Simple Arbitrage Betting and a Sports Betting Arbitrage Calculator

Sports betting is one of the most common ways people enjoy professional sports. Some people like having more of a stake in their home team’s success. Some folks think they can outsmart the bookies and make some money. Whatever the reason, people love betting on sports.

However, unsurprisingly, sports betting is a very mathematical subject. There are many, many people who make a living sports betting. Interestingly, most of these people would describe themselves as more mathematically inclined than sports inclined. That is, those who are most successful at sports betting are those that understand the mathematics at play. This article will serve as an introduction to the mathematics behind sports betting and include an interpretation as linear programming. In particular, we’ll also show how to find arbitrage betting plans using linear programming while including an arbitrage betting calculator. Follow this link to go directly to our sports betting arbitrage calculator

Sports Betting Basics: The Role of the House

There is something that I always like to remind myself when it comes to sports betting. When setting and updating odds, payouts, lines, etc. bookies are NOT making predictions about sports. I did not misspeak: after they start taking bets, bookies do not care about the sports at all.

The house wants to make money no matter who wins, plain and simple. The way the house does this is two steps:

1. Adjust the odds to have roughly a 5% advantage on either side of the bet
2. Move the lines, odds, over/under etc. to ensure that the payout is roughly the same no matter who wins.

Combining these two ideas together means that the house will take roughly a 5-10% profit no matter the outcome. Bookies are in the business of taking bets NOT making bets.

Let me work an example. On a popular sports betting website, the payouts as of September 14^th 2020 for predicting the NBA’s western conference champion are: Lakers (-160), Clippers (+160), Nuggets (+900). These numbers mean that if you bet 160 on the Lakers and you hit your bet, you receive 260 (The – means you bet that much to profit 100). If you bet 100 on the Clippers, you also get 260 (The plus means you profit that much on a 100 bet). Let me say it another way, to receive $100 from the bookie (your original + profit) on a successful bet, you would need to wager:

1. $61 on the Lakers ($61 + $61/1.6 =$100)
2. $38 on the Clippers ($38+1.6*$38=$100)
3. $10 on the Nuggets ($10 + 9*$10 =$100)

If I am a Vegas bookie, I want to pay out roughly the same amount no matter who wins. That means for every $10 bet on the Nuggets, I want about $38 bet on the Clippers and $61 bet on the Lakers. Suppose exactly this happens. Then, no matter what happens I pay out $100. But, since I took in $61+$38+$10=$109 on bets, I profit $9 no matter who wins. To put this another way, the bookies don’t care who wins.

Bookies Adjust the Lines

The previous section highlighted that a bookie’s goal is to adjust the odds and the lines so that, no matter who ends up winning or which side ends up paying out, the amount of money paid is the same. Above, we talked about money line betting, but here we’ll talk about something similar: the line. For example, consider the Bengals – Browns matchup on September 17, 2020. Currently the Browns are favored by 6.5 points. These lines are typically -110 meaning if you bet $110 and win, you get back $210.

In the case of the pure line, the bookie wants exactly the same amount of money bet on each team. Suppose $500 is bet on both teams. Then, no matter what, the bookie takes in $1000 and pays out $955. This is the exact same calculation as above. What if the odds-makers set the initial line at Bengals +6.5 but more people end up betting on the Bengals? What if $1000 is bet on the Bengals on the first day while only $500 is bet on the Browns’ side? Then,

1. If the Browns cover the spread, the bookie took in $1500 and only pays out $955.
2. If the Bengals cover, the bookie took in $1500 but will pay out $1910. They lose money.

The bookie doesn’t want to be in the business of taking risks, they just want to make money by providing their betting services. So what can they do? They want to encourage more people to bet on the Browns in order to even out the money. Maybe they’ll give the Browns a smaller advantage. If they move the line from Browns -6.5 to Browns -5.5, then the Browns are more likely to cover the spread. Theoretically, this will move more of the betting from the Bengals side to the Browns side.

The bookies will continue this process of moving the lines until, no matter what happens, the two sides have roughly the same payout so that, no matter what happens, the bookie makes money.

Even though bookies take sports bets, it doesn’t mean that bookies are making sports bets. If anything, bookies are accountants.

Making Money Sports’ Betting

There are really two main ways to make money sports betting and they both hinge on the same idea. The purpose of the entire above conversation was to hammer home the point that bookies lines don’t reflect what they think about the relevant sporting events. Rather, the lines reflect what the general population thinks about sports events and how they bet their money. Because people are inherently biased, the lines are too.

Sports betting needs to take this into account. You aren’t betting against the bookie, you are betting against everyone else. With this in mind, I will identify two ways to make money sports’ betting:

1. Arbitrage Betting
2. Mathematical modeling to predict probabilities of events

While I will discuss both these ideas, I will focus much more on the second event and highlight a mathematical technique for an optimal betting strategy given an estimation of true probabilities of events. Let us begin by discussing sports arbitrage betting before displaying our arbitrage betting calculator

Sports Betting Arbitrage

Sports arbitrage betting is the process of betting on both sides of a game. Because of the intricacies of the existence of multiple bookies and the lines moving, it is often profitable to bet both sides. This even holds true with the built in 10% advantage the bookies’ give themselves. Let us deal with two examples. Example 1 is based on different bookies setting different lines.

Suppose you are betting on a hypothetical Giants – Cowboys game. You notice that a book in NY favors the giants more than one in Las Vegas. If you think carefully about our above discussion, this is probably because more New Yorkers want to bet on their own team so money came in on the Giants. Therefore, the NY sportsbook wants to encourage more betting on the Cowboys’ side so they move the line to favor the Giants by more.

Suppose the NY sports book has the Giants at -155 and the Cowboys at +135. The Vegas line sees the game as a tossup so it has both teams at -110. What happens if you bet on the Cowboys in NY but against the Cowboys in Vegas? Because you did your homework and understand that you want to receive roughly the same amount back no matter who wins, you bet $40 on the Boys in NY and $50 against them in Vegas. What happens then?

1. If the Cowboys win, your $40 NY bet hits at +135 and you receive $94. You profit $4.
2. If the Cowboys lose, your $50 Vegas bet hits at -100 which nets you $95. You profit $5.

Because of the different odds offered, you were able to game the system to profit no matter what happened. Later, we’ll include a sports betting arbitrage calculator to do the mathematics for you.

Four Steps to Sports Betting Arbitrage

The first step in arbitrage betting and building an arbitrage betting calculator is to find two lines at different sportsbooks that are sufficiently far from each other. Notice that in the previous case, +135 and -110 are sufficiently different. NOTE: this is the hard part, modern books try to safeguard against arbitrage betting and so will often match other books closely. But, if the lines are different there is a chance you can swing it.

• Step 1: Determine at which book you will place the ‘to win’ bet and the ‘to lose bet’. You want to place the ‘to win’ bet at the book which gives a larger reward for the team winning. This correspond to the larger + number or the smaller – number. If the numbers are mixed, you place the ‘to win’ bet at the book offering +x.
• Step 2: Determine how much you have to bet at each book in order to get back $100. This includes the original bet. If you bet on +x, then to bet $100 you need to bet (10,000)/(100+x) to get back $100 on a successful bet. If you bet –x, then you need to bet (100x)/(100+x) to get back $100.
• Step 3: Add together the two bets from step 2. If the number is larger than $100, then you CANNOT make money on arbitrage betting here. This is because you will spend more than $100 to win $100. If the number is less than $100, congrats you found free money.
• Step 4: Place your bets in the same proportion as computed in step 2. In my example I computed a $40 bet to win $100 in NY and a $50 bet to win $100 in LV. If I want to make more money, I can increase my bets but I need to keep the proportion the same. Therefore, I can bet $80 and $100 or $400 and $500 and still profit.

The next section, our sports betting arbitrage calculator, computes steps 2 through 4 above for you.

Sports Betting Arbitrage Calculator

Notes on Usage of Arbitrage Betting Calculator

Below is our sports betting arbitrage calculator. To use this for arbitrage betting, you simply enter the two lines you intend to bet on and the total bet and the calculator tells you how much to place on each wager. It will also compute your profit in the last line. It is important that it you guarantee a win on either Line 1 or Line 2. That is, Line 1 and Line 2 should usually be betting on a specific team to win in one case and lose in the other. For instance, you can bet on the NFL playoffs and have Line 1 be a ‘Chiefs win’ and Line 2 be ‘Chiefs lose’.

Our Arbitrage Betting Calculator

While the sports betting arbitrage calculator above tells you how to place your bets, it is important that you are able to place your bets quickly to avoid the lines changing. If this happens, your sure thing turns into an actual bet.

Mathematical Modeling for Probability Predictions

The second way to make money on sports betting is to identify lines that are extremely far from where they should be. The way to do this is through mathematical modeling and being able to accurately guess the probability of one team or another winning. Let me illustrate this with an admittedly contrived example.

Suppose some book is taking bets on a dice game. However, they are offering +300 if the dice shows up 1-5 but +600 if the dice shows up six. Because I know a six shows up exactly 1/6 of the time, I actually expect to make money if I bet on six. Quickly: If I bet $100 on 6, I get back $700 with probability 1/6 and get back $0 with probability 5/6. Therefore, I expect to profit in the long run.

The reason I expect to make money is because the +600 did not accurately reflect the probability that a six shows up. Whenever the payout does not reflect accurately the probability of winning, there is money to be made.

As we’ve discussed, the lines move constantly. The lines reflect public perception, not winning probabilities. Therefore, it is entirely conceivably that the lines do not accurately reflect the probability that given team wins. In fact, it is likely this is the case. Therefore, if we have a system, a mathematical model, that can accurately give the probability of a given team winning, we can identify when the Vegas lines are wrong and when we can profit.

Optimal Sports Betting with Linear Programming

In this section we answer the question ‘If I know the probabilities of every team winning and the payout a sportsbook is offering, how should I place my bets’? If you aren’t risk adverse, the answer is: Place all your money on a single team. The way to choose the team and the way to prove that this answer is correct is through the lens of linear programming. The rest of this article will be technical and heavy on the mathematics.

Suppose there are number of games for me to bet on. Suppose my model predicts the home team has probabilities p₁,…,p_n of winning and that the sportsbook returns x₁,…,x_n on a successful one dollar bet. The question is, if I bet a fixed amount broken up into y₁,…,y_n on each game, how can I best distribute my bets to maximize my expected winnings?

It turns out that we can write this as a linear program:

We know from the theory surrounding the Simplex Algorithm, the solution must occur when some y_i=const. Knowing this, the optimal team to bet on can be observed to be that team for which the expected return, x_i\cdot p_i is largest.

To put this another way, the optimal betting strategy is to place all your money on the team that the sportsbook has underrated the most.

Calculating Expected Positive Payoffs

If we want to decide who to bet on, we need to be able to convert from +y and -y into the x_i values in the last section. These numbers are precisely the percentage return on a correct bet. It is easy to convert in the +y case. Here, you may set x_i = 1+\frac{y}{100}. For example, on a +300 bet, you get $400 back on a $100 bet. You multiply your money by 4.

In the -y case, the proportion is given by x_i = 1+ \frac{100}{y} . So, for a -200 bet, you get 150% of your money back on a correct wager.

If you have a good guess for the probabilities of each team winning for a given day, your best bet is when the payout exceeds the probability by the most. That is, we pick the game for which x_i \cdot p_i is largest.

Safer Betting Strategies

While linear programming was the technique of choice in the ‘optimal’ case of maximizing expected value, this technique only works if we can absorb arbitrarily large losses. In theory, the optimal betting strategy is to use the Kelly Criterion. The Kelly Criterion suggests a proportion of your entire bankroll to bet in order to maximize long term growth given probabilities and payouts. Using the probability of an event and the expected payout, you can compute the optimal betting amount very easily. This technique has the benefits of:

• Never resulting in a bankrupt (assuming your money is infinitely divisible, which is not quite true. It is close though)
• Results in the best average rate of growth in the long run

Conclusions

This article was meant to serve as a very basic introduction to sports betting. My main goal was to highlight the value that a deep knowledge of mathematics has on your ability to wager intelligently. Along the way, we provide a sports betting arbitrage calculator for you to use.

From sports betting arbitrage to mathematical modelling, understanding the mathematics of sports betting can actually lead one to make money in the long run. The funny thing is, though, that making money of sports betting is typically more about understanding the market than understanding the sports.