Learning outcomes
- Test a two-outcome price pair for theoretical arbitrage.
- Calculate equal-payout stakes and net theoretical return.
- Explain how mismatched rules or partial execution can destroy the apparent advantage.
1. What arbitrage means
A two-sided arbitrage exists when prices for all mutually exclusive and collectively exhaustive outcomes of the same event can be combined so the theoretical payout exceeds the total amount staked. It is a price relationship, not a prediction about which team wins. The mathematical result assumes all legs are accepted and graded consistently.
2. A fully worked hypothetical
Suppose an exactly matched, two-outcome market is quoted at +110 on Side A and +110 on Side B at separate venues. Each dollar staked returns $2.10 total if it wins. The reciprocal-price test is 1/2.10 + 1/2.10 = 0.95238. Because that sum is below 1.00, it is a theoretical arbitrage.
With $100 total allocated equally—$50 on each side—the winning side returns $105, for a theoretical $5 net profit, or 5% of total stake, regardless of which side wins. This is a simplified illustration with no commissions and no grading disagreement.
For unequal prices, distribute stakes to equalize gross payout. If total capital is T and decimal odds are dA and dB, the stake on Side A is T × (1/dA)/[(1/dA)+(1/dB)]. The remainder goes to Side B. The equalized gross payout is T/[(1/dA)+(1/dB)].
3. Why a screen can show a false arb
An apparent arb can be invalid when one side includes overtime but the other does not, when a soccer market includes a draw that was omitted, or when a bet is voided at one venue but settled at another. A stale quote can disappear between legs. Max-stake restrictions, fees, foreign exchange, payout caps, and stake rounding change the economics. The percentage on the screen is not the net realizable return until these assumptions have been checked.
4. Guided calculation workflow
First confirm outcomes cover the complete same market. Convert odds to decimal. Add reciprocal decimal prices. If the sum is under one, compute equal-payout stakes for a hypothetical bankroll and calculate every settlement outcome. Then reduce the result by known costs and test what happens if one order is rejected or repriced. If the worst-case scenario breaches the risk budget, pass.
5. The difference from a middle
An arbitrage seeks a positive theoretical payout in all correctly matched outcomes. A spread middle may produce a large gain on a narrow result but a modest loss otherwise. Neither label overrides settlement and execution risk.
Practice
Question: Both sides show +110 and you allocate $100 equally. What is the theoretical net result? Answer: Each winning leg returns $105; net is +$5, before costs.
Question: One leg is accepted but the other is rejected. Is the position still an arbitrage? Answer: No. There is now unhedged outcome risk.
Recap
Mathematical arb means a reciprocal-price sum under 1 across exhaustive identical outcomes. Equalize payouts, not necessarily stakes. Verify all execution and settlement assumptions.
Last reviewed: Pending editorial review.
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