Matched Betting Maths

7/26/2019 Matched Betting Maths

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1 Backing and Laying

Let SB be the back stake and OB be the back odds in decimal format.

Let SL be the lay stake (that is, the stake we accept from the punter on the other side of thetransaction) and OL be the lay odds (that is, the odds at which the punter on the other side of thetransaction places their bet) in decimal format.

Let CB be the back commission in percent, and set CB = C

B

100. This is usually zero.

Let CL be the lay commission in percent, and set CL= C

L

100. This is usually non-zero.

(Note that we always have the following: SB, SL >0, OB, OL > 1, and CB, CL 0.)

1.1 Back/Lay Qualifying

If the back bet wins, our profit PB is:

PB =SB(OB 1)(1 CB) SL(OL 1)

that is, the winnings from the back bet, adjusted for commission, minus the payout for the lay bet.

If the lay bet wins, our profit PL is:

PL=SL(1 CL) SB

that is, the lay stake, adjusted for commission, minus the back stake.

For our profit to be the same regardless of outcome, we need to solve:

PL= PB

= SL(1 CL) SB =SB(OB 1)(1 CB) SL(OL 1)

= SL(1 CL) + SL(OL 1) =SB(OB 1)(1 CB) + SB

= SL SLCL+ SLOL SL= (SBOB SB)(1 CB) + SB

= SLOL SLCL=SBOB SB SBOBCB+ SBCB+ SB

= SL(OL CL) =SBOB SBOBCB+ SBCB

= SL= SB(OB OBCB+ CB)

OL CL

In most situations (e.g. when backing at a bookie) CB = 0, so the above becomes:

SL= SBOB

OL CL

As we should already know the values ofSB,OB,OL,CB, andCL, the value ofSLis easily calculated.


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1.2 Back/Lay Free SR


PB =SB+ SB(OB 1)(1 CB) SL(OL 1)

that is, the stake and winnings from the back bet, adjusted for commission, minus the payout for

the lay bet. (Here we assume that commission is strictly only applied to winnings.) We include theback stake in our profit as it is returned to us with the winnings even though we did not put it forthourself.

If the lay bet wins, our profit PL is:PL=SL(1 CL)

There is no back stake in this expression, since the back bet is free.


PL= PB

= SL(1 CL) = SB+ SB(OB 1)(1 CB) SL(OL 1)

= SL(1 CL) + SL(OL 1) =SB+ SB(OB 1)(1 CB)

= SL(OL CL) =SB+ SB(OB 1 OBCB+ CB)

= SL(OL CL) =SB(OB OBCB+ CB)

= SL= SB(OB OBCB+ CB)

OL CL

IfCB = 0, then the above becomes:

SL= SBOBOL CL

As we should already know the values ofSB,OB,OL,CB, andCL, the value ofSLis easily calculated.

1.3 Back/Lay Free SNR



This is the same as with a qualifying bet: In a qualifying bet, although the back stake is returnedto us with the winnings, since we put it forth in the first place, it is not included in our profit. In afree SNR bet, the back stake is effectively put forth on our behalf, but we do not receive that stakealong with the winnings, so it cannot possibly be included in our profit. In both cases, our profitfrom a winning back bet is the same. Meanwhile, what occurs with a losing lay bet is the same inboth cases.

If the lay bet wins, our profit PL is:PL=SL(1 CL)

This is the same as with a free SR bet: there is no back stake in this expression, since the back betis free.


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PL= PB

= SL(1 CL) =SB(OB 1)(1 CB) SL(OL 1)

= SL(1 CL) + SL(OL 1) =SB(OB 1)(1 CB)

= SL SLCL+ SLOL SL= (SBOB SB)(1 CB)= SLOL SLCL= SBOB SB SBOBCB+ SBCB

= SL(OL CL) = SB(OB 1 OBCB+ CB)

= SL= SB(OB 1 OBCB+ CB)

OL CL


SL= SB(OB 1)

OL CL

Long Odds for Free SNR Bets

To see why longer odds generally result in greater profits for free SNR bets, consider an idealisedsituation in which the back and lay odds are equal and back and lay commission are zero:OB =OL= O (say) and CB =CL= 0. [Note the difference between O and 0!]

Let our profit from this idealised matched bet be P. Then we have:

P =PL=SL=SBO 1

O

Since O >1, O1O

>0, and since SB >0 and is fixed, to increase Pwe must increase O1O

. Now, toincrease O1

O we must increaseO; therefore, to increase Pwe must increase O.

1.4 Back/Lay Qualifying with Free SNR on Lose

Let Rbe the expected return (in monetary terms) from the free SNR bet; that is, the profit we mayreasonably expect from it given the competitiveness of the available odds and the value of the bet.

If the back bet (that is, the first, qualifying bet) wins, our profit PB is:


Since we have won the back bet, we do not get any refund, and we are done.

If the lay bet wins, the back bet loses and we are granted a free SNR bet. Our profit PL then is:

PL=SL(1 CL) SB+ R

that is, the lay stake, adjusted for commission, minus the back stake, plus the expected return fromthe free SNR bet.


PL= PB


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= SL(1 CL) SB+ R= SB(OB 1)(1 CB) SL(OL 1)

= SL(1 CL) + SL(OL 1) =SB(OB 1)(1 CB) + SB R

= SL SLCL+ SLOL SL= (SBOB SB)(1 CB) + SB R

= SLOL SLCL= SBOB SB SBOBCB+ SBCB+ SB R

= SL(OL CL) = SBOB SBOBCB+ SBCB R

= SL= SB(OB OBCB+ CB) R

OL CL


SL= SBOB R

OL CL

Long Odds for Free SNR on Lose Bets

To see why longer odds generally result in greater profits for bets with a free SNR refund on lose,

consider an (extremely!) idealised situation in which: the back and lay odds are equal; back and laycommission are zero; the value of the free SNR bet is equal to the value of the qualifying bet; andwe may reasonably expect to recoup 100% of the value of an SNR bet. That is:

OB =OL=O

CB =CL= 0

R= SB

Let our profit from this (extremely!) idealised matched bet be P. Then we have:

P= PL= SL SB+ SB =SL= SBO SB

O =SB

O 1

O

As with free SNR bets, to increase P, we must increase O.

2 Dutching

We place ndifferent back bets. We assume that the bets are on mutually exclusive selections which

together cover all possible outcomes.

Let Si be the stake and Oi be the odds in decimal format for the ith bet, with i Z, 1 i n.(That is, S1 is the stake and O1 is the odds for the first bet, S2 is the stake and O2 is the odds forthe second bet, and so on up to Sn and On, the stake and odds for the nth and last bet.)

Let Ci be the commission in percent for the ith bet, and set Ci= C

i


One of the n bets will win, and the other n 1 bets will lose. If, say, the ith bet wins, then ourprofit Pi is:

Pi=Si(Oi 1)(1 Ci)

j=i

Sj

that is, the winnings from the ith bet, adjusted for commission, minus the stakes for the other bets.


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P1 = P2=. . .= Pi = P(i+1) = . . .= P(n1)=Pn

= S1(O1 1)(1 C1) S2 S3 . . . Sn

= S2(O2 1)(1 C2) S1 S3 . . . Sn

= . . .

=Si(Oi 1)(1 Ci) S1 . . . S(i1) S(i+1) S(i+2) . . . Sn

=S(i+1)(O(i+1) 1)(1 C(i+1)) S1 . . . S(i1) Si S(i+2) . . . Sn

= . . .

=S(n1)(O(n1) 1)(1 C(n1)) S1 . . . S(n2) Sn

=Sn(On 1)(1 Cn) S1 . . . S(n2) S(n1)

Considering the above equalities pairwise, we can see that i Z, 1 i < n, we have:

Si(Oi 1)(1 Ci) + Si = S(i+1)(O(i+1) 1)(1 C(i+1)) + S(i+1)

= Si(Oi 1 OiCi+ Ci) + Si= S(i+1)(O(i+1) 1 O(i+1)C(i+1)+ C(i+1)) + S(i+1)

= Si(Oi OiCi+ Ci) = S(i+1)(O(i+1) O(i+1)C(i+1)+ C(i+1))

We can then generalise the above i, j Z, 1 i, j n:

Si(Oi OiCi+ Ci) =Sj(Oj OjCj+ Cj)

= Si= Sj(Oj OjCj+ Cj)

Oi OiCi+ Ci

IfCi= 0 i Z, 1 i n, then the above becomes:

SiOi=SjOj = Si= SjOj

Oi

We must fix the stake for one of the bets say, the jth bet. Then, as we should already know thevalues ofOi and Ci i Z, 1 i n, the values ofSi i Z, 1 i n, i=j are easily calculated.

3 Multiples

Let SB be the back stake on an n-fold multiple bet with combined back odds in decimal format ofOBM.

Let OBi be the back odds in decimal format for the ith leg of the bet, with i Z, 1 i n. (Thatis, OB1 is the back odds for the first leg, OB2 is the back odds for the second leg, and so on up toOBn, the back odds for the nth and last leg.) We should have OBM =OB1OB2 . . . OBn.

Let SLi be the lay stake and OLi be the lay odds in decimal format for the ith leg of the bet.

Let CB be the back commission in percent, and set CB = C

B


Let CLi be the lay commission in percent for the ith leg of the bet, and set CLi = C

Li

100. This is

usually non-zero.

We assume that the multiple bet wins if and only if all its legs win, and that the events being bet onin each leg are non-overlapping and so can be laid separately. We also assume, for simplicity, thatthe lay odds for all legs are known at the start and do not change thereafter.


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First Leg

Lose

If the back bet loses on this leg, then we lose SB, the back stake, and we gain SL1(1 CL1), the laystake for this leg, adjusted for commission.

Our net profit for this leg is P1lose =SL1(1 CL1) SB.Our total net profit is:

P =P1lose = SL1(1 CL1) SB

and we are done.

Win

If the back bet wins on this leg, then we lose SL1(OL1 1), the payout for the lay on this leg.

Our net profit for this leg is P1win =SL1(OL1 1).Our net profit so far is P1win =SL1(OL1 1), and we continue to the next leg.

Second Leg

Lose

If the back bet loses on this leg, then we lose SB, the back stake, and we gain SL2(1 CL2), the laystake for this leg, adjusted for commission.

Our net profit for this leg is P2lose =SL2(1 CL2) SB.

Our total net profit is:

P =P1win+ P2lose =SL1(OL1 1) + SL2(1 CL2) SB

and we are done.

Win

If the back bet wins on this leg, then we lose SL2(OL2 1), the payout for the lay on this leg.Our net profit for this leg is P2win =SL2(OL2 1).

Our net profit so far is P1win+ P2win =SL1(OL1 1) SL2(OL2 1), and we continue to the nextleg.

ith Leg

Lose

If the back bet loses on this leg, then we lose SB, the back stake, and we gain SLi(1 CLi), the laystake for this leg, adjusted for commission.


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Our net profit for this leg is Pilose = SLi(1 CLi) SB.

Our total net profit is:P= P1win+ P2win+ . . . + P(i1)win+ Pilose

=SL1(OL1 1) SL2(OL2 1) . . . SL(i1)(OL(i1) 1) + SLi(1 CLi) SB

and we are done.

Win

If the back bet wins on this leg, then we lose SLi(OLi 1), the payout for the lay on this leg.

Our net profit for this leg is Piwin =SLi(OLi 1).

Our net profit so far is P1win+ P2win+ . . . + P(i1)win+ Piwin =SL1(OL1 1)SL2(OL2 1) . . .SL(i1)(OL(i1) 1) SLi(OLi 1), and we continue to the next leg.

nth Leg

Lose

If the back bet loses on this leg, then we lose SB, the back stake, and we gain SLn(1 CLn), the laystake for this leg, adjusted for commission.

Our net profit for this leg is Pnlose =SLn(1 CLn) SB.

Our total net profit is:P =P1win+ P2win+ . . . + Piwin+ . . . + Pnlose

=SL1(OL1 1) SL2(OL2 1) . . . SLi(OLi 1) . . . + SLn(1 CLn) SB

and we are done.

Win

If the back bet wins on this leg, then we lose SLn(OLn 1), the payout for the lay on this leg, andwe gain SB(OBM 1)(1 CB), the winnings from the multiple bet, adjusted for commission.

Our net profit for this leg is Pnwin =SLn(OLn 1) + SB(OBM 1)(1 CB).

Our total net profit is:P = P1win+ P2win+ . . . + Piwin+ . . . + Pnwin

=SL1(OL1 1) SL2(OL2 1) . . . SLi(OLi 1) . . . SLn(OLn 1) + SB(OBM 1)(1CB)

and we are done.


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Matching the Bet


P =P1lose

=P1win+ P2lose

= . . .

=P1win+ P2win+ . . . + P(i1)win+ Pilose

=P1win+ P2win+ . . . + P(i1)win+ Piwin+ P(i+1)lose

= . . .

=P1win+ P2win+ . . . + Piwin+ . . . + Pnlose

=P1win+ P2win+ . . . + Piwin+ . . . + Pnwin

That is: P =SL1(1 CL1) SB

=SL1(OL1 1) + SL2(1 CL2) SB

= . . .

=SL1(OL1 1) SL2(OL2 1) . . . SL(i1)(OL(i1) 1) + SLi(1 CLi) SB

=SL1(OL1 1)SL2(OL2 1) . . .SL(i1)(OL(i1) 1)SLi(OLi 1) + SL(i+1)(1CL(i+1))SB

= . . .

=SL1(OL1 1) SL2(OL2 1) . . . SLi(OLi 1) . . . + SLn(1 CLn) SB


Observe that the above is easily solved pairwise.

First, we have:SL1(1 CL1) SB =SL1(OL1 1) + SL2(1 CL2) SB

= SL1(1 CL1) + SL1(OL1 1) =SL2(1 CL2)

= SL1(OL1 CL1) =SL2(1 CL2)

= SL1 = SL2(1 CL2)

OL1 CL1

And in general, i Z, 1 i < n, we have:

SL1(OL1 1) SL2(OL2 1) . . . SL(i1)(OL(i1) 1) + SLi(1 CLi) SB

=SL1(OL1 1)SL2(OL2 1) . . .SL(i1)(OL(i1) 1)SLi(OLi 1) + SL(i+1)(1CL(i+1))SB

= SLi(1 CLi) + SLi(OLi 1) =SL(i+1)(1 CL(i+1))

= SLi(OLi CLi) =SL(i+1)(1 CL(i+1))

= SLi =SL

(i+1)

(1 CL(i+1)

)

OLi CLi


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And finally, we have:

SL1(OL1 1) SL2(OL2 1) . . . SLi(OLi 1) . . . + SLn(1 CLn) SB


= SLn(1 CLn) + SLn(OLn 1) =SB(OBM 1)(1 CB) + SB

= SLn(OLn CLn) =SB(OBM 1 OBMCB+ CB) + SB

= SLn(OLn CLn) =SB(OBM OBMCB+ CB)

= SLn = SB(OBM OBMCB+ CB)

OLn CLn


SLn = SBOBMOLn CLn

As we should already know the values ofSB, OBi , OLi , CB, and CLi i Z, 1 i n, the values of

SLii Z, 1 i n are easily calculated.

Matched Betting Maths

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