Introduction to Markov Chains

[manistra]

Hi. I found this community very interesting  and i am willing also to contribute to it.
I will try to share with you results of my study, on possibilities of Markov models and Markov chains application in various
types of games of luck. This study will be focused on roulette game mostly.
I am going to use some probability basics for sake of explaining ideas (hope you dont mind it)

Markov chains are mathematic model beeing studied in context of probability, binary sequence prediction, lossless data
compression algorithms, genetics etc.  For some basic info about Markov and its models pls goggle it a little.

There is common agreement on academic level that, while markov chains can be applied on systems with random variables, it can
not be directly applied with systems with independent events, such as pure games of luck.
That beeing said, someone can write off Markov model instantly before we even start to study it
with roulette. By definition, no deterministic system can exist without a measurable state space.
A series of coin flips or dice rolls have no measurable state space because they are independent events.  In other words,
random and independent events have no relevant previous or continuing context and therefore they constitute
stochastic processes. But, there exist other reasons why i applied it, (yes i am curious person, but that is not ONLY reason)
And the reason is: roulette is NOT pure random game. Before you start laughing hard, please define me RANDOM system ? How we
can define system as pure, non deterministic random system ?

I believe that roulette is PSEUDO random generator.
It has input variables (spin rpm, elevation, force of ball injection and 100 other super sensitive variables, but it is
deterministic system, although for normal observer, it remains purely stohastic). Therefore, this system is 100% reliable for
our research, as any other deterministic system. So I was guessing if there exist any possibility that we can use some
technique to reverse-engineer roulette spin process without any use of forbidden devices (visual tracking systems, ballistics
systems etc) This is probably impossible task that lies ahead of us
BUT i believe that there exist real possibility to use mathematical models (and software) legally to find "dealer bias" or
"dealer mark" (not wheel bias) on roulette. This is field where I believe Markov chains can be of big help, and where we can
get some profit from it.

In next post, i will show you an example how I applied Markov chain analysis on roulette. It is applied on
real Wiesbaden spins (not applicable to RNG generators)

[manistra]

In math, Markov chain is being called upon Andrey Markov, and it denotes array of system states. In each moment system state

can transform in the new state or it can be left in the same. Changes of state are called "Transitions".

A Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov property, namely that, given the present

state, the future and past states are independent. Formally,

       

The possible values of Xi form a countable set S called the state space of the chain.

Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of going from one

state to the other states.

Now lets assume we have observed roulette spin outcomes and we registered following state array:

R, R, B, B, B, R, B, B, R, R, R, B, B, B, R, B. (*)

In my tests, i divided real wheel layout on left and right side. When ball stopped into left side, it is R, otherwise it is B. when 0 hits, it is a loss.

In many problems, we want to predict future states on next observation or later observations. We can not determine exactly

next state, but according to observed states, we will try to define probability window that system will find itself in

particular state.

For this observation we ask following questions:

1. If observed system is now in state R, what is probability that in next observation state will be R ?
2. If observed system is now in state R, what is probability that in next observation state will be B ?
3. If observed system is now in state B, what is probability that in next observation state will be B ?
4. If observed system is now in state B, what is probability that in next observation state will be R ?

Lets try to answer this questions. From state array (*) we can see that system was in state R 7 times. In 3 cases, system

state left
in state R, and in 4 cases it transformed to B. Therefore we can conclude:


P (,,System was in state R and stayed in state R") =

P (,,System was in state R and transformed to  B")=

Also
P (,,System was in state B and stayed in state B") =


P (,,System was in state B and stayed in state B") =


Information about transformation probability from one state into another can be presented in matrix algebra
This matrix is called matrix of transformation states and usuallay is marked with capital P, and its elements are indexes
pi..j from array of all possible states (as we are observing binary system we have 2 possible states)
Matrix P can sometimes be presented in hierarhical way (trees) when we are using markov models for analysis
of random walk events


For each state Trensformation

probability is conditional probability that system will be in state i if system was in state k

If we apply this on our example than we can mark

           

Lets calculate probability that system

will be in state R or state B on next observation. Applying total probability formula we are getting next system linear

algebra:

           

If we mark vector p1 as out start state vector and woth p2 our next observation state vector, then system can be written in

next
short form:

p2 = p1P

If P is transformation matrix for Markov process, and p1 vector of starting state, then

p2 = p1P

is vector for next observation.

In our example it is:

               

To conclude, if system is in start state R with probability on first observation
and with probability in state B on first observation, than we can
claim that system will be in state R with probability on our next observation

I applied this analysis with c program on Wiespaden spins from year 2004 (10 continuous states tracked in row, 1000 sessions, 80 spins max per session, progression 1,2,4,8,16,32,64,128,  max loss 20, max gain 50, no bet if prob < 0,75 otherwise bet.)
with following results:


           


I can not say more about perspective of this thing, but results are there and I believe it deserves further attention.

Stay tuned for more results and new Markov trickery!

[hermes]

Too much math! Most of us are not geniuses in math but if somebody could make a software of Markov Chain where the math will play in background and only give us a sing "to bet or not to bet" (Shakespeare) that would be a winning situation for all of us.
Thanks
Hermes

[manistra]

I agree this seems to be complex, but IMHO, it is not so complex math.  I have software for Markov analysis but it is
not ready to go public as it in early development phase. Currently i am working on implementation of Markov on
other non binary decisions (dozens, streets etc). When i will have anything that works i will certainly publish it in Software section in this forum. Anyway, don't expect that you will ever have software that will always tell you winning decisions. It is not possible, with Markov or any other software.  8-)

[hermes]

I don't expect from software miracles but if it would be a 50% right it is excellent. Progression would take care of the other 50%.
I am looking forward for your software.
Thanks, Hermes

[admin]

Manistra, feel free to send it to vlsroulette @ gmail.com

I'll host it gladly  [smiley=thumbsup.gif]

[kawa4711]

Hi, Manistra,
I`ve found in a german speaking forum some interesting ideas from @Edgardo and @ winkel (he found a lot of interesting roulette phenomens like this:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Quote:
If in 3 coups the first and the 3rd Number have a difference of 0, 1 or 2
then these Numbers will be bet.
we bet the lower number -2 and the higher number +2 places

for example

if there had been
22
12
20
we found 22 and 20
we place bet on 18 19 20 21 22 23 24 (difference 2 set 7 No.)

if there had been
21
12
20
we would bet 18 19 20 21 22 23 (difference 1 set 6 No.)

if there had been
20
12
20
we would bet 18 19 20 21 22 (difference 0 set 5 No.)

bet only the first following coup

br
winkel
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


in this excellent thread of @ Edgardo, who has detected a markov-like phenomen:

nolinks://nolinks.roulette-board.de/index.php?showtopic=6186&hl

best regards

kawa4711

[JLP]

Hi Edgardo & Kawa,

Edgardo seguís usando el sistema posteado en el foro Alemán???
Qué resultados te dió??
==========================================

The system is simple and it seems silly but anyone

Having statistics can verify.

It consists of the following.

It begins to write a series and the first number doubles when on average between 8 and the ball 10. We play their neighbours 2 without playing No.

Which turned and write on one side -2 without charge is still playing
Scoring from -2 to double by shooting until the second No. apartir now staking 4 chips by shooting 2 neighbours first and the second 2 and is still scoring accumulated reaching -36 chips.

If you arrive (and if it did not levy comes fast about 9 or 10 balls). Reinstatement begin missed 9 balls and hoping to re-double the first No. and repeat the operation siguiente.Si charged also becomes start.

On average collected should double efforts to losses and
will win that an average of 18 tokens for every 36 that are lost when there comes the juego.Pero it curious that this mathematical relationship of
2-1 is not so in reality the game in our favor between 3 and 4 to 1.
Unbelievably when the ball turns a No. looking again that same No.
But as Figure 2 predominates on the widely 3 will start
Leaving neighbors of those No. which doubled its value and is very short
Term why not interested in the management lasts so few shots because the
Roulette is cortísimas gusts. (repeat not seek pointers).

The only problem we have with this system is that there is a hole
Which is the salidor and that will give us a lot of anger in this triple when the ball between us, but not to get nervous that in the long run
One result will have a monthly benefit favor. between 400 to 1000 chips. Playing sessions 108 balls per day, we will demand preferentemente.Eso
About 3 and a half.

Greetings to all readers and hope reviews and tests performed on actual statistics.

========================================

El Sistema es sencillo y parece tonto pero todo aquel
que tenga estadisticas podrá verificarlo.
Consiste en lo siguiente.
Es el antijuego del Sist.Fortaleza.
Se comienza a anotar una serie y cuando dobla el primer Número en promedio entre la bola 8 y la 10.Se juegan sus 2 vecinos sin jugar al Nº
que dobló y se anota en un costado -2 si no se cobra se sigue jugando
anotando de a -2 por tiro hasta que doble el segundo Nº apartir de ahora jugamos 4 fichas por tiro 2 vecinos del primero y 2 del segundo y se sigue anotando el acumulado hasta llegar a -36 fichas.
Si se llega (y esto si no se cobra viene rápido unos 9 o 10 bolas).Se vuelve a empezar dejando pasar 9 bolas y esperando que vuelva a doblar el primer Nº y volver a repetir la operación siguiente.Si se cobra también se vuelve a empezar.
En promedio las gestiones cobradas deberian duplicar a las perdidas ya
que se ganaria un promedio de 18 fichas por cada 36 que se pierden cuando no viene el juego.Pero lo curioso que esta relación matemática de
2-1 no es así en la realidad del juego y nos favorece entre 3 y 4 a 1.
Increiblemente la bola cuando dobla un Nº busca otra vez ese mismo Nº
pero como la figura 2 predomina sobre la 3 ampliamente van a empezar a
salir los vecinos de esos Nº que doblaron y lo que vale es el muy corto
plazo por eso no interesa que la gestión dure tan pocos tiros ya que la
ruleta es de rachas cortísimas.(REPITO NO BUSQUEN PUNTEROS).
El único problema que tenemos con este sistema es que queda un agujero
que es el salidor y que nos va a dar mucha bronca cuando triplique y este la bola entre los nuestros; pero a no ponerse nervioso que a la larga en el
resultado mensual tendremos un beneficio a favor.Entre 400 a 1000 fichas.
Jugar sesiones de 108 bolas por dia preferentemente.Eso nos demandará
aprox 3 y media.
Saludos a todos los lectores y espero comentarios y pruebas realizadas sobre estadísticas reales.


12
20
 15
 20 20 14*1
 28 -2
 6 -2
25   -2
 6 -2 6 34*27
12   -4
1    -4 19
 24 *****
23  
16  
 26
 31
 31
32  
25  
18  
25   25 2*17
 10 -2
 15 -2
 17 -2 29
1    ***
36  
21  
 17
 13
 29
27  
32  
1    
 15 15 32*19
 4 -2
27   -2 27 13*6
3    -4
21   -4 21 4*2
12   -6
0 -6
 20 -6
14   -6 -36
 8 ****
14  
 24
 6
7    
 33
27  
 20
5    
12   ****
9    
 20 20 14*1
14   -2 33
 11 ****
0
12  
9    
 24
25  
 2
 31
 10
7    
36  
 35
5    
 4
 29
 29 29 18*7
1    -2
 28 -2
 24 -2
21   -2
9    -2
 6 -2
 22 -2
5    -2 5 10*24
 6 -4 6 34*27
 35 -6
27   -6 3
 24 ***
 4
 6
 28
12  
14  
 29
23  
9    
 15
 35 35 12*3
 22 -2
12   -2 33
 20 ***
 24
 13
 29
 26
 31
 6
3    
 10
 22 22 9*18
18   -2 33
7    
1    
 20
34  
9    
3    
 20
 28
 26

Ganancia total 114
===========================================

Cheers,
JLP.-

[JLP]

Hi,

Here is a test of the neighbours system :

DUBLINET
DEALER TOM

4
8
12
19
11
23
5
17
17  ---->bet 34-25
5    ---->bet 24-10
31
10  W
--------
8   ---->bet 23-30
27
12 ---->bet 35-28
9
30  W
--------
23  ---->bet 10-8
4    ---->bet 21-19
19  W
--------
32
2
0
9
33
27  ---->bet 13-6
29  ---->bet 7-18
24
20
33
31
34
18  W
--------
4
2    ---->bet 25-21
34  ---->bet 6-17
13
9
33
23
12
25  W
--------
10
25  ---->bet 17-2
36
19
34  ---->bet 6-17
33  ---->bet 1/16
1   W

Cheers,
JLP.-

[MattyMattz]

Hey JLP,
this looks interesting but I can't figure out how we get a betting trigger.  I understand that we bet the neighbours once the trigger is met, but I'm lost when it comes to finding that trigger.

Thanks,
Matt

[JLP]

Hey JLP,
this looks interesting but I can't figure out how we get a betting trigger.  I understand that we bet the neighbours once the trigger is met, but I'm lost when it comes to finding that trigger.

Thanks,
Matt

Hello Matt,

The trigger here is when 1 number repeats.
When this happens we bet the 2 neighbours of that number that have repeated.
You see in the example that 2 numbers have repeated, so bet neighbours of that 2 numbers ----> bet 4 numbers.
When we win we stop and wait for another number or numbers to repeat and follow.

Cheers,
JLP.-

[MattyMattz]

Thanks JLP (I can't believe I didn't notice that...)

This is very intriguing... I will speak/study more on it later, as I'm heading out the door right now.

Thanks again,
Matt

[Edgardo]

Estimado y solidario colega JLP, disculpe que no le respondi antes, ya que recien veo la consulta suya.

En el Foro Aleman, como dices, empezamos a trabajar en las cadenas de Markov y su relación, con las dobles repeticiones de los números, pero lamentablemente, solo se hizo público el comiezo de la discución, luego lo siguieron trabajando en privado, y no me dejaron ingresar, por lo que no se en que estarán, o si ya ha llegado a alguna conclusión.

De todas maneras, el tema se veía muy prometedor.

Un saludo cordial, Edgardo.

[kawa4711]

Hi, to all
to answer some questions:

@winkels rouletteideas were these:

Quote:
If in 3 coups the first and the 3rd Number have a difference of 0, 1 or 2  
then these Numbers will be bet.  
we bet the lower number -2 and the higher number +2 places  

for example  

if there had been  
22  
12  
20  
we found 22 and 20  
we place bet on 18 19 20 21 22 23 24 (difference 2 set 7 No.)  

if there had been  
21  
12  
20  
we would bet 18 19 20 21 22 23 (difference 1 set 6 No.)  

if there had been  
20  
12  
20  
we would bet 18 19 20 21 22 (difference 0 set 5 No.)  

bet only the first following coup  

br  
winkel  

This means, that it doesn`t matter if the number is a neighbour of another number on the green (table) like, 17,18 or on the wheel like 0, 26

Its easier to follow the number on the table like 17, 18 or 20,22 etc.

If a number is thrown
for example  

if there had been  
22  
12  
20  
we found 22 and 20  
we place bet on 18 19 20 21 22 23 24 (difference 2 set 7 No.)  

the trigger is 20,22 because there is only one number missing between 20 and 22 (its the 21) and the numbers 20 and 22 are within 3 coups.


if there had been  
21  
12  
20  
we would bet 18 19 20 21 22 23 (difference 1 set 6 No.)  

here the trigger is 21 and 22 (there are no numbers between the 2 neighbours and the 2 neighbournumbers are thrown within 3 coups

if there had been  
20  
12  
20  
we would bet 18 19 20 21 22 (difference 0 set 5 No.)  

so the trigger is 20 (2 times the same number) within 3 coups.

Conclusio:

If in 3 coups the first and the 3rd Number have a difference of 0, 1 or 2  
then these Numbers will be bet.  
we bet the lower number -2 and the higher number +2 places

I hope I ve explained the idea correctly

Best regards

kawa4711

[TwoCatSam]

kawa

You have explained this method perfectly.  For once, I can find no question to ask.  This should be easy to run through a few numbers and see what shakes out.

Sam

[TwoCatSam]

-