Labouchère system
The Labouchère system, also called the cancellation system or split martingale, is a gambling strategy used in roulette. The user of such a strategy decides before playing how much money they want to win, and writes down a list of positive numbers that sum to the predetermined amount. With each bet, the player stakes an amount equal to the sum of the first and last numbers on the list. If only one number remains, that number is the amount of the stake. If bet is successful, the two amounts are removed from the list. If the bet is unsuccessful, the amount lost is appended to the end of the list. This process continues until either the list is completely crossed out, at which point the desired amount of money has been won, or until the player runs out of money to wager.[1]
The theory behind this strategy is that since the player is crossing two numbers off of the list (win) for every number added (loss) that the player can complete the list, (crossing out all numbers) thereby winning the desired amount even though the player does not need to win as much as expected for this to occur.
It should be mentioned that the Labouchere System is meant to be applied to even money Roulette propositions such as Even/Odd, Red/Black or 1-18/19-36. When any of these bets are made in the game of Roulette, a spin resulting in a, "0," or, "00," results in a loss, so even though the payout is even money, the odds are clearly not 50/50. The Labouchere System attempts to offset these odds.
If a player were to play any one of the above propositions, there are eighteen individual results which result in a win for that player and twenty individual results that result in a loss for that player. The player has an 18/38 chance of success betting any of the above propositions, which is around 47.37%.
Theoretically, because the player is canceling out two numbers on the list for every win, and adding only one number for every loss, the player needs to have his proposition come at least 33.34% to eventually complete the list. For example, if the list starts with seven numbers and the player wins five times and loses three (62.5% winning percentage) the list is completed and the player wins the desired amount, if the list starts with seven numbers and the player wins 43,600 times and loses 87,193 times (33.34% winning percentage) the list completes and the player wins.
A formula to understand this is as follows:
Where x = Number of Wins
y = Number of Losses
Z = Numbers Originally on the List
When
( y + z ) / 2 ≤ X
The result is the list being completed.
Assuming a player bets nothing but black (red/black proposition) and black can be expected to hit 47.37% of the time, but the system only requires that it hit 33.34% of the time, it can be said that black only need hit approximately 70.38% of the time (33.34/47.37) it can generally be expected to in order for the system to prevail.
An obvious downfall to the system is bankroll, because the more losses sustained by the player, the greater the amount being bet on each turn (as well as the greater the amount lost overall) is. Consider the following list:
10 10 20 20 20 10 10
If a player were to bet black and lose four times in a row, the amounts bet would be: $20, $30, $40, and $50. By taking these four consecutive losses, the player has already lost $140 and is betting $60 more on the next bet. Consecutive losses, or an inordinate amount of losses to wins can also cause table limits to come into play.
Occasionally, a player following this system will come to a point where he can no longer make the next bet as demanded by the system due to table limits. One work-around for this problem is simply to move to a higher limit table, or a player can take the next number that should be bet, divide it by two and simply add it to the list twice. The problem with the latter option is that every time a player commits such a play, it will infinitesimally increase the percentage of spins a player must win to complete the system. The reason this is so is because the player is adding two numbers (which both will be crossed out in the event of wins) where only one loss was sustained.
To prove this, if a player were to play the Labouchere System the same way with the exception being that the player always added half of the wager lost to the bottom of the list twice for every wager lost where:
x = Number of Wins
y = Number of Losses
Z = Numbers Originally on the List
When:
y + (z/2) ≤ x
The result is the list being completed.
The player would actually have to win in excess of 50% of the time (the actual percentage of wins necessary, given x and y, being dependent on z) in order to complete the list, or more than the player could actually be expected to win.
References
- ^ Burrell, Brian. Merriam-Webster's Guide to Everyday Math. Merriam-Webster.
- Tijms, Henk (2004). "Probabilities in everyday life". Understanding probability: chance rules in everyday life. Cambridge University Press. pp. 91–93. ISBN 0-521-54036-4.
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.
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.
| Variable | Explanation |
| s1 | Stake in outcome 1 |
| s2 | Stake in outcome 2 |
| o1 | Odds for outcome 1 |
| o2 | Odds for outcome 2 |
| r1 | Return if outcome 1 occurs |
| r2 | Return if outcome 2 occurs |
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
| Bookmaker 1 | Bookmaker2 | |
| Outcome 1 | 1.25 | 1.43 |
| Outcome 2 | 3.9 | 2.85 |
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.
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