Arbitrage betting

This type of arbitrage takes advantage of different odds offered by different bookmakers. For an example of an event with only two possible outcomes (e.g., a tennis match in which either Federer wins or Henman wins), the two 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 in both fractional and decimal format:

Fractional odds:

Decimal odds:

The bookmaker's return rate is ${\displaystyle 1-(1.25*3.9)/(1.25+3.9)=5.34\%}$, which is the amount the bookmaker earns on offering bets at some event. Bookmaker 1 will in this example expect to earn 5.34% on bets on the tennis game. For an individual bookmaker, the sum of the inverse of all outcomes of an event will always be greater than 1. ${\displaystyle 1.25^{-1}+3.9^{-1}=1.056}$ and ${\displaystyle 1.43^{-1}+2.85^{-1}=1.051}$

Inverse of decimal odds:

	Bookmaker 1	Bookmaker2
Outcome 1	${\displaystyle 1.25^{-1}=0.8}$	${\displaystyle 1.43^{-1}=0.699}$
Outcome 2	${\displaystyle 3.9^{-1}=0.256}$	${\displaystyle 2.85^{-1}=0.351}$

The idea of arbitrage betting 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:

${\displaystyle 1.43^{-1}+3.9^{-1}=0.956}$

Placing a bet of $100 on the most likely outcome with the lowest odds (outcome 1 with bookmaker 2) and a bet of $36.67 ${\displaystyle (\$100\cdot (1.43/3.9))}$ on outcome 2 at bookmaker 1 would ensure the bettor a profit. When there are more than two possible outcomes the value of the subsequent bets can be calculated with respect to the lowest quoted odds.

In case outcome 1 comes out, one could collect ${\displaystyle r_{1}=\$100*1.43=\$143}$ from bookmaker 2. In case outcome 2 comes out, one could collect ${\displaystyle r_{2}=\$36.67\cdot 3.9=\$143}$ 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 ${\displaystyle o_{1}}$ and ${\displaystyle o_{2}}$, where ${\displaystyle o_{1}^{-1}+o_{2}^{-1}<1}$. If one wishes to place stake ${\displaystyle s_{1}}$ at outcome 1, then one should place ${\displaystyle s_{2}=s_{1}\cdot o_{1}/o_{2}}$ 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 1/1 and a 2/1, by covering the 1/1 with $500 and the 2/1 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, arbitrage calculators can handle calculations for both book arbitrage ("back/back" or "lay/lay") and "back/lay" arbitrage opportunities on an intra-exchange or inter-exchange basis, and are free.

For arbitrages involving three outcomes (e.g. a game which can be won, lost or drawn) having the odds ${\displaystyle o_{1}}$ for Outcome 1, ${\displaystyle o_{2}}$ for outcome 2 and ${\displaystyle o_{3}}$ for outcome 3 with their respective bids being ${\displaystyle b_{1}}$, ${\displaystyle b_{2}}$ and ${\displaystyle b_{3}}$ and sum of the bids being B.

The amount required to bet on each possibility in order to ensure profit can be calculated by

${\displaystyle b_{1}=B/(1+(o_{1}/o_{2})+(o_{1}/o_{3}))}$

${\displaystyle b_{2}=B/(1+(o_{2}/o_{1})+(o_{2}/o_{3}))}$

${\displaystyle b_{3}=B/(1+(o_{3}/o_{1})+(o_{3}/o_{2}))}$

Betting exchanges such as Smarkets 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 many 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 will fluctuate 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 favorable difference in the odds will guarantee some profit.

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, also known as matched betting, is a form of sports arbitraging where the bettor hedges or backs their bets as usual, but since they received the bonus, a small loss can be allowed on each wager (2–5%), which comes off their 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.

By signing up to various bookmakers, it is 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 arbitrages, which is where a middleman service is useful. As many bookmakers require a certain turnover of the bonus amount, matching money from different bookmakers against each other enables the player to in effect quickly "play free" the money of the losing bookmaker and in effect transfer it to the winning bookmaker. By avoiding most of the turnover requirements in this way the player can usually expect a 70-80% return on investment.

As well as spending time physically matching odds from various bet sites to exchanges, the other draw back with bonus bagging and arbitrage 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 minimised.

Shop arbitrage (also known as sharbing or shop-arbing) is the process of using a betting shop's coupons and a betting exchange to create an arbitrage position. This is made possible because online prices change quickly to close these positions and betting shops are slower to change the prices on their printed coupons.