according to a model that only uses long-term form; no adjustments for more recent performance or for course fit are made. Suppose that bettor A uses a pure course history model (i.e. they derive probability estimates only from performance at the relevant course), while bettor B uses a slightly worse version of the bookmaker's long-term form model. If we assume that long-term form gets you 95% of the way to the "true" probabilities, while course history gives you the remaining 5%, then in this example bettor B will have a more accurate model than bettor A, but bettor A will be more profitable placing bets against this bookmaker. The reasoning is the same as above: bettor B's long-term from model does not add any predictive value to the market odds, while bettor A's pure course history model does.

The table below attempts to illustrate this with a simplified example:

The purpose of this example is to try to simply illustrate what has been said in words so far. Read the model descriptions in the notes underneath the table. Suppose that each model (e.g. "Equal", "Random", etc.) places a 1 unit bet whenever their probability estimate is greater than the "Book" estimate; green cells indicate where this occurs. I have tried to choose 2 examples that are "representative" of each model's output given how they have been described. In practice, each model's output would be random, and only in the long-term would we expect their profit to converge to that described in the table. The key takeaways here are that both the "Equal" and "Course History" models will profit above and beyond what would be implied by betting randomly. Conversely, the "Random" model, which is just adding noise to the bookmaker's odds, will get a long-term return roughly equal to 0 (which is what's expected when there is no bookmaker margin).

Further, it's clear that the "Course History" model fits the data worse than "Random": it assigns the higher win probability to the wrong golfer on the Phil / DJ bet. However, as said previously, what is important is whether a model provides information above and beyond what the market odds provide; given that the market odds never incorporate course history, it will be true on average that the "Truth" lies in between the "Book" and "Course History" probability estimates (given our setup, it will be much closer to "Book"). Finally, the point of including the "Bad" model here was to point out that it requires something strange for a model to consistently deviate from the "Book" in the opposite direction of the truth, which is the only way to have long-term returns below what you could achieve betting randomly.

The general principle is that if the bettor's model plus the market odds together predict probabilities better than the market odds alone, the bettor can generate long-term value above that implied by the bookmaker's margin. Now, let's return to the question posed at the start of the article: what is the benefit to formally incorporating the bookmaker's odds into your betting process?

Let's continue with the course history model example, as it illustrates the point clearly. In either of the matches above, the "Course History" model is indicating massive value. In theory, the bookmaker could add a huge margin to their fair probability estimates (e.g. offering Phil at implied odds of 70%) and the course history model would still "see value" and place a bet. Evidently, given Phil's true win probability is 40%, this would be a big negative expected value play. However, if a sample of historical "Book" odds were collected, and their margin was removed (which might not be easy to do), then the bettor could use the historical outcomes of the matches to assess how much predictive value his model is adding to the "Book" probability estimates. For example, suppose we find that a weighted average of 95% "Book" and 5% "Course History" fits the data best. Then, to asses whether a bet should be placed, we would apply this weighted average. For example, in the DJ/Phil match above, our estimates that combines "Book" and "Course History" would be 61% and 39% for DJ and Phil, respectively. Now this combined model would still indicate a bet on Phil should be made in the case where there is no margin added, but if the offered odds increased to 40% (i.e an 8% margin) there would no longer be value. By incorporating the bookmaker's odds into the modelling process, the percieved value from the course history model has changed from positive (when its model estimate is used alone) to negative when it is combined with the bookmaker's fair estimate.

To conclude, using a sample of historical odds in your modelling process can be an effective way to set an informative threshold rule. If, historically, a weighted average of the market's fair odds and your model's odds have outperformed the market's fair odds by themselves, you can apply this weighting to assess whether the deviation of your model from the market is large enough to indicate value. For example, if a 50-50 weighting of the market's estimate and your estimate was the most accurate, then you would need at least a 16% deviation in your model's odds from the market's fair odds to overcome an 8% margin. In the DJ/Phil example, if an 8% margin was applied equally to each player, then the "Book" odds would become roughly 68% and 40%; this would then require the bettor's model to provide odds of at least 43% on Phil to be comfortable that they were placing a positive expected value bet.

As it relates to our betting process, this will be something to look into in the future. Currently we are in the early process of building a database of golf odds from various sportsbooks. Our goal in modelling golf has always been to fit the data as well as possible. This is partly due to the fact that we use our model for many purposes apart from betting. This discussion, which closely follows the linked Benter paper, indicates that this is not the most important factor if the model's only goal is to be profitable. Without using historical odds, there is no way to for us to assess whether our model estimates add value to the public odds. That being said, the value-added of our model likely lies in our assessment of "player form", as opposed to course-specific factors. Additionally, it seems as though course history and course fit are already priced into most market odds. Taken together, we hope this indicates that the large deviations of our model from the market partially reflects real predictive value.