Advanced Sticky Bonus Questions

[gaming_mouse]

I just read the FAQ post on sticky bonuses at bonuswhores.com ( link to article ), and I'm curious if anyone (Bruce? pzhon?) can show me the derivations of some of the forumulas....

Following the naming convention on the linked to article, let:

D=deposit
B=bonus
X=your target (you aim for $X or $0=bust)
BR=your whole gambling bankroll

First question is about your chance of success, which (for a fair game) is (B+D)/X. I can see that this formula is true when I plug in some different numbers (eg, if X = 2*(B+D), you just bet everything and clearly you have a .5 chance of winning). But what I cannot figure out how to prove is 1) the formula in the general case and 2) how to show the forumula does not depend on your bet sizes or betting strategy. I'd like to see how these two things are done.

Second question is about the following quote from the linked to article:

[ QUOTE ]

Since there is no house advantage, for any X, you can extract more EV out of the bonus. Let's say you've reached a value of X, and you set your goal to reach X+dX or bust trying. Simple calculation shows that for this bet:

EV=B/X * dX
SD=(X-B)/sqrt(X) * sqrt(dX)


[/ QUOTE ]
How were these two equations arrived at?

I believe the last question is a simple one, but I've never seen ROR (I think that's what coming into play here) described like this before. The author notes that:

[ QUOTE ]

Using arguments similar to the ones justifying 300BB bankroll requirement in poker, my personal criterion for bankroll is: (I am conservative when it comes to bankroll)

BR = 4*Var/EV

I think this is equivalent to requiring ~500BB bankroll for limit hold'em.


[/ QUOTE ]

First, does this mean that with a BR of (4*Var/EV), you will have the same chance of busting out while playing sticky bonuses indefinitely as would a winning player (based on some reasonable winrate like 1BB/100 or 2BB/100) at limit holdem? If not, what does it mean?

Second, where does this formula come from? Is this just a recasting of the ROR formula given, eg, by bruce in this thread, or it something else entirely.

Thanks for any info/proofs,
gm

[pzhon]

[ QUOTE ]
I just read the FAQ post on sticky bonuses at bonuswhores.com ( link to article ), and I'm curious if anyone (Bruce? pzhon?) can show me the derivations of some of the forumulas....

[/ QUOTE ]
I haven't read all of the links yet, but I can answer the questions.

[ QUOTE ]

D=deposit
B=bonus
X=your target (you aim for $X or $0=bust)
BR=your whole gambling bankroll

First question is about your chance of success, which (for a fair game) is (B+D)/X. I can see that this formula is true when I plug in some different numbers (eg, if X = 2*(B+D), you just bet everything and clearly you have a .5 chance of winning). But what I cannot figure out how to prove is 1) the formula in the general case and 2) how to show the forumula does not depend on your bet sizes or betting strategy. I'd like to see how these two things are done.

[/ QUOTE ]
Since there is no house advantage, your expected ending amount is the same as your starting amount, B+D. If you end up with either 0 or X, you have X with probability (B+D)/X.


[ QUOTE ]
[ QUOTE ]

Since there is no house advantage, for any X, you can extract more EV out of the bonus. Let's say you've reached a value of X, and you set your goal to reach X+dX or bust trying. Simple calculation shows that for this bet:

EV=B/X * dX
SD=(X-B)/sqrt(X) * sqrt(dX)


[/ QUOTE ]
How were these two equations arrived at?

[/ QUOTE ]
You can compute the EV as above, but you start with X and your new target is X+dX, so you win with probability X/(X+dX). The improvement from ending up at X+dX rather than X is dX while the loss from ending up at 0 is X-B. You can compute the SD from that.


[ QUOTE ]
[ QUOTE ]

Using arguments similar to the ones justifying 300BB bankroll requirement in poker, my personal criterion for bankroll is: (I am conservative when it comes to bankroll)

BR = 4*Var/EV

I think this is equivalent to requiring ~500BB bankroll for limit hold'em.


[/ QUOTE ]

First, does this mean that with a BR of (4*Var/EV), you will have the same chance of busting out while playing sticky bonuses indefinitely as would a winning player (based on some reasonable winrate like 1BB/100 or 2BB/100) at limit holdem?

[/ QUOTE ]
Yes. The SD of limit Hold'em is roughly 15 BB/100.

[ QUOTE ]

Second, where does this formula come from?

[/ QUOTE ]
In some sense, this comes from the Central Limit Theorem.

[BruceZ]

Here is a link to the derivation of the bankroll formula for games with a known mean and variance. The central limit theorem allows us to model the game with coin flips, and the coin flip formula comes from a random walk. 4*var/ev would give a ROR of 0.03%.

[gaming_mouse]

[ QUOTE ]
Since there is no house advantage, your expected ending amount is the same as your starting amount, B+D. If you end up with either 0 or X, you have X with probability (B+D)/X.

[/ QUOTE ]

very elegant... i love it. thanks for your answers.

now how do you incorporate the house edge into this argument? that is, some casinos don't offer the nearly fair games like BJ switch, but instead offer only games like "no peek" blackjack where the HA is not negligible (.55% in that case).

Would you first calculate the simple expected end amount of playing out the wagering requirement as

E_end_amount = (B+D) - WR*HA

And now set the EV of reaching your target X or bust as equal to that:

X*(P_reach_target) = B + D - WR*HA

Or perhaps, using the logic of another post you made about blackjack, only subtract of .9*WR*HA?

So that the chance of winning is:

(B + D - WR*HA)/X

This of course assumes that you can reach your target and WR requirement at approximately the same time, but from my understanding of this bonuswhores.com post you can do this with high probability by using the following optimal bet size:

bet=[T-C]^2/(4*WR)

where T is the target, C is your current balance, WR is how much you have left to play.

Which brings me to my last question: How is this forumula derived and just how effective is the strategy of using its recommended bet sizes? Eg, what is the distribution of your projected distance from the target (or from the WR) those times you hit the target using this strategy?

Thanks again,
gm

[gaming_mouse]

[ QUOTE ]
Here is a link to the derivation of the bankroll formula for games with a known mean and variance. The central limit theorem allows us to model the game with coin flips, and the coin flip formula comes from a random walk. 4*var/ev would give a ROR of 0.03%.

[/ QUOTE ]

Thanks for the link Bruce.

[gaming_mouse]

[ QUOTE ]
[ QUOTE ]
Here is a link to the derivation of the bankroll formula for games with a known mean and variance. The central limit theorem allows us to model the game with coin flips, and the coin flip formula comes from a random walk. 4*var/ev would give a ROR of 0.03%.

[/ QUOTE ]

Thanks for the link Bruce.

[/ QUOTE ]

I just read the proof. It is very, very pretty. Nice work.

[pzhon]

[ QUOTE ]

now how do you incorporate the house edge into this argument? ... Would you first calculate the simple expected end amount of playing out the wagering requirement as

E_end_amount = (B+D) - WR*HA

[/ QUOTE ]
No. An important point for analyzing bonuses is that busting out relieves you of the wagering requirement. You also have to worry about completing the wagering requirement while you are not close to busting out or reaching your target.

[ QUOTE ]
This of course assumes that you can reach your target and WR requirement at approximately the same time, but from my understanding of this bonuswhores.com post you can do this with high probability by using the following optimal bet size:

bet=[T-C]^2/(4*WR)

where T is the target, C is your current balance, WR is how much you have left to play.

[/ QUOTE ]
First, that formula is wrong. Look at what it suggests when the wagering requirement is 0! It doesn't take into account the house advantage or your balance (which is an upper bound on the amount you can bet).

Second, the idea behind it may be that after n fair bets, the standard deviation is sqrt(n)*bet size (assuming coin-tosses, not blackjack). If you use a constant bet size of (T-C)^2/(4WR), it is about a 2-standard deviation result to be above the target after you reach the wagering requirement, which happens after (4 WR^2)/(T-C)^2 bets.

I think a better strategy is to aim for T+WR'*HA, where WR' is the remaining wagering requirement after betting. Either bet everything, or bet just what you need to reach T+WR'*HA. Once you reach T+WR'*HA, grind out the wagering requirement with tiny bets.

I believe my strategy decreases the average amount wagered, hence increases the probability of success. The exact performance is messy, but should be easy to test by experiment.

[gaming_mouse]

[ QUOTE ]
[ QUOTE ]

now how do you incorporate the house edge into this argument? ... Would you first calculate the simple expected end amount of playing out the wagering requirement as

E_end_amount = (B+D) - WR*HA

[/ QUOTE ]
No. An important point for analyzing bonuses is that busting out relieves you of the wagering requirement. You also have to worry about completing the wagering requirement while you are not close to busting out or reaching your target.

[/ QUOTE ]
So how would you determine your chance of winning in these cases?

To take a concrete example, consider a 50% sticky bonus on $1000 which requires a 35*(D+B) WR (a total of $52,500). The best available game to complete this bonus is European Blackjack, with a house edge of .55% (so the house expects to get back $289). Given a target of, say, $2000, what is the chance that you win?

And how would the concern of completing the WR somewhere in between busting out and hitting your target come into play? What can you do about this concern?

[ QUOTE ]

I think a better strategy is to aim for T+WR'*HA, where WR' is the remaining wagering requirement after betting. Either bet everything, or bet just what you need to reach T+WR'*HA. Once you reach T+WR'*HA, grind out the wagering requirement with tiny bets.

I believe my strategy decreases the average amount wagered, hence increases the probability of success. The exact performance is messy, but should be easy to test by experiment.

[/ QUOTE ]

This makes sense.

[pzhon]

[ QUOTE ]

So how would you determine your chance of winning in these cases?


[/ QUOTE ]
This is really messy, particularly if there is a maximum bet size. If you are interested, perhaps you should run a simulation.

For the specific case of blackjack, there is a problem that you should not bet more than half of your balance, or else you will be unable to double down or split, and this greatly increases the house advantage. Also, you will overshoot frequently, and it is up to you how to treat overshooting.

[ QUOTE ]
And how would the concern of completing the WR somewhere in between busting out and hitting your target come into play? What can you do about this concern?

[/ QUOTE ]
I see no harm in betting large enough amounts that you are very likely to bust out or hit your target+WR'*HA quickly. You have to expose yourself to the risk of busting out, or else the sticky bonus is worthless. If you bust out, you should do so rapidly, so you decrease the average amount wagered.

[Izverg04]

Nice to see my post discussed here.

[ QUOTE ]

First, does this mean that with a BR of (4*Var/EV), you will have the same chance of busting out while playing sticky bonuses indefinitely as would a winning player (based on some reasonable winrate like 1BB/100 or 2BB/100) at limit holdem? If not, what does it mean?

[/ QUOTE ]
Not quite. If you follow my derivation, I derive the target by pricing marginal variance once you reach the target by using the same threshold Var/EV ratio as you would playing poker. However, on the way towards reaching your target, you make far better risk vs. reward bets.

The result is that, using my recommendation for target setting, you have a smaller risk-of-ruin doing sticky bonuses than you would playing poker.