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Arbitrage betting in theory

There is a number of potential arbitrage deals. Below is an explanation of some of them including formulas and risks associated with these arbitrage deals. The table below introduces a number of variables that will be used to formalise the arbitrage models.

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 0^-1₁ + 0^-1₂ < 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.

Back-lay sports arbitrage

Betting exchanges have opened up a new range of arbitrage possibilities since on the exchanges it is possible to lay (i.e. to bet against) as well as to back an outcome. Arbitrage using only the back or lay side might occur on betting exchanges. It is in principle the same as the arbitrage using different bookmakers. Arbitrage using back and lay side is possible if a lay bet on one exchange provides shorter odds than a back bet on another exchange or bookmaker. However, the commission charged by the bookmakers and exchanges must be included into calculations.

Back-lay sports arbitrage is often called scalping or trading. Scalping is not actually arbitrage, but short term trading. In the context of sports arbitrage betting a scalping trader or scalper looks to make lots of small profits, which in time can add up. In theory a trader could turn a small investment into large profits by re-investing his earlier profits into future bets so as to generate exponential growth. Scalping relies on liquidity in the markets and that the odds fill flucuate around a mean point. A key advantage to scalping on one exchange is that most exchanges charge commission only on the net winnings in a particular event, thus ensuring that even the smallest favourable difference in the odds will guarantee some profit.

Bonus sports arbitrage

Many bookmakers offer first time users a signup bonus in the range $10 – $200 for depositing an initial amount. They typically demand that this amount is wagered a number of times before the bonus can be withdrawn. Bonus sport arbitraging is a form of sports arbitraging where you hedge or back your bets as usual, but since you received the bonus, a small loss can be allowed on each wager (2-5 %), which comes off your profit. In this way the bookmakers wagering demand can be met and the initial deposit and sign up bonus can be withdrawn with little loss.

The advantage over usual betting arbitrage is that it is a lot easier to find bets with an acceptable loss, instead of an actual profit. Since most bookmakers offer these bonuses this can potentially be exploited to harvest the sign up bonuses.

Making money

By signing up to various bookmakers, it’s possible to turn these ‘free’ bets into cash fairly quickly, and either making a small arbitrage, or in the majority of cases, making a small loss on each bet, or trade. However, it is relatively time consuming to find close matched bets or arbs, which is where an arb / close matched bet service is useful.

Drawbacks

As well as spending time physically matching odds from various bet sites to exchanges, the other draw back with bonus bagging / arb trading in this sense is that often the free bets are ‘non-stake returned’. This effectively reduces the odds, in decimal format, by 1. Therefore, in order to reduce ‘losses’ on the free bet, it is necessary to place a bet with high odds, so that the percentage difference of the decrease in odds is minimalised.

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