What is arbitrage in sports betting?
Arbitrage is a situation in which the odds offered by different bookmakers for all options of the same event are such that a player can wager on every option and guarantee a profit regardless of the outcome. This is possible when the sum of the reciprocals of the best available odds across different bookmakers is less than 1 — equivalently, when the sum of the implied probabilities is less than 100%.
Under normal conditions, a single bookmaker's odds will always have implied probabilities summing to more than 100%, because the bookmaker embeds a margin. However, different bookmakers may disagree on the probabilities of an event, and one bookmaker might offer substantially better odds on one option while another bookmaker offers better odds on another option. When the best odds from different bookmakers are combined, the total implied probability can drop below 100%, creating an arbitrage opportunity.
Identifying an arbitrage opportunity
To determine whether an arbitrage opportunity exists, take the highest odd available for each option across all bookmakers and compute the sum of their reciprocals:
d₁: best available odd for option 1
d₂: best available odd for option 2
dₙ: best available odd for option n
A = (1/d₁) + (1/d₂) + ... + (1/dₙ)
If A < 1, an arbitrage opportunity exists. The quantity (1 − A) represents the guaranteed profit as a fraction of the total amount wagered.
For example, consider a match with two options. Bookmaker X offers 2.15 for option 1, and bookmaker Y offers 2.05 for option 2. But bookmaker Z offers 1.95 for option 1 and 2.10 for option 2. The best odds across all bookmakers are:
Option 1: 2.15 (from bookmaker X)
Option 2: 2.10 (from bookmaker Z)
A = (1/2.15) + (1/2.10) ≈ .4651 + .4762 ≈ .9413
Since .9413 < 1, this is an arbitrage opportunity with a guaranteed profit of approximately 1 − .9413 = .0587 = 5.87% of the total wagered amount.
Calculating the stakes
To guarantee the same profit regardless of the outcome, the player must distribute the total wager across the options in proportion to the reciprocals of the odds. The stake for each option is:
S: total amount to wager
sₖ: stake on option k
sₖ = S · (1/dₖ) / A
Using the example above with a total wager of ¤1000:
s₁ = ¤1000 · (1/2.15) / .9413 ≈ ¤1000 · .4651 / .9413 ≈ ¤494.13
s₂ = ¤1000 · (1/2.10) / .9413 ≈ ¤1000 · .4762 / .9413 ≈ ¤505.87
Verification:
If option 1 wins: ¤494.13 · 2.15 = ¤1062.38
If option 2 wins: ¤505.87 · 2.10 = ¤1062.33
In both cases, the return is approximately ¤1062, which is a profit of approximately ¤62 on a ¤1000 total wager, consistent with the 5.87% calculated above. The minor difference is due to rounding.
Practical considerations
Arbitrage opportunities, while theoretically risk-free, are subject to several practical constraints.
First, odds change rapidly. An opportunity that exists at one moment may disappear within seconds as bookmakers adjust their odds. The player must be able to place the required wagers at all the relevant bookmakers before any of the odds change.
Second, bookmakers may limit or close the accounts of players they identify as arbitrage bettors. Bookmakers can detect arbitrage activity by observing patterns such as unusual bet sizes, bets placed immediately after odds changes, or consistent wagering on only the highest-odds option across markets.
Third, the profit margins in arbitrage are typically small, often between 1% and 3%. After accounting for the time required to identify opportunities, the capital needed across multiple bookmaker accounts, and the risk of odds changing between placing the individual wagers, the effective profitability may be lower than the theoretical calculation suggests.
Fourth, some bookmakers impose maximum bet limits that may prevent the player from placing the full required stake on one side of the arbitrage.
Despite these limitations, arbitrage remains a mathematically well-defined concept and its principles are foundational to understanding how odds relate to probabilities and how discrepancies between bookmakers can be quantified and exploited.