# Investing – yes investing – in Sports Competitions

The recent Supreme Court’s ruling that the federal ban on betting is unconstitutional has created a lot of discussion on the implications of this decision. Is betting bad? Is betting good? Are people going to be exploited by the bookmakers? All of these questions are legitimate questions and concernts and certainly deserve attention. However, from a purely mathematical/statistical perspective betting is no more different than investing – yest investing – in the stock market. Why don’t people oppose to people betting on which company’s shares will go up? In fact, I personally think (and I will short show you why) that you have a better shot at getting a better Return on Investment (ROI) from sports betting as compared to the stock market. Of course, you need to be disciplined, and I think this is where the objections of many people come with respect to legalizing betting. But what aspect of your life you really do not need to be disciplined?

Let’s see why betting is not really different than gambling..ummm investing in the stock market. First we will briefly introduce the Kelly’s growth criteria. Kelly assumes that our goal through betting is to maximize the expected long-run percentage growth of our portfolio measured on a per gamble basis. Without getting into the details of derivation, the Kelly’s criterion determines the optimal bet fraction of your bankroll as:

$f = \dfrac{p}{LoseMultiplier}-\dfrac{q}{WinMultiplier}$

where:

• $p$ is the probability we win the bet
• $q$ is the probability we lose the bet
• $LoseMultiplier$ is our loss per ­$\$1 bet on a losing bet
• $WinMultiplier$ is our profit on $\$1 on a winning bet
• $f$ is the fraction of your bankroll to wager on this bet

So basically, if you know the probability of winning the bet $p$ then being disciplined and betting a fraction $f$ of your bankroll will maximize in the long-run the expected growth of your portfolio. The Kelly criterion helps strike the right balance between risk and safety and most importantly in an easy manner. Of course, if $f < 0$ you do not put any wager on the bet! Now of course it comes with drawbacks. For one, you need to be able to work out the real probabilities of bets. However, as I will explain later, for specific types of bets (i.e., the moneyline bets) this is no different than evaluating the calibration of your prediction model! Secondly, the Kelly criterion is inherently aggressive, since it can lead the bettor to wager even half of his total bankroll. However, one can tweak the approach and decide apriori a maximum wager $W$ on each bet (which needs to remain fixed for ALL bets), which basically means that the final amount bet would be $f\cdot W$.  Let us see each of these points in more detail.

First let’s explain the moneyline bet which we will use from now on. Consider you want to bet on the Steeler’s season opener with the Browns. You might see something along the lines Steelers -260 and Browns +250. What this means is that if you take the Steelers to win you need to “risk” $260 to win$100 if Pittsburgh wins, while if you take the Browns you can win $250 if you risk$100 and Cleveland wins. Now that we got this out of the way let’s get into the investment strategy.

Real probability of winning a bet:  Let us consider that you are betting on a moneyline bet.  Say that your model predicts Steelers are going to win with a probability 80%. If your model is well-calibrated (i.e., the probability obtained is the true win probability for each team), then this is the probability of winning a moneyline bet on the Steelers. If you bet on the Browns, then you have a 20% probability of winning the bet. Therefore, if you trust your model — or even better if you have evaluated your model and is well-calibrated — you have the parameters $p$ and $q$ for the Kelly criterion. The only thing that remains is to calculate $LoseMultiplier$  and $WinMultiplier$. The $LoseMultiplier$ is always 1, since when betting on the moneyline you are losing exactly what you have wagered. The $WinMultiplier$ depends on whether you bet on the favorite or the underdog. If the line for a favorite bet is $-x$, then your $WinMultiplier = \dfrac{100}{x}$ if you bet on the favorite. For a bet on the underdog with a line $y$, your $WinMultiplier = \dfrac{y}{100}$.   Now you have all the parameters to calculate $f$.

Applying Kelly’s criterion: One of the problems with Kelly criterion as alluded to above is its aggressiveness, $f$ represents the fraction of the total bankroll that Kelly suggests to bet. To make things a little less aggressive, one can set a fixed maximum amount of money to wager on a bet (e.g., $50). It is crucial for the strategy suggested here that this maximum wager per bet $\Gamma$ does not change (i.e., be disciplined to temptetations)!! So assuming a max wager per bet of $\Gamma$, for each bet we calculate $f_{favorite}$ and $f_{dog}$. If$latex f_{favorite}>0$we bet on the favorite $f_{favorite}\cdot \Gamma$, otherwise we pass on this bet. Similarly, if $f_{underdog} > 0$, we bet $f_{underdog}\cdot \Gamma$ on the underdog, while if$f_{underdog} < 0$ we do not bet on the underdog. Let’s put this strategy into play — through simulations!!! I collected Vegas moneylines for the NFL seasons 2009-2015 and I used our well-calibrated football prediction matchup (FPM) model. I will not get into the details of the model, but in brief it is a combination of bootstrap and a logistic regression model, while details can be found here. Using the above strategy the following figure presents the Return on Investment (RoI) for each season. RoI is defined as: $RoI = \dfrac{MoneyWon - MoneyLost}{Total Wager}$ The blue line presents the return on$1 of wager for the each season separately, while the red line is the cumulative/rolling return since 2009 and up to the corresponding season. Overall, through these 7 seasons you would have won 25 cents on every \$1 you wager.

Furthermore, the average wager on a favorite was $0.25\cdot \Gamma$, while the average wager on an underdog was $0.32\cdot \Gamma$. Finally, the average win multipliers were: $WinMultiplier_{favorite} = 0.54$ and $WinMultiplier_{underdog} = 2.6$.

What about the stock market? A 25% return on investment certainly sounds better than the 7% long-term return to the stock market. There are many similarities on the way the two markets operate but also some important differences that might make specific aspects of sports betting more appealing to some. In both markets the goal is to outsmart some one else (and in both cases we need to be disciplined). In the case of betting we try to outsmart the bookmaker (or your friend if you are placing a friendly wager), while in the case of the stock market we are trying to outsmart another person that will buy from (sell to) us stocks of company X. In the case of (moneyline) betting as explained above outsmarting means making better probabilistic predictions compared to the bookmaker, while for the stock market it means making better predictions on whether the price of a stock will increase or not. Both seem very similar tasks in principles but I have never tried to predict the stock market and there are very good reasons for this (with the main one that I obviously might not be that smart). However, I can guarantee you that a simple model like the one I presented for the NFL prediction will do poorly in predicting the movement of the stock market. The main difference between the two prediction tasks, is that sports games are kind of a closed system, i.e., the only variables that really matter are the players and their performance, while the stock market is a wide-open system where an event 4000 miles away might impct the stock market movement. Simply put, everything that affects the outcome of a game can be measured to some degree, while the price of a stock can be affected by any possible news – even if its fake news. Most glaringly, even for anticipated news we seem to have as good of a grasp as Malkiel’s blindfolded monkey. For example, after news like Brexit and the latest US presidential elections, pundits made sure to inform us about the upcoming crash in the stock market, which we by now know how it ended up. Of course, all of this does not mean that you cannot do well in the stock market; certainly you can and there are indeed experts that have done well fairly consistently, but from a purely predictability point of view, I think it should be obvious that predicting sports outcomes might be easier. And while we are on this topic, Lopez, Matthews and Baumer, have put together a nice analysis on the predictabiltiy and luck for different sports.

What does this all mean? This does not guarantee you gains! Let’s be clear about this. But it shows that if you are disciplined you can have a good return. Do you need to be a math wiz to predict games? No!! But you have to have a good understanding of what probability means (and this is another benefit of legalizing betting; people might understand probabilities better when they lose approximately 25% of the time on a -300 bet!). You could potentially use some of the prediction models that are out there (e.g., FiveThirtyEight’s which seems to be also well-calibrated). If you decide to go down Kelly’s road you should also remember that you need to be patient! You might lose more bets than you win! But this is not what ultimately defines return on investement. Kelly’s formula tries to find the optimal wager given your (or your model’s) belief on a game and the implied belief of the bookmaker from the moneyline. Also think about it — if you bet on a 20% underdog you will lose more than win (in terms of number of bets), but if your probabilities are correct, you will win more money than you lose – again if you are disciplined(!), and this discipline involves avoiding any sort of biases (e.g., for your favorite team).

And just to be clear; I am neither endorsing gambling, nor I say you should use the approach I described 🙂 I am just saying that there is no reason to believe that betting on sports is different than investing in the stockmarket! I am very interested to see how this  progresses.