Arbitrage Stake-Split Maths, Explained

What does an equal-return stake split do?

Stake-split maths assigns a share of a total amount to each mutually exclusive outcome so that each winning branch returns the same amount. It does not make a price set executable or remove the risk that an operator changes or rejects a quote.

First run the implied-probability test described in arbitrage betting explained. If the sum is not below one after fees, there is no theoretical arbitrage to split.

The two-outcome formula

For total stake T and decimal prices O1 and O2:

• S1 = T × O2 / (O1 + O2)
• S2 = T × O1 / (O1 + O2)

The returns are equal because S1 × O1 = S2 × O2.

As an illustrative calculation, let T = €100, O1 = 2.10, and O2 = 2.00. The stakes are €48.78 and €51.22. Each branch returns approximately €102.44 before commission, rounding, taxes, or any different operator rules.

The three-outcome formula

For prices O1, O2, and O3, first calculate q = 1/O1 + 1/O2 + 1/O3. When q < 1, equal-return stakes are more simply expressed as:

• S1 = (T / q) / O1
• S2 = (T / q) / O2
• S3 = (T / q) / O3

This form makes the check visible: T / q is the target gross return for each branch. It works only when all three selections are mutually exclusive, cover every possible settlement outcome, and use compatible rules.

What changes an arbitrage split in practice?

Decimal prices, stake increments, minimums, exchange commission, and currency conversion can all change the outcome. Round one stake to the permitted increment, recalculate the other stakes, then compare the lowest possible net return with the total committed amount. Do not call a pair an arbitrage from headline prices alone.

This is particularly important for thin markets, where a displayed amount may not be accepted. Limits, liquidity, and signal quality provides context for interpreting that constraint.

How do you keep an arbitrage calculation auditable?

Record the original quote, accepted quote, stakes, fees, and any rule differences. A useful bet log makes it possible to distinguish a calculation error from a price or settlement change later.

Review price relationships in PhotonOdds

The Arbitrage view can help organise cross-book price comparisons. It does not replace an operator's accepted quote, terms, or your own financial judgement.

Try the numbers in the arbitrage stake-split calculator — the same equal-return formula, with the implied total and the fees-and-limits caveats made explicit.

18+ only. Betting carries risk. PhotonOdds provides analytical and educational tools, not a promise of profit or a recommendation to place a bet. If gambling is causing harm, see Responsible Gambling.