Advanced Sticky Bonus Questions

[gaming_mouse]

I just read the FAQ post on sticky bonuses at bonuswhores.com ( link to article (http://www.bonuswhores.com/phpBB2/viewtopic.php?t=2711&postdays=0&postorder=asc&star t=0&sid=3cbd9d0370c77427bb6c1ec2ba701caa) ), 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 (http://archiveserver.twoplustwo.com/showflat.php?Cat=&Number=207100&page=&view=&sb=5&o =&fpart=all&vc=1), 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 (http://www.bonuswhores.com/phpBB2/viewtopic.php?t=2711&postdays=0&postorder=asc&star t=0&sid=3cbd9d0370c77427bb6c1ec2ba701caa) ), 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 (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=683150&page=0&view=ex panded&sb=5&o=14&fpart=2#Post682045683150) 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 (http://www.bonuswhores.com/phpBB2/viewtopic.php?t=7689&postdays=0&postorder=asc&star t=0&sid=0e6d1fa8d9cfd50d2824918da82ebfc5) 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 (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=683150&page=0&view=ex panded&sb=5&o=14&fpart=2#Post682045683150) 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 (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=683150&page=0&view=ex panded&sb=5&o=14&fpart=2#Post682045683150) 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 (http://www.bonuswhores.com/phpBB2/viewtopic.php?t=7689&postdays=0&postorder=asc&star t=0&sid=0e6d1fa8d9cfd50d2824918da82ebfc5) 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?

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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.

[Izverg04]

[ 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 ]
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.


[/ QUOTE ]

The wagering requirement and the +EV gained by busting early should be taken into account the same way whether the bonus is sticky or cashable.

When you bust, you don't have to complete the wagering requirements saving yourself WR'*HA. So you deal with this gain exactly the same way as you would with B'=WR'*HA sticky bonus. In the context of sticky bonuses, you recalculate the target as

T=(B+B') + sqrt(BR*(B+B'))/2.

In practice, sticky bonuses are usually offered by Playtech-powered casinos, which offer zero-HA games, so this is not a concern.

In the context of zero-HA games, pzhon's advice to go immediately for the target and then grind it out with tiny bets is not optimal, since all you are concerned with now is speed. It's much faster on average to reach the target with $20 bets than making a couple of $200 bets and grinding out $1 bets when you win.

[Izverg04]

Finally, there are a couple of things that are not quite right with my analysis of the sticky targets. I haven't posted these before, because in the end, my formula is just a recommendatation, and one that I am pretty happy with.

First, the threshold Var/EV, and requirement that BR=n*Var/EV, where 2<~n<~4, is derived using risk-of-ruin arguments. Here I am using the same pricing system for marginal variance, that is not really related to your risk of ruin. If you wanted, you could derive a higher target by dealing with the sticky-bonus deal as a single bet, and requiring that it doesn't exceed your threshold Var/EV. That is not what I am doing. I am comparing two wagers, one a sticky bonus with target X, and the other a sticky bonus with target X+dX, and look at which one is more attractive.

I believe I've shown before that pricing marginal variance is different from pricing overall variance in this recent post:

Advanced Risk Management (http://http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=2406261&page=&view=&sb=5& o=&vc=1)

If I used the ideas in that post, I would make a higher target recommendation.

Second, the BR=n*Var/EV is derived using a normal approximation. In calculating the target I am looking at a +EV bet where you risk a lot (Target-bonus) to gain a little (dTarget). We know that for the same EV and Var these bets are less attractive than coin flips for which normal approximation would hold. A converse situation (risking a little to win a big jackpot) is discussed here:

Risk of ruin when central limit theorem hasn't kicked in (http://archiveserver.twoplustwo.com/showflat.php?Cat=&Number=1406721&page=&view=&sb=5& o=&vc=1)

If I used this concept, I would make a lower target recommendation.

Oh well, I guess the two corrections cancel somewhat, and so I keep my recommendation.

[gaming_mouse]

Izverg,

Thanks for the responses. I'm still reading over them as well as over the articles you linked to, but I'll start with my first question:

So we need another parameter of merit: RTTmax -- your risk tolerance threshold for an infinite hourly wage.

If you know that your RTT(winrate=$200/hr)=$20,000 and RTTmax(winrate->inf)=$40,000 (for example) then you can quickly figure out that you should give up 50% of EV to eliminate variance. In general, you will get an earning equivalency graph that will look like this (blue line corresponds to the values in the example).

This confused me. If your bankroll was determined by some massive simulation of billions of hands played out over the course of a minute, say, then you will be either busted out or essentially infinitely wealthy. So what is RTTMax?

[Izverg04]

[ QUOTE ]
So what is RTTMax?

[/ QUOTE ]
Here is a very basic gambling question. You have two "wagers" that take you the same time to play through, and the first one provides more value with more risk: EV1>EV2; Var1>Var2. You determine that both of them are within your risk tolerance (or "bankroll"), that is

Var1/EV1<RTT; Var2/EV2<RTT.

You can choose only one. How do you choose?

Since EV and variance are additive, the question can be reduced to whether the bet that doesn't take any of your time and has EV'=EV2-EV1 and Var'=Var2-Var1 is attractive to you. I am saying that it is attractive when

Var'/EV'<RTTmax, where RTTmax<RTT.

RTTmax can be interpreted as the risk tolerance treshold for an infinite wage.

[gaming_mouse]

[ QUOTE ]


Var1/EV1<RTT; Var2/EV2<RTT.

You can choose only one. How do you choose?

Since EV and variance are additive, the question can be reduced to whether the bet that doesn't take any of your time and has EV'=EV2-EV1 and Var'=Var2-Var1 is attractive to you. I am saying that it is attractive when

Var'/EV'<RTTmax, where RTTmax<RTT.

[/ QUOTE ]

Good explanation. I follow everything so far.

[ QUOTE ]
RTTmax can be interpreted as the risk tolerance treshold for an infinite wage.

[/ QUOTE ]

This loses me. Why can it be interpreted this way? Just because it takes no time? You are still only winning EV'...
so calling it an infinite wage confuses me. Am I missing something?

[Izverg04]

[ QUOTE ]

[ QUOTE ]
RTTmax can be interpreted as the risk tolerance treshold for an infinite wage.

[/ QUOTE ]

This loses me. Why can it be interpreted this way? Just because it takes no time? You are still only winning EV'...
so calling it an infinite wage confuses me. Am I missing something?

[/ QUOTE ]
By wage I mean $/hr, so it's infinite. I am making this interpretation because I see only one key difference between bet with EV1 and EV' -- the first one takes time and the second one doesn't.

(There is also the difference that wager EV' is made only in conjunction with wager EV1, but I am assuming that after making wager EV1 your risk tolerance doesn't change)