Minimum k-cut

Miraclebets are a particular case arising on betting markets due to either bookmakers' different opinions on event outcomes or plain errors. By placing one bet per each outcome with different betting companies, the bettor can guarantee to make a profit. As long as different Bookmakers are used for betting the Bookmakers do not have a problem with this.

A typical Miraclebet is around 5%, sometimes less, however 5%-10% are also normal and during some special events they might reach 20%.

Miraclebets in practice

In practice, finding these overlaps takes an enormous amount of luck or significant computational power. You'd need to analyse millions of odds with different bookmakers before you found any guaranteed profit.

Miraclebets in theory

Below is an explanation of a Miraclebet, including formulas associated with them. The table below introduces a number of variables that will be used to formalise the model.

Variable	Explanation
$s_1$	Stake in outcome 1
$s_2$	Stake in outcome 2
$o_1$	Odds for outcome 1
$o_2$	Odds for outcome 2
$r_1$	Return if outcome 1 occurs
$r_2$	Return if outcome 2 occurs

Miraclebets

Miraclebets 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 $r_1=\$100*1.43=\$143$ from bookmaker 2. In case outcome 2 comes out, one could collect $r_2=\$36.67*3.9=\$143$ 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 $o_1$ and $o_2$, where $o_1^{-1}+o_2^{-1}<1$. If one wishes to place stake $s_1$ at outcome 1, then one should place $s_2=s_1*o_1/o_2$ 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 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".

See also

• Advantage gambling
• Dutch book
• Mathematics of bookmaking