Sports Arbitrage Betting on 3+ Teams for Guaranteed Profits

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In this article, we will continue our prior discussion about sports arbitrage betting. Last time we gave an introduction to sports arbitrage betting, worked some examples, and provided a calculator. Here, we are going to discuss the topic of multi-team arbitrage betting.

This concept is really just an extension of the head-to-head arbitrage example we talked about last time when there are more than two possible outcomes. For example, instead of picking the winner of a game, we could

  • Pick which NFL team wins a division (out of 4), or
  • Pick which NBA team wins the title, or
  • Pick a golfer to win a major.

This article will be a bit about the theory of arbitrage betting and how we can, given the bookmakers’ assigned odds, determine when arbitrage is possible. As such, it will be quite mathematical.

A Sports Arbitrage Betting Refresher

Quickly, let us remind ourselves what sports arbitrage betting is. Sports arbitrage betting is the process of identifying a series of bets so that, no matter the outcome of the sporting event, the better is guaranteed to make a profit. Typically, sportsbooks give themselves anywhere from a 5-10% advantage so that they profit a bit on all bets placed. Thus, (in general) the only way that we can find arbitrage opportunities is to use multiple books that have different odds.

Is this possible or even likely? Remember that the sportsbooks update their odds to reflect the amount of bets that are placed on each team/option. It is quite unlikely that bets are placed at different books in exactly equal proportions. Thus, the odds are different books are almost always slightly different. If the odds are different enough (5-10% on average), then we will have arbitrage opportunities.

Mathematical Foundations

In order to formulate the math we require, let us define some parameters. Suppose there are n options to bet on and that exactly one of these bets is guaranteed to hit (for example, picking the super bowl champion there are 32 options, exactly 1 is guaranteed to payout). Then, suppose that we bet x_i on option i. Further we define \rho_i to be the win multiplier for a successful bet on i.

For example, if we bet on a team with +700 or 7:1 odds, then betting $1 results in profiting $7 dollars so that we are given $8 back. Therefore, on a +700 bet, we have the value \rho_i =8 because successful bets multiply your input by 8.

Distributions of Funds for Optimal Sports Arbitrage betting

This focus of this article is on the theory of sports arbitrage betting and so will read like mathematical paper from here on out.

Our goal: Suppose there are n events of which exactly one is guaranteed to happen. We are given multiplicative factors \rho_1,\dots , \rho_n that describe the amount of return on a successful bet on event i. We want to determine bet amounts x_1,\dots , x_n to bet on each event so that we are guaranteed a profit.

Guaranteeing a profit means that no matter the outcome we win more than the total amount we bet. Translating this:

  • For every event, the amount we win – given by \rho_i x_i – is larger than the total amount bet – given by \sum_{i=1}^n x_i .

If we want to make sure that in every circumstance we profit, we want to maximize the minimum winnings on any given bet. We can solve an optimization problem that gives the following result (a theorem!)

  • If we want to assign the x_i values so that we maximize our minimum winnings, we should assign x_i = \frac{k}{\rho_i} where k is a parameter free to choose that determines our total amount bet.

This k is the eventual payout we receive.

An Example of Optimal Funds Distributions

Suppose if we are betting on a set of events that have odds given +200, +300, +400, and +500 and we are guaranteed that exactly one of these events will happen. Then if we pick k=300,

  • The +200 bet has \rho_1=3 so we bet $100 on this event to pay out $300.
  • The +300 bet has \rho_2=4 so we bet $75 on this event to pay out $300.
  • The +400 bet has \rho_3=5 so we bet $60 on this event to pay out $300.
  • The +500 bet has \rho_4=6 so we bet $50 on this event to pay out $300.

Notice that no matter what happens we get the same amount of money back ($300). Our total bet amount is $100+$75+$60+$50 = $285 so that, again no matter what happens we profit $15. Here, an arbitrage bet is available.

What happens if there is a fifth event that is also at +500 odds? Then, we have an additional event that we must bet $50 on in order to profit. However in this case, no matter what we win $300, but we bet a total of $335. Here, because the total amount bet is larger than the guaranteed winnings, there is no way to bet on these 5 events to guarantee a profit.

The theorem in the prior section says that, no matter what other way we might distribute our $335 amongst the bets, there is no way to increase the payout on all the events simultaneously. To put that another way, if we want to increase the payout on a few of the bets we must decrease the payouts on other bets. To put that a third way, there is no way to increase the payouts so we have a guaranteed profit no matter the outcome.

How to find Sports Arbitrage Betting Opportunities

We want to identify, given the set of \rho_1, \dots , \rho_n , when a sports arbitrage betting opportunity is available. This is a fairly straightforward calculation. We want the guaranteed winning amount (which will be the same across all events) to be larger than the total amount bet. Per the discussion above, we choose x_i = \frac{k}{\rho_i}. So, the total payout is \rho_i \cdot x_i = k. Then, the total amount bet is

\sum_{i=1}^n x_i =\sum_{i=1}^n \frac{k}{\rho_i}

This is in order to win $k. Therefore, we profit if the winnings are larger than the proposed wager. In math terms, this is 1>\sum_{i=1}^n \frac{1}{\rho_i}

This gives us a way to effectively check if a sports arbitrage betting opportunity is available in a multi-team bet.

We may simply look at a multi-team betting scenario and compute the sum of \frac{1}{\rho_i} for each possible outcome.

The effect of multiple books

Sportsbooks cook the odds to guarantee themselves a profit. It is not uncommon for a sportsbook to guarantee themselves a 5 or 10% profit on any bet offered (this is sometimes called the vig). Therefore, it is almost impossible to find sports arbitrage betting opportunities at a single sportsbook. However, because books typically have slightly different odds, we can use this to our advantage. If a certain book has a higher payout for a certain team, we should place our bet at that book.

Right now, if you were to use the above method to bet on the super bowl winner at bovada, you would be guaranteed roughly a 20% loss. However, if you include the odds at betonline and always place your bet at the book that has the better payout, you can reduce this to guarantee roughly a 10-15% loss. Adding more books will let you decrease this to the point that you are actually guaranteed a profit.

Almost-Arbitrage

If we are betting on the super bowl winner, in order to be guaranteed a payout we need to bet on every single outcome. However, this includes placing bets that are almost guaranteed to fail. For example, roughly 2% of our bets are placed on the Jets, Giants, Bengals, Redskins, Broncos, Falcons, and Jaguars. If we omit our bets on these teams, we can reduce our expected loss without really risking anything. It seems almost a guarantee that none of these teams will win a championship, so betting on them seems a waste.

A Caution

Sportsbooks want to make a profit and they know of many common techniques to guarantee a profit. Because sportsbooks are private entities, they are free to reject any bet they want. If a sportsbook thinks you are profiting too consistently, then they may close your account and decline any bets you make. If the book thinks you are arbitrage betting, they will close your account.

One of the most surefire ways to tip the book that you are sports betting is to place bets on 10 of the 32 NFL teams to win a championship in very specific amounts. Our method requires the better to bet specific amounts on every event. The book can tell you are doing this. In order to successfully arbitrage bet, one needs to use many, many books so as to hide the fact that you are doing this type of betting.

Published by jdashbrock