# Sports Betting Arbitrage

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### Introduction

Before covering arbitrage in sports betting we will first discuss what arbitrage is in general. The general discussion is fairly technical, but you can opt to skip directly to arbitrage in a sports betting context.

### An arbitrage opportunity

In formal terms, an arbitrage opportunity exists when you can make a series of transactions or trades such that you incur no negative net cash flow at any probabilistic state and a positive cash flow in at least one state. This is a pretty heavy sentence, but hopefully the picture will become clearer as you read on. In the simplest terms an arbitrage opportunity infers (at least the possibility of) a risk-free profit at no cost.1

Note that being risk-free alone does not make a strategy an arbitrage opportunity. You can buy an Australian government bond and earn a risk-free interest on it, but this isn’t arbitrage because the transaction involves a negative cash flow. This negative cash flow took place when you purchased the bond. Today’s cash flow is -\$10,000 and next year’s cash flow is +\$10,500. You have made a guaranteed profit of \$500, but you had to spend money in the first place.

The following is the most simplistic example of arbitrage. Suppose you can borrow money at 4% interest per annum and can earn 5% in interest per annum from a risk-free government bond. This would create an arbitrage opportunity that can be exploited as follows:

• Today:
Borrow \$10,000 at 4% interest p.a.
Use the borrowed \$10,000 to purchase a government bond that pays 5% interest
Net cash flow = + \$10,000 (from borrowing) – \$10,000 (from buying bond) = \$0
• One year from now:
Receive \$10,500 back from the maturity of the bond (\$10,000 + 5% interest)
Pay back \$10,400 for the loan (\$10,000 + 4% interest)
Net cash flow = + \$10,500 (from maturity of bond) – \$10,400 (from paying back loan) = \$100

Here you have earned \$100 risk-free with no investment. You’ve made money from nothing. This is the definition of arbitrage.

To tweak this example slightly, suppose you had an investment with an uncertain interest rate. Suppose you had a 50% chance of earning 5% interest and a 50% chance of earning 4% interest. If the former occurs you make \$100, and if the latter occurs you would make no profit. This is still an arbitrage opportunity because the transaction involves no negative cash flow and the chance of a positive cash flow.

### Statistical arbitrage

The example above concerned deterministic arbitrage because there was a guarantee of no negative cash flow at any point. In contrast, with statistical arbitrage, the strategy earns you a net profit if employed repeatedly, but there is a possibility of a negative cash flow each time the strategy is employed. With statistical arbitrage there is a mispricing of one or more assets/investments based on the expected value of these assets/investments. Taking advantage of it once doesn’t guarantee a profit and no loss, however if you repeat the strategy over and over you would expect to earn a profit.2

If we return to our previous example, suppose you had a 90% chance of earning 5% interest and a 10% chance of earning 3% interest. This means that you have a 90% chance of making \$100 in profit, and a 10% chance of losing \$100 (because the interest on the investment would not be sufficient to pay off the 4% loan). A deterministic arbitrage opportunity does not exist, because there is a possibility of incurring a negative cash flow (-\$100, 10% of the time). However if you could repeat this strategy over and over again, you would earn \$100, 90% of the time, and lose \$100, 10% of the time. Your expected payoff per transaction is the weighted average of the two possible outcomes: (\$100 * 90%) + (-\$100 * 10%) = \$80. The expectation is that you would earn a net profit, hence statistical arbitrage is possible. The idea is that if you play the strategy often enough you can be confident of making a net profit. This profit is possible because one of the assets (either the loan or the bond) is mispriced.

In a gambling context, a casino has a statistical arbitrage advantage in every game it offers. This is known as the house advantage. While the gambler can win on a single spin of the roulette wheel, if he or she plays often enough, the house should win out.

## Sports betting arbitrage

Arbitrage betting involves the placing of wagers on a sporting event to make a profit, or at least make no loss, regardless of the event outcome. Sports betting arbitrage opportunities typically arise when two or more bookmakers offer disparate odds. Note that an arbitrage opportunity is distinct from a hedging opportunity. A hedging opportunity typically arises due to varying odds over time rather than inter-agency discrepancies. For example you could place a wager on one team at 2.10 odds, only to observe later that the odds on the opposing team have lengthened to 2.20. You can hedge your bets by placing a second bet on the opposing team to lock in a guaranteed profit. For example, the odds on Australia to beat England in a test match will change each day depending on the previous day’s results. Hedging opportunities frequently arise during live betting, where the odds drastically change with each goal or try scored. Hedging isn’t the same as arbitrage, because when you place your initial bet you are not certain if a hedging opportunity will arise.

### A two-outcome arbitrage example

Recall from our discussion of bookmaker margins that the margin is calculated by taking the sum of the reciprocals of the decimal odds. Consider the odds below for two bookmakers on the same event.

Outcome Odds
Bookmaker 1 Bookmaker 2 Best Available
Chicago Bulls 2.20 2.62 2.62
Miami Heat 1.68 1.50 1.68

Margin 1.05 1.05 0.9769

Here the margins are 105% for both bookmakers. Note, however, that if you had an account with both bookmakers, you are effectively offered the odds of 2.62 and 1.68, because you will only consider the highest odds available. The margin for this combination is 1/2.62 + 1/1.68 = 97.69%.

If the calculated margin using the best available odds is less than 100%, then an arbitrage opportunity exists. If we set our bets to pay out the same profit regardless of the outcome, our profit margin would be 1/0.9769 – 1 = 2.36%.

Given that an opportunity exists, how much should we bet on each outcome? Numerous combinations are possible, but we will consider three key strategies:

1. Unbiased arbitrage: A strategy to make the same profit regardless of the outcome
2. Biased arbitrage: A strategy to makes a profit if Chicago wins, with no loss if Miami wins
3. Biased arbitrage: A strategy to make a profit if Miami wins, with no loss if Chicago wins

In the following formulas we will use the general variables:

w1 = wager (bet amount / stake) on outcome 1, in dollars
w2 = wager (bet amount / stake) on outcome 2, in dollars
W = your total wager on the market, which equals w1 + w2

σ1 = best available odds for outcome 1
σ2 = best available odds for outcome 2

### Strategy 1 – unbiased arbitrage

We will look at this using two approaches. First, suppose we wish to bet a certain amount W on the market in total. We want to know how much of W should be wagered on outcome 1 and how much should be wagered on outcome 2. For unbiased arbitrage we can calculate the bet amounts as follows:

w1 = W / (σ1/σ2 + 1)
w2 = W / (σ2/σ1 + 1)

Using our Chicago (outcome 1) vs. Miami (outcome 2) example, suppose we want to bet \$100 in total (W = \$100), and our best available odds are as follows: σ1 = 2.62, σ2 = 1.68. We should place the following bets on Chicago and Miami:

Chicago: w1 = \$100 / (2.62/1.68 + 1) = \$39.07
Miami: w2 = \$100 / (1.68/2.62 + 1) = \$60.93
Total bet: W = \$39.53 + \$60.93 = \$100

If Chicago wins our profit is (\$39.07 * 2.62) – \$100 = \$2.36
If Miami wins our profit is (\$60.47 * 1.70) – \$100 = \$2.36

Regardless of the event outcome our profit is \$2.36, which equates to a profit margin of 2.36%, as calculated earlier (one divided by the margin, minus 1).

Now suppose we wish to bet an amount w2 on Miami. We want to know how much to bet on Chicago (w1) to set up an unbiased arbitrage profit. If we replace W with w1 + w2 in the previous formulas and simplify, we get:

w1 = w2 * (σ2 / σ1)

Likewise, if we wish to bet w1 on Chicago, and want to know how much to bet on Miami (w2), we use the formula:

w2 = w1 * (σ1 / σ2)

For example, suppose we plan to bet \$100 on Chicago. How much should we bet on Miami to secure an unbiased arbitrage?

w2 = w1 * (σ1 / σ2) = \$100 * (2.62 / 1.68) = \$155.95

This makes for a total bet of W = \$100 + \$155.95 = \$255.95

If Chicago wins our profit is (\$100 * 2.62) – \$255.95 = \$6.05
If Miami wins our profit is (\$155.95 * 1.68) – \$255.95 = \$6.05

Note that \$6.05 / \$255.95 = 2.36%, as before.

### Strategies 2 & 3 – biased arbitrage

Biased arbitrage can be used when we have a bias as to which outcome we believe will occur, but want to avoid losing money if we get it wrong. Suppose we believe that Chicago will beat Miami and want to improve on the 2.36% profit that is achievable using unbiased arbitrage. As before, we will look at how to calculate our bet amounts using two approaches.

First, suppose we wish to bet a certain amount W on the market, and we believe Chicago (outcome 1) will win. We want to know how much of W should be wagered on Chicago, and how much should be wagered on Miami. We calculate the bet amounts as follows:

w2 = W / σ2
w1 = Ww2

Suppose we wish to bet \$100 in total (W = \$100), and our odds are as follows: σ1 = 2.62, σ2 = 1.68. If we believe Chicago (outcome 1) will win we should place the following bets:

Miami: w2 = \$100 / 1.68 = \$59.52
Chicago: w1 = \$100 – \$59.52 = \$40.48
Total bet = \$100

If Chicago wins our profit is (\$40.48 * 2.62) – \$100 = \$6.06 (6.06% profit margin)
If Miami wins our profit is (\$59.52 * 1.68) – \$100 = -\$0.01

Recall that with unbiased arbitrage we would receive \$2.36 regardless of the outcome, whereas now we receive \$6.06 if we backed the correct team, with no profit if we get it wrong. Note here that if Miami wins we actually lose \$0.01, which is due to the rounding effect of only being able to place wagers in one cent increments.

Conversely, we could use a biased arbitrage strategy where we predict Miami will win. We then calculate the bet amounts as follows:

w1 = W / σ1
w2 = Ww1

If we wish to bet \$100 in total (W = \$100), we should place the following bets on Chicago and Miami:

Chicago: w1 = \$100 / 2.62 = \$38.17
Miami: w2 = \$100 – \$38.17 = \$61.83
Total bet = \$100

If Chicago wins our profit is (\$38.17 * 2.62) – \$100 = \$0.01
If Miami wins our profit is (\$61.83 * 1.68) – \$100 = \$3.87 (3.87% profit margin)

It is of interest to note that the potential profit margin for having a bias towards the underdog exceeds the potential if you have a bias towards the favourite. Note here that if Chicago wins we win \$0.01, which is due to the rounding effect of only being able to place wagers rounded to the nearest cent.

Now, maintaining our belief that Miami will win, suppose we wish to bet w2 on Miami, and want to know how much to bet on Chicago to set up an unbiased arbitrage profit. If we replace W with w1 + w2 in the previous formulas and simplify, we get:

w1 = w2 / (σ1 – 1)

For example, suppose we wish to bet w2 = \$100 on Miami and we believe Miami will win.. How much should we bet on Chicago (w1) to set up a biased arbitrage?

w1 = \$100 / (2.62 – 1) = \$61.73

This makes for a total bet of \$100 + \$61.73 = \$161.73

If Chicago wins our profit is (\$61.73 * 2.62) – \$161.73 = \$0.00
If Miami wins our profit is (\$100 * 1.68) – \$161.73 = \$6.27 (3.88% profit margin)

## Three-outcome betting and beyond

### Unbiased arbitrage with more than three selections

The examples above all concerned arbitrage opportunities for betting markets where there are only two possible outcomes. When there are three or more outcomes, the test for whether an arbitrage opportunity remains the same. If the sum of the reciprocals of the odds is less than 1, then an opportunity exits.

For a three-outcome event, such as the winner of a football game, if an arbitrage opportunity exists the optimal unbiased betting strategy is as follows:

Variables:

w1 = bet (in dollars) on outcome 1 (home team win).
w2 = bet (in dollars) on outcome 2 (draw)
w3 = bet (in dollars) on outcome 3 (away team win)
W = w1 + w2 + w3 = combined bet amount

σ1 = odds for outcome 1 (home team win).
σ2 = odds for outcome 2 (draw)
σ3 = odds for outcome 3 (away team win)

Calculate the optimal bets on each outcome as follows:

w1 = W / (1 + σ1/σ2 + σ1/σ3)
w2 = W / (1 + σ2/σ1 + σ2/σ3)
w3 = W / (1 + σ3/σ1 + σ3/σ2)

From this we can extrapolate the formula to k possible outcomes:

wi = W / ( ∑ oi/σj )       Where ∑ is summed for j = 1 to k where k is the number of possible outcomes. Note that when j = i, σi/σj = 1.

### Biased arbitrage with more than three selections

If we are confident of a particular result we can employ a biased arbitrage strategy to make a larger profit if the pick is correct, with no loss if it isn’t. Returning to the three-outcome example, suppose we believe outcome 1 will occur. If W is the total amount we wish to spend, then our new optimal bets are:

w2 = W / σ2
w3 = W / σ3
w1 = Ww2w3 = W * [1 – 1/σ2 – 1/σ3]

From this we can extrapolate the formula to k possible outcomes. Let m, where 1 < m < k, denote the outcome that we believe will occur.

wi = W / σi            For im
wm = W * [1 – ∑(1 / σi) + 1 / σm]      Where ∑ is summed for j = 1 to k where k is the number of possible outcomes.

Below is an example of the possible outcomes for three different strategies on the following football game:

Outcome Agency 1 Agency 2 Best Available
Everton 1.80 2.30 2.30
Draw 3.35 3.25 3.35
Tottenham 4.40 2.95 4.40
Margin 1.08 1.08 0.96

The possible outcomes for this example are displayed below. The no arbitrage bet involves a simple wager of \$1,000 on Everton to win.

Outcome No Arbitrage
(Everton Win)
Unbiased
Arbitrage
Biased Arbitrage
(Everton Win)
Everton wager \$1,000 \$452.63 \$474.22
Draw wager \$0.00 \$310.76 \$298.51
Tottenham wager \$0.00 \$236.60 \$227.27

Profit if Everton wins \$1,300 \$41.06 \$91.71
Profit if draw -\$1,000 \$41.06 \$0.00
Profit if Tottenham wins -\$1,000 \$41.05 \$0.00

Note that the arbitrage strategies result in far lower profits, but are substantially less risky than placing a single wager.

An arbitrage calculator can be found in the tools section. This is useful both as an educational tool and as a strategy calculator when opportunities arise.

## Arbitrage betting in practice

### Arbitrage opportunities

During the course of sports betting you will encounter the following sources of arbitrage opportunities:

• Free bets
Many bookmakers offer free bets, typically as a sign up bonus. When you receive a free bet you are effectively given an arbitrage opportunity because you can place a wager with no downside risk. You could set up an arbitrage opportunity where you back one team using the free bet and back the second team using funds in your account balance. For example if you deposit \$50 and are given a \$100 free bet, you could place \$50 on team A at 1.92 odds, and place the \$100 free bet on team B at 1.92 odds. If team A wins you will have a \$96 balance, and if team B wins you will have a \$92 balance. With many bookmakers you would have met your turnover requirements, which means you can then withdraw at least \$92 after depositing \$50, for a \$42 profit. The drawback of this strategy, however, is that some bookmakers will restrict your ability to make opposing wagers using free bets. A different strategy is to open accounts with two bookmakers, and place the two free bets on the opposite outcomes of an event. One of the free bets is then guaranteed to win.
• Odds mispricing
For those who don’t believe this ever occurs, trust us, it does. In June 2011 a bookmaker offered 1.79 on team A and 4.00 on team B in an AFL head to head market. This created an unbiased arbitrage opportunity for a guaranteed 23.66% return. Other bookmakers at the time had odds of 1.25 and 4.00 for teams A and B, respectively. This mispricing was corrected within a few minutes, but all bets placed at those odds were honoured.
• Contrasting odds between bookmakers
This occurs virtually every day. The examples used in this section all concerned contrasting odds between bookmakers. Our website features Live arbitrage opportunities, most of which rely on memberships with European bookmakers. Taking advantage of these opportunities regularly requires holding active bookmaker memberships with a large number of services around the world.

### Risks involved with arbitrage betting

While arbitrage in theory is a risk-free practice, in reality there are a number of risks. Below are a few of them.3

• Odds shifts:
An academic survey found that arbitrage opportunities in online sports betting have a median lifespan of 15 minutes.4 If you experience a delay in placing your bets, the odds could shift sufficiently to remove the arbitrage opportunity before you have finished. This is a genuine risk, given you have to submit bets across a number of bookmakers. You should always position yourself to submit your bets within a span of one minute. Refresh the odds at each bookmaker just before you submit the set of wagers to ensure you can still get the odds you saw previously quoted.
• Bet cancellations due to bookmaker error:
The arbitrage opportunity may exist simply because of a clerical error on behalf of the bookmaker. While they are rare, bookmaker errors do occur. In the terms and conditions for most bookmakers you will find that they reserve the right to void all bets that were placed at erroneous odds. It’s a bit harsh, but the reality is the bookmaker can undo their errors, but punters cannot! If you do get one or more wagers cancelled prior to the event, the best course of action is to lock in a new bet at the available odds. Due to the bookmaker margin this will lock in a small guaranteed loss, but at least you won’t expose your remaining bets to heavy losses. This risk also highlights the importance of checking in on your betting accounts prior to the event to verify that each bet will stand.
• Calculation or bet submission errors on your behalf:
If you place enough bets you will eventually make errors, either in calculating your strategy or during the bet submission process. As an example of a potential error (one made by this author!), suppose you intend to make a small throw-away bet of \$1.00 on an outside contender to be the first try scorer at 71.00 odds. In the haste of submitting the bet and recording it in your tracking spreadsheet, you find that you’ve accidentally submitted a \$71.00 wager! It sounds like a silly error, but these things do happen, so double check all calculations and pay close attention when submitting bets.
• Exceeding betting limits:
If you try to submit high value bets to take advantage of an opportunity, you may find that one or more of your bets exceeds the maximum accepted wager on the market, and therefore won’t be accepted. If you have already submitted other wagers this can throw your strategy into jeopardy and expose you to a large loss. One strategy to reduce this risk is to submit your bets in descending order of magnitude. Your first bet is the most likely to be rejected, and if it is, you have not committed yourself (other than by depositing funds with that bookmaker) to the strategy.
• Turnover requirements:
Most bookmakers have turnover requirements on deposited funds and free bet winnings. You will likely have to subject your winnings to additional risk prior to withdrawing the funds. It is important that you be aware of the turnover requirements for each bookmaker before depositing funds. Some bookmakers require that funds be turned over once. Others require that funds be turned over once at 1.50 odds or more. Some bookmakers demand that funds be turned over up twelve times! Also, for bonus funds and free bet winnings, make sure you are aware of any time restrictions on turning the proceeds over. Some bookmakers will void your free bet winnings if you haven’t met the turnover requirements within, say, 90 days.

### Notes and Sources

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