What’s the Optimal Betting Strategy?

Once the predictions are made, how does one determine the quality of the bets? How many of the bets are worth taking? As seen by the Gambling Performances page, the simplest way would be to calculate the difference between the ego predictions and the Win Total line for each organization, and then sort the bets by that metric in descending order. By doing so, one would think that the bets with the best chance of winning are at the top, and the bets with the lowest chance of winning are at the bottom. However, is it worth taking all 32 bets? If not, then how would one know which bets to take and which to avoid?

The goal of a bettor is to have a betting portfolio that maximizes Units, as that is viewed by many as the universal measure of gambling success. While Return on Investment is a valuable measure, the amount of money (or portion of bankroll) that goes into those Returns also matters. Units is a function with two variables: Return on Investment (ROI) and Number of Bets. The formula is [Total Units = ROI * Number of Bets]. In terms of Win Total Futures, one could maximize Number of Bets by taking all 32 bets, but the Return on Investment would theoretically decrease as one would be taking on riskier bets with a higher chance of losing. Or one could try to maximize ROI by taking fewer, theoretically safer bets, but obviously Number of Bets would decrease as a result. One’s strategy could depend on several factors, two main ones being Betting Style and Risk Aversion. Betting Style here refers to how one bets in terms of Fixed Bankroll and Variable Bankroll. Fixed Bankroll means that the bettor has a set amount of money to put on each bet, and Variable Bankroll means that he or she alters the amount of money that they put in depending on the specific bet. The more bets that a Fixed Bankroll bettor adds to their portfolio, the larger the total amount of money that he or she is putting into their portfolio. For a Variable Bankroll bettor, the number of bets is not as relevant (as you could put any amount on any of the bets), so for this section, I’m going to assume that everyone is a Fixed Bankroll bettor. Part of this reason is that sportsbooks are suspicious of gamblers that use various, specific amounts for each bet, so allocating a flat amount across bets should help the successful gambler fly under the radar. Sportsbooks are known for having a history of limiting successful gamblers in order to stem losses, so the Fixed Bankroll method is most applicable for the widest spectrum of gamblers. Risk Aversion for a Fixed Bankroll gambler would then be how far down the chart that one would go in order to build their portfolio of bets. The more bets one adds, the more money that they are gambling with, and therefore, the risker that their portfolio becomes overall. For a Variable Bankroll gambler, risk is much more complicated in terms of how much of one’s bankroll goes towards a specific bet and how many bets one makes.

Before I go into the actual content of this page, I just want to stop down and explain how I view this model and its relationship to gambling. To me, there is little difference between investing in stocks and gambling on sports. In the same way that an investor uses some information (typically some research done by the investor or a stock broker) to make an investment in a stock or a group of stocks (or other types of investments), a sports bettor is using information (typically “gut feeling” for a casual gambler) in order to gamble on a bet or a series of bets. The key difference, as alluded to in the previous sentence, is that sports gambling has a reputation of most participants using little to no actual research, which is one reason why people are much better at losing money in gambling than in the stock market. This distinction makes people think that investing in stocks is responsible, while gambling your money on sports is irresponsible. But the core “game” is still the same in many ways. Banks (Sportsbooks) set a value level for a company (a Win Total line for a NFL organization) that can go up or down based on the public’s perception of how the company (NFL organization) will actually perform. If one thinks that a company (NFL organization) is being undervalued, he or she purchase stock in the company (take the Over bet). If the investor turns out to be right, then he or she gains value (the bet pays out). If the investor is wrong, then he or she will lose value (the bet does not pay out). Once again, investing can be (and often is) much more complex, but at its roots, it is very similar to the type of gambling that is being conducted via this project. In this project, when I say “portfolio” like an investment, I am referring to a set of bets. And when I say “bets” like a gambler, I am referring to investments. So what are the ways to determine the optimal portfolio of bets?

Method #1 – All 32

Method #1 – Take all 32 bets each year. It may seem counterintuitive, but this method is the least risk averse, or, in other words, the riskiest for a Fixed Bankroll bettor. This would require the bettor to put in the maximum amount of money into the portfolio, taking on riskier and riskier bets as they build up to the total 32. On the other hand, for a Variable Bankroll gambler, it could be the most risk averse option, as they can spread their bankroll across the highest number of bets. This is, of course, assuming that the Fixed Bankroll bettor has a variable total bankroll, and that the Variable Bankroll bettor has a fixed total bankroll. The Fixed Bankroll bettor bets the same amount per bet, so increasing the number of bets adds money to the portfolio. The Variable Bankroll bettor has a set amount of total bankroll that they are willing to put into the portfolio and is spreading that total amount across those bets in different individual allocations. The lower the number of bets, the less money the Fixed Bankroll bettor puts into the portfolio, which contains (theoretically) more reliable bets, lessening the risk. The lower the number of bets, the more money (or weight) that the Variable bettor puts on individual bets, putting his or her eggs in fewer baskets, therefore increasing the risk. The results of this betting method can be seen on the website homepage in Green on the left side of the chart. A sign of the riskiness of the style is that when things go well (like in 2020) the Units are higher than the other methods, and when things don’t go well (like in 2023), the Units are lower than the other methods. Overall, however, it is unlikely that this method would be optimal as over the first 4 seasons. The Binomial Probability method, explained below, outperforms the “All 32” method.

Method #2 – (ego – Line) Absolute Difference

Method #2a – Take a specific number of bets that are sorted by the absolute difference between the ego prediction and the line from the sportsbook. For example, one could simply take the top 20 bets based on the (ego – Line) difference. The results of taking the top 20 bets each season can be seen in the chart on the homepage in orange on the right side of the chart. Using a cutoff of the top 20 is not optimal given the 4 seasons of data, as seen by a lower Total Units compared to Method #1, but it does allow one to compare the results of this project to the leading hedge funds around the world via the website HedgeFollow. Looking at the chart below, the optimal number of bets each season would be 16, resulting an an average Rate of Return of 24.30% and Total Units of 15.55.

While the 16 bets/season result in #2b does not outperform the #2a equivalent, I should note that the 0.80 cutoff in the table above yielded higher results than any number of bets using #2a. I should also reiterate that Method #2b uses a fixed methodology where the cutoff chosen is the same for all of the seasons. If one did change the cutoff parameter from season-to-season, the results could vary wildly, both in the positive and negative direction. Below is a table of the optimal portfolio using Method #2b. While not impossible to pick the specific portfolios, it would be extremely unlikely to pick the ones below to generate this result, especially given the range of the Number of Bets.

Method #3a – Take a specific number of bets sorted by binomial probabilities. This Method is much more complex than the first two and will require some explanations and assumptions. If one looks at each individual game as a binomial event, in that either the team wins or loses (assuming no ties), then one can calculate what the probability of that team winning any certain amount of games is in a 17-game season (or any length of season). To do so, one must interpret the ego prediction (the predicted number of wins in the season) as the probability that the team wins any specific game. For example, the 2023 Steelers had an ego of 9.93, so one could say that their probability to win any specific game would be 58.41% (9.93/17). The assumption here is that the Steelers have the same chance to win every game (binomial event) equally, which is not realistic in the actual world. Using the formula: p(r) = (ⁿC_r)*(p^r)*(1-p)^(n-r) , one can calculate the binomial probability, or p(r), that the Steelers will win [r] games in a season with [n] total games. Here ⁿC_r represents the binomial coefficient of Pascal’s Triangle given any specific [n] and [r]. For example, using the formula above, the Steelers have a 17.2% chance to win 9 games, or, in other words, go 9-8. The sum of the p(r) column in the chart below denotes the running total of the individual binomial probabilities either above or below a certain number of wins, the direction of which is determined by the relationship between ego and the line set by the sportsbooks. Because the Steelers had Win Total of 8.5 set by the sportsbooks, and a 9.93 ego, one would sum the binomial probability of each win total above and including 9 wins. The chart below shows that the Steelers had a 17.2% chance to be exactly 9 games, and they had a 76.1% chance that they would win 9 or more games in the season. That probability was calculated by summing the individual probabilities above (i.e. p(r=17) + p(r=16) + p(r=15) + …), including the probability of 9 wins. 76.1% is the binomial probability that the Steelers Over 8.5 Wins bet will be successful. If the ego prediction had been below the win total line set by the sportsbooks, then one would add the individual probabilities below the Win Total line, up to and including the Win Total just below the line. For example, if the line had been 10.5 Wins, then one would have taken Under 10.5 Wins, and the binomial probability of the Steelers winning 10 or fewer games would have been 60.5%.

As a result, one could organize bets in favorability not by the absolute difference between the ego and the line as shown above in Methods 1 and 2, but rather by the binomial probability of each team to win their respective bets.

However, not much changes with this Method, as shown by the chart below. While the portfolio of bets does not change, the reorganization of the bets alters the running ROI and running Total Units for various Number of Bets values.

Method #3b – Use a cutoff based on specific binomial probabilities. Similar to the absolute difference between ego and the line metric in Method #2, where one could take a certain number of bets using the difference to sort the options (Method #2a) or use the difference as a cutoff mark and only take options with a certain difference or greater (Method #2b), one could use a certain cutoff value of binomial probability instead of a static Number of Bets each season.

In the table above, one can see that this method can yield more lucrative results, but the results are much less stable compared to Method #3a. In that method, part of the smoothness of the result has to do with a finite set of possible cutoffs (1 to 32) with each addition to the portfolio of bets being control by a set increment. With Method #3b, there are far more possibilities in terms of the number of bets in the portfolio of bets that one could create. There are an infinite amount of values between 100% and 50%, but changing the cutoff won’t change what the binomial probabilities of the bets are, and due to the cutoff being constant across the 4 seasons, there is a finite set of possible bets. The table above shows just 16 of them. Take the cutoff of 54% as an example. It has a higher Total Units than any of the options in Method #3a. But if one took the cutoff of 55% or 53%, the results would go back to being in line with the other methods (albeit perhaps marginally better). On the other hand, one could have picked 60% and had worse results than similar options in Method #3a (a cutoff of 24/season for a total of 96 bets generated 13.41 Total Units, and a cutoff of 25/season for a total of 100 bets generated 11.03 Total Units). The issue with all of these Methods is that one never knows what the optimal cutoff would be. One could argue that about gambling in general. But what this Method allows is for more clarity to the gambler about the likelihood of success compared to the absolute difference between the ego prediction and the sportsbooks’ lines.

The cutoff of 52.4% is bolded for a specific reason. If one has a portfolio of bets each with odds of -110, then he or she needs to win 52.4% of those bets in order to not lose any money. Winning any percentage of bets below that value would result in a loss of money and vice versa. If your portfolio has an average odds of -110 then it gets more complicated, because it then matters what the odds of the winning bets are. In the long run, let’s assume that this rule of 52.4% holds. The logic follows that if one take bets that, on average, have a 52.4% chance of winning or better, then in the long term, one should theoretically be making money. For this project, the average odds are actually better (or, in other words, cheaper) than -110, so that breakeven probability of 52.4% is technically lower in this case.

Finally, before moving on to the final Method, I should reiterate that these methods are being shown with a fixed methodology season-to-season. For example, the bettor is betting using the 54% cutoff each season with no variation. However, if one did use variation, then the results could be even better. The optimal portfolio across the four seasons can be seen in the table below.

Method #4 – (Binomial Probability – Implied Probability) Difference

Method #4a – Use a cutoff of a specific number of bets sorted by the difference between the binomial probabilities and the implied probabilities determined by the odds from the sportsbooks. When a sportsbook sets the odds for bet, there is an implied probability that the sportsbook believes that the bet will win. For example, when a bet has odds of -110, the implied probability is 52.4%. But when a sportsbook sets both sides of a bet with odds of -110, then each bet has a 50% probability to win. How could that be possible? It’s because both Person A who takes one side and Person B who takes the other side believe that their side has a better chance than 50% to win. If someone did truly think that it was a 50/50 then that person would be foolishly taking a bet with a negative probability difference (50% – 52.4% = -2.4%). In other words, the “vig” (the reason why 50/50 bets are -110/-110 instead of 100/100) gives the sportsbooks a ~10% profit in the long term no matter what happens (as long as they correctly set bets to receive 50% of the total money collected on one side of the bet and 50% of the total money collected on the other side). That’s the reason why sportsbooks offer seemingly amazing deals to get people to start gambling with their sportsbook. Once the gambler is hooked, then he or she becomes a money printing machine for the sportsbooks. Someone might be reading this section and think, “But what if I’m really knowledgeable because I watch so much sports? I sometimes make money gambling, so what about that?” Well, there are a couple flaws to that mentality. First, being a sports fan simply does not translate to being a successful gambler. One could not watch enough sports piecemeal in order to get a full picture of whatever sports event upon which he or she is gambling, certainly not at a level that the sportsbook are. Sportsbooks collect and analyze endless data to generate the most accurate probabilities of events possible. Second, humans are more likely to remember their successes more than their failures. So while someone might remember one time that they hit a big payout on a seemingly uncertain bet or portfolio of bets, he or she is much less likely to remember the larger portion of bets that they lost (especially if those losses were individually by a lower amount). In other words, the casual gambler does not track their bets, because that would take work, and casual gambler is not using gambling as an opportunity to do more work. They are gambling to have fun; it’s a leisure activity. And third, even if someone was able to overcome the vig and not lose any money over the long term, that person is in the Top 1% of gamblers, while the bottom 99% are systematically going to lose significantly over time. Most people thinks that they are in the Top 1%, while the simple truth is that many are not.

Anyways, back to Method #4a. If every bet has binomial probability set by the model and an implied probability set by the sportsbook, then one could take the difference between the two probabilities and sort the bets by the difference. Intuitively, I would think that this method makes sense to sort the bets this way, as gambling success is not just whether the bettor thinks a bet will win, but rather, whether the bettor thinks that a bet has a better chance of winning than the sportsbook do.

Method #4b – A cutoff of a specific difference between the binomial probabilities and the implied probabilities determined by the odds from the sportsbooks. Just like in Methods #2b and #3b, one could use a cutoff of a certain value rather than taking the same Number of Bets each season. Once again, there are a large number of possible combinations, but below is a table with some possible options and their results.

The results in the table for Method #3b are much better and more stable than the results in the table above. However, the optimal portfolio via Method #4b using a variety of cutoffs instead of a fixed cutoff has better results than the Methods #2b and #3b. Below is a comparison of all three Methods across the four seasons.

Conclusion – So What’s the Best Method?

So which method is best? The honest, boring answer is that it depends. Or, in other words, I’m not sure. At least yet. In terms of the “a” methods, as I mentioned above, I believed that Method #4a would be the optimal method, because it sorted the bets by difference in probability of success. Unlike in Method #3a, Method #4a incorporated the odds from the sportsbooks. Therefore, this Method ideally would push not only the bets that were the most likely to succeed to the top but also the ones that paid out the most if they did succeed. Looking at the top of the table below, it looks like I was correct in my theory.

As one can can see in the table above, each of the Methods has its advantage. Method #2a has the largest sum of Total Units, meaning that it is the Method with the highest average Total Units across the board. Method #3a has the most Number of Bets with the highest Total Units, especially for the Bet Numbers higher than 12. For Bet Numbers 13-32, #3a has 14 highest Total Units compared to #2a’s 10 and #4a’s 1. Additionally, for those bets, Method #3a’s sum of Total Units (262.74) edges out #2a (262.40) and easily beats #4a (195.80). Method #4a’s strength is in the lower Bet Numbers, as theorized above. The bets with the more lucrative payouts are pushed higher on the chart. For Bets Numbers 1-12, Method #4a has 9 highest Total Units compared to #2a’s 3 and #3a’s 0. For those Bet Numbers, #4a has a sum of Total Units of 71.80, larger than #2a’s 56.59 and #3a’s 47.43. Another advantage for Method #4a is that it does not have any Bet Numbers with negative Total Units, unlike #2a and #3a which are negative for Bet Numbers 1-3.

To bring it full circle to the beginning on this page, the “best Method” is going to depend on the individual person. The Risk Aversion and Betting Style of a person will determine what the ideal Method is. Again, it’s hard to show the results for a Variable Bankroll, because there are an infinite array of results. Instead, for this situation, let’s assume that Variable Bankroll refers to not having any limit on any specific bets instead of the actual definition, which is the ability to bet different amounts across the portfolio. If the bettor has a Fixed Bankroll (capping their individual amount to bet on each bet evenly), either by decision or limitation by a sportsbook, I would say that #2a is the Method with the least amount of risk, because it has the highest average Total Units across the range of Bet Numbers. Betting any Betting Number using Method #2a would be the safest way maximize the investment in the portfolio, as the more bets added to the portfolio, the more of the bankroll one can put into the portfolio. On the other hand, if one has a Variable Bankroll (once again, not in the traditional sense, but instead meaning that one does not have any limitations and is willing to bet their entire bankroll on a smaller number of bets), then #4a would be the optimal Method. Rather than spreading one’s money across all 32 bets, like in #2a, concentrating one’s bankroll on a specific number of bets an lead to a much higher return. For example, betting the top 11 bets in each of the four seasons generated an ROI of 31.44%, a higher ROI than the 10.16% for all 32 in #2a. Even putting all of your money on just one bet per season, the riskiest version of gambling here, would have generated a positive result using #4a. Finally, if a bettor has a Variable Bankroll and knows that he or she is uncomfortable putting all of their Bankroll on 12 or fewer bets, then Method #3a could be the optimal way to set up the portfolio. #3a has the individual maximum Total Units across the three Methods (Bet Number of 16 yielded 15.75 Total Units) and has a significant advantage if the bettor is comfortable concentrating his or her bankroll usage between Bet Numbers 13-27.

As with the Variable Bankroll above, it’s difficult to show the full array of results of the “b” Methods due to the high number of possible combinations, but above is a set of various results from those Methods. While #2b looks like it’s the most consistent, #3b has the highest optimal of the three. As with the “a” Methods, Method #4b has a better top of the table, as lucrative bets that are most likely to succeed are pushed to the top. The downside of that Method is that results after Bet Number 12 suffer. What I can also show is the optimal results changing the Bet Number for each of the seasons for each of the Methods (#2b, #3b, #4b) as shown above on this page.

The truth is that, for most people, the most realistic Method in practice is none of the above. They are either unable or unwilling to track bets or have any sort of consistency when gambling. Bettors are humans, and humans change, especially when it comes to preferences. What Bankroll Style that a bettor used last season might not be the same as this season. Something like Risk Aversion is fluid and could change depending on recent performances of the model. Biases can cloud judgement. “A social media influencer won a ton of money on a parlay? Okay, I’ll make some parlay bets for a week.” “If I had just bet the top 21 instead of the top 20 then I would have made a lot more money this season. Well then, next season I’m betting the top 21.”

These differences in preferences and people’s refusal to put work into something that should be fun could help create a demand for a sort of “gambling consultancy.” If everyone’s gambling preferences were different then someone could provide clients with personalized gambling advice like a financial analyst gives investment advice to clients. Understanding a client’s Bankroll situation (in terms of both size on investment and style preference) combined with his or her Risk Aversion will allow an analyst to provide the optimal portfolio of bets, not strictly in terms of returning the optimal results, but instead optimal for that specific client.

Up to Date Charts (2020 – 2024)

The set of bets below uses the data from above to optimize the portfolio of bets. For Bets Numbers 1-12, it uses Method #4a. For Bet Numbers 13-16, it uses #3a. For Bet Numbers 17-32, it uses #4a.