Non poker bonuses at internet casinos (Homer?)

[JTG51]

I've seen Homer post about this subject in the past, but I'd love to hear input from anyone.

I'm curious about how much internet casino bonuses are worth. Actually, I'm curious how to figure it out. I've got three questions.

For simplicity, assume the casino is offering a $100 bonus on a $100 deposit, you can play a game that gives the house a 0.5% edge, and you have to play 20 times the deposit plus the bonus ($4000 in this case).

1) How much is this bonus worth? Is it simply $100 - $4000 * .005, or $80?

2) What is the chance of going broke before reaching $4000 of play? I imagine the size of the average bet changes the answer. Does 4000 $1 bets give a significantly lower ROR than 800 $5 bets?

3) Any other tips to maximize the EV of these bonuses (besides play the game with the smallest house edge)?

Thanks.

[Homer]

I've seen Homer post about this subject in the past, but I'd love to hear input from anyone.

But wouldn't you prefer Homer's input? /images/graemlins/grin.gif

For simplicity, assume the casino is offering a $100 bonus on a $100 deposit, you can play a game that gives the house a 0.5% edge, and you have to play 20 times the deposit plus the bonus ($4000 in this case).

That's pretty standard these days.

1) How much is this bonus worth? Is it simply $100 - $4000 * .005, or $80?

Yep. Well, actually, this assumes a zero risk of ruin (which is approximately right if you are conservative and bet small amounts).

2) What is the chance of going broke before reaching $4000 of play? I imagine the size of the average bet changes the answer. Does 4000 $1 bets give a significantly lower ROR than 800 $5 bets?

Let's see, I have to find the ROR equation here, hang on.

Okay, it's: ROR = exp(-2uB/sigma^2)

SD = 1.15 units (don't ask me why, but this is the SD of a hand of blackjack)
u = EV = .005 units
B = Bankroll = 4000/(bet size) units

So, for a $1 bet size, ROR = exp(-2*.005*4000/1.15^2) = 7.32E-14 (Uhh, really friggin small)

For a $5 bet size, ROR = .23%

For a $10 bet size, ROR = 4.86%

3) Any other tips to maximize the EV of these bonuses (besides play the game with the smallest house edge)?

I guess try to optimize the bet size. As the bet size goes up, your overall EV goes down, but the time played goes down as well, which means your EV/hr goes up. If time isn't a concern, minimize your bet size, but if it is, maximize your EV/hr.

I'm kind of out of it now, so maybe I'll think about this some more tomorrow.

-- Homer

[JTG51]

Great, thanks Homer.

I'm surprised how small the ROR is. You don't happen to know the SD of 9/6 Jacks or Better video poker, do you? I assume it's higher than BJ since a large part of the EV is based on very rare hands. Sound reasonable?

[GuyOnTilt]

My question is, why would a winning hold'em player like yourself want to spend the hours that clearing this bonus would take for a gain of $80, when your hourly EV is certainly going to be much higher playing poker?

[Homer]

Oh man, I screwed this up. You're supposed to enter bankroll as negative in the ROR equation. Also, your EV is negative and I put that in as positive. Lastly, what the formula I used gives is your long-term ROR. You want the short-term ROR since you only have to play a fixed number of hands. I need to dig up the thread in which I posted that equation.

I'll try to get to it tonight. If not, I'll do it tomorrow. PM me if I forget.

-- Homer

p.s. - Come play 3/6 triple draw lowball on UB!

[Homer]

GOT, a $100/$100 bonus with a $4000 playthrough can be cleared in about 3 hrs, using $10 bets. Yes, it is possible to play 133 hands/hr if you know basic strategy by heart. This comes out to $25 an hour or so (possibly a little less depending on how using the larger bet size effects the ROR).

Plus, it's nice to do something different once in a while.

-- Homer

[Homer]

You don't happen to know the SD of 9/6 Jacks or Better video poker, do you?

This site kicks ass:

http://www.gamblingtools.net/vp/

Variance of 9/6 JOB is 19.51 units/hand, so SD is 4.41 units/hand. As I mentioned, in blackjack it is only 1.15 units per hand.

I assume it's higher than BJ since a large part of the EV is based on very rare hands. Sound reasonable?

Nice logic. As you can see, you're right.

-- Homer

[JTG51]

Just a change of pace mostly. I don't like to play poker every day, but I do like to make money every day.

[JTG51]

Yes, it is possible to play 133 hands/hr if you know basic strategy by heart.

I'm actually a little surprised. I assumed I'd be able to play quite a bit faster than that at either BJ or video poker.

[JTG51]

PM me if I forget.

OK.

p.s. - Come play 3/6 triple draw lowball on UB!

Yeah, I'm sure that game needs another sucker!

All this lowball talk is making me think it's time for me to start playing at UB.

[Homer]

Here's a cut and paste from one of my old posts:

<font color="blue">BEGIN</font>

Stating it another way, if I have hourly rate x, hourly standard deviation y and hours played z, what's the chance I'll lose 146 BB's (as in my case) or any other number.

Here is the short-term risk of ruin formula from Blackjack Attack. I see no reason why this formula cannot be applied here:

r = N((B-ev)/sd) + e^((2*ev*B)/var) * N((B+ev)/sd)

N() = cumulative normal distribution function
e = base of natural logarithm system (~2.7183)
B = short-term bankroll, expressed as negative number
ev = expected value
sd = standard deviation
var = variance (square of sd)

Example: Your hourly rate is 1 BB/hr, your hourly standard deviation is 10 BB/hr, you intend to play for 40 hrs and have a 146 BB bankroll. What is the probability of going broke?

B = 146
ev = 1*40 = 40
sd = 10*sqrt(40) = 63.25
var = 4000

r = N((-146-40)/63.25) + e^((2*40*-146)/4000) * N((-146+40)/63.25)

r = N(-2.94) + e^(-2.92) * N(-1.68)

r = .415%

<font color="blue">END</font>

Now, you want to know what your ROR playing blackjack is for various bet sizes, given a $4000 playthrough requirement and $200 bankroll.

r = N((B-ev)/sd) + e^((2*ev*B)/var) * N((B+ev)/sd)

B = -200/BS
ev = -.005*4000/BS
sd = 1.15*sqrt(4000/BS)
var = sd^2

(BS = bet size)
<font class="small">Code:</font><hr /><pre>
BetSize Bankroll EV SD Var ROR EV(w/bonus) EV/hr
1 -200.0 -20.00 72.73 5290.00 1.2% $77.54 $1.94
2 -100.0 -10.00 51.43 2645.00 7.5% $65.08 $3.25
3 -66.7 -6.67 41.99 1763.33 14.3% $51.33 $3.85
4 -50.0 -5.00 36.37 1322.50 20.3% $39.37 $3.94
5 -40.0 -4.00 32.53 1058.00 25.3% $29.33 $3.67
6 -33.3 -3.33 29.69 881.67 29.6% $20.86 $3.13
7 -28.6 -2.86 27.49 755.71 33.2% $13.65 $2.39
8 -25.0 -2.50 25.71 661.25 36.3% $7.43 $1.49
9 -22.2 -2.22 24.24 587.78 39.0% $1.99 $0.45
10 -20.0 -2.00 23.00 529.00 41.4% -$2.80 -$0.70
11 -18.2 -1.82 21.93 480.91 43.5% -$7.06 -$1.94
12 -16.7 -1.67 21.00 440.83 45.4% -$10.89 -$3.27
13 -15.4 -1.54 20.17 406.92 47.2% -$14.35 -$4.66
14 -14.3 -1.43 19.44 377.86 48.7% -$17.50 -$6.12
15 -13.3 -1.33 18.78 352.67 50.2% -$20.38 -$7.64
16 -12.5 -1.25 18.18 330.63 51.5% -$23.02 -$9.21
17 -11.8 -1.18 17.64 311.18 52.7% -$25.46 -$10.82
18 -11.1 -1.11 17.14 293.89 53.9% -$27.73 -$12.48
19 -10.5 -1.05 16.69 278.42 54.9% -$29.83 -$14.17
20 -10.0 -1.00 16.26 264.50 55.9% -$31.80 -$15.90
</pre><hr />

Hope I did this right, who knows.

It seems that your EV/hr is maximized when you bet $4/hand (given this bankroll and playthrough requirements), while of course your overall EV is maximized when you bet $1/hand. The deal becomes -EV when you bet $10/hand or more.

Will rest on this and think about it some more tomorrow.

-- Homer

[Homer]

I'm actually a little surprised. I assumed I'd be able to play quite a bit faster than that at either BJ or video poker.

Could be. I haven't played in a while, so my estimate may very well be low. I know that at some places the software moves slowly, so you can't get that many hands in per hour.

BTW, professional VP players can play like 600-800 hands per hour (I believe that's the number I heard).

-- Homer

[Homer]

Here are some figures for playthrough requirement (in terms of number of times deposit+bonus) versus optimum bet size, in terms of maximizing EV/hr (for a $100/$100 bonus):
<font class="small">Code:</font><hr /><pre>PT BetSize
3 $37
5 $21
10 $9
15 $6
20 $4
30 $2</pre><hr />
-- Homer

[JTG51]

I'm not sure I understand the EV column on your chart. The values in that column ignore ROR, correct? It seems to me that the EV should be -20 for any bet size. Why does the size of the bet change the EV when you are betting a total of $4000 each time? I should expect to lose $20 on one $4000 bet, and on 4000 $1 bets, shouldn't I

[Homer]

The 2nd through 5th columns are in terms of units (when bet size is 1, bankroll is 200 units, EV is -20 units, SD is 72.73 units).

As I said I may be doing something wrong here.

-- Homer

[JTG51]

Ah, makes sense now. I guess I should have noticed the lack of dollar signs. Thanks.