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I've developed a model for predicting the probability that each horse will win a race.

The output of the model is the predicted probability that each horse will win, the sum of all the probabilities will be 1.

If I were to look at bookmaker prices for a horse race they would look something like this:

3.9, 5.4, 3.95, 6.7, 9, 14

If we took the sum of the probabilities that each horse would win, we'd get something like this:

1/3.9 + 1/5.4 + 1/3.95 + 1/6.7 + 1/9 + 1/14 = 1.027

The odds come out to be over than 1 due to the bookmaker overround.

This is problematic for me because I need to compare the odds offered at the bookmaker and the probability from my model to determine if there is any value placing a bet.

In order to compare them accurately I will need to either

  1. Remove the overround from the bookmaker prices (deflate the probabilities)
  2. Inflate the output of my model to include the bookmaker's overround

Can anyone suggest how to do this accurately from a theoretical point of view?

I believe that doing this linearly - applying the same factor to each horse - is incorrect.

I believe that each horse should take a percentage of the over round based on their odds of winning, the horses that are more likely to win should take a bigger proportion of the overround. Does this sound more correct?

Thanks for your help.

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    $\begingroup$ You don't need to do either. You can compute the expected return on your bet given the bookmaker's odds (which determines the payout), under your probability model. If the expected return is high enough (at least positive, presumably, though if you're risk averse it would need to be more), you should bet. $\endgroup$
    – Glen_b
    Commented Jun 3, 2014 at 8:11

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You are correct to reason that the bookmaker has probably not applied the overround equally to all runners. It's more likely to be a function of that runner's contribution to the book and the amount of money they expect to take on it.

If you have a sufficient amount of data on bookmaker prices and race winners, and assume a specific relationship between a runner's odds and its share of the overround then you can find the parameters of that relationship which maximize the likelihood - in other words, the adjustment to the raw bookmaker probabilities that result in the most accurate predictions of the winners.

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    $\begingroup$ Thanks for your answer. I will look into some stats and build a model for it, I thought it might be way more simple than that. I just have one question. The favourite-longshot bias means that there will be "too much" money bet on longshots and "not enough" bet on favourites, why would this mean that favourites would have a larger share of the overround? Do bookies expect to make the bulk of their money on the longshots? $\endgroup$
    – janderson
    Commented Jun 3, 2014 at 9:05
  • $\begingroup$ My reasoning behind weighting more of the overround to the favourites is that they take up more of the book percentage. I didn't really consider the bookies trying to profit maximise. $\endgroup$
    – janderson
    Commented Jun 3, 2014 at 9:09
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    $\begingroup$ @Watson Sorry for the confusion. My background is in gambling but I have little experience of horse racing. Different sports exhibit different manifestations of favourite-longshot bias and it sounds like I have mistakenly flipped it. Your point about greater percentage of the book is also a good one - either way if you follow this approach you should gain some insight into how they divide the overround. I've edited my answer to reflect these points. $\endgroup$
    – M. Berk
    Commented Jun 3, 2014 at 9:33
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To determine if there is any value placing a bet just take your model's % of a horse and multiply it to the odds offered. If for example your model says horse A has a 25% of winning and the odds offered is 4,10 then you have value. But this is not all. You have to test your model's outcomes. What i mean is check what is the % of winning on the real deal. Here is an example. Your model has predicted a 15% of winning for 105 horses and in reality there are 11 winners out of these 105 horses which equals to 10,5%. So your model overestimates and even if you would get value odds for the initial 15% prediction in reality thats a big loss cause of your model's fault. I hope you get the message.

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