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Arbitrage using bookmakers

This type of arbitrage takes advantage of different odds offered by different bookmakers. Assume the following situation:

We consider an event with 2 possible outcomes (e.g. a tennis match - either Federer wins or Henman wins), the idea can be generalized to events with more outcomes, but we use this as an example.

The 2 bookmakers have different ideas of who has the best chances of winning. They offer the following Fixed-odds gambling on the outcomes of the event

For an individual bookmaker, the sum of the inverse of all outcomes of an event will always be greater than 1. 1.25 ^{− 1} + 3.9 ^{− 1} = 1.056 and 1.43 ^{− 1} + 2.85 ^{− 1} = 1.051

The fraction above 1, is the bookmakers return rate, the amount the bookmaker earns on offering bets at some event. Bookmaker 1 will in this example expect to earn 5.6% on bets on the tennis game. Usually these gaps will be in the order 8 - 12%.

The idea is to find odds at different bookmakers, where the sum of the inverse of all the outcomes are below 1. Meaning that the bookmakers disagree on the chances of the outcomes. This discrepancy can be used to obtain a profit.

For instance if one places a bet on outcome 1 at bookmaker 2 and outcome 2 at bookmaker 1:

1.43 ^{− 1} + 3.9 ^{− 1} = 0.956

Placing a bet of 100$ on outcome 1 with bookmaker 2 and a bet of $100 * 1.43 / 3.9 = 36.67 on outcome 2 at bookmaker 1 would ensure the bettor a profit.

In case outcome 1 comes out, one could collect r₁ = $100 * 1.43 = $143 from bookmaker 2. In case outcome 2 comes out, one could collect r₂ = $36.67 * 3.9 = $143 from bookmaker 1. One would have invested $136.67, but have collected $143, a profit of $6.33 (%4.6) no matter the outcome of the event.

So for 2 odds o₁ and o₂, where o₁^-1+ o₂^-1 < 1. If one wishes to place stake s₁ at outcome 1, then one should place s₂ = s₁ * o₁ / o₂ at outcome 2, to even out the odds, and receive the same return no matter the outcome of the event.

Or in other words, if there are two outcomes, a 2/1 and a 3/1, by covering the 2/1 with $500 and the 3/1 with $333, one is guaranteed to win $1000 at a cost of $833, giving a 20% profit. More often profits exists around the 4% mark or less.

Reducing the risk of human error is vital being that the mathematical formula is sound and only external factors add "risk". Numerous online arbitrage calculator tools exist to help bettors get the math right. For example, the Arb Cruncher sports betting calculator handles calculations for both book arbitrage (back/back or lay/lay) and back/lay arbitrage opportunities on an intra-exchange or inter-exchange basis, and is free.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.