Is this an AP opportunity?

There is a local casino where I live that has a blackjack varient that I have never heard of before, called "players edge" its kind of like spanish 21 but it has some differences (not exactly the same as players edge 21, which the wizard has written about)

the rules are almost the same as Spanish 21 H17 redouble, except that they don't pay the multi-card 21, 7-7-7, 6-7-8 and super bonus's

but there is a "special" move that you can make, called "buy a card", which I think makes the house edge swing in favor of the player

if you have doubled your hand, and your hand has still has 4 cards or less, you can pay $1 and the dealer will let you hit your doubled hand!

example: you are delt an 11 (8-3) against a dealer 10 and you double, the dealer pulls an ace putting your new hand total at 12 vs 10, you pay $1 and the dealer pulls a 2 giving you 14 vs 10, you pay another $1 and the dealer pulls a 7 giving you 21, with a hand total of 21 you automatically win the hand,

The cost to buy a card is static at $1 if you are betting $3 or $300. you can buy 2 cards per hand as long as the total cards is 4 or less when you buy them.

I don't know how to calculate the EV for this BJ variant but I think its in the favor of the player, does anyone know how to calculate this?

Bondy3,

Clearly, the EV is dependent on your wager. For example, if you bet $3, then buying another card is the better move only if the Hit EV is 33% better than the Stand EV: say, St.EV is -0.4, white H.EV is -0.05. On the other hand, if you're betting $300, then the H.EV need be only 0.33% better than the St.EV to make buying a card the better option.

Thus, any discussion of the EV of this rule must include the initial wager.

Hope this helps!

Dog Hand

Edit: changed EV for $300 bet.

Quote: DogHand

Bondy3,

Clearly, the EV is dependent on your wager. For example, if you bet $3, then buying another card is the better move only if the Hit EV is 33% better than the Stand EV: say, St.EV is -0.4, white H.EV is -0.05. On the other hand, if you're betting $300, then the H.EV need be only 3% better than the St.EV to make buying a card the better option.

Thus, any discussion of the EV of this rule must include the initial wager.

Hope this helps!

Dog Hand

lets assume im flat betting $25 as that is what my bankroll will allow.

Quote: Bondy3

does anyone know how to calculate this?

Yes, but not me. I can tell you that at $3 a bet you will likely never use the option, but at $300 you will use it on many of your double type hands. Will the added EV on those few hands (a few out of a hundred) make up for the house edge on the base game.... That's a definite maybe....

My educated guess is that the game is not player advantage... Especially if the name of the game is something like players edge, I'd be pretty sure the mathematician behind the game double checked his work to not have a player positive game =P.

To the others points, you might as well look at it from a MAX bet scenario, to even see if the player can gain an edge from the static $1 fee. Next, you said you can use the extra card after a double, up to 4 cards... Wouldn't this mean you can only ever use the option once? 2 initial cards + double card = 3 cards... Thus, you can only invoke this rule once to get a total of 4 cards? Or were you saying you could ALSO take this option on 4 cards?

I think the Fee, while small, and static (so impacting larger bets less) still doesn't bring the edge positive. Imagine if you had to Ante $1 every hand to play, we know the house edge would absolutely go up. Yes, drawing an extra card can help a lot at times, but even in your scenario (8-3 double ace, use rule to get another card vs dealer 10)... you're still drawing on 12v10 which is still negative. Getting the extra card is LESS negative than before, but you're still sitting on a negative expectation.

Quote: Romes

My educated guess is that the game is not player advantage... Especially if the name of the game is something like players edge, I'd be pretty sure the mathematician behind the game double checked his work to not have a player positive game =P.

For what it's worth, I have seen operators put some crazy things into table games without doing any math or realizing the actual cost of a rule change. Just remember the 2:1 blackjack promotions.
If this is not a legitimate game from one of the known vendors, it may not have been vetted properly.

For the record, here are the regular playing rules:

https://wizardofodds.com/games/players-edge-21/

Quote: Ibeatyouraces

For the record, here are the regular playing rules:

https://wizardofodds.com/games/players-edge-21/

This game is different, there is no "buy a card" option in that game, Also that game has matched 20 pays even money always. They are different games

Quote: Bondy3

This game is different, there is no "buy a card" option in that game, Also that game has matched 20 pays even money always. They are different games

It's the same game with tweaked rules.

Remember this rule of thumb also. If the casino is offering something for a price, it's probably not a good bet.

Quote: Ibeatyouraces

It's the same game with tweaked rules.

Remember this rule of thumb also. If the casino is offering something for a price, it's probably not a good bet.

alright let me rephrase, how does this "buy a card" effect the house edge? (assume it cost 1% of the bet to buy a card, to make the math simple)

How do you make a 1% bet? Are you telling us that if you make an initial bet of $10, you can buy this card(s) for only 10 cents each? If so, how is that handled?

Quote: Ibeatyouraces

How do you make a 1% bet? Are you telling us that if you make an initial bet of $10, you can buy this card(s) for only 10 cents each? If so, how is that handled?

I was thinking just to assume its 1 black chip to bet and 1 white chip to buy the card (black = $100 and white = $1)

Also, I went back and found out that you can only buy 1 card per hand, and once you buy a card you are giving up your ability to surrender the hand (or to hit/double)

Quote: DogHand

On the other hand, if you're betting $300, then the H.EV need be only 0.33% better than the St.EV to make buying a card the better option.

It's actually only half that, because it comes after a double, so the cost to draw a fourth or fifth card only adds 0.17% to cost of the hand. Where the majority of edge from this comes is that you can now double eights and nines and hit again if you draw a stiff. I'm not too familiar with Spanish 21, but most of them have a similar house edge as common blackjack games ~.66%. Given the changes on the 6-7-8 etc, as well as the small cost to take another card, being able to take extra cards after a double would have to be worth 1% or so. My intuition is that it would be worth looking into to see. And it may actually make counting worth while ... as the count gets higher, your doubles become even more likely to be good, while the dealer showing a 4, 5, 6 becomes even more likely to bust

Quote: TomG

It's actually only half that, because it comes after a double, so the cost to draw a fourth or fifth card only adds 0.17% to cost of the hand. Where the majority of edge from this comes is that you can now double eights and nines and hit again if you draw a stiff. I'm not too familiar with Spanish 21, but most of them have a similar house edge as common blackjack games ~.66%. Given the changes on the 6-7-8 etc, as well as the small cost to take another card, being able to take extra cards after a double would have to be worth 1% or so. My intuition is that it would be worth looking into to see. And it may actually make counting worth while ... as the count gets higher, your doubles become even more likely to be good, while the dealer showing a 4, 5, 6 becomes even more likely to bust

I was hoping it would be more than just 1%

I actually broke down and wrote a sim,

ran 10 billion hands, put the cut card at 40 cards from the end of the shoe, (good penetration)

I ran this for a $5,$25,$100 and $100000 bets, (the 100k bet is just to make the cost of buying a card zero with respect to the bet)

the house edge was as follows (in percent of original bet)

0.415 $5
0.410 $25
0.404 $100
0.399 $100k

the house advantage of the normal game is about 0.421, (not allowing this "buy a card" at all)

Did you run the sim in CVData?

Quote: Bondy3

....
the rules are almost the same as Spanish 21 H17 redouble, except that they don't pay the multi-card 21, 7-7-7, 6-7-8 and super bonus's

but there is a "special" move that you can make, called "buy a card", which I think makes the house edge swing in favor of the player

if you have doubled your hand, and your hand has still has 4 cards or less, you can pay $1 and the dealer will let you hit your doubled hand!

example: you are delt an 11 (8-3) against a dealer 10 and you double, the dealer pulls an ace putting your new hand total at 12 vs 10, you pay $1 and the dealer pulls a 2 giving you 14 vs 10, you pay another $1 and the dealer pulls a 7 giving you 21, with a hand total of 21 you automatically win the hand,

The cost to buy a card is static at $1 if you are betting $3 or $300. you can buy 2 cards per hand as long as the total cards is 4 or less when you buy them.

I don't know how to calculate the EV for this BJ variant but I think its in the favor of the player, does anyone know how to calculate this?

I thought this was an interesting game so i wanted to look into it further, but it was too hard to answer fully (for me), so I have partial answers for both examples below
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my answers were worked out using:
Rules from the OP
1. Spanish deck
2. H17
3. doesn't pay 'traditional' spanish 21 bonuses eg multi-card 21, 7-7-7 etc
4. player 21 always wins

rule assumptions/changes that make the math 'easier for me'
1. infinite deck
2. Dealer checks for BJ (I assumed this, since WoV is an American website)
3. double down rescue NOT allowed
4. no splits

Please note: any other rules not mentioned above were not needed for working out the difference in EV (or only would have had a negligible impact)
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NB: P = Player, D = Dealer, ~ = about, and gross means: 'if the player didn't have to buy for a $1, in other words if it was free to hit ONCE after doubling

From the OP's example (part 1) P12 vs D10 is worth:
~ -53.83% EV if you stand
~ -32.95% EV if you hit
for a gross EV improvement of 20.88% in the player's favor for that hand (since you can only have one extra hit in my scenario)

OP example (part 2) P14 vs D10 is worth:
~ -53.83% EV if you stand
~ -40.65% EV if you hit
for a gross EV improvement of 13.18% in the player's favor for that hand (since you can only have one extra hit in my scenario)

NB: the chance of pulling an automatic win (player 21) is included in the hit EV's for both of the above examples

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It would be interesting to find out what the correct strategy would be for a:
P11 vs D Ace and D10 (with and without double down rescue)
P10 vs D9 (with and without double down rescue)
P9 vs D 2 to 5
P soft12 to soft18 vs D 2 to 7
and any other hands you can think of that could potentially be a good 'hit after double' scenario
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i would love to know the chance of getting an 11 double turning into a 3-card 12 for 6 and 8 decks

My guess would be about 1/18 for getting a 2 card 11 and 1/216 for getting a 2-card 11 into a 3-card 12 for an infinite spanish deck (please correct me if this is wrong)

hope this helps you and other people in working out a better/more correct strategy
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important update:

if you are allowed to double down rescue, my answers will probably be of little help