# Is there an arbitrage in these odds?

Checks whether the best odds on every outcome of one event add up to an arbitrage, and splits a total stake across the outcomes so each pays the same, with the profit or loss.

- Page: https://www.acalculator.org/game/arbitrage-calculator
- JSON spec: https://www.acalculator.org/game/arbitrage-calculator.json
- Version: 659b62b8720a

## Default answer

Example with the default inputs (Odds format American, Outcomes [Odds +110; Odds +105], Total stake $1,000.00): Spreading $1,000.00 across the outcomes pays $1,037.35 whichever wins, a profit of $37.35.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| format | Odds format | The format of every outcome’s odds: American, decimal, or fractional. |
| outcomes | Outcomes | The best odds on each outcome of the event, one row per outcome. List every outcome. |
| total | Total stake | The total amount to spread across the outcomes. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| profit | Profit whichever outcome wins | The payout minus the total stake; negative when there is no arbitrage. |
| roi | Return on the total stake | Profit ÷ total stake, in percent: 100 ÷ (sum of implied probabilities) − 1. |
| arbitrage | Arbitrage? | Yes when the equal payout is above the total stake (the implied probabilities add up to less than 100%). |
| book | Sum of implied probabilities | The implied probabilities of all outcomes (100 ÷ decimal odds) added up. |
| payout | Payout whichever outcome wins | Total stake ÷ (sum of 1 ÷ decimal odds); the same for every outcome. |
| stake1 | Stake on outcome 1 | The part of the total stake to bet on outcome 1: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake2 | Stake on outcome 2 | The part of the total stake to bet on outcome 2: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake3 | Stake on outcome 3 | The part of the total stake to bet on outcome 3: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake4 | Stake on outcome 4 | The part of the total stake to bet on outcome 4: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake5 | Stake on outcome 5 | The part of the total stake to bet on outcome 5: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake6 | Stake on outcome 6 | The part of the total stake to bet on outcome 6: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake7 | Stake on outcome 7 | The part of the total stake to bet on outcome 7: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake8 | Stake on outcome 8 | The part of the total stake to bet on outcome 8: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake9 | Stake on outcome 9 | The part of the total stake to bet on outcome 9: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |
| stake10 | Stake on outcome 10 | The part of the total stake to bet on outcome 10: total × (1 ÷ its decimal odds) ÷ (sum of 1 ÷ decimal odds). |

## Method

With decimal odds dᵢ and total stake T: book B = Σ 1/dᵢ; stake on outcome i = T × (1/dᵢ) ÷ B; every outcome pays T ÷ B; profit = T ÷ B − T. An arbitrage exists when B < 1.

## Assumptions

- The outcomes listed cover every way the event can end, and exactly one of them wins.
- Each bet is placed at the odds shown, with no limits, fees or cancelled bets. In practice odds move and bookmakers may void or limit such bets.
- Stakes are exact; round them to what the bookmaker accepts, which changes the payouts slightly.
- Results are mathematics, not betting advice.

## Worked examples

1. format = american, outcomes = {"odds":"+110"} or {"odds":"+105"}, total = $1,000.00 gives book = 96.399535%, arbitrage = Yes, stake1 = $493.98, stake2 = $506.02, payout = $1,037.35, profit = $37.35. Source: hand calculation in content.mdx: 10/21 + 20/41 = 830/861; stakes 1000 × (10/21) ÷ (830/861).
2. format = decimal, outcomes = {"odds":"2.5"} or {"odds":"3.6"}, total = $100.00 gives book = 92.777778%, arbitrage = Yes, stake1 = $43.11, payout = $107.78. Source: hand calculation in content.mdx: 0.4 + 5/18 + 0.25 = 167/180; 100 × 180/167.
3. format = american, outcomes = {"odds":"-110"} or {"odds":"-110"}, total = $220.00 gives arbitrage = No, stake1 = $110.00, payout = $210.00, profit = -$10.00. Source: hand calculation in content.mdx: 22/21 > 1, so no arbitrage; 220 × 21/22 = 210.

## FAQ

### How do I know if odds are an arbitrage?

Add up 1 ÷ decimal odds over every outcome. If the sum is below 1 (below 100% in implied probability), the odds are an arbitrage. For +110 and +105 on the two sides of a game, the sum is 10/21 + 20/41 = 96.40%, so they are.

### How do I split the stakes in an arbitrage?

Bet on each outcome in proportion to 1 ÷ its decimal odds. With a 1,000 total on +110 and +105, bet 1,000 × (10/21) ÷ (830/861) = 493.98 on the first and 506.02 on the second. Each pays 1,037.35.

### How much profit does an arbitrage make?

The payout is the total stake ÷ the sum of 1 ÷ decimal odds, whichever outcome wins. The profit is the payout minus the total. At a 96.40% sum, the return is 1 ÷ 0.9640 − 1 = 3.73%.

### What if the sum is over 100%?

Then there is no arbitrage. The same split still pays the same whichever outcome wins, but it loses money: at −110 on both sides (104.76%), a 220 total pays back 210.

### Is arbitrage betting risk-free?

Only in the arithmetic. In practice odds can change before every bet is placed, a bookmaker can void or limit a bet, and rounded stakes change the payouts. Many bookmakers restrict accounts that bet this way.

## Sources

- Wikipedia, Arbitrage betting: https://en.wikipedia.org/wiki/Arbitrage_betting
- Wikipedia, Fixed-odds betting (odds formats and overround): https://en.wikipedia.org/wiki/Fixed-odds_betting
- National Council on Problem Gambling, help and treatment (1-800-MY-RESET): https://www.ncpgambling.org/help-treatment/