Arbitrage: spot the gap, understand the catch

01One game. Different prices. Every outcome covered.

Arbitrage is a pricing relationship, not a prediction about who wins. If different books offer prices that let correctly sized stakes cover every mutually exclusive result and return more than the total staked, the quoted market has a mathematical arbitrage opportunity.

All examples here use made-up training prices, not live offers. They assume every bet is accepted at the shown odds and settles under identical rules. Real execution can fail; this is education, not a recommendation to place bets.

02The test: do the implied probabilities add to less than 100%?

Convert American odds to decimal: +110 becomes 2.10; -125 becomes 1.80. For positive American odds A, decimal odds = 1 + A/100. For negative odds, decimal odds = 1 + 100/|A|. Decimal odds include the returned stake.

For each possible result, calculate 1 ÷ decimal odds. Add those figures. Below 1 means a mathematical pricing gap; exactly 1 means break-even before costs; above 1 means the prices do not create an arbitrage. This test is valid only when the outcomes fully cover the same market.

Let S be that sum and B the total example stake. Allocate B × (1/decimal odds) ÷ S to each result. Before rounding or costs, the equal return is B ÷ S and the return on total stake is (1 ÷ S) − 1. Do not mistake 1 − S for that return.

03Three outcomes means three bets—not two

A hypothetical 90-minute soccer market offers Home 3.10, Draw 3.60 and Away 3.10 across different books. The sum is approximately 92.29%, below 100%. A $100 training allocation is about $34.95 / $30.10 / $34.95. Depending on the result, the gross return is about $108.35–$108.36, before costs. Omitting the draw would leave an uncovered outcome.

A three-way regulation result is not the same market as a two-way team-to-advance bet. Extra time, penalties, pushes, retirements and void rules must match. Attractive-looking odds cannot repair different contracts.

04Why a real opportunity can still lose money

The second price can move after the first bet is accepted. A book may accept only part of a stake, apply a limit or reject a bet. That leaves an exposed position rather than a completed arbitrage. Do not assume a displayed quote is an accepted ticket.

Fees, commission, currency conversion and cent rounding can consume a small gap. A void or different settlement rule can remove the coverage. Check the actual market rules and terms at each operator; account restrictions and applicable taxes also matter.

This differs from positive expected value, which depends on a probability estimate, and from a middle, which targets an overlap between lines. Neither automatically provides a positive return in every outcome. Never chase an unmatched leg by increasing stakes to recover a loss.