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

[mr.ore]

"The Target win and Bankroll, as mentioned in this article, are useless artificial concepts."

I absolutely disagree with this opinion and also with the example.

1) If you bring in your playing roulette "target win" and your "bankroll size", you can much better plan your strategy. If you know, that you are willing to gamble say 10% of your month salary, and you will gamble for one year, then you know, how much you will bet, and so you know your "bankroll size". Now you might decide, that you want to win 10% of that bankroll, and prepare yourself a plan how to play, which you will follow all the year, each month you will play your optimal strategy to reach your year target optimally, one step after another in a long term plan. You also have your gambling more under control, because once you decide to play this optimal strategy(suppose you could found it) for that FIXED SIZE TARGET, you cannot change it, because it would not be optimal. You also know, that you will not bet more than you planned during the year. You know FOR SURE, that you will either reach the target, or lose all money you planned to gamble with. Gambling is not rational, but if you must gamble(according to prospect theory gambling might be "rational" for "poor" people, because their utility function is not linear, and they are willing to risk more to gain less, "poor" is relative of course), than supposing that you are (semi)rational player, then you would still play BEST RESPONSE to the unfavourable game.

2) When player wants to win 50 units, then there must exist optimal strategy how to do so, supposing that his initial bankroll size is fixed. So instead of winning it by one unit, he should try to get there directly. It might seem to be less intuitive, like the example of betting on EC and on loss on a dozen, instead of betting on two dozens, and the gameplay will be much longer, but it is better to play like this. Actually, with each placed unit you also pay for the "entertainment" the casino offers you, so the longer the play, the more is your gain. It is better to play longer to reach same target with higher probability than to play short session with lower probability of winning. You can trade your time for small difference in probability, what is favourable for you in more aspects.

[mr.ore]

Martingale and antimartingale exposed! They are two sides of the same coin. There is the real coin, optimal betting policy to win 32 units on even chances:

Bankroll       option              Pwin 32u
  0    LOST
  1    bet 1 units on 1:1    0.027249
  2    bet 2 units on 1:1    0.056012
  3    bet 3 units on 1:1    0.084775
  4    bet 4 units on 1:1    0.115136
  5    bet 5 units on 1:1    0.143899
  6    bet 6 units on 1:1    0.174260
  7    bet 7 units on 1:1    0.204621
  8    bet 8 units on 1:1    0.236669
  9    bet 9 units on 1:1    0.265432
10    bet 10 units on 1:1    0.295793
11    bet 11 units on 1:1    0.326154
12    bet 12 units on 1:1    0.358202
13    bet 13 units on 1:1    0.388563
14    bet 14 units on 1:1    0.420611
15    bet 15 units on 1:1    0.452658
16    bet 16 units on 1:1    0.486486
17    bet 15 units on 1:1    0.515250
18    bet 14 units on 1:1    0.545611
19    bet 13 units on 1:1    0.575972
20    bet 12 units on 1:1    0.608019
21    bet 11 units on 1:1    0.638380
22    bet 10 units on 1:1    0.670428
23    bet 9 units on 1:1    0.702476
24    bet 8 units on 1:1    0.736304
25    bet 7 units on 1:1    0.766665
26    bet 6 units on 1:1    0.798713
27    bet 5 units on 1:1    0.830760
28    bet 4 units on 1:1    0.864588
29    bet 3 units on 1:1    0.896636
30    bet 2 units on 1:1    0.930464
31    bet 1 units on 1:1    0.964293
32    WON

If you are playing capped Martingale and your initial bankroll is 31 units, then you bet 1 unit and if you lose than your next bankroll size is 30 units and you bet 2 units, if you lose again than your bankroll is 28 units and you bet 4 units and so on until you win or broke. If your bankroll is lower than half the target, this strategy leads to antimartingale, which when it crosses the border of half the target switches to martingale. This strategy is optimal for even chances games (supposing you are playing on option with lowest house edge, like don't pass in craps, which has lowest house edge from all craps options), and no strategy can do better.

[mr.ore]

Let's say that gambler's starting bankroll is 31 units, and he is playing even chance on european roulette without le'partage or imprison rule, and his aim is to win 8 units => 31+8 = 39 target. What is better? To martingale there with 8x one unit win, or play directly the optimal bold strategy for EVEN CHANCES(not optimal for roulette, only for that chance)?

A) Martingale 8x one unit win:

Martingale is actually a chain of optimal bold policies with target=start bankroll+1, and after each win we use new policy and new target. In the following computation I assume, that player bets either another step in martingale progression, or rest of his bankroll, so that he either win in one hit, or get as near the target as possible. As the number of units he has increase, so his probability to reach the target. So if the player has 32 units starting bankroll and he loses 1+2+4+8+16=31 units, he bet his last unit and try to anti-martingale it to 16 units and then bet those 16 units to reach 32 units and then he bets one unit to reach target 33 units.

The probability to win 8 unit with starting bankroll b of size 31 is

P(win32|b=31)*P(win33|b=32)*P(win34|b=33)*P(win35|b=34)*
P(win36|b=35)*P(win37|b=36)*P(win38|b=37)*P(win39|b=38) =

0.964293*0.965232*0.966161*0.967040*
0.967903*0.968719*0.969481*0.970221 = 0.76696278824249834551

B) Bold policy for even chance and target = 39, initial bankroll 31 units:

P(win39|b=31) = 0.780085

Difference: 0.780085 - 0.76696278824249834551 = 0,01312221175750165449

Conclusion: To use bold policy is better than martingale, chances of reaching the target are greater with that. For such a small target as 8 units the difference between two strategies is so big as 1,31%!!! And the longer the gambler plays, the greater the difference.

[kav]

Extremely interesting posts the last two. Thanks.
Please try to be as explanatory and detailed as possible.

[mr.ore]

Once again for those who did not get it:

Bold policy on even chance is played like this:

Your starting bankroll is SB, your target win is T, T>SB, your actual bankroll is B:
1. B = SB
2. while B > 0 and B < T do
  2.1 bet X units on any EC, X = min_from( (T-B), B )
  2.2 On loss our B is reduced by X, on win B is increased by X
3. end while

So simply said in words - you have some bankroll and some realistic target you want to win. You substract your actual bankroll from target, and bet it. If you win, you reached your target and it's done. If you lose, but you still have some money, then you again either bet the difference, or if it is not enough to win with one hit, then you bet all you have. And so you continue until you reach the target or lose it all. This is even more aggressive than Martingale, and this is mathematically proved optimal way of betting on even chances. Like it or not, that's the truth - nothing better exists for even chances. In roulette you have more odds to select from or even you can combine chances and get different odds like 3:1 when betting on 9 numbers and follow progression in risk bold policy, but for even chances, this is proved to be optimal strategy to maximize probability that you will win your target, since all you can change is your bet size.

It also means that holy grail cannot exist, because nothing is better than this. No system can have better probability of pushing player from SB to T. All what the other systems do is that they play more timid and are winning less units per spin, so it takes longer to win. They always lose with higher probability before reaching target than this optimal strategy, which on average bets as few units as possible to reach the target. Because house edge is relatively small in roulette, many players do not notice this. But if this were played say on stupid "miniroulette" with house edge 10% against "normal" system, it would be much more visible what is better.

Martingale is actually special case of this when T=SB+1. It's obvious why is this better than Martingale - with martingale you are willing to risk huge money to win just one unit, and you are even doing so when thinks are going bad and you are losing. If you are winning, than you are winning only one unit, but you could have won more when it was doing well and you had prepared your big bankroll to survive long series, but you were not enough bold to play like this.

In roulette to be bold means to bet on options with odds which will bring you with as small unit as possible as near to the target as possible(not exactly true but very near, there are these stupid cycles...), and when the options are all used, then you start progression on one number. Because there are not all odds to bring you to your target as near as possible in roulette, there is some space to optimization. The only optimal policy in roulette is for target 3 units and less.

Actually, this all means that best is to play progression on one number, because the options with lower payout are just shortcut to end your progression and increase probability to reach the target. Nice thing is, that this can be used to develop some not optimal but nice systems with balanced progression in risk and bet size and maybe this could help to overcome some table limits. That's what I also want from this, not absolutely perfect percentages, they are just a nice bonus.

But the think I'm doing right now is actually nothing more than hunting hundredths of percents, but it's fun anyway. Many are trying to develop holy grail, but only few are trying to improve their chances as much as possible. I do prefer to hunt for small fish who I know it exists. Instead of those HG hunters, I got at least some results.

[betatester]

Hello forum.
Hello @mr. ore.
Very, very interesting articles.
They make me think a lot.

May a put a question for you, that I can't calculate for myself...?

Let's gets suppose that we have a system (I don't) but anyway, that provides us a "prediction" for next bet.
This hypothetical system points to a location on the layout getting a "favourite" straight number (35/1), or a split (17/1), or a street (11/1), or a square (8/1), or a double street (5/1), or a dozen (or column) (2/1) or any of the 6 E. Chances, or all together, for every next bet.
This "locations" are only kind of predictions and we can imagine that sometimes those predictions will overlap one over each other.
So for next spin we may have a prediction for number 31, split (31/34), street (31/33), but for (13/18) double street, 3 doz, 1 col, red.

If we have a restricted capital of 36 units and we fixed a goal of +1 at LEAST, to finish a session, how do you think it may look (in terms of probability) a kind of "policy" that bets 2 units in each spin, 1 unit in the "favourite" number for that next spin (35/1) and 1 unit in another of the fractional chances, with that "plan".

Capital = 36 units.
Goal: Reach a Target = 36 + 1= 37 (+1 unit at least)
Plan:
                                              If win (n/1)   if win (35/1)   if win both   if lost both   k
1   1   bet (2/1)      1   bet (35/1)      1      34      37      -2      34
2   1   bet (5/1)      1   bet (35/1)      2      32      38      -4      32
3   1   bet (8/1)      1   bet (35/1)      3      30      39      -6      30
4   1   bet (11/1)      1   bet (35/1)      4      28      40      -8      28
5   1   bet (11/1)      1   bet (35/1)      2      26      38      -10      26
6   1   bet (17/1)      1   bet (35/1)      6      24      42      -12      24
7   1   bet (17/1)      1   bet (35/1)      4      22      40      -14      22
8   1   bet (17/1)      1   bet (35/1)      2      20      38      -16      20
9   1   bet (17/1)      1   bet (35/1)      0      18      36      -18      18
10   1   bet (17/1)      1   bet (35/1)      -2      16      34      -20      16
11   1   bet (17/1)      1   bet (35/1)      -4      14      32      -22      14
12   1   bet (17/1)      1   bet (35/1)      -6      12      30      -24      12
13   1   bet (17/1)      1   bet (35/1)      -8      10      28      -26      10
14   1   bet (17/1)      1   bet (35/1)      -10      8      26      -28      8
15   1   bet (17/1)      1   bet (35/1)      -12      6      24      -30      6
16   1   bet (17/1)      1   bet (35/1)      -14      4      22      -32      4
17   1   bet (17/1)      1   bet (35/1)      -16      2      20      -34      2
18   1   bet (17/1)      1   bet (35/1)      -18      0      18      -36      0
                                                                       -36 End

As you can see, now there is a "new" chance to calculate in case both bet wins.
And also I'll maintain the prediction on the single number, just because is the max. return for a single 1 unit bet/prediction.
We are in positive if hit any of the first 8 bets.
In the 9 we have to jump if we only touch the 17/1 bet.
And so if our first hit (in any of both) occurs after the 9 time bet.
We jump accordingly to our k.

Just take a look.
Does it makes better or worst that any other 2 units x 18 times bet policy (being one of those a fixed 1/37) for those 36 units capital?

If those reflexions call your attention I'll be very glad to know the answer.

Al the best.
Betatester.

[betatester]

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