Advanced Sticky Bonus Questions

[Izverg04]

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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

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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.


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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

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

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]

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So what is RTTMax?

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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]

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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.

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Good explanation. I follow everything so far.

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

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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]

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

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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?

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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)