Side Bet

I have been playing a lucrative blackjack side bet. The standard deviation, so I have read, about 6.7. So variance is about 45. The advantage is 5x that of blackjack. When the bet becomes positive EV (which is somewhat infrequent), I have been betting about 1/2 normal unit per TC. I play BJ at about 1/4 Kelly, so in essence I normally underbet in blackjack to cut my risk. However, I have no idea whether I am overbetting the side bet, nor what the optimal bet should be. I do not want to give many details. Any help, appreciated. Thanks.

While I don't have the exact math at hand I feel like most exploitable side bets usually have limits of like $1-$25. Thus, even if you did Kelly on any kind of 'real' bankroll I would think that you would find a $25 bet is below Kelly. A 1% Kelly bet is for a 1% advantage, and exploitable side bets quite often carry more than a 1% advantage when they hit a certain point. Therefore, when the actual count index for your 5x blackjack advantage is hit, if you're going to exploit the side bet you should be max betting it ($25 I'd assume).

If for some reason the max on the side bet is way more (I'd think you should be max betting it even at $50 as well), or infinite (let's hope) then you could simply do the Kelly math like you would in blackjack to see what the bet should be:

You already know how to calculate Kelly. You might have to run a sim to find out how frequent this 5x blackjack positive advantage occurs, but then you can play with the bet to check out the gain (blackjack example):

True Count (ignore for side bet) = 5
Advantage = 2% (assuming .5% HE and standard .5% for each true count)
Frequency = 1.65 (this is where you'd have to run a sim to find your frequency)
Bet = $160 (this is the number that Kelly from your bankroll will give you. You can play with to see your gain per hand if you wanted to under bet a full Kelly bet... again this is pretty much just IF they let you bet more than 25)

Gain Per Hand = (Bet * Frequency) * Advantage = (160 * 1.65) * .02 = 7.0488

Quote: Sonuvabish

I have been playing a lucrative blackjack side bet. The standard deviation, so I have read, about 6.7. So variance is about 45. The advantage is 5x that of blackjack. When the bet becomes positive EV (which is somewhat infrequent), I have been betting about 1/2 normal unit per TC. I play BJ at about 1/4 Kelly, so in essence I normally underbet in blackjack to cut my risk. However, I have no idea whether I am overbetting the side bet, nor what the optimal bet should be. I do not want to give many details. Any help, appreciated. Thanks.

Here's my idea to find your answer:

Find the covariance between the main blackjack bet and the side bet. You can do this with a simulation, or even by dealing many rounds and recording the results. Then consider the combination of main bet and side bet as one bet with a size of 1 unit. Let f = the size the side bet and 1-f = the size of the main bet. Then the EV of the hand = f * EV side + (1-f)* EV main. And variance = f^2 * variance side + 2 * f * (1-f) * covariance + (1-f)^2 * variance main. Your total bet would be 1/4 Kelly which would be 0.25 * EV total / variance total. The profit for the bet is the product of this bet size and the EV total. Vary the value of f to maximize the profit.

Quote: Romes

While I don't have the exact math at hand I feel like most exploitable side bets usually have limits of like $1-$25. Thus, even if you did Kelly on any kind of 'real' bankroll I would think that you would find a $25 bet is below Kelly. A 1% Kelly bet is for a 1% advantage, and exploitable side bets quite often carry more than a 1% advantage when they hit a certain point. Therefore, when the actual count index for your 5x blackjack advantage is hit, if you're going to exploit the side bet you should be max betting it ($25 I'd assume).

If for some reason the max on the side bet is way more (I'd think you should be max betting it even at $50 as well), or infinite (let's hope) then you could simply do the Kelly math like you would in blackjack to see what the bet should be:

You already know how to calculate Kelly. You might have to run a sim to find out how frequent this 5x blackjack positive advantage occurs, but then you can play with the bet to check out the gain (blackjack example):

True Count (ignore for side bet) = 5
Advantage = 2% (assuming .5% HE and standard .5% for each true count)
Frequency = 1.65 (this is where you'd have to run a sim to find your frequency)
Bet = $160 (this is the number that Kelly from your bankroll will give you. You can play with to see your gain per hand if you wanted to under bet a full Kelly bet... again this is pretty much just IF they let you bet more than 25)

Gain Per Hand = (Bet * Frequency) * Advantage = (160 * 1.65) * .02 = 7.0488

Using the Wizard's formula, I should bet this game at approximately 1/5 of a unit. Thank you for somehow leading me in this direction. I'm not overbetting at about 2/5 which is where I'm really at, but it is completely disproportionate to my blackjack play. I don't want that kind of risk, and it is an obvious flag to the casino. I am going to reduce it to 1/4.