Talk:Arbitrage betting

There's something wrong with the formulae.

shouldn't return be:

r1 = s1o1 - (s1 + s2);

r2 = s2o2 - (s1 + s2);

Math

As they stand, the figures here don't seem to make any sense to me. Surely the math looks like this, right?

outlay: $130 ( $100 stake to b1, $30 to b2 )

outcome 1: b1 pays $130, b2 pays $0 ( break even )

outcome 2: b1 pays $0, b2 pays $90 ( $40 _loss_ )

Has someone been dicking with the numbers?

- 219.165.164.126 06:06, 18 April 2006 (UTC)

Sports Arbitrage

I think there are several errors with the explanation

Firstly the point of the Sports Arbitrage is to find games where the Bookies odds for all the outcomes add up to less than 1 (a certainty). This discrepency is where the money is made. If the sum of all the outcomes i.e. win, lose, or draw is less than one then that must mean that there is a probability for another type of outcome - the arbitrage. Of course this probability has no physical meaning it is the result of incorrect odds being applied to the game.

If in the example given

	Bookmaker1	Bookmaker2
Outcome1	1.3	1.5
Outcome2	4.3	3

The probabilities for a Bookmaker B of Outcome O are given by

$P_{BO}={\frac {1}{DecimalOdds_{BO}}}$

For (Bookmaker 1, Outcome 1) and (Bookmaker 2, Outcome 2) which is the example given in the text

${\begin{matrix}P_{Total}&=&{\frac {1}{1.3}}+{\frac {1}{3}}\\&=&0.77+0.33\\&=&1\end{matrix}}$

So there is no arbitrage for the example given!

There is however an arbitrage for (Bookmaker 1, Outcome2) and (Bookmaker 2, Outcome 1)

${\begin{matrix}P_{Total}&=&{\frac {1}{4.3}}+{\frac {1}{1.5}}\\&=&0.233+0.667\\&=&0.9\end{matrix}}$

For the punter there are 2 possible outcomes and each outcome gives a return R for a stake S. We want to know what stakes are required to generate the same return on each outcome - a no loss situation!

$R_{12}\,$	$=(O_{12}-1)*S_{12}-S_{21}\,$
$R_{21}\,$	$=(O_{21}-1)*S_{21}-S_{12}\,$
$R_{12}\,$	$=R_{21}\,$

$(O_{12}-1)*S_{12}-S_{21}\,$

$=(O_{21}-1)*S_{21}-S_{12}\,$

$O_{12}*S_{12}\,$

$=O_{21}*S_{21}\,$

${\frac {S_{12}}{S_{21}}}\,$

$={\frac {O_{21}}{O_{12}}}\,$

$S_{12}+S_{21}\,$

$=Stake\,$

multiply out and simplifying we get

$S_{12}\,$

$={\frac {O_{21}*Stake}{(O_{12}+O_{21})}}\,$

$S_{21}\,$

$={\frac {O_{12}*Stake}{(O_{12}+O_{21})}}\,$

$S_{21}\,$

$={\frac {4.3*Stake}{(4.3+1.5)}}\,$

$=0.7414*Stake\,$

$S_{12}\,$

$={\frac {1.5*Stake}{(4.3+1.5)}}\,$

$=0.2586*Stake\,$

For a Stake = £100, put £74.14 on Outcome 1 with Bookmaker 2 and put £25.86 on Outcome 2 with Bookmaker 1.

If Outcome 1 occurs:

Return = (1.5 - 1)*74.14 - 25.86

      = £11.20

If Outcome 2 occurs: Return = (4.3 - 1)*25.86 - 74.14

      = £11.20