No-limit Accumulation

[London Colin]

A very back-to-basics question coming up -

Suppose you have a goal of 2, 4, or 8 (etc.) times your starting BR, with no max bet. Am I right in assuming that it is optimal to repeatedly go all-in, until you either lose a hand or reach your goal? (Rather than betting half your BR, in order to be able to split when desirable.)

 

[gronbog]

I've done some work using my simulator on this and, for a game with a house edge, this does turn out to be the optimal strategy. You want to reduce your total exposure to the house edge by making as few bets as possible. The value of reaching your goal with fewer hands is much greater than the penalty incurred for not being able to double or split.

For games with a player edge, it turns out that betting less than all-in becomes optimal. From my brief experiments, there is a sweet spot that depends on the size of the player edge. As the player edge increases, the amount you bet decreases. The idea is that you want to increase the total amount you bet during your session (by making more bets, abeit smaller) to get to your goal, while at the same time managing the risk.

See https://www.blackjacktournaments.co...charity-port-perry-on-canada.8400/#post-48809 and S.Yama's reponse here https://www.blackjacktournaments.co...charity-port-perry-on-canada.8400/#post-48810. Both are from this thread: https://www.blackjacktournaments.co...blue-heron-charity-port-perry-on-canada.8400/

In all cases, if you find yourself in a position where you can use a progression to win the remaining amount needed, then you should do so.

 

[The_Professional]

Very interesting and thanks for sharing as usual. So, in this case counting might be helpful in accumulation tournament. Some of these tournaments have 2 rounds of 20 or so hands each. If you only need to double your bank role to advance, one strategy would be to bet minimum until you have an edge and then bet large on those few hand till you get to your pre-set goal. If BJ pays 2:1, then it is even more powerful approach as the players eye could be much higher.

 

[gronbog]

I have generally been in the "don't bother counting during tournaments" camp, but in the particular case you mention (i.e. you only need to double your bet), I have counted and made the all-in bet during a good count when possible in an attempt to squeeze out the extra fraction of a percentage point in the probability of success. The event discussed in the links I posted above is one of those situations. I also like to make the bet late in the round so as to limit the chance of other players copying the strategy when it is successful. Having said that, when it is not successful, it is very effective in convincing other players to never attempt it!

 

[London Colin]

I've done some work using my simulator on this and, for a game with a house edge, this does turn out to be the optimal strategy. You want to reduce your total exposure to the house edge by making as few bets as possible. The value of reaching your goal with fewer hands is much greater than the penalty incurred for not being able to double or split.

Thanks for confirming that. It was my working assumption, but I suddenly realised there might be some room for doubt.

Might it actually depend on the particular game, though? -

In a multi-deck game, the house edge goes from about 0.5% to about 2.5% when you switch from basic strategy to a no-split/no-double strategy. So the implication seems to be that the average total amount bet goes up by more than a factor of five, if you split your bankroll for each bet rather than going all in. (i.e. 0.5% of the increased action is greater than 2.5% of the lesser amount.)

But in a single-deck game, the jump in house edge might be more like from 0.1% to 2%, a twentyfold increase. Could that make bankroll splitting the way to go?

For games with a player edge, it turns out that betting less than all-in becomes optimal. From my brief experiments, there is a sweet spot that depends on the size of the player edge. As the player edge increases, the amount you bet decreases. The idea is that you want to increase the total amount you bet during your session (by making more bets, abeit smaller) to get to your goal, while at the same time managing the risk.

At first glance, that seems a little counter-intuitive. One would assume the thing to do would be to bet a fraction of your bankroll in proportion to your edge, effectively seeking to grow your bankroll just as you would if this were advantage play with your own money.

But I think I see what you are saying - the point is that, assuming you are not yet running out of hands, the goal is not to grow your bankroll as quickly as possible, but rather to grow it with as low a risk of ruin as possible. So long as you can keep on betting, you will get to your goal eventually.

In practice, all the tournaments I come across are very short (e.g., 10 hands). I'm wondering if that fact demands the all-in strategy, regardless of any other considerations.

 

[gronbog]

Might it actually depend on the particular game, though? -

In a multi-deck game, the house edge goes from about 0.5% to about 2.5% when you switch from basic strategy to a no-split/no-double strategy. So the implication seems to be that the average total amount bet goes up by more than a factor of five, if you split your bankroll for each bet rather than going all in. (i.e. 0.5% of the increased action is greater than 2.5% of the lesser amount.)

But in a single-deck game, the jump in house edge might be more like from 0.1% to 2%, a twentyfold increase. Could that make bankroll splitting the way to go?

I've re-read this a couple of times now, but I don't understand what you're trying to say here. How does the ratio of the house edges for the two strategies relate to the total amount bet during the tournament session?

At first glance, that seems a little counter-intuitive. One would assume the thing to do would be to bet a fraction of your bankroll in proportion to your edge, effectively seeking to grow your bankroll just as you would if this were advantage play with your own money.

But I think I see what you are saying - the point is that, assuming you are not yet running out of hands, the goal is not to grow your bankroll as quickly as possible, but rather to grow it with as low a risk of ruin as possible. So long as you can keep on betting, you will get to your goal eventually.

In practice, all the tournaments I come across are very short (e.g., 10 hands). I'm wondering if that fact demands the all-in strategy, regardless of any other considerations.

OK, I can see that, as usual, my explanation may have been misleading. But you have touched on a point that, had I included it, might have made things clearer. My mistakes were in focusing on the house/player edge and my use of the word "risk".

There are, in fact, at least 3 factors at play here. There is interplay among them but in overall order of importance, they are

1. The number of hands we have to work with.
   
2. The house/player edge of the game
   
3. The variance of the game

Let's look at how they interact. For the sake of the discussion, and in keeping with your original post, let's assume that the goal is double your starting bankroll or more.

Number of hands and House/Player Edge
The number of hands available is the most important factor in determining how much to bet, but the existence and size of a player/house edge also matters

On one extreme, if we had only one hand to work with, then the only reasonable strategy would be to go all-in on that hand. Our probability of success would the probability of winning the hand (~43%). Betting less and depending on a blackjack (~4%) or a winning double/split (~30%) would have a much lower probability of success. Note that this is true regardless of the house or player edge of the game or the variance of the game.

On the other extreme, if we had an infinite number of hands then we need to consider the house/player edge:

• For a game with a house edge, we are going to need positive variance in order to succeed. The way to maximize the variance of a session is to bet the highest amount possible on each hand using the fewest number of hands to either succeed or fail. The lower your bets, the closer your end result will tend to be to the negative expectation. The more hands you play, the closer your result will tend to be to the negative expectation. It may not be obvious, but by betting as much as possible on each hand, one tends to lower the total amount bet during the session, which is the point I was trying to make in my original post. This is because it minimizes the number hands played to realize success or failure. In the case of no-limit, the most common case is that the total amount bet will be the amount of your starting bankroll. Possibly 2 or 3 times that, if you push before winning or losing. As you bet less, it becomes more likely that your total amount bet will exceed these amounts since you are increasingly more likely to go though more cycles of wins/losses/pushes before finally succeeding or failing. The other extreme case here is betting minimum which would maximize the number of hands played and thus the total amount bet before the most likely outcome in which the house edge grinds you down to zero.
  
  
• For a game with a player edge, or a game, like blackjack, for which a player edge can be created, as you point out, we would bet to reduce our risk of ruin and grind it out until we either reached the goal or busted out. Our probability of success would be an extremely high (1 - ROR).

Real world situations exist in the middle of these two extremes. There is a fixed number of hands to work with. Unless your goal is less than the EV of the session at a given bet size, this factor alone makes maximizing variance the most important aspect of your strategy regardless of any house or player edge.

• For a house edge game, you still want to do this by betting as much as possible on each hand using as few hands as possible to either succeed or fail.
  
  
• For a player edge game, you do this by betting an amount large enough to make reaching the goal possible while at the same time small enough to maximize the total amount bet during the session. There is an optimal amount which balances these two goals and this is what I meant when I referred to balancing the total amount bet against risk in my original post. Sorry for any confusion with the traditional notion of risk of ruin. A smaller bet increases the risk of failure. Going all-in reduces the risk of failure, but it is not the optimal bet. This is because, in a player edge game, multiple chances to reach the goal have value in the form of increased variance.
  
  In the thread in which I analysed the spanish 21 event I was playing, the success rates of my various bots suggested that for that particular session of 25 hands, no limit betting and that particular player edge and the typical advancing threshold for that session, the optimal bet for players betting a fixed percentage of their bankrolls was 75%, although this was not the most successful strategy overall.

Variance of the Game

This factor should be considered in conjunction with the nature of the house/player edge. A game may have a player edge, but it may be due to some special payout which is relatively rare. For example, say suited blackjacks pay 10 to 1. The game technically has a player edge, but for a short session of 10, 20 or 25 hands, you should probably treat it as if the game has a house edge due to the low frequency of the event which generates the player edge. This game has higher variance than normal.

 

[The_Professional]

"This is because, in a player edge game, multiple chances to reach the goal have value in the form of increased variance".
Nice post indeed. Should the sentence above say "in the form of REDUCED variance" or I got that part wrong?

 

[London Colin]

I've re-read this a couple of times now, but I don't understand what you're trying to say here. How does the ratio of the house edges for the two strategies relate to the total amount bet during the tournament session?

What I meant was that the split-bankroll strategy presumably produces a higher total amount bet (on average), because it gives rise to a 'random walk' - a potentially lengthy sequence of winning and losing bets, with the bankroll ebbing and flowing until it reaches zero or the goal (or we run out of hands).

But the different strategy also produces a different house edge, and I was taking the phrase - "You want to reduce your total exposure to the house edge" - in essence to mean that you want to reduce the product of the house edge and the total action. So by that definition you could have reduced exposure to the house edge by virtue of more action combined with a lower house edge.

On reflection, and in the light all the helpful stuff you add below, I'm not sure how relevant this is. (Certainly, I can see that my above definition is flawed, or at least incomplete, since it makes no reference to variance.)

But, for what it's worth, to put my half-baked notion in its most general form, imagine comparing the goal-reaching prospects of two different games, with radically different house edges. The games needn't be BJ, and let's say there is a max bet, with the starting bankroll being several max bets.

You can imagine two straight-line graphs representing the expectations of each bankroll dwindling to zero over time, with one being a much steeper decline than the other. And superimposed on these can be random walks representing possible actual results, dipping under and rising above the straight-line paths.

So does the more shallow expected decline of one bankroll mean more opportunity (a higher probability) that the random walk may rise above the goal, compared to the steeper version from which recovery is more difficult?

And if the bet size on the shallow version is reduced, that will clearly hurt our chances in terms of variance (the random walk will find it harder to move any distance away from the straight-line path). But if we are offered a trade-off - accept a lower max bet in return for the lower house edge - is there a right price for such a trade?

I'm well aware that I may be talking utter nonsense here! As always, thanks for humouring me.

OK, I can see that, as usual, my explanation may have been misleading. But you have touched on a point that, had I included it, might have made things clearer. My mistakes were in focusing on the house/player edge and my use of the word "risk".

There are, in fact, at least 3 factors at play here. There is interplay among them but in overall order of importance, they are

1. The number of hands we have to work with.
   
2. The house/player edge of the game
   
3. The variance of the game

Let's look at how they interact. For the sake of the discussion, and in keeping with your original post, let's assume that the goal is double your starting bankroll or more.

Number of hands and House/Player Edge
The number of hands available is the most important factor in determining how much to bet, but the existence and size of a player/house edge also matters

On one extreme, if we had only one hand to work with, then the only reasonable strategy would be to go all-in on that hand. Our probability of success would the probability of winning the hand (~43%). Betting less and depending on a blackjack (~4%) or a winning double/split (~30%) would have a much lower probability of success. Note that this is true regardless of the house or player edge of the game or the variance of the game.

On the other extreme, if we had an infinite number of hands then we need to consider the house/player edge:

• For a game with a house edge, we are going to need positive variance in order to succeed. The way to maximize the variance of a session is to bet the highest amount possible on each hand using the fewest number of hands to either succeed or fail. The lower your bets, the closer your end result will tend to be to the negative expectation. The more hands you play, the closer your result will tend to be to the negative expectation. It may not be obvious, but by betting as much as possible on each hand, one tends to lower the total amount bet during the session, which is the point I was trying to make in my original post. This is because it minimizes the number hands played to realize success or failure. In the case of no-limit, the most common case is that the total amount bet will be the amount of your starting bankroll. Possibly 2 or 3 times that, if you push before winning or losing. As you bet less, it becomes more likely that your total amount bet will exceed these amounts since you are increasingly more likely to go though more cycles of wins/losses/pushes before finally succeeding or failing. The other extreme case here is betting minimum which would maximize the number of hands played and thus the total amount bet before the most likely outcome in which the house edge grinds you down to zero.
  
  
• For a game with a player edge, or a game, like blackjack, for which a player edge can be created, as you point out, we would bet to reduce our risk of ruin and grind it out until we either reached the goal or busted out. Our probability of success would be an extremely high (1 - ROR).

Real world situations exist in the middle of these two extremes. There is a fixed number of hands to work with. Unless your goal is less than the EV of the session at a given bet size, this factor alone makes maximizing variance the most important aspect of your strategy regardless of any house or player edge.

• For a house edge game, you still want to do this by betting as much as possible on each hand using as few hands as possible to either succeed or fail.
  
  
• For a player edge game, you do this by betting an amount large enough to make reaching the goal possible while at the same time small enough to maximize the total amount bet during the session. There is an optimal amount which balances these two goals and this is what I meant when I referred to balancing the total amount bet against risk in my original post. Sorry for any confusion with the traditional notion of risk of ruin. A smaller bet increases the risk of failure. Going all-in reduces the risk of failure, but it is not the optimal bet. This is because, in a player edge game, multiple chances to reach the goal have value in the form of increased variance.
  
  In the thread in which I analysed the spanish 21 event I was playing, the success rates of my various bots suggested that for that particular session of 25 hands, no limit betting and that particular player edge and the typical advancing threshold for that session, the optimal bet for players betting a fixed percentage of their bankrolls was 75%, although this was not the most successful strategy overall.

Variance of the Game

This factor should be considered in conjunction with the nature of the house/player edge. A game may have a player edge, but it may be due to some special payout which is relatively rare. For example, say suited blackjacks pay 10 to 1. The game technically has a player edge, but for a short session of 10, 20 or 25 hands, you should probably treat it as if the game has a house edge due to the low frequency of the event which generates the player edge. This game has higher variance than normal.

 

[gronbog]

"This is because, in a player edge game, multiple chances to reach the goal have value in the form of increased variance".
Nice post indeed. Should the sentence above say "in the form of REDUCED variance" or I got that part wrong?

Hmmm, in re-reading that, even I am wondering what the heck that was supposed to mean. Another mysterious statement on my part that could use some explanation.

First of all, I did mean "increased variance". Recall that this is the case of a fixed number of hands, so we're trying to increase variance in order to succeed. And what I was really trying to say is that in a player edge game, using a smaller bet and a larger number of hands to reach the goal has value because of increased positive variance due to an overall higher total amount bet, on average, than when going all-in.

 

[gronbog]

Colin, it's late here and I want to read what you have written a little more carefully before replying.

 

[gronbog]

What I meant was that the split-bankroll strategy presumably produces a higher total amount bet (on average), because it gives rise to a 'random walk' - a potentially lengthy sequence of winning and losing bets, with the bankroll ebbing and flowing until it reaches zero or the goal (or we run out of hands).

Agreed -- this is a better way to explain the concept than the way I tried to.

But the different strategy also produces a different house edge, and I was taking the phrase - "You want to reduce your total exposure to the house edge" - in essence to mean that you want to reduce the product of the house edge and the total action. So by that definition you could have reduced exposure to the house edge by virtue of more action combined with a lower house edge.

OK, I see what you mean now, and why I couldn't grok it. By "overall exposure" I was referring only to the total amount bet during the session.

On reflection, and in the light all the helpful stuff you add below, I'm not sure how relevant this is. (Certainly, I can see that my above definition is flawed, or at least incomplete, since it makes no reference to variance.)

But, for what it's worth, to put my half-baked notion in its most general form, imagine comparing the goal-reaching prospects of two different games, with radically different house edges. The games needn't be BJ, and let's say there is a max bet, with the starting bankroll being several max bets.

You can imagine two straight-line graphs representing the expectations of each bankroll dwindling to zero over time, with one being a much steeper decline than the other. And superimposed on these can be random walks representing possible actual results, dipping under and rising above the straight-line paths.

So does the more shallow expected decline of one bankroll mean more opportunity (a higher probability) that the random walk may rise above the goal, compared to the steeper version from which recovery is more difficult?

I would think so. This might also be a good way to visualize why you bet max to increase variance in a house edge game while you can afford to bet less than that in a player-edge game where the lines would rising toward the goal instead of dwindling toward zero.

And if the bet size on the shallow version is reduced, that will clearly hurt our chances in terms of variance (the random walk will find it harder to move any distance away from the straight-line path). But if we are offered a trade-off - accept a lower max bet in return for the lower house edge - is there a right price for such a trade?

I would think so. But I think that the trade off is only possible after the house edge decreases past zero and becomes a player edge. I believe that for all house edges, no matter how small, and even for a house edge of zero, the larger the max, the better.

I'm well aware that I may be talking utter nonsense here! As always, thanks for humouring me.

Not at all. I always enjoy exchanging ideas with you!

 

[S. Yama]

This is a very interesting subject with some great posts. Can I be humored a bit, too?

Let me make a few posts with some details where, as usual, the devil is hiding.
Depending on varieties of goals and rules no one-strategy is right, and the optimal play requires to change strategies, sometimes drastically opposite, in the midst of the game, based on the results of the latest played round.

In practice, the differences are relatively small, as long as our reasonable betting results in reaching the goal or busting out. I call it avoiding ending in “limbo”.

The most interesting aspect to me is the bet size, at different stages of the game, and present bankroll, and the goal score, depending on the game edge (and what constitutes the game edge is a whole another subject effecting the strategies in different ways).

This is all influenced by many factors, sometimes overlapping, canceling, or busting up the effect, and always taken together,
and as gronbog stated, it includes:
A) The number of hands we have to work with.
B) The house/player edge of the game.
C) The variance of the game.
I would also add
D) Ratio of the goal to starting bankrolls as a specific fraction.
E) Ratio of minimum to maximum bets allowed.

The least problem we have with the “A”.
On a positive side is that the more rounds we get to play the easier it is to reach the goal or bust out and not end up in the “limbo”.
The more rounds played the better chances that players betting small/medium to medium “incidentally” reach the goal, but in negative games their small/medium and medium bets lower the goal; and in player advantage games we have the chance to put more money into a positive EV. The other negative aspect of more rounds is that even if we reach the goal in some part of the game the following rounds may bring us down from the goal.

The game edge (“B), in some cases, has to be analyzed by specific rules and their frequencies that cause it. Usually, for a blackjack game it is referred to as winnings minus losing per original bet, but it also could be the net gain/loss per average bet. I find it most useful for other calculations to use the numbers for winnings minus losing per decision.

The variance of the game (“C”) causes most confusion as it is open to many interpretations, and is closely related to what causes the game edge by a particular game rules. For sure we can not use the typical meaning of it as bj variance (1.15), as we can create rules that can lower it (like surrender) or increase it (multiple original bet payouts/loses) and yet have opposite effect on desired strategies.

Some goal scores may require use of uneven fractions of our bankroll at some point in the game (“D”) that may cause overreaching the goal, thus having (almost) the same chances as somewhat higher goal, if we have to use a progression betting to reach it.

Similar effect (however small) may have a high ratio of minimum to maximum bet (“E”), where loosing a bunch of min. bets before we decide to make a big bet(s) may reduce the number of progression steps for an additionally necessary won round after we win the big bet(s). However, winning of a similar number of minimum bets may not necessary increase the number of progression step to win that same additional needed bet.
Now, please note, that the opposite of what I wrote above may be the case, as well, based on the “D”,

More on relationship between A-E later, but for tomorrow I will try to show how a “straightforward” game edge to desired goal and number of hands work together, using concrete numbers.

Till later,
S. Yama

 

[S. Yama]

To understand clearly what’s going on with betting in accumulation tournaments I like to visualize it. Let’s firstly simplify it a bit and describe the terms.
Let’s say we have a 10% player’s edge game, but for simplicity there are no 1.5 payouts for blackjack and doubles/splits. This 10% edge with the usual 8.5% pushes would come to 50.75% winnings and 40.75% loses. Note that if we get to play all pushed hands till they win or lose winnings are .5075/.915 and the edge per decision is (.5075-.4075)/(.5075+.4075) = .1093
The game edge per decision is 10.93% and the wins are 55.465%. If we need to double our bankroll going all in will succeed 55.465% of the times.

Let’s see what happens if we try to double up our bankroll using smaller bets, assuming that there is no time/rounds limits.

Betting half of our bankroll.
It helped me to imagine a point (or a particle) on a graph of five lines (like in a musical score sheet) starting on the middle line, representing 1 original bankroll. If we win our first bet we move up one line (to one and one half of the original bankroll), push – staying on it, and losing we drop down one line. When we move up two lines we reached our goal of doubling up but if we drop down two lines from the original middle line we get eliminated.
We need to find out the percentage of success for all resolved games. Keep in mind that once we go back to the middle of the line after the first initial hand the ratio of success for them is exactly the same as the ones resolved, and it can be omitted.
In the first decision we move up “w” times, representing percentage of winning, and move down “l”- representing percentage of losing.
After the second decision we will have w * w (winning two in a row), and reaching the goal; and l*l (losing two in a row) and busting out. The success ratio is w(sq.)/[(w(sq.)+l(sq.)].
The rest of the unresolved situations w*l plus l*w brings us back to the middle (the original brl) and we know that if we play until it is resolved it will be the same ratio w*w.
To make sure, let’s continue observing what happens to “wl” and “lw”
After the third decision we have ”wl-w” and “wl-l” plus “lw-w” and “lw-l”
After the fourth decision we reach to goal with “wlww” and “lwww”, and go to the middle with “wlwl”, “wllw”, “lwwl”, and “lwlw”. We can see that our success comes from “wlww”, which has the ratio of winning w3l/(w3l+wl3), which is the same as the first success of w2/(w2+l2), as do all other resolution at following rounds. We can conclude that for betting half the original bankroll this is the ratio of success for all cases.

If we bet always one third of the original bankroll we can observe similar propensities.
We reach the goal with w3/(w3+l3).

Here is the table of success for betting always-fixed parts of the original bankroll for a game with the specific edge (per decision):
Goal 2x-brl Bet all-in Bet half brl Bet third brl Bet quarter brl
(1% edge): 50.5% 51.0% 51.5% 52.0%
(2% edge): 51.% 52.0% 53.0% 54.0%
(5% edge): 52.5% 54.99% 57.45% 59.89%
(10% edge): 55.0% 59.90% 64.61% 69.05%

Seems like an easy walk, the smaller the bet the better the chances for doubling the bankroll. But that is true only if our goal is relatively low and we have unlimited time/rounds to play.
For example, in a standard, basic strategy game, betting half of the original bankroll with the goal of doubling up, after 9 hands, 4% of the results would linger in the limbo, not busted out and not reaching the goal – negating the small gains from betting smaller fractions of the bankroll. And if we use even smaller fraction of starting bankroll the unresolved situations rise fast.
Also, if our goal is higher the unresolved situation increase quickly (for example, betting half with the goal 4 times the original brl, the limbo numbers are 36% after nine hands and 13% after 19 hands). But, perhaps more importantly, with higher goals the benefits of smaller bets decrease and become undesired strategies.

S. Yama

PS
using w[sup]3[/sup] have not worked for me as superscript - what's the procedure? Thx

 

[S. Yama]

From the previous post we know that we can use formulas to calculate success based on goal, game edge (per decision), and chosen fraction of our original bankroll.

Let’s look how it works for a higher goal, e.g. four times the starting bankroll, and for illustration I picked 10% edge (55% winnings).

Going all-in.
The formula is w*w/(w*w+l); .55*.55/(.55*.55+.45)= 40.2%

If we always use half of the original bankroll, it is (w6+(w7l+w8l2+...+wnl(n-6)))/l2
A quick explanation, we need to win 6 “half bankroll” bets, so the original wager has to be won 6 times, hence “w” to the sixth power, and then for all other decisions it goes down to equal wins and loses or ticks up one loss and two wins, hence the sequence where we use one power up for both wins and loses. The original bet is only lost when we lose two in a row, others go “to the middle” and can be omitted. The above sequence for 50/50 chances adds 1/3 of the previous w-power and changes very little what powers it applies to until there is a big difference in the game edge.
Using the exact calculation betting half the bankroll until we bust or quadruple brl is triumphant only 15.4%.
If we always bet a third of the original bankroll the formula is (w9+(w10l+...+wnl(n-9)))/(l3+(l4w+...+lnw (n-3)) and we succeed 4.8%.
But wait...
If we significantly reduce our constant bet as a fraction of the original bet (keep in mind that we playing with no rounds/time limit), something interesting is happening.
For betting one fifth the success grows to 14.4%, by the time it is 1/7.33 it is as good as betting all-in 40.7%, and betting one tenth of the original brl we reach the goal 77.4% of the times!

This is what I think gronbog and Colin talked using phrases “random walk” and “shallow path”. When we make big bets in a positive game then we go quickly up and down but more up. With medium/ bets “the points on the graph” go up and down, but dispersion takes them down to the bust line and they don’t have a chance to go up later on as the game is finished. With the small bets there is an increased chance that the upward trend will take them to the goal before they hit the bust line.

Now, we need to look at how game edge influences the betting size and success ratio...

S. Yama

 

[hopinglarry]

This is wonderful work on your part.

I have not played in a lot of accumulation tournaments. Just in general, 2-1 BJs are the most common change I have seen in BJ tournaments. This does make a positive game. Most of the time it has been 20-25 hands total. If there is a max bet it has usually been what ever the starting BR was.

Most of the accumulation tournaments have not been just round of accumulation, X number and then table winners for later rounds. Instead they have been the X number advancing and the BR for the next round would be however many chips you advanced with. This has a tendency to cause large BR multipliers.

 

[gronbog]

Yes -- excellent posts by S. Yama, once again.

I have not played in a lot of accumulation tournaments. Just in general, 2-1 BJs are the most common change I have seen in BJ tournaments. This does make a positive game. Most of the time it has been 20-25 hands total.

This is a player-edge game, but given the 20-25 hands, I would probably treat it as if it was not, because the player edge is entirely dependent on being dealt 1 or more blackjacks during the short session.

 

[S. Yama]

Thank you hopinglarry and gronbog, it’s fun.

To finish my last post I would like to add another observation.
Though, with the player edge of 10%, betting all-in is better than smaller fraction until we bet about one eighth of the original bankroll, with significantly higher advantage, numbers change. We need about 40% edge to make betting half the bankroll a better choice than all-in, for a goal of 4 times the brl. (and 41% to make one-third brl betting better).
With the edge of 50% all-in succeeds 69%, ½ - 78%, and 1/3 – 83%.

Those numbers are only illustrative of underlying processes, and can not be use in real life as we don’t get to play games with such huge winnings and it would take many rounds (sometimes almost forever) to play them until we reach the goal of bust out.

I will be gone for a few weeks very soon, so one more sub-subject.
I think, that if a game is modified for a tournament and offers some multi-bet payoff bonuses they should be included in assessing the game edge. It may have not a full value only when our bet hits a premium payout and brings us above the goal. Note that, for example, if we would disregard bj paying 2 to 1, we should also disregard benefits coming from being able to split, win double downs, etc..
Let’s make a game where we can’t double down and all pushes lose (that’s about 10% disadvantage), but to make it up blackjacks pay automatically 10 to 1. Our goal is to almost double up our bankroll, sessions are short 15 rounds, and the minimum bet is very small.
If we use all-in strategy, which applies to negative games, our chances are (let’s say) 44%.

A better plan would be to bet one ninth on the first hand. If no bj then we bet just enough to reach the goal, hoping for a bj, and the minimum if we get it – goal reached.
We can play 14 hands like this if we are staying close to the original brl, then bet all-in on the fifteenth round.
If we fall somewhat down we need to go all-in in later rounds, but mostly 10-13th round (depending how much we are down), and, after winning that all-in, repeat the strategy, or use a progression, to reach the goal. Don’t know the exact success but it could be in the high fifties.

S. Yama