An arbitrage bet locks in a profit regardless of which side wins by taking the best available price on every outcome of the same market from different bookmakers. The maths is the same as dutching. The opportunity is different, though, because it does not come from choosing which outcomes you want to back. It comes from two bookmakers briefly disagreeing about the same event, and that disagreement usually does not last very long.
The formula
Add the implied probability of the best available price for every outcome, using the best price you can find across the bookmakers you are willing to use. If the total is below 100%, an arbitrage opportunity exists:
book = (1 ÷ best price, outcome 1) + (1 ÷ best price, outcome 2) + …
Stake on each outcome in proportion to its implied probability, just as you would with dutching. The payout is then the same regardless of which outcome wins.
Worked example
Book A prices one side of a two-way market at 2.10. Book B prices the other side at 2.05:
- Implied probabilities - 47.6% and 48.8%
- Book - 96.4%
- Stakes on $1,000 total - $493.98 / $506.02
- Payout, either outcome - $1,037.35
- Result - a guaranteed +$37.35, a 3.73% return, regardless of the outcome
The calculation is the same as dutching. The difference is where the 3.6 percentage point gap comes from. It is not a judgement about which outcome is undervalued. It comes from two bookmakers offering different prices on the same real-world event at the same time.
Why it has a shelf life
That disagreement can disappear quickly, and it has nothing to do with the size of your own stake. Bookmakers use automated systems to adjust prices as money comes in on either side. A price that is out of line with the wider market attracts bets, which pushes it back towards the market consensus.
Odds comparison and arbitrage services can also put the same opportunities in front of thousands of users at once. As a result, an opportunity that might once have lasted for hours can disappear within minutes.
The price you see is only a snapshot. By the time both legs are placed, one of the prices may already have changed.
Two-way markets versus three-way markets
A two-way market, such as tennis or most US sports, needs exactly two legs for the arbitrage to work, usually at two different bookmakers.
A three-way market, such as football's 1X2, needs three legs and often three different bookmakers because it is uncommon for the best price on all three outcomes to be available at the same book.
A three-way arbitrage on 2.35, 3.60 and 3.70, spread across three bookmakers:
- Implied probabilities - 42.6%, 27.8%, 27.0%
- Book - 97.4%
- Stakes on $600 total - $262.25 / $171.19 / $166.56
- Result - a guaranteed +$16.28, a 2.71% return
The maths works in exactly the same way with three outcomes. Add the implied probabilities and stake proportionally.
The practical risk is different. Three legs at three bookmakers mean three separate chances for a price to move, three separate stake limits to deal with, and three separate accounts whose risk systems may notice the pattern.
A three-way arbitrage with the same percentage edge as a two-way opportunity therefore carries more practical execution risk. The maths may be the same, but getting all three bets placed at the required prices is more difficult.
The risk is not the math. It is getting both legs on
The formula assumes that every stake is placed at the price used to calculate it. In practice, three things can easily break that assumption:
- One leg's price moves before you place it. Place the first leg, and by the time you reach the second bookmaker, its price may have shifted - the book that looked like 96.4% a minute ago might now be back over 100%, and you are left deciding whether to take a worse combined price or walk away with one leg already placed and unhedged.
- Stake limits. Many bookmakers cap how much a given account can place on a specific market, especially one already showing signs of sharp, one-sided money. A calculated stake of $493.98 is worthless if the book only lets you place $100 of it.
- Account restrictions. Bookmakers that identify an account as consistently taking arbitrage-shaped bets - placing stakes that only make sense in combination with a bet elsewhere - commonly restrict that account's stakes on future bets, sometimes sharply, regardless of how the individual bet in question is settled. This is a standard, widely documented practice in the industry, not a rare edge case, and it is worth knowing about before treating arbitrage as a strategy to scale rather than an occasional opportunity.
None of this changes the arithmetic of an individual arbitrage opportunity. The formula is just as reliable as the dutching formula it is based on.
The difference is that the calculated return assumes perfect execution. In a real market, prices can move, stakes can be limited, and accounts can be restricted. Those factors can make the practical result less attractive than the theoretical ROI suggests.
What the calculated ROI does not include
A 2.71% return sounds attractive on its own, but it is a return on the stake for one specific opportunity. It is not an annualised or guaranteed rate.
A more useful comparison is how much time and attention it took to find the mismatch, place every leg correctly, and deal with any account restrictions afterwards.
As an occasional opportunity, the maths works exactly as calculated. As a strategy that you want to repeat at scale, the shrinking time windows and account restrictions become much more important than the arithmetic itself.
That is why arbitrage generally works better as something you take advantage of when a genuine mismatch appears, rather than something you expect to run continuously.
Check the prices in front of you with the arbitrage calculator, and consider placing the leg you are least confident will remain available first. That is usually the price most likely to move before you can place the other leg.