Understanding and Exploitation of Sports Betting Odds and Promotions

Overview

Profiting from sports betting is a traditionally difficult endeavor, leaving most people in the red. To carve a slice of profits, one must search for and exploit imperfections in lines. Doing so often requires large amounts of time and energy, either to build a model, gain new information and intuition. With the advent of online sportsbook, however, the pie has become much larger and more accessible. Books regularly run promotions resulting in profit boosts and free bets to attract and retain users on their platform. While these promotions can be seen as fun and maybe a little profitable, if a bettor understands the math behind their bets, they can identify the profitable promotions to participate in and also maximize the profitability of these promotions they partake in. In this post, we’ll dive into the math and probability behind bets and then apply them to strategies to maximize profit for common promotions.

First, let’s establish some basic notation around the components of a bet. For an example, let’s say you come across a bet that reads as “Marlins win +150”:

• Odds: The odds are +150, meaning for every $100 you wager, you win $150 if your bet wins. If the odds are negative (e.g. Marlins win -150), that means you have to bet $150 to win $100 if your bet wins. In this text, I’ll convert odds to a ratio to make the equations work with both + and - odds.
• Wager: This is how much of your money you put into the bet. In the examples above, if you had a wager of $100, you’d win $150.
• Winnings: This is how much net profit you make if your bet wins. In the example above, if you had a wager of $100, your winnings would be $150 if you win
• Profit: This is your net profit. If you win, this is equal to your winnings. If you lose, this is equal to your wager * -1.

As you build more complex bets, this formula may become more complex. To start, here’s some basic notation and formulas:

Understanding the House Cut

When looking at both sides of a bet, you’ll see that there’s a gap between the odds. For example, you say you see a bet that reads: “Marlins win +150, Marlins lose -200” If you placed a wager of $100 on them winning and also bet $167 on them losing, you’d end up losing money regardlesss of the outcome:

• If Marlins win, your net profit = $150 - $167 = -$17
• If Marlins lose, your net profit = -$100 + $84 = -$17 In fact, if you changed the wagers on each side, you could never guarantee a profit regardless of the outcome.

This discrepancy gives the sportsbook some edge and is often called the vig. I define vig as your losses/total dollars wagered if you take both sides of the bet such that the your loss is equal in both outcomes of the bet (as demonstrated in covered profit formula). The amount of vig differs by sportsbook, sport, type of bet, and odds. Due to the way odds are displayed, it isn’t easy to calculate vig besides plugging the odds into a somewhat complex calculation (shown below). Calculating the exact vig may be possible for a few bets, but if you need to quickly scan hundreds of bets, it’s much more useful to build some intuition around these numbers. The graphs here try to show an important relationship between vig and odds. As the odds get longer (i.e. one event becomes more likely to happen than the other), the difference between both sides of the bets also increase to maintain a constant vig. For example, if you saw +100/-109 (9% increase) and +700/-856 (22% increase), you might assume the first bet is a better deal. However, they both have a vig of 2%.

Formulas for the profitability of a covered bet and the vig of a bet:

Visualization of different odds and vig amounts:

Optimizing a Profit Boost

Now that we have some basic concepts down, let’s consider a common promotion called a profit boost. Tactically, a sportsbook will offer a profit boost of x% for a bet up to $y. That boost is multiplied against the winnings of a bet.

Going back to that example bet of “Marlins win +150”, if you bet $100, you’d normally expect to have winnings of $150. If you had a profit boost of 20%, your winnings would be $150*(1+20%) = $180.

To guarantee a profit with this promotion, you can take both sides of the bet (ensure you follow the sportsbook’s terms and conditions, which usually state you cannot take both sides of the bet on the same sportsbook). To maximize profit in this scenario, we can look at the covered profit equation and attempt to maximize its value. Based on the formula below, we mathematically achieve this by picking a long bet that has a low vig (to maximize the converted odds of both bets per for the formula:

Visually this can be shown on the heatmap below:

Intuitively, this makes sense because we want to maximize the impact of the profit boost. Since the profit boost gets applied on the winnings, we want a bet with high winnings, which translates into a bet with high odds.

Finally, these promotions will have restrictions on them, such as limits on the amount of the bet and the amount of winnings. Within those bounds, you’ll still want to apply the goals above of picking the longest possible bet that has a low vig.

1. maximize the absolute value of the boosted profit

2. minimize the the vig of the bet, so you can hedge against the boosted bet

3. Of the total profit equation, only the profit gets boosted, so we want to pick a bet that has a bigger proportion of the profit. This means we want to pick a longer bet (e.g. +500, +600, etc)

4. Minimizing the vig will enable us to hedge against the bet while still maintaining profit. Typically, vig increases for longer bets, so it is important to strike a balance between how long the bet is and how low the vig is.

Optimizing a Free Bet

Another common promotion is a free bet. With these free bets, you can place a bet without using any of your actual money. If you win, you’ll receive the winnings from the bet, but you won’t receive the amount of the free bet itself–an important distinction when figuring out how to maximize the value of a free bet.

As a result, the profit equation now looks like this:

To maximize your profit of your free bet, you’ll want to maximize the conversion of it. In other words, you want to convert the highest percent of the free bet amount into profit. The equation below calculates your free bet conversion. (insert covered free bet conversion)

Similar to maximizing the value of a profit boost, you will want to follow the same principles of picking a long bet with a low vig. To visualize this, the heatmap below shows different conversion rates based on how long the bet is and the vig. Typically you can find odds that enable a conversion rate in the high 60s or higher.

Optimizing a Risk-Free Bet

Connected with a free bet, many sportsbooks offer a sign-up offer that makes your first bet risk-free, up to some amount. If you win your bet, nothing special happens. If you lose your bet, however, you typically receive a free bet in the amount of your winnings (some books offer site credit in lieu of a free bet)

To maximize your profit for this promotion, you’ll need to combine a few concepts above. First, you’ll want to hedge your first bet, taking into account the value of the free bet if you lose. If you do lose, you’ll want to convert your free bet.

Mathematically, it looks like this:

First you’ll need to estimate what conversion rate of a free bet you can achieve on the sportsbook offering this sign up promotion. A conservative percentage is 67%.

Next, when choosing your first bet, you’ll want to follow the same principles as when placing a boosted wager as the profitability formula is the same except for the expected conversion rate, which is a fixed number.

To determine how much to cover your bet, you can follow this equation:

Finally, if you lose your first bet (which is likely since it is a long bet), you’ll receive a free bet. Follow the strategy above for maximizing the value of this free bet.

Putting everything together, with some reasonable assumptions around the bets you can find (assuming you can find a bet of +200 with 0% vig) you can expect a profit of 40-50% of the amount of the risk-free bet.

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