Mon 09 October 2017

Filed under deep learning

#### Part 3: Prediction and Evaluation

I considered one major betting strategy during this final phase of the project. It is as follows:

If
$\text{predicted probability} > \text{some decision threshold}$, and
$\text{predicted probability} > \text{sportsbook probability}$
then place bet

This strategy can be thought of as conservative approach. We include a parameter that indicates if we are confident that we have an edge on the casino or sportsbook or not.

Allow me a quick foray into the structure of gambling. Sportsbook odds are set, not by information, but by popular sentiment as it is revealed by action. If some Boxer A gets a large amount of action, then the sportsbook will consider Boxer A to be more likely to win, and consequently, payout on this outcome is reduced. Thus, my model is attempting to answer the question, "When does prediction based on historic data result in a confidence higher than the confidence of the sportsbook -- which is a function of popular sentiment?". In this sense, at it's most stripped down, the model is trying make a totally impartial, data driven decision, and looks for opportunities when public perception is not aligned with historically based signal.

The strategy also has the opportunity to benefit from it's built-in safeguard. Suppose the sportsbook places some Boxer A at a 10% chance of winning, and the model predicts an 11% chance of winning. Without the safeguard, the model would decide to place the bet. Wagering on such low probabilities is something we don't want. Instead, we want to see action whenever the algorithm is confident above some appropriate threshold.

To be implemented, first comes acquistion of more data. Historic sportsbook odds needed to be collected so that we could compare them with model's.

In boxing, the bookmaker's odds come structured into a form which is referred to as the moneyline. The moneyline is a little confusing at first. Generally, one fighter who considered favored to win is assigned a negative number. The other fighter is considered the underdog, and is assigned a positive number.

The best way to remember how to read the moneyline is to always start with the image of a one-hundred dollar bill in your mind.

• A negative number (assocated with the favored fighter) shows how much money you need to bet to win a profit of $100. • A positive number (associated with the underdog) shows how much profit a winning wager of$100 would yield.

So if the moneyline has Boxer A at -130, we know that Boxer A is expected to win. Further, you know you'd have to place a \$130 bet on this fighter to win \$100.
For Boxer B, the moneyline might have them set at +110. This mean Boxer B is the underdog, and if you placed a \$100 bet on this fighter, you'd win \$110.

Since Keras is returning the probability of a boxer winning, we now need to convert the moneyline into regular probabilities so we can compare apples to apples.

To do this, first the actual numeric values (-130 and +110, on this example above), must be converted to what are referred to as implied probabilities (more about implied probabilities in a second). The formula is as follows:

$\text{Implied probability for 'negative' moneyline} = \frac{ - ( \text{'negative' moneyline value})}{- ( \text{'negative' moneyline value} ) + 100}$
and

$\text{Implied probability for 'positive' moneyline} = \frac{100}{\text{'positive' moneyline value} + 100}$

Thus, -130 is converted to 0.56, and +110 is converted to 0.50.

But what is an implied probability anyway?

Implied probability is our usual notion of probability which has actually been modified by what is called either vigorish, or juice. Both of these terms refer to a built-in edge, by the bookmaker, on the true odds. The modification shifts the moneyline is such a way that the sportsbooks can make their profit. Usually, the vig amounts to 20 points. It's basically the casino's cut. Fortunately, it's easy to remove the vigorish using this simple formula:

Take one of your implied probability. Divide it by the sum of both of your implied probabilities.

Thus:

$\text{Actual probability } = \frac{\text{Implied probability A}}{\text{Implied probability A} + \text{Implied probability B}}$

With the math settled, I began searching for a set of historic moneylines for records which I could use in my test set. Using a variety of sources (including laborious searching of the Wayback Machine, and locating an actual broker for assistance), I collected a set of 728 moneylines. After munging, the final size was 679.

We now bring our attention back to decision thresholds. What would be the optimal value where our $\text{model probability} > \text{some decision threshold}$? To determine this, it was merely a matter of looping through thresholds from $[ 0, 1 ]$ by 0.1 and collecting the resulting classifications.

The outputs collects at each of these varying decision thresholds were as follows:

1. Total number of wagers that satisified the criteria and thus were placed
2. Number of wagers placed which won
3. Number of wagers placed which lost
4. A tabulation of the balance resulting from money won via successful wagers and money lost via unsuccessful wagers
5. ROI (return on investment): simply $\frac{\text{balance}}{\text{total investment}}$

We assumed that each bet placed was a \$100 bet. Every loss will incur a deduction of \$100, whereas each winning bet will earn a deposit depending on the sportsbook odds. This means if the model can predict 'easy' matches, it can win a smaller amount of money, but if the matches are harder to predict, the model can earn more.

Here is a plot showing the outcome for #1 on the list above: This gives us an intuition on where to place our decision threshold. We are interested in the point where the blue line and the green line are closest together, which means we will win the highest proportion of our placed bets. Simultaneously we would like this point to be as high as possible along the y axis, meaning the model chose to place a high net number of bets.

The chart below shows the model will place the proportionately largest number of winning bets around an 0.85 to 0.94 decision threshold.  It looks like roughly 0.90 is a reasonable place to set the threshold.

Now that we've got a feel for where our threshold might be, we can look forward to the bottom line -- did we turn a profit?

Here is a plot showing the outcome for #5 (ROI) on the list above: This chart shows the ROI values at the regions of interest we saw in the first plot: This chart shows we can get an ROI of roughly 22.5% if we set the decision threshold to 0.90. We could push it up to 24.8% if we choose 0.94 as the threshold, but notice the precipitous drop starting at 0.95 on the plot above. Better to be wary of the presence of variance and/or noise and choose to focus on a more median value.

To unpack the ROI value, we can look at the actual dollar amount earned: We are seeing that when we stick with 0.90 we earn \$292.00, from a net investment of \$1,300.

Finally, let's see what the plot looks like when we view most of these results simultaneously: Here we can see the sweet spot of 0.90 clearly.

However, modeling like this isn't always deterministic. Small variations in inputs such that occur during randomly selecting test and validation sets can ripple through the pipeline and give different results.

I reran the project from top to bottom a few times, and found that indeed the threshold is a pretty fragile spot. These three images show that the 0.9 threshold might not always be the optimum. To visually get a feel for the variance, I made a plot of these 100 models: That's quite a lot of varaiance. How to mitigate this? The idea is akin to ensembling the model with many different versions of itself. I decided to create 100 models, evaluate each of them, and take average for ROI, bets placed, cash balance, etc.

Here's a plot of the average: Notice how much smoother the line is than on the other plots. It seems like the choice of 0.90 is still pretty conservative. But in gambling, perhaps conservative is okay.

With the bulk of the work now done, we can claim success.

The outcome can be summed up in this elevator pitch sized statement:

"The model, with a decision threshold of 0.90, chose to place thirteen bets, winning all but one. With an initial investment of \$1,300, it won \$292, which represents a ROI of 22.5%."

This system performed much better than I expected. My initial hope was simply to build a MLP which had accuracy better than a coinflip. But the fact that the model can turn a profit when swimming with the Vegas sharks is very exciting.

The next step is to put the model to use, see how it performs over a few month time period, and then see what improvements can be made. This is the true test -- putting my real money where my mouth is! I will report back with a Part 4 of this blog post series when I have had enough wagering experience to infer what can be improved.

Thank you for reading this far. Please comment if you have the inclination!

Sat 16 September 2017

Filed under deep learning

Modeling

Fri 15 September 2017

Filed under deep learning

Characteristics of the dataset

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