Interesting reading regarding betting policies. Could this improve our gambling?

[mr.ore]

There are some math articles I have found regarding roulette. It is quite "dark side" reading, but all stated there is true, if you manage to understand it. I believe that some gambling system might be based on this info. Not a winning one, but mathematically optimal one. The fact that any gambling system will lose does not mean that there is no optimal way to play. If we are going to lose anyway, let's do so at least optimally.

[mr.ore]

How to gamble if you must. Valid only for EC games where no multiple bets are possible at same time, not absolutely "optimal" for roulette, but close.

The result from reading this is this simple system:

1. your starting bankroll is SB, your target win is T, T>SB, your actual bankroll is B, Odds = 35 for number
2. B = SB
2. while B >= 0 and B < T do
  2.1 bet X units on any number, X = round_up ( min_from( (T-B)/Odds, B ) )
  2.2 On loss our B is reduced by X
3. end while

It is played on number to minimize amount you stake to win target amount, since amount of lost units due to house edge is equale placed_units*house_edge and so it is best to bet on number. In step 2.1 we bet so much that one win will reach our target, therefore the round up. If our bankroll is lower than this, we bet it all.

This is mathematically proven optimal way of betting on one number and yes - it's more crazy than martingale  . The game shall be resolved fast...

[mr.ore]

I have read several times the math article I posted first in order to understand at least something what is there, and found a few interesting ideas, which I am now almost sure that they can lead to a "roulette basic strategy". Many games have an optimal basic strategy how to play them, like blackjack. I have been looking for such a basic strategy for roulette, but I have found none, in some places they write that the strategy does not exist. It is weird! There MUST be mathematically optimal strategy, which gives the gambler best chance to reach his goal. All those basic strategies in other games minimizes house edge, which is often bad, if you do not play that basic strategy. In roulette(european), there is no way how to play with any other house edge then 1/37. But there are another things - many betting options with different odds, and that is what is to be exploited, according to the article.

In the article I posted, there is stated, that the optimal betting  policy is a bold policy. What is the betting policy? It is a set of rules, which tells on which odds and how many units we must bet according to our actual bankroll and our target bankroll. Actual bankroll is greater than zero and lower than target, and for any it's value there must be one rule. There exists optimal way how to reach the target bankroll! So, in order to create some basic strategy, we will have to choose our target reasonably with respect to our initial bankroll. For each such situation (game of roulette + table spread + (initial bankroll) + target) there exists at least ONE optimal strategy how to reach the target, and bunch of less optimal strategies, and also at least one worst strategy.

Optimal strategy is the bold strategy. What is it? Let our actual bankroll be I, and target N, then we bet on an option with odds N-I : 1. This strategy is proved to be mathematically optimal(we can't play any better) for unfavorable games like roulette, but the proof assumes, that there always exists a bet with needed odds. It is not the real case in the game of roulette, the only options we can choose from are EC, dozens/columns, lane, street, split and number with odds 1:1,2:1,5:1,8:1,11:1,17:1,35:1 respectively. All those bets are very simple optimal strategies. It is also possible to combine those to get another mixed options, like betting on five numbers, which have non-integer odds.


Optimal betting policy to reach target N is obtained this way:

I) we make a table with numbered rows  from minimal bet and ending with N-1. The line will contain rule how to bet according to that bankroll.

ii) For each line I, we make a list L_i of ALL possible bets/options we can do with bankroll of size I with respect to table and spread, also combined ones like "x units on odds 35:1", "x units on odds 0.5:1 (two dozens)"... Let's call this operation "generation G(I)". Note that the options do not contain rules like "bet on red", but only rules "bet on odds x:1". Policy does only tells us on what odds we have to bet, then it is to our bet selection what will we bet on.

iii) For each line I, we remove from the L all options, which, if they won, would lead to our bankroll being greater than the target N. Let's call this operation "removal R(L_i)".

iv) For each line I, we choose one optimal rule, which will tell us, on what odds bet. I do not know yet how to choose it, in the article there is an unequation which I do not understand how it works, and which decides, whether the rule is optimal. For simple cases like target N=1,2,3,4,5 this can be done without difficult math on paper by drawing all possibilities. Let's call this operation "selection S(L_i)".

I will later try to create a program which will try all all possibilities and find them by "brute force" for some very large N. The problem is exponential, so it will take time...

5. It is done, we have the optimal strategy.

So - some simple optimal bold strategies: We supposse minimal bet 1 unit on anything and no table limits.

target N=2 units:

We may have bankroll I, 0<I<2. For I=1: generation G generates L={all posibilities we can do with one unit}={1:1,2:1,5:1,8:1,11:1,17:1,35:1}, then we remove from it anything what would win more than N=2 with operation R, and L={1:1}. Operation S which selects the best options have only one option to choose from, so the result is L={1:1}. So the table and only posibility is to bet on chance with 1:1.

Bankroll          option                              probability to reach N=2
 units
   1      bet 1 unit on 1:1 odds             0,48648648648648648649

This is the optimal strategy to win 1 unit with bankroll of 1 unit. Simple. Why so much work with something so simple? What is important is the process of thinking, and not the result.


target N=3:

We have bankroll I, 0<I<3.

Table draft :
I=1: a) generation G(1) = L_1 = {bet 1 unit on 1:1,2:1,5:1,8:1,11:1,17:1,35:1},
      b) after reduction R(L_1), L_1 = {bet 1 unit on 1:1, 2:1}, because we can bet on odds 1:1 and 2:1
          without winning more than N=3 units.
      c) after selection S(L_1), L_1 = {bet 1 unit on 2:1}, because according to bold policy, we choose
          odds N-I:1 = 3-1:1 = 2:1
      d) so rule for bankroll I=1 is : "bet 1 unit on option with odds 2:1"

I=2: a) G(2) = L_2 = {<all bets possible with two units>}
       b) after reduction R(L_2), L_2 = {bet 1 unit on 1:1; bet 2 units on 0.5:1}
       c) selection S(L_1):
           Bold policy says, that we should choose odds N-I:1 = 3-2:1 = 1:1. But what about the rule "bet 2 units on 0.5:1"?
           If we now bet 1 unit on EC 1:1, with p=18/37 we will directly reach the target N. If we lose, our bankroll will be
           1, and in the next spin, we will apply the rule "bet 1 unit on option with odds 2:1". So to put it together, if we
           choose now to bet 1 unit on 1:1 and in case we lost we bet 1 remaining unit on 2:1, our probability of reaching the
           target N=3 is p = 18/37 + 19/37*12/37 = 0,65303140978816654492. If we decided to "bet 2 units on 0.5:1",
           probability to reach N would be 24/37 = 0,64864864864864864865, which is worse than the other rule!
           Generally in selection S, we must test, what would happened if some rule lost and what is the  
           probability that in the end we still reach N and choose rule with highest probability of this happening.
          That's why this method will find optimal strategy to reach the target N and win. No other way of play can do better.
       d) rule for I=2: "bet 1 units on option with odds 1:1"

So the optimal betting policy to win 3 units is:

Bankroll                             option                                         probability to reach N=3
 units
     1            "bet 1 unit on option with odds 2:1"             12/37 = 0,32432432432432432432
     2            "bet 1 unit on option with odds 1:1"             18/37 + 19/37*12/37 = 0,65303140978816654492


So simply said - if we have 1 unit and we want to have 3 units, we bet on dozen or column, if we have 2 units and we want to have 3 units, we bet first on even chance, and if we lost, we will then bet remaining 1 unit on dozen or column. This is nothing spectacular. But - how many of us did otherwise with last unit or two? He who betted his last unit on even chance, and after a win he left it there and continued to play, committed a great blasphemy against Gods of math!

Let's examine other policies we can have for target N=3 units:

Bankroll                         option                                          probability to reach N=3
 units
     1       "bet 1 unit on option with odds 1:1"                     18/37*24/37 = 0,31555880204528853178
     2       "bet 2 units on 0.5:1"                                          24/37 = 0,64864864864864864865

As we can see, this policy leads to lower probability to reach N=3 with both possible sizes of bankroll. No one should ever consider playing like this. I did such a "move" few times before I knew this, and now I can be only sorry for that

Bankroll                         option                                          probability to reach N=3
 units
     1       "bet 1 unit on option with odds 1:1"                          X
     2       "bet 1 unit on option with odds 1:1"                          Y

Now there is problem how to compute X and Y.

X = 18/37*18/37 + 18/37*19/37*18/37*18/37 + 18/37*19/37*18/37*19/37*18/37*18/37 + ...
  = 18/37*18/37 + 18/37* 18/37*(18/37*19/37 + 18/37*19/37*18/37*19/37 + ...)
  = 18/37*18/37 * (1 + 18/37*19/37 + 18/37*19/37*18/37*19/37 + ...)
  = 18/37*18/37 * (1 + (18/37*19/37)^1 + (18/37*19/37)^2 + ...)
  = (18/37)^2 * U
   = (18/37)*Y

Y = 18/37 + 19/37*18/37*18/37 + 19/37*18/37*19/37*18/37*18/37+ ...
  = 18/37 + 18/37*18/37*(19/37 + 19/37*18/37*19/37 + 19/37*18/37*19/37*18/37*19/37 + ...) =
  = 18/37 + 18/37*(18/37*19/37 + 18/37*19/37*18/37*19/37 + 18/37*19/37*18/37*19/37*18/37*19/37 + ...)
  = 18/37 * (1 + 18/37*19/37 + 18/37*19/37*18/37*19/37 + ...)
  = 18/37 * (1 + (18/37*19/37)^1 + (18/37*19/37)^2 + ...)
  = (18/37) * U
   = X / (18/37)
   = (37/18)*X

U = (1 + (18/37*19/37)^1 + (18/37*19/37)^2 + ...)

If you look at this policy, we can have endless cycle of winning and losing that one unit before we lost bankroll or get to 3 units. Since we might be betting more units to reach that target, it is obvious it is less optimal, because of house edge. But to compute those probabilities is difficult. This simple case could be computed with formula for sum of a series, but I'm not going into it right now. What is important to see is the fact, that in betting policies, there can be cycles which might be much more difficult to compute. But to compute the right probabilities is important in order to find the optimal way. The only solution is to program a computer to compute it. Also while writing this post I realized another important think - Y can be simply computed from X. When I will be programming my planner for this, this might be worth to look into how to generally formulate probabilities from other probabilities to make the computation faster. I'm writing this for me anyway, but it may be usefull for someone with same ideas searching for what others tried. It is interesting, how many new ideas can be found, when someone tries to formulate his existing ideas with words.

So we wanted a policy for just N=3 and there are so many. Why to bother with this? The results are nothing new, it's obvious and it is not needed to do so much defining to found this, it can do anyone with calculator. But the advantage of doing things in a difficult and general way is that we can decide, whether it is worth to continue this way. It's obvious, that there can exists better and worse roulette system, therefore it is worth it for a gambler to spend his time on roulette system development, at least to avoid bad moves, and increase his chances a little.

What is more important - I think this way should really lead to mathematically optimal system for roulette, some kind of basic strategy. Of course it will lose in the end, maybe quite often, but it will be the best way to play, the best response the player can have. I want people on this forum to tell me their opinion on this, I may be mistaken and there might be somethink simple I overlooked.

Another think - this policy does not tell us, how to do bet selection. Mathematically it does not matter where you bet in the end, but my experience is, that good bet selection can be useful. I do not know, how to create good bet selection, idea is, that I choose events to bet against - for example if I would be betting on a street, I might decide to bet against repetition, and bet any other street, and while there would be no repetition, I would have little advantage because of more hits. I would like someone to recommend me a good bet selection when betting on any simple chance - ie. for even chances, dozens&columns, lanes, corners, streets, splits and numbers. If the method I proposed above will be useful, a good selection method would be needed.

I made an experiment with unoptimal betting policy, which flat bets all simple chances in roulette, and there were long runs when it was winning, on which I believe it could be possible to capitalize. Not optimal strategy, but also good idea. The above method could be also used to find optimal flat betting policies, in that article I posted first there is one optimized (not optimal) flat betting policy to win 10 units at one of the last papers and one one optimized (not optimal) simple bold strategy. "Simple" means that hedging is not allowed. Article states, that combining chances that can happen simultaneously can lead to better probabilities which will increase probability to reach target, and also that there are methods to make simple strategies better.

As I understand, it leads to this:

Flat betting:

Bankroll          option                          
 units
   1      bet 1 unit on 8:1 odds
   2      bet 1 unit on 8:1 odds
   3      bet 1 unit on 5:1 odds
   4      bet 1 unit on 2:1 odds
   5      bet 1 unit on 5:1 odds
   6      bet 1 unit on 2:1 odds
   7      bet 1 unit on 1:1 odds
   8      bet 1 unit on 1:1 odds
   9      bet 1 unit on 1:1 odds


Betting with progression:

Bankroll          option                          
 units
   1      bet 1 unit on 8:1 odds
   2      bet 1 unit on 8:1 odds
   3      bet 1 unit on 5:1 odds
   4      bet 3 units on 2:1 odds
   5      bet 1 unit on 5:1 odds
   6      bet 2 units on 2:1 odds
   7      bet 3 units on 1:1 odds
   8      bet 2 units on 1:1 odds
   9      bet 1 unit on 1:1 odds

I feel those betting policies are very near to parachutes, which can be found in Manrique's section of this forum. Parachutes also allow for progression both in odds and bet size. It's because of this similarity that the math article caught my attention - those parachutes and ideas how to use them I personally consider to be one of the best things on this forum. But for those betting policies, there might be methodology how to develop them. Then gambler would decide to use just one for his entire life - before he would ever considered to lay a bet on the table, he would decided himself, how much he will bet in his entire lifetime, this would be his bankroll, and then he would also decide, how much he wants to win in his life, this would be his target. Then he would let the computer to find an optimal policy - his custom basic strategy - to reach his lifetime target. If he lost his planned bankroll, he would never play roulette again in his life, since that would not be mathematically optimal - he should have planned better... If he wants to win X+Y, then two policies to win first X, and then later Y, when played one after another, might not be as optimal as policy to reach X+Y directly...

And now some simple non-optimal parachute-like flat betting policy to win 35 units, for which I want people on this forum to propose a good bet selection, since when the optimal one will be found, the bet selection would not be much different:

Bankroll          option
 units

 0    LOST
 1    bet 1 unit on odds 17:1
 2    bet 1 unit on odds 17:1
 3    bet 1 unit on odds 17:1
 4    bet 1 unit on odds 17:1
 5    bet 1 unit on odds 17:1
 6    bet 1 unit on odds 17:1
 7    bet 1 unit on odds 17:1
 8    bet 1 unit on odds 17:1
 9    bet 1 unit on odds 17:1
10    bet 1 unit on odds 17:1
11    bet 1 unit on odds 17:1
12    bet 1 unit on odds 17:1
13    bet 1 unit on odds 17:1
14    bet 1 unit on odds 17:1
15    bet 1 unit on odds 17:1
16    bet 1 unit on odds 17:1
17    bet 1 unit on odds 17:1
18    bet 1 unit on odds 17:1
19    bet 1 unit on odds 11:1
20    bet 1 unit on odds 11:1
21    bet 1 unit on odds 11:1
22    bet 1 unit on odds 11:1
23    bet 1 unit on odds 11:1
24    bet 1 unit on odds 11:1
25    bet 1 unit on odds 8:1
26    bet 1 unit on odds 8:1
27    bet 1 unit on odds 8:1
28    bet 1 unit on odds 5:1
29    bet 1 unit on odds 5:1
30    bet 1 unit on odds 5:1
31    bet 1 unit on odds 2:1
32    bet 1 unit on odds 2:1
33    bet 1 unit on odds 2:1
34    bet 1 unit on odds 1:1
35    WON

In the pictures in attachments this was played with starting bankroll of 34 units, and after lost there were a reset. It is far from optimal way which I do not know yet how to find, but it is flat betting and quite consistent with long runs (could it be possible to capitalize on them?) which are normally asociated with progressions.

Last word - if someone uses flat betting system where he increases or decreases on how many numbers he bets, so it is very light progression and main part of progression is in risk, then he should know that it is good and mathematically quite correct, because it is what should be done according to math.

[mr.ore]

Maybe I have an idea, how to compute probability that betting policy will reach target bankroll N for each possible value of bankroll I, 0<I<N.

Let Pwin(I) be probability that we will reach target N with bankroll of size I, that's what we want to compute.

Let PB(I) be probability that our bet with odds O(I) and units U(I) we bet if our bankroll size is I will win.

Let's explore some facts about this situation:

FACT 1: Bet can either win, or lose.


If our bankroll is 0, we cannot reach target N, so probability of doing so is 0. This leads to FACT 2.

FACT 2: Pwin(0) = 0


If our bankroll is 1, we cannot lose, so probability of reaching target N is 1. This leads to the FACT 3.

FACT 3: Pwin(N) = 1


Let's have bankroll I and bet according to our betting policy:

If the bet wins this spin, in the next spin our bankroll would be I+O(I)*U(I), and probability we will reach target N would be PB(I)*Pwin(I+O(I)*U(I)), so we multiply probability that our single bet will win with probability, that we will reach target with bankroll we will have after that win.

But we can also lose our bet, then our bankroll in the next spin would be I-U(I). The probability we would still reach the target after loss is then (1-PB(I))*Pwin(I-U(I)).

From FACT 1 we know that bet can either win or lose, so the probability, that we will reach target N with bankroll I is sum of probabilities that we will reach it with bankroll after a win and that we will reach it with bankroll after a loss. So it leads to the FACT 4.

FACT 4: Pwin(I) = PB(I)*Pwin(I+O(I)*U(I)) + (1-PB(I))*Pwin(I-U(I))

If our bankroll I=1, then Pwin(1) = PB(1)*Pwin(1+O(1)*U(1)) + (1-PB(1))*Pwin(1-U(1)). Because we have only one unit, U(1)=1, so Pwin(1) = PB(1)*Pwin(1+O(1)*1) + (1-PB(1))*Pwin(1-1). Pwin(1-1) = Pwin(0) = 0, as we know from FACT 2. This leads to the FACT 5.

FACT 5: Pwin(1) = PB(1)*Pwin(1+O(1))

FACT 6: Optimal bold policy to win N units if our bankroll is I is to bet on odds N-I:1. That means, that each simple bet in roulette is optimal strategy. It means, that we know some rules of the optimal policy apriori and so we also know probabilities PB for them.

FACT 7: Derived from FACT3, FACT 4 and FACT 6.

   O(N- 1) = 1, U(N-1) = 1, PB(N-1) = 18/37
   Pwin(N-1) = (18/37)*Pwin(N-1+1*1) + (1-(18/37))*Pwin(N-1-1)
   Pwin(N-1) = (18/37)*1 + (1-(18/37))*Pwin(N-2)
   
   O(N- 2) = 2, U(N-2) = 1, PB(N-2) = 12/37
   Pwin(N-2) = (12/37)*Pwin(N-2+2*1) + (1-(12/37))*Pwin(N-2-1)
   Pwin(N-2) = (12/37)*1 + (1-(12/37))*Pwin(N-3)
   
   O(N- 5) = 5, U(N-5) = 1, PB(N-5) =  6/37
   Pwin(N-5) = (6/37)*Pwin(N-5+5*1) + (1-(6/37))*Pwin(N-5-1)
   Pwin(N-5) = (6/37)*1 + (1-(6/37))*Pwin(N-6)
   
   O(N-8) = 8, U(N-8) = 1, PB(N-8) =  4/37
   Pwin(N-8) = (4/37)*Pwin(N-8+8*1) + (1-(4/37))*Pwin(N-8-1)
   Pwin(N-8) = (4/37)*1 + (1-(4/37))*Pwin(N-9)
   
   O(N-11) = 11, U(N-11) = 1, PB(N-11) =  3/37
   Pwin(N-11) = (3/37)*Pwin(N-11+11*1) + (1-(3/37))*Pwin(N-11-1)
   Pwin(N-11) = (3/37)*1 + (1-(3/37))*Pwin(N-12)
   
   O(N-17) = 17, U(N-17) = 1, PB(N-17) =  2/37
   Pwin(N-17) = (2/37)*Pwin(N-17+17*1) + (1-(2/37))*Pwin(N-17-1)
   Pwin(N-17) = (2/37)*1 + (1-(2/37))*Pwin(N-18)
   
   O(N-35) = 35, U(N-35) = 1, PB(N-35) =  1/37
   Pwin(N-35) = (1/37)*Pwin(N-35+35*1) + (1-(1/37))*Pwin(N-35-1)
   Pwin(N-35) = (1/37)*1 + (1-(1/37))*Pwin(N-36)
   
   
If our bankroll size is 1, U(1) = 1, and we can make only simple flat bets, thats all our options.
From this we can partialy compute probability of reaching target N from 1 unit. So our odds can by only basic ones and not some derived ones.

FACT 8: For Pwin(1), the rule will look like one of the following ones:
   
   O(1) = 1, U(1) = 1, PB(1) = 18/37
   Pwin(1) = (18/37)*Pwin(1+1*1) + (1-(18/37))*Pwin(1-1)
   Pwin(1) = (18/37)*Pwin(2)
   
   O(1) = 2, U(1) = 1, PB(1) = 12/37
   Pwin(1) = (12/37)*Pwin(3)
   
   O(1) = 5, U(1) = 1, PB(1) =  6/37
   Pwin(1) = (6/37)*Pwin(7)
   
   O(1) = 8, U(1) = 1, PB(1) =  4/37
   Pwin(1) = (4/37)*Pwin(9)
   
   O(1) = 11, U(1) = 1, PB(1) =  3/37
   Pwin(1) = (3/37)*Pwin(11)
   
   O(1) = 17, U(1) = 1, PB(1) =  2/37
   Pwin(1) = (4/37)*Pwin(18)
   
   O(1) = 35, U(1) = 1, PB(1) =  1/37
   Pwin(1) = (1/37)*Pwin(36)


Let's try to use the above facts to compute the what is the probability, that we will win 3 units flat betting.

Bankroll                         option                             probability to reach N=3
 units
     0                                                                     0
     1       "bet 1 unit on option with odds 1:1"                          Pwin(1)
     2       "bet 1 unit on option with odds 1:1"                          Pwin(2)
     3                                                                     1

Pwin(1) = (18/37)*Pwin(2)
Pwin(2) = (18/37) + (19/37)*Pwin(1)

Pwin(1)*(37/18) = Pwin(2)
Pwin(1)*(37/18) = (18/37) + (19/37) *  Pwin(1)
Pwin(1)*(37/18) - (19/37)*Pwin(1) = (18/37)
Pwin(1)*((37/18)-(19/37)) = (18/37)
Pwin(1)*(37*37-19*18)/(18*37) = (18*37)
Pwin(1) = (18/37)*(18*37)/(37*37-19*18) = (37*18*18)/(37*1027) = (18*18)/(1027)
--------------------------------
Pwin(1) = 0,31548198636806231743

Pwin(2) = (18/37) + (19/37)*0,31548198636806231743
--------------------------------
Pwin(2) = 0,64849074975657254138

This is the worst policy to win 3 units.

The problem is, that when I was writing this that I was almost sure, that it would lead to some method of dynamic programming, that when I know some probabilities, I can compute the others from the known ones. But presence of cycles does not allow this. These rules CAN be useful for something...

I really thought this will lead to something, I will look into it tomorrow again. Maybe, that there is only finite number of types of cycles in the policy, each of them can be computed manually, and this used to make an algorithm to compute the probabilities.

I see another problem with those policies - even if we will be able to compute probabilities, as the size of target bankroll N will be bigger, the possible number of bets we could do in state I would grow exponentially. So it might not be possible to compute policies for big wins. I do think, that this might be even NP-full problem. Now I can see, why basic strategy for roulette does not exists, they had not enough CPU power back in these days. Even now, it will be hard to find some big ones.

But to find optimal flat betting strategy is possible, we bet just such option, whose odds will bring us as near to target as possible. If we are under 35 units loss, we might try to combine more bets to win more in case of win. For example if we bet one unit on number and one on split sharing one number with it, we win 52 units, and from time to time the number on split without unit also wins, so it has lower variance. This is not optimal, but might be useful. It is just to design one table for target bankroll...

[GARNabby]

Thank you, i will try to read this soon.

[mr.ore]

So I started to program it blindly, and realized, that the FACT 4 can be used in stupid recursive function, which is fast enough for small targets, it takes only half a second to compute it with reasonable accuracy for table of size 8, but time of this solution grows exponentially. Also have programmed stupid Monte Carlo method, but it is much slower and accuracy sucks.

Good thing is that now I can compute probability that some flat betting policy will reach the target.

Example of optimal flat betting system to win 8 units. I am almost sure there is no more optimal one.
 
Bankroll    Option                         probability to reach N=8
 units

     0        LOST
     1        bet 1 units on 5:1        0.119065
     2        bet 1 units on 5:1        0.239789
     3        bet 1 units on 5:1        0.363060
     4        bet 1 units on 2:1        0.483437
     5        bet 1 units on 2:1        0.606709
     6        bet 1 units on 2:1        0.734248
     7        bet 1 units on 1:1        0.863526
     8        WON

It is played like this, example:

The player wants to win 4 units, and is willing to risk 4 units, but he wants to flat bet. Then his bankroll is 4 units, and target is 8 units, and probability to reach that 8 units 0.483437. He looks to the table above, and because his bankroll is 4, he bets on option with odds 2:1 - dozen or column. He loses, and his banroll is 3 units, and according to the table he must bet on option with odds 5:1 - that is lane, so he bet there and again loses. He again loses, his bankroll is now 2 units, and again he got to bet lane, which wins this time. Now his bankroll is seven, and table says, that he should bet on option with odds 1:1, so he bets even chance. He bet red and loses, his bankroll is now 6 units, and strategy dictates to bet on dozen or column. He do so and wins, and his target is reached.

Let's compare above strategy with betting 4 units on even chance.

18/37-0.483437=0.48648648648648648649-0.483437 = 0.00304948648648648649

So to flat bet is about by 0,3% worse than betting all on even chance. So flat betting is not optimal strategy for roulette. But there is also an advantage - the game is prolonged, and table limits does not matter so much. Of course there is limit, when you have for example table for flat betting to win 256 units, then probability to win that with 128 units bankroll is some 41%, and the larger the table, the lower the chance to win.

If you flat bet this way, you trade some probability to win and low variance for long play and no table limits. Yet this flat betting stays at least optimal in the set of flat betting strategies to reach the target of given size.

[mr.ore]

What about optimized simple bold policy from the article?

Bankroll    Option                         probability to reach N=10
 units  

     0        LOST
     1        bet 1 units on 8:1        0.096347
     2        bet 1 units on 8:1        0.194039
     3        bet 1 units on 5:1        0.290382
     4        bet 3 units on 2:1        0.389424
     5        bet 1 units on 5:1        0.488436
     6        bet 2 units on 2:1        0.587449
     7        bet 3 units on 1:1        0.686461
     8        bet 2 units on 1:1        0.788149
     9        bet 1 units on 1:1        0.891212
   10        WON

What is the probability, that player with 5  units will double them? P = 0.488436. What is better, this strategy or to bet 5 units on even chance?

0.488436-18/37 = 0.488436-0.48648648648648648649 = 0,00194951351351351351

It seems that this strategy has 0.19% advantage to even chance, when betting 5 units. If the computation was correct enough, and I believe in it because the difference is only in the third number behind decimal point, then this is a little gem hidden in math article. Advantage is only very small, but it is there! It is definitely possible to improve playing roulette with math!

[NoBody]

Hi Mr.Ore,

Thanks for sharing with us.

Have you check out the 36 unit parachute by victor?

nolinks://vlsroulette.com/money-management/36-unit-parachute/

It is similar with the "non-optimal parachute-like flat betting policy to win 35 units".

All the best to you.

Regards,
NoBody ^.^

[mr.ore]

That's nice one parachute Actually, any parachute is special case of policy, where one hit gets all, so the player does not need to know his bankroll status, only count spins. The main difference is the fact, that policy targets certain win amount, while parachute also minimizes bet size but the target is being changed with each spin. It's hard to say what is better, maybe computer simulation over many spins would tell us. We are going to continue playing anyway, so it is not that bad that the parachute lowers a little probability of winning by increasing target amount. I will try to create parachute to win 1 unit or losing all, which can be also seen as a policy without cycles, and try to run simulation. Wow - maybe that parachutes are really better when there are no cycles possible.

[mr.ore]

Well, VLSroulette writes there in that 36 unit parachute:

If you apply the 20% retun rule, then win target should be: 36 units x 20% = 7,2 units... well, make it 8 units as win target.

I think that if the parachute is being used for target betting, it is not as optimal as policy, which is designed to reach the target (semi)optimally. Parachute can give some extras from time to time, but it is not best way of play if you want maximize the probability that today you will win your 8 units and leave. If the player uses that parachute to win 8 units and is willing to risk 36 units and want to flatbet, he could also use policy to reach 36+8=44 units and start at line 36.

Alternative to that parachute would be this policy:

Bankroll       option              Pwin 44u
  0     LOST
  1     bet 1 units on 35:1    0.018949
  2     bet 1 units on 35:1    0.038281
  3     bet 1 units on 35:1    0.057155
  4     bet 1 units on 35:1    0.076667
  5     bet 1 units on 35:1    0.097042
  6     bet 1 units on 35:1    0.116532
  7     bet 1 units on 35:1    0.136910
  8     bet 1 units on 35:1    0.158927
  9     bet 1 units on 35:1    0.179328
10    bet 1 units on 17:1    0.195099
11    bet 1 units on 17:1    0.212060
12    bet 1 units on 17:1    0.226813
13    bet 1 units on 17:1    0.242247
14    bet 1 units on 17:1    0.260284
15    bet 1 units on 17:1    0.272672
16    bet 1 units on 17:1    0.288603
17    bet 1 units on 17:1    0.308162
18    bet 1 units on 17:1    0.320433
19    bet 1 units on 17:1    0.337095
20    bet 1 units on 17:1    0.358563
21    bet 1 units on 17:1    0.370941
22    bet 1 units on 17:1    0.389256
23    bet 1 units on 17:1    0.413110
24    bet 1 units on 17:1    0.428051
25    bet 1 units on 17:1    0.447125
26    bet 1 units on 17:1    0.474390
27    bet 1 units on 17:1    0.488004
28    bet 1 units on 11:1    0.508865
29    bet 1 units on 11:1    0.534948
30    bet 1 units on 11:1    0.553364
31    bet 1 units on 11:1    0.575939
32    bet 1 units on 11:1    0.606392
33    bet 1 units on 11:1    0.625433
34    bet 1 units on   8:1    0.650450
35    bet 1 units on   8:1    0.682999
36    bet 1 units on   8:1    0.706536
37    bet 1 units on   5:1    0.734260
38    bet 1 units on   5:1    0.769493
39    bet 1 units on   5:1    0.798502
40    bet 1 units on   2:1    0.830552
41    bet 1 units on   2:1    0.869788
42    bet 1 units on   2:1    0.907996
43    bet 1 units on   1:1    0.951528
44   WON

With this strategy the player have chance of 70,65% to win his 8 units flat betting. But this strategy is not optimal, because with so big bankroll there might be much better options to bet on, with betting more units on some or even betting multiple options, I don't know. What I'm trying to is to program application to generate strategies with algorithm which is briefly explained in the math article.

Another think - if someone decide to use those policies, do it at your own risk, there MIGHT be error in my code or even algorithm of computing that probabilities. For small cases it works nice, the results are same as those above which I computed with pencil and paper. They do not differ from results I got with another very slow method, so my personal confidence in it is big, but do your own testing before using that.

[NoBody]

Hi Mr.Ore,

Thanks for the example and explanation.

Now I see the differences of parachute and policy more clearly.

It is all maths, and hope we can optimise our luck with the numbers.

All the best!

Regards,
NoBody ^.^

[mr.ore]

Today, something bad and also great happened to me. The situation is, that while my grandmother was in a hospital because she is old and sick, my uncle stole all family money and lost it in slots. My father, who is his brother, had no guts to give him a lesson. I was really drunk and when I heard the story, I decided to first in my life to beat someone. The f*****g b***h stole the money and he even didn't  acknowledged his fault, accusing others! He have already stolen money from his mother anyway, so he deserved that he got from me!!!

Today is the first time, that me, a nice math person, hit someone!!! He pissed me off!!!

Now I am drunk, but even so, I would like to WARN all people out there.

NEVER STOLE MONEY FROM YOUR FAMILY!!! IF YOU DO SO, THEN YOU ARE f*****g b***h AND YOU DESERVE TO DIE!!!

If you want to gamble, then make a bankroll from your salary and prepare yourself a strategy, then go ONCE A YEAR to gamble, absolutely optimally. NEVER EVER GAMBLE WITH OTHER'S MONEY!!!

While I have never gambled with someone else's money, I have decided, after seing where gambling can lead, that THIS YEAR I WILL NOT GAMBLE ANYTHING, absolute STOP!!!!

I will help you people out there with my math skills, but I BEG YOU!!! DO NOT STEAL FROM YOUR FAMILY, AND GIVE A PART OF YOUR WIN TO HELP SOME OF YOUR FRIENDS, IF THEY ARE IN TROUBLE!!! IF YOU HELP YOUR FRIENDS, YOU KNOW, THAT YOUR MONEY WERE SPEND OPTIMALLY!!!

Sorry for this post, but I am drunk, and I NEED to warn ALL OF YOU!!! BEWARE OF YOURSELVES!!!

[GARNabby]

Today, something bad and also great happened to me. The situation is, that while my grandmother was in a hospital because she is old and sick, my uncle stole all family money and lost it in slots. My father, who is his brother, had no guts to give him a lesson. I was really drunk and when I heard the story, I decided to first in my life to beat someone. The f*****g b***h stole the money and he even didn't  acknowledged his fault, accusing others! He have already stolen money from his mother anyway, so he deserved that he got from me!!!

Today is the first time, that me, a nice math person, hit someone!!! He pissed me off!!!

Now I am drunk, but even so, I would like to WARN all people out there.

NEVER STOLE MONEY FROM YOUR FAMILY!!! IF YOU DO SO, THEN YOU ARE f*****g b***h AND YOU DESERVE TO DIE!!!

If you want to gamble, then make a bankroll from your salary and prepare yourself a strategy, then go ONCE A YEAR to gamble, absolutely optimally. NEVER EVER GAMBLE WITH OTHER'S MANY!!!

While I have never gambled with someone else's money, I have decided, after seing where gambling can lead, that THIS YEAR I WILL NOT GAMBLE ANYTHING, absolute STOP!!!!

I will help you people out there with my math skills, but I BEG YOU!!! DO NOT STEAL FROM YOUR FAMILY, AND GIVE A PART OF YOUR WIN TO HELP SOME OF YOUR FRIENDS, IF THEY ARE IN TROUBLE!!! IF YOU HELP YOUR FRIENDS, YOU KNOW, THAT YOUR MONEY WERE SPEND OPTIMALLY!!!

Sorry for this post, but I am drunk, and I NEED to warn ALL OF YOU!!! BEWARE OF YOURSELVES!!!

Good luck to you, and your family.

This gambling stuff can really lead to a lot of problems, especially the slots... takes only a few minutes there.

[kav]

Hello Ore,

It's nice to have a math person among us. Good job.
I have studied physics long ago, but I hate math and have forgotten most of the "high match". But I'm still decent with probabilities and simple math.

You raise many interesting issues.
I always wondered why probabilities professors don't do more papers on gambling and roulette.
It's a fact that they approach the roulette issue from a totally different point of view than a gambler.
Anyway, I have two main points to make:

1) Math professors do not gasp that the most important issue in roulette is the deviation from normal distribution of results.  This is the most important thing. Everything else (house edge, bold betting etc.) is almost irrelevant. Any theory or method that does not focus in this fact is misleading. As far as roulette goes math academics are stupid and narrow-minded. With all their theories and complex equations they never reached even near the most mathematically sound betting system there is. Which is the Martingale. In real life Martingale can be a disaster, due to extreme capital requirements and table betting limits. But theoretically it is a mathematical system that beats roulette. The way professors approach roulette, they would have never come close to even think about such a system. Shame on them.

2) The specific approach you have studied in this thread, is mostly useless, due to the reasons already mentioned. This analysis gives too much importance to Target win. But there is no target win for a gambler, in the sense it is mentioned in the article. There would be a "Target win" only f a gambler knew how much he wants to win till the last day of his gambling life and there would be a "Bankroll" only if he knew and had in hand all the money he would ever gamble. No gambler does. As I have showed in a previous post our gambling life is one long line and gambling sessions are connected. The Target win and Bankroll, as mentioned in this article, are useless artificial concepts. To give you an example: If you have let's say 10 units you can define a Target win of 11 units total. After you reach that you define a new Target of 12 units. After you reach this you go to a new target of 13 units etc. till you reach a Target of 50 units. On the other hand you could say from the start that your Target is 50 units. According to the article you mention there is a huge difference between the two cases. But it is an artificial difference. We all know that after a gambler makes his 10 units into 11, he is not going to stop gambling forever - a new win target will be made. This fact alone is sufficient to render the article irrelevant to true roulette gambling.

Nice to meet you. I enjoy your posts.

[mr.ore]

If you have read the article redblack.pdf, you would now, that Martingale is an optimal bold policy to win one unit in even chances. It is a special case of more general idea.

Either way, I don't think that all this is useless, because from the information I wrote there in this thread you can already improve your gambling.

If you are playing one of those two dozens systems like LWs & divisor on them, you can instead of that play one unit on even chance and if you lost, then one unit on dozen/column. Mathematically the chance of win si 65%, while chance to win on two dozens is only 64%. This way you are trading your time for better percentage and it also halves impact of table limits on your play.

You can also use some of the flat betting schemes to create flat betting "martingale", when you start with bankroll 1 unit before target. The characteristics of this resembles those progression system, but you can play longer and never have to increase the unit size.

There is more to the policies - I did not inspected yet how to exploit the fact that if you bet on 9 numbers and one of them win, your odds are (35-8)/9 = 3:1. With enough units, you can have odds which are not present in game by default. There is no SIMPLE option you can bet with those odds, but if you wish to do progression in risk instead of bet size, what you should anyway, this can be useful. My simulations shows, that even if I add some non-optimal risk-in progression level, the lenght of runs increases, and one could capitalize on this.

With math, a player can better plan his sessions. If player does know he will never stop, he should play with maximal unit size on one number, no other method of play might lead to such a success in long term (and big failure if unlucky). By target playing he sacrifies some of his longer term chances and probability for short term profits which comes faster, but with lower probability. He can still split his play to better planned sessions. For example to win that 5 units he can use the bold policy to win 10 units described above instead of playing 5 units on even chance directly. It would be better for him, and even if not optimal yet, still better than EC, and those small percentages MULTIPLY and within a few hundreds spins the difference might be huge.

[mr.ore]

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