Blackjack "Cover Your Losses" Promotion

I have an account with an online poker room that is connected to a casino and is having a promotion that will cover my blackjack losses this weekend. The stipulation is that the losses are returned to me in the form of a bonus that must be cleared by generating rake at their poker tables and expires after 30 days. I estimate that I can clear as much as $500 in bonus over the next 30 days, so assume that any money I lose up to $500 this weekend is returned to me. I can tolerate to lose as much as $2000 at blackjack this weekend.

What is my optimal strategy for maximizing the return from this promotion? How much should I bet and how much would I need to be up before I stop? I was thinking I could bet $500 the first hand (and double and split as necessary since I can tolerate risking $2000) and stop if I lose, but would I continue betting if I won?

The rules here are 4 decks, dealer stands S17, doubling after split allowed, no hitting split aces, can only split to two hands, and no surrendering for a return of ~99.6%. Thanks in advance to anyone willing and knowledgeable enough to help with this.

it does seem to me that you would stop once you had the $500 in losses, going over that being OK if you lost a double-down. Why lose more on purpose?

As for winning, you should just set a $ amount ahead of time to quit, that would be the same as usual. If you were looking to manage a losing session and win instead, well, hell, those are the kind of problems you like to have!

Quote: odiousgambit

it does seem to me that you would stop once you had the $500 in losses, going over that being OK if you lost a double-down. Why lose more on purpose?

As for winning, you should just set a $ amount ahead of time to quit, that would be the same as usual. If you were looking to manage a losing session and win instead, well, hell, those are the kind of problems you like to have!

I agree that it makes sense to stop betting if I reach $500 in losses and that I shouldn't start with a bet that would put me at more than $500 if I lost it, unless I'm doubling or splitting. I'm still not sure about when to stop if I'm winning, I don't normally play blackjack and I don't really set goals for winning. Ideally, I'd like to continue betting until it stops being +EV to do so.

For example, suppose I bet $500 and win. Lets assume for simplicity that blackjack is a coinflip. If I were to bet $1000, Half the time I'd lose and my $500 win gets wiped out, the other half of the time I win $1000, which should result in +$250 in expected value. The max bet at this site happens to be $3000, so supposing I were up $2500 it seems I could bet $3000 and that would result in +$250 expected value as well. This logic seems to suggest that the best strategy is to bet however much I'm up plus $500 until I lose. However, I'm just speculating and I don't know if that's actually the most +EV strategy.

Theoretically you will get the most EV by playing with a strategy that maximizes the probability to claim the cashback. So you could set a target so that 99% of time you will lose the $500 and claim the cashback and 1% of time you will win something big. But you also need to factor in the 0.4% house edge of the game as well as the time and effort to turn the cashback into withdrawable funds and possible losses while doing so, so I wouldn't probably go to more than a $5000 win target.

Also if the maximum cashback is 100% to $500 I wouldn't bet more than $250 at the start to avoid losses exceeding $500 in case you have to split/double down. If I lost the first $250 bet, I would then bet $125 and so on.

Quote: sangaman

Ideally, I'd like to continue betting until it stops being +EV to do so.

If you are currently W dollars ahead, you should obviously be betting W+500 to maximize your EV.
If you bet W+500, the EV of the bet is
0.498*(2W+500) = 0.996W + 249.
As long as this value is greater than W, you are in the positive territory. Therefore, your stop criteria is $249/0.004 = $62250

Quote: weaselman

If you are currently W dollars ahead, you should obviously be betting W+500 to maximize your EV.
If you bet W+500, the EV of the bet is
0.498*(2W+500) = 0.996W + 249.
As long as this value is greater than W, you are in the positive territory. Therefore, your stop criteria is $249/0.004 = $62250

Alright that makes sense. The max bet is $3000 so I'd have to stop there, but I could use bet $500, then $1000, then $2000 assuming I win each hand and stop if I lose any of them. I'd win $3500 roughly 1/8th of the time that way for $437.50 EV. I'd double if it's advantageous to do so, so occasionally I'd win more than $3500 or lose more than the $500 I can recover.

I suppose it'd be optimal to continue betting even after I'm up $3500 despite the fact that I can't bet more than $3000, right? I don't know if I'd actually keep going at that point because I wouldn't want variance to get out of hand, but would another a $3000 bet at that point be theoretically correct?

Quote: sangaman

I suppose it'd be optimal to continue betting even after I'm up $3500 despite the fact that I can't bet more than $3000, right?

Well, it depends on what you call "optimal". It is not a +EV bet. If you are up $3500, and bet $3000, you will not be collecting the bonus if you lose, and therefore are betting at a regular -0.4% disadvantage.

Quote: sangaman

Alright that makes sense. The max bet is $3000 so I'd have to stop there, but I could use bet $500, then $1000, then $2000 assuming I win each hand and stop if I lose any of them. I'd win $3500 roughly 1/8th of the time that way for $437.50 EV. I'd double if it's advantageous to do so, so occasionally I'd win more than $3500 or lose more than the $500 I can recover.

Actually it would hurt your EV to make a bet where you could end up losing more than $500 if you doubled down and lost. So at $4000 balance it would more optimal to bet half of that, ie. $2000 to make sure you can never end up losing more than what the cashback amount is going to be.

Quote: sangaman

I suppose it'd be optimal to continue betting even after I'm up $3500 despite the fact that I can't bet more than $3000, right? I don't know if I'd actually keep going at that point because I wouldn't want variance to get out of hand, but would another a $3000 bet at that point be theoretically correct?

Yes, ignoring the problems with doubling down (assuming it's an 50/50 bet) yes it would be theoretically correct to bet $3000 at $4000 balance, but it might only increase the EV by a modest amount. So it boils down to whether the risk of losing $3000 is worth the modest increase in EV to you.

Quote: weaselman

Well, it depends on what you call "optimal". It is not a +EV bet. If you are up $3500, and bet $3000, you will not be collecting the bonus if you lose, and therefore are betting at a regular -0.4% disadvantage.

Not true because he still has $1000 left which he can bet again and claim the cashback if it loses, so a $3000 bet at $4000 balance would be still +EV.

Quote: Jufo81

Not true because he still has $500 left which he can bet again or claim the cashback, so $3000 bet would be still +EV.

How is it +EV? He has $3500, bets $3000, and ends up with $3488 on average.

Quote: weaselman

How is it +EV? He has $3500, bets $3000, and ends up with $3488 on average.

EDIT: Corrected values

Simplifying it as a coin flip game with 0.4% house edge so the probability to win is 0.498 and the probability to lose is 0.502:

Probability to reach $7000 from $4000 balance ($500 initial deposit plus $3500 in winnings) is unknown P. It satisfies the following recursive equation:

P = 0.498 + 0.502*0.498^2*P

(either $3000 bet wins or $3000 bet loses followed by two consecutive winning bets of $1000 and $2000 which makes us reach $4000 balance once again)

-> P = 0.5688

Probability to end up with only $500 cashback from $4000 balance = 1 - 0.5688 = 0.4312

EV = $500*0.4312 + $7000*0.5688 = $4197.3

which is $197.3 more than the current balance of $4000.

Collecting $500 cashback will bring you back to zero, not +500. $7000*0.56 = 3920 < 4000

Quote: weaselman

Collecting $500 cashback will bring you back to zero, not +500. $7000*0.56 = 3920 < 4000

My numbers were expressed as total values and not relative to profit/loss. Expressing it as profit/loss:

Starting balance +$3500 (not $4000 mind you)

Probability to reach +$6500 from +$3500 remains unchanged: P = 0.5688 (see corrections I made in previous post)

EV = $0*(1-0.5688) + $6500*0.5688 = $3697.2

which is $197.2 more than the current profit of +$3500. The value is exactly the same as I got in my previous post.

Quote: sangaman

I suppose it'd be optimal to continue betting even after I'm up $3500 despite the fact that I can't bet more than $3000, right? I don't know if I'd actually keep going at that point because I wouldn't want variance to get out of hand, but would another a $3000 bet at that point be theoretically correct?

To answer this more precisely with numbers, assuming a coin flip game (0.4% house edge) the EV with finishing at $4000 target ($3500 profit) is:

EV = 0.498^3*$3500 = $432.3

The EV of finishing at $7000 target (a $3000 bet at $4000 balance) is:

EV = 0.498^3*0.5688*$6500 = $456.6

So the answer is: yes the EV increases by $24.4 by shooting to a $7000 target instead of $4000. And in case you are already sitting at $4000 balance then shooting for $7000 is worth +$197.2 to you per previous calculations.

Quote: Jufo81

Not true because he still has $1000 left which he can bet again and claim the cashback if it loses, so a $3000 bet at $4000 balance would be still +EV.

Quote: weaselman

How is it +EV? He has $3500, bets $3000, and ends up with $3488 on average.

I think Jufo is right. If I'm up $4k and bet $3k, I lose .004 * 4000 or $16 EV. However, the times that I lose and go down to $1000, I can then make a bet of $1500 which as we've shown before has an EV of +$250. So I'm losing $16 EV by making the $3000 bet, but when I lose (a little more than half the time) I can make up for it with a +$250 EV bet that more than compensates for the $16. If you assume I lose half the time, betting $3000 when up $4000 and betting $1500 after a loss should have an EV of 0.5 * 250 - 16 = $109.

Quote: Jufo81

Actually it would hurt your EV to make a bet where you could end up losing more than $500 if you doubled down and lost. So at $4000 balance it would more optimal to bet half of that, ie. $2000 to make sure you can never end up losing more than what the cashback amount is going to be.

Why would it hurt my EV? Sure, I'm exposing myself to a risk of actual losses, but I'm also increasing my upside if I double and win. I think I want to minimize the house edge as much as possible, and I'm only going to double down when I have the edge. I don't think it's any different than doubling when none of your losses are covered, you'll want to double any time you have the edge and it's a better option than hitting or standing. Maybe I'm missing something.

Quote: sangaman

I think Jufo is right. If I'm up $4k and bet $3k, I lose .004 * 4000 or $16 EV. However, the times that I lose and go down to $1000, I can then make a bet of $1500 which as we've shown before has an EV of +$250. So I'm losing $16 EV by making the $3000 bet, but when I lose (a little more than half the time) I can make up for it with a +$250 EV bet that more than compensates for the $16. If you assume I lose half the time, betting $3000 when up $4000 and betting $1500 after a loss should have an EV of 0.5 * 250 - 16 = $109.

Yes, but if you lose the $3000 hand, you will only have $1000 left (your $500 initial deposit plus $500 remainining winnings) so you should bet $1000 (not $1500) followed by $2000 in an attempt to reach $4000 again and you would be exactly back to the same balance you were at. The EV calculations I made above assumed betting like this.

Quote: sangaman

Why would it hurt my EV? Sure, I'm exposing myself to a risk of actual losses, but I'm also increasing my upside if I double and win. I think I want to minimize the house edge as much as possible, and I'm only going to double down when I have the edge. I don't think it's any different than doubling when none of your losses are covered, you'll want to double any time you have the edge and it's a better option than hitting or standing. Maybe I'm missing something.

It's a bit difficult to explain but the EV does decrease rapidly unless your total losses are covered (and they wouldn't be for any losses that are over $500). Alternatively you could always hit instead of doubling and never split but then you are in fact playing a game with ~2% house edge.

Sorry, made that last post before looking at the posts on the second page. Jufo, your equations make sense to me. Starting from 0, it doesn't seem that bad to me to sacrifice $24.40 of EV to reduce variance and practically double the chances I walk away a winner from this promotion. However, if I get to $3500 profit, continuing to $6500 is worth $197.20 and that seems too juicy to walk away from. I imagine that if I got to $6500 it would be similarly appealing to shoot for $9500. It's a bit of a paradox.

Realistically, what would you guys do if you were offered this same promotion? At what point would you stop? I'm leaning towards stopping at $3500 profit. Betting $3000 would be a bit gut-wrenching, and I wouldn't be able to double after split since that would put too much of my bankroll at risk so the house edge bumps up a bit. Maybe that is too conservative, but it would be nice to have a realistic chance of walking away a winner.

Thanks to all who have replied.

Quote: Jufo81

Yes, but if you lose the $3000 hand, you will only have $1000 left (your $500 initial deposit plus $500 remainining winnings) so you should bet $1000 (not $1500) followed by $2000 in an attempt to reach $4000 again and you would be exactly back to the same balance you were at. The EV calculations I made above assumed betting like this.

Oops, you are right. It would be a $1000 bet after a losing $3000 bet, not $1500, but the EV is still ~$250 for that particular bet.

Quote: weaselman

How is it +EV? He has $3500, bets $3000, and ends up with $3488 on average.

It is not money you play, you play position on rebate.

The fact that a $500 balance allows you to bet $1000 with an advantage, softens the loss of $3000 on a $3500 position.
Hence a $3000 bet on $3500 might still be favourable.

Quote: sangaman

Sorry, made that last post before looking at the posts on the second page. Jufo, your equations make sense to me. Starting from 0, it doesn't seem that bad to me to sacrifice $24.40 of EV to reduce variance and practically double the chances I walk away a winner from this promotion. However, if I get to $3500 profit, continuing to $6500 is worth $197.20 and that seems too juicy to walk away from. I imagine that if I got to $6500 it would be similarly appealing to shoot for $9500. It's a bit of a paradox.

Yeah, with rebate offers like this situations arise where you have to weigh the risk vs. reward, ie. are you willing to risk $4000 already obtained winnings for a $197 more EV? So it boils down to what is a meaningful sum of money to you. Someone with a large bankroll who could repeat this same offer many times would probably shoot for a much higher target than someone who can only do it once.

Quote: sangaman

Realistically, what would you guys do if you were offered this same promotion? At what point would you stop? I'm leaning towards stopping at $3500 profit. Betting $3000 would be a bit gut-wrenching, and I wouldn't be able to double after split since that would put too much of my bankroll at risk so the house edge bumps up a bit. Maybe that is too conservative, but it would be nice to have a realistic chance of walking away a winner.

What also factors in is how much effort is it to turn that $500 rebate into cash and can you do it without a loss, because with a high target you are most likely going to claim the rebate. And note that in blackjack the odds for win/lose are not even close to 50/50 but rather 47.5%/52.5% so to have three consecutive wins to reach a $4000 balance is not 1/8 shot but rather 0.475^3 = 10.7%. Of course if one or more of those hands happens to include a Blackjack/double down/split, you might already be at a higher balance than $4000 after winning just three hands. But personally I wouldn't push my luck to shoot for a higher than 1 in 10 shot to win.

If you do decide to make the $3000 bet from a $4000 balance and end up losing, you can console yourself by thinking about Wizard's signature:

"It's not whether you win or lose; it's whether or not you had a good bet." :-p