Martingale Comparison

8/18/2019 Martingale Comparison

1/14

Probability of win

Trial 1 Trial 2 Trial 3

Outcome Outcome Outcome

1 Win 0.5 1 Win 0.5 1 Win

1 Win 0.5 1 Win 0.5 1 Loss

1 Win 0.5 1 Loss -0.5 2 Win

1 Loss -0.5 2 Win 0.5 1 Win

1 Win 0.5 1 Loss -0.5 2 Loss

1 Loss -0.5 2 Win 0.5 1 Loss

1 Loss -0.5 2 Loss -0.5 4 Win

1 Loss -0.5 2 Loss -0.5 4 Loss

Total expectancy using Martingale

Trial 1 Trial 2 Trial 3

Outcome Outcome Outcome

1 Win 0.5 1 Win 0.5 1 Win

1 Win 0.5 1 Win 0.5 1 Loss

1 Win 0.5 1 Loss -0.5 1 Win

1 Loss -0.5 1 Win 0.5 1 Win

1 Win 0.5 1 Loss -0.5 1 Loss

1 Loss -0.5 1 Win 0.5 1 Loss

1 Loss -0.5 1 Loss -0.5 1 Win

1 Loss -0.5 1 Loss -0.5 1 Loss

Total expectancy using fixed size betting

Bet sizeb1 Probabilityp1 Bet sizeb2 Probabilityp2 Bet sizeb3

Bet sizeb1

Probabilityp1

Bet sizeb2

Probabilityp2

Bet sizeb3

The above illustrates, over 3 trials, that the Martingale offers no ege in eOnly by changing the win probability in the yellow cell, can we alter e!pect


2/14

50

#inal e!pectancy

b1p1 $ b2p2 $ b3p3

0.5 1.5

-0.5 0.5

0.5 1.0

0.5 1.0

-0.5 -1.0

-0.5 0.0

0.5 0.5

-0.5 -!.5

0.0

#inal e!pectancy

b1p1 $ b2p2 $ b3p3

0.5 1.5

-0.5 0.5

0.5 0.5

0.5 0.5

-0.5 -0.5

-0.5 -0.5

0.5 -0.5

-0.5 -1.5

0.0

Probabilityp3

Probabilityp3

pectancy over flat betting"ancy"


3/14

1 2 3 % & ' 1 2 3 % & ' 1 2 3 % & ' Perm(

) )))))) ) ) ) ) ) ) ) ) ) ) ) ) 1 2 3 % & ' 1

1 )))))1 ) ) ) ) ) 1 ) ) ) ) ) 1 1 2 3 % & & 2

2 ))))1) ) ) ) ) 1 ) ) ) ) ) 1 1 1 2 3 % % & 3

3 ))))11 ) ) ) ) 1 1 ) ) ) ) 1 2 1 2 3 % % % 4

% )))1)) ) ) ) 1 ) ) ) ) ) 1 1 1 1 2 3 3 % & 5

& )))1)1 ) ) ) 1 ) 1 ) ) ) 1 1 2 1 2 3 3 % % 6

' )))11) ) ) ) 1 1 ) ) ) ) 1 2 2 1 2 3 3 3 % 7

* )))111 ) ) ) 1 1 1 ) ) ) 1 2 3 1 2 3 3 3 3 8

+ ))1))) ) ) 1 ) ) ) ) ) 1 1 1 1 1 2 2 3 % & 9

))1))1 ) ) 1 ) ) 1 ) ) 1 1 1 2 1 2 2 3 % % 10

1) ))1)1) ) ) 1 ) 1 ) ) ) 1 1 2 2 1 2 2 3 3 % 11

11 ))1)11 ) ) 1 ) 1 1 ) ) 1 1 2 3 1 2 2 3 3 3 12

12 ))11)) ) ) 1 1 ) ) ) ) 1 2 2 2 1 2 2 2 3 % 1313 ))11)1 ) ) 1 1 ) 1 ) ) 1 2 2 3 1 2 2 2 3 3 14


4/14

1% ))111) ) ) 1 1 1 ) ) ) 1 2 3 3 1 2 2 2 2 3 15

1& ))1111 ) ) 1 1 1 1 ) ) 1 2 3 % 1 2 2 2 2 2 16

1' )1)))) ) 1 ) ) ) ) ) 1 1 1 1 1 1 1 2 3 % & 17

1* )1)))1 ) 1 ) ) ) 1 ) 1 1 1 1 2 1 1 2 3 % % 18

1+ )1))1) ) 1 ) ) 1 ) ) 1 1 1 2 2 1 1 2 3 3 % 19

1 )1))11 ) 1 ) ) 1 1 ) 1 1 1 2 3 1 1 2 3 3 3 20

2) )1)1)) ) 1 ) 1 ) ) ) 1 1 2 2 2 1 1 2 2 3 % 21

21 )1)1)1 ) 1 ) 1 ) 1 ) 1 1 2 2 3 1 1 2 2 3 3 22

22 )1)11) ) 1 ) 1 1 ) ) 1 1 2 3 3 1 1 2 2 2 3 23

23 )1)111 ) 1 ) 1 1 1 ) 1 1 2 3 % 1 1 2 2 2 2 24

2% )11))) ) 1 1 ) ) ) ) 1 2 2 2 2 1 1 1 2 3 % 25

2& )11))1 ) 1 1 ) ) 1 ) 1 2 2 2 3 1 1 1 2 3 3 26

2' )11)1) ) 1 1 ) 1 ) ) 1 2 2 3 3 1 1 1 2 2 3 27

2* )11)11 ) 1 1 ) 1 1 ) 1 2 2 3 % 1 1 1 2 2 2 28

2+ )111)) ) 1 1 1 ) ) ) 1 2 3 3 3 1 1 1 1 2 3 29

2 )111)1 ) 1 1 1 ) 1 ) 1 2 3 3 % 1 1 1 1 2 2 30

3) )1111) ) 1 1 1 1 ) ) 1 2 3 % % 1 1 1 1 1 2 31

31 )11111 ) 1 1 1 1 1 ) 1 2 3 % & 1 1 1 1 1 1 32

32 1))))) 1 ) ) ) ) ) 1 1 1 1 1 1 ) 1 2 3 % & 33

33 1))))1 1 ) ) ) ) 1 1 1 1 1 1 2 ) 1 2 3 % % 34

3% 1)))1) 1 ) ) ) 1 ) 1 1 1 1 2 2 ) 1 2 3 3 % 35

3& 1)))11 1 ) ) ) 1 1 1 1 1 1 2 3 ) 1 2 3 3 3 36

3' 1))1)) 1 ) ) 1 ) ) 1 1 1 2 2 2 ) 1 2 2 3 % 37

3* 1))1)1 1 ) ) 1 ) 1 1 1 1 2 2 3 ) 1 2 2 3 3 38

3+ 1))11) 1 ) ) 1 1 ) 1 1 1 2 3 3 ) 1 2 2 2 3 39

3 1))111 1 ) ) 1 1 1 1 1 1 2 3 % ) 1 2 2 2 2 40

%) 1)1))) 1 ) 1 ) ) ) 1 1 2 2 2 2 ) 1 1 2 3 % 41

%1 1)1))1 1 ) 1 ) ) 1 1 1 2 2 2 3 ) 1 1 2 3 3 42

%2 1)1)1) 1 ) 1 ) 1 ) 1 1 2 2 3 3 ) 1 1 2 2 3 43

%3 1)1)11 1 ) 1 ) 1 1 1 1 2 2 3 % ) 1 1 2 2 2 44

%% 1)11)) 1 ) 1 1 ) ) 1 1 2 3 3 3 ) 1 1 1 2 3 45

%& 1)11)1 1 ) 1 1 ) 1 1 1 2 3 3 % ) 1 1 1 2 2 46

%' 1)111) 1 ) 1 1 1 ) 1 1 2 3 % % ) 1 1 1 1 2 47

%* 1)1111 1 ) 1 1 1 1 1 1 2 3 % & ) 1 1 1 1 1 48

%+ 11)))) 1 1 ) ) ) ) 1 2 2 2 2 2 ) ) 1 2 3 % 49

% 11)))1 1 1 ) ) ) 1 1 2 2 2 2 3 ) ) 1 2 3 3 50

&) 11))1) 1 1 ) ) 1 ) 1 2 2 2 3 3 ) ) 1 2 2 3 51

&1 11))11 1 1 ) ) 1 1 1 2 2 2 3 % ) ) 1 2 2 2 52

&2 11)1)) 1 1 ) 1 ) ) 1 2 2 3 3 3 ) ) 1 1 2 3 53

&3 11)1)1 1 1 ) 1 ) 1 1 2 2 3 3 % ) ) 1 1 2 2 54


5/14

&% 11)11) 1 1 ) 1 1 ) 1 2 2 3 % % ) ) 1 1 1 2 55

&& 11)111 1 1 ) 1 1 1 1 2 2 3 % & ) ) 1 1 1 1 56

&' 111))) 1 1 1 ) ) ) 1 2 3 3 3 3 ) ) ) 1 2 3 57

&* 111))1 1 1 1 ) ) 1 1 2 3 3 3 % ) ) ) 1 2 2 58

&+ 111)1) 1 1 1 ) 1 ) 1 2 3 3 % % ) ) ) 1 1 2 59

& 111)11 1 1 1 ) 1 1 1 2 3 3 % & ) ) ) 1 1 1 60

') 1111)) 1 1 1 1 ) ) 1 2 3 % % % ) ) ) ) 1 2 61

'1 1111)1 1 1 1 1 ) 1 1 2 3 % % & ) ) ) ) 1 1 62

'2 11111) 1 1 1 1 1 ) 1 2 3 % & & ) ) ) ) ) 1 63

'3 111111 1 1 1 1 1 1 1 2 3 % & ' ) ) ) ) ) ) 64

(( ( ( ( ( ((

(( ( ( ( ( ((


6/14

"a#e $ara#eters

Ma!imum number of trials -1.'/ '

%xpectancy

0verage win size 1

0verage loss size 1 4 or Profit #actor

Probability of win &)")5 6elly value

&etting syste# 'MM(

7tarting bet size 1

8pon a win, multiply prior bet size by ) an then a

8pon a loss, multiply prior bet size by 2 an then a

9uit after how many wins: ' -' never ;uit/

9uit after how many losses: ' -' never ;uit/

Trial 1 Trial 2

Outcomes Bet size Outcome !pectancy Bet size Outcome !pectancy Bet size

LLLLLL 1 Loss -0.50 2 Loss -0.50 4

LLLLLW 1 Loss -0.50 2 Loss -0.50 4

LLLLWL 1 Loss -0.50 2 Loss -0.50 4

LLLLWW 1 Loss -0.50 2 Loss -0.50 4

LLLWLL 1 Loss -0.50 2 Loss -0.50 4

LLLWLW 1 Loss -0.50 2 Loss -0.50 4

LLLWWL 1 Loss -0.50 2 Loss -0.50 4

LLLWWW 1 Loss -0.50 2 Loss -0.50 4

LLWLLL 1 Loss -0.50 2 Loss -0.50 4

LLWLLW 1 Loss -0.50 2 Loss -0.50 4

LLWLWL 1 Loss -0.50 2 Loss -0.50 4

LLWLWW 1 Loss -0.50 2 Loss -0.50 4

LLWWLL 1 Loss -0.50 2 Loss -0.50 4

LLWWLW 1 Loss -0.50 2 Loss -0.50 4

The table below illustrates, over ' -or less/ trials, that the Martingale oany betting system an playing strategy, an convince yourself that it!pectancy ivie by ave bet size> -gray bac=g

To simulate Martingale betting@ upon a win, multiply bet size by ), anTo simulate anti-Martingale betting@ upon a win, multiply bet size by 2To simulate d)*le#bert betting@ upon a win, multiply bet size by 1, anTo simulate flat betting@ upon a win, multiply bet size by 1, an then a


7/14

LLWWWL 1 Loss -0.50 2 Loss -0.50 4

LLWWWW 1 Loss -0.50 2 Loss -0.50 4

LWLLLL 1 Loss -0.50 2 Win 0.50 1

LWLLLW 1 Loss -0.50 2 Win 0.50 1

LWLLWL 1 Loss -0.50 2 Win 0.50 1

LWLLWW 1 Loss -0.50 2 Win 0.50 1

LWLWLL 1 Loss -0.50 2 Win 0.50 1

LWLWLW 1 Loss -0.50 2 Win 0.50 1

LWLWWL 1 Loss -0.50 2 Win 0.50 1

LWLWWW 1 Loss -0.50 2 Win 0.50 1

LWWLLL 1 Loss -0.50 2 Win 0.50 1

LWWLLW 1 Loss -0.50 2 Win 0.50 1

LWWLWL 1 Loss -0.50 2 Win 0.50 1

LWWLWW 1 Loss -0.50 2 Win 0.50 1

LWWWLL 1 Loss -0.50 2 Win 0.50 1

LWWWLW 1 Loss -0.50 2 Win 0.50 1

LWWWWL 1 Loss -0.50 2 Win 0.50 1

LWWWWW 1 Loss -0.50 2 Win 0.50 1

WLLLLL 1 Win 0.50 1 Loss -0.50 2

WLLLLW 1 Win 0.50 1 Loss -0.50 2

WLLLWL 1 Win 0.50 1 Loss -0.50 2

WLLLWW 1 Win 0.50 1 Loss -0.50 2

WLLWLL 1 Win 0.50 1 Loss -0.50 2

WLLWLW 1 Win 0.50 1 Loss -0.50 2

WLLWWL 1 Win 0.50 1 Loss -0.50 2

WLLWWW 1 Win 0.50 1 Loss -0.50 2

WLWLLL 1 Win 0.50 1 Loss -0.50 2

WLWLLW 1 Win 0.50 1 Loss -0.50 2

WLWLWL 1 Win 0.50 1 Loss -0.50 2

WLWLWW 1 Win 0.50 1 Loss -0.50 2

WLWWLL 1 Win 0.50 1 Loss -0.50 2

WLWWLW 1 Win 0.50 1 Loss -0.50 2

WLWWWL 1 Win 0.50 1 Loss -0.50 2

WLWWWW 1 Win 0.50 1 Loss -0.50 2

WWLLLL 1 Win 0.50 1 Win 0.50 1

WWLLLW 1 Win 0.50 1 Win 0.50 1

WWLLWL 1 Win 0.50 1 Win 0.50 1

WWLLWW 1 Win 0.50 1 Win 0.50 1

WWLWLL 1 Win 0.50 1 Win 0.50 1

WWLWLW 1 Win 0.50 1 Win 0.50 1


8/14

WWLWWL 1 Win 0.50 1 Win 0.50 1

WWLWWW 1 Win 0.50 1 Win 0.50 1

WWWLLL 1 Win 0.50 1 Win 0.50 1

WWWLLW 1 Win 0.50 1 Win 0.50 1

WWWLWL 1 Win 0.50 1 Win 0.50 1

WWWLWW 1 Win 0.50 1 Win 0.50 1

WWWWLL 1 Win 0.50 1 Win 0.50 1

WWWWLW 1 Win 0.50 1 Win 0.50 1

WWWWWL 1 Win 0.50 1 Win 0.50 1

WWWWWW 1 Win 0.50 1 Win 0.50 1


9/14

Total overall e!pectancy 0.00

0verage bet size 2.25

1 !pectancy ivie by ave bet size 0.001")) Percentage of positive outcomes ,

)"))

1

)

Trial 3 Trial % Trial &

Outcome !pectancy Bet size Outcome !pectancy Bet size Outcome !pectancy

Loss -0.50 Loss -0.50 1 Loss -0.50

Loss -0.50 Loss -0.50 1 Loss -0.50

Loss -0.50 Loss -0.50 1 Win 0.50

Loss -0.50 Loss -0.50 1 Win 0.50

Loss -0.50 Win 0.50 1 Loss -0.50

Loss -0.50 Win 0.50 1 Loss -0.50

Loss -0.50 Win 0.50 1 Win 0.50

Loss -0.50 Win 0.50 1 Win 0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

fers no ege in e!pectancy over flat betting" #eel welcome to e!periment by changiimpossible to alter e!pectancy, no matter what the probability of a win is, an no moun/" The table coul be e!tene from ' to any number of trials ? the result woul

then a 1 -reverts bet size bac= to 1/A upon a loss, multiply bet size by 2 an a, an then a ) -oubles your bet/A upon a loss, multiply bet size by ) an a 1 -ra .1 -reuces your bet/A upon a loss, multiply bet size by 1 an a 1 -increases

) -retains bet size/A upon a loss, multiply bet size by 1 an a ) -retains bet size


10/14

Win 0.50 1 Win 0.50 1 Win 0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Loss -0.50 2 Loss -0.50 4 Loss -0.50

Loss -0.50 2 Loss -0.50 4 Loss -0.50

Loss -0.50 2 Loss -0.50 4 Win 0.50

Loss -0.50 2 Loss -0.50 4 Win 0.50

Loss -0.50 2 Win 0.50 1 Loss -0.50

Loss -0.50 2 Win 0.50 1 Loss -0.50

Loss -0.50 2 Win 0.50 1 Win 0.50

Loss -0.50 2 Win 0.50 1 Win 0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Loss -0.50 4 Loss -0.50 Loss -0.50

Loss -0.50 4 Loss -0.50 Loss -0.50

Loss -0.50 4 Loss -0.50 Win 0.50

Loss -0.50 4 Loss -0.50 Win 0.50Loss -0.50 4 Win 0.50 1 Loss -0.50

Loss -0.50 4 Win 0.50 1 Loss -0.50

Loss -0.50 4 Win 0.50 1 Win 0.50

Loss -0.50 4 Win 0.50 1 Win 0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Loss -0.50 2 Loss -0.50 4 Loss -0.50

Loss -0.50 2 Loss -0.50 4 Loss -0.50

Loss -0.50 2 Loss -0.50 4 Win 0.50

Loss -0.50 2 Loss -0.50 4 Win 0.50

Loss -0.50 2 Win 0.50 1 Loss -0.50

Loss -0.50 2 Win 0.50 1 Loss -0.50


11/14

Loss -0.50 2 Win 0.50 1 Win 0.50

Loss -0.50 2 Win 0.50 1 Win 0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Loss -0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Loss -0.50 2 Win 0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Loss -0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Win 0.50 1 Win 0.50 1 Win 0.50

Total o+erall expectancy


12/14

Trial '

Bet size Outcome !pectancy

!2 Loss -0.50 ?31"&)

!2 Win 0.50 )"&)

1 Loss -0.50 )"))

1 Win 0.50 1"))

2 Loss -0.50 ?1"))

2 Win 0.50 1"))

1 Loss -0.50 )"&)

1 Win 0.50 1"&)

4 Loss -0.50 ?3"))

4 Win 0.50 1"))

1 Loss -0.50 )"&)

1 Win 0.50 1"&)

2 Loss -0.50 ?)"&)

2 Win 0.50 1"&)

g the values in the yellow cells to simulatetter what strategy you use"always be the same"

-oubles your bet/"verts bet size bac= to 1/" your bet/""

#inale!pectancy


13/14

1 Loss -0.50 1"))

1 Win 0.50 2"))

Loss -0.50 ?*"))

Win 0.50 1"))

1 Loss -0.50 )"&)

1 Win 0.50 1"&)

2 Loss -0.50 ?)"&)

2 Win 0.50 1"&)

1 Loss -0.50 1"))

1 Win 0.50 2"))

4 Loss -0.50 ?2"&)

4 Win 0.50 1"&)

1 Loss -0.50 1"))

1 Win 0.50 2"))

2 Loss -0.50 )"))

2 Win 0.50 2"))

1 Loss -0.50 1"&)

1 Win 0.50 2"&)

1 Loss -0.50 ?1&"))

1 Win 0.50 1"))

1 Loss -0.50 )"&)

1 Win 0.50 1"&)2 Loss -0.50 ?)"&)

2 Win 0.50 1"&)

1 Loss -0.50 1"))

1 Win 0.50 2"))

4 Loss -0.50 ?2"&)

4 Win 0.50 1"&)

1 Loss -0.50 1"))

1 Win 0.50 2"))

2 Loss -0.50 )"))

2 Win 0.50 2"))

1 Loss -0.50 1"&)

1 Win 0.50 2"&)

Loss -0.50 ?'"&)

Win 0.50 1"&)

1 Loss -0.50 1"))

1 Win 0.50 2"))

2 Loss -0.50 )"))

2 Win 0.50 2"))


14/14

1 Loss -0.50 1"&)

1 Win 0.50 2"&)

4 Loss -0.50 ?2"))

4 Win 0.50 2"))

1 Loss -0.50 1"&)

1 Win 0.50 2"&)

2 Loss -0.50 )"&)

2 Win 0.50 2"&)

1 Loss -0.50 2"))

1 Win 0.50 3"))

0.00

Documents

Martingale Comparison