Arbitrage on a three-way market: splitting stakes across win, draw and loss

In a three-way market you must stake all three outcomes: home win, draw and away win. Add 1 ÷ decimal price for each. If the total is under 1, split the stake in proportion. Prices of 2.40, 3.50 and 3.60 total 0.9802, a 2.02% margin, and every result returns the same amount.

The arbitrage calculator handles two or three outcomes, so you can check the figures here with your own prices.

Why a draw changes the shape of the problem

On a two-way bet one side or the other must win, so two stakes cover everything. A soccer match settled on the 90 minutes can end in a draw, so a bet only on the two teams leaves the draw uncovered. Covering it requires a third stake, and the 100% test needs a third term in the sum.

The stakes on 1,000

• Home: 1,000 × 0.41667 ÷ 0.98016 = 425.10, returning 425.10 × 2.40 = 1,020.24
• Draw: 1,000 × 0.28571 ÷ 0.98016 = 291.50, returning 291.50 × 3.50 = 1,020.24
• Away: 1,000 × 0.27778 ÷ 0.98016 = 283.40, returning 283.40 × 3.60 = 1,020.24

Each outcome returns 1,020.24 from a 1,000 outlay, a gain of 20.24. The stakes are not equal, since the draw price is longer than the home price and needs a smaller share to return the same amount. The three prices came from different places by construction: if all three came from one place, their sum would be over 1 because every place builds in its own margin.

Checking the sum with a different split

You can verify the stakes by changing the total. With T = 500, the stakes are exactly half: 212.55, 145.75 and 141.70, and each outcome returns 510.12. The ratios between the stakes do not change with the total, only their size. A useful check is that stake × price is the same on all three lines. If it is not, one of the stakes was computed from the wrong price.

Where these arbitrages come from

One place's home price, a second place's draw price and a third place's away price can each be the most generous of the three while the three still sum to less than 1. That requires disagreement between the places on at least one outcome. The more outcomes there are, the more combinations to check, which is why three-way and many-way markets show up in arbitrage lists as often as two-way ones.

Same rules on every leg

All three bets must be on the same market with the same definition. The standard 1X2 settles on the 90 minutes plus stoppage time. A "to qualify" or "to win the match" price that includes extra time and penalties is a different, two-way market, and mixing it with 1X2 prices breaks the arithmetic. Check that every leg names the same match, the same period and the same rules for abandonment before treating the sum as a result.

Try your own prices. Enter the three decimal prices into the arbitrage calculator to see whether the sum falls under 100% and what each stake would be.