Table Games Fun push on 22 math/strategy problem

For anyone that likes to play with odds and simulations, a scenario I have been mulling over.

In both the LVA and ACG coupon books, the D presents a coupon for 'push on 22 at blackjack'. It can't be used on splits and doubles.

Now since these are one time use coupons, this is very much just an academic exercise. But it is still an interesting problem trying to figure out the optimal way to take advantage of this coupon , and what it actually does to the house edge.

First, the strategic consideration

• Do you stick to basic strategy, and just use the coupon should a 22 arise?
• Do you stick to basic strategy, except hit all 12's (even against bust cards) until you use it?
• Do you do a version of that, except not hit against a certain set of bust cards on the theory that your advantage is so high anyway?
• Do you hit all 12's, but also potentially hit some 13s, etc when there is a 'weak' bust card like a 2?

I'll probably try to do some calculation on some of that, and we should be able to calculate it. Although option 3 could be up in the air, depending on the below

Next, the expected value consideration

Obviously, you are going to gain 1 unit if and when you finally get the 22 to push, So the easiest calculation would be adding one unit to the expected value of the average amount of hands it would take to reach 22.

But this isn't totally accurate. If you change strategy on all the interverning hands, you could turn some other losses along the way into wins or pushes. And since you altered strategy, it is possible that the hand you push on would normally not be a hit, and thus its expectation may not be the full -1 unit.


To get any kind of real answer on this one will likely take a measure of simulation. Any guesses? Anyone know of a BJ simulator that could be customized for running this?

 

I used these coupons last year. I stood with a 12 against the dealer's draw card and hit a 12 against a dealer's stand card knowing I couldn't lose. It worked well for me.

 

Interesting.

My first math check was on the most extreme bust card - 6 (I will check 5 as well since with stand on soft 17, 5 is in theory worse for the dealer if I recall).

For a dealer 6 vs a 12, the book play is to stand, but it is still negative expected value (-.15416). The dealer only busts 42.29% of the time, and you lose every other time.

Normally, hitting a 12 is even worse, because you would bust 30% of the time, and not beat them any of the times in between.

Holding this coupon, if you hit the 12 it becomes positive expectation of 0.137026

So you would hit 12 against a 6 for sure. I assume this will therefor apply to all 12's but I will run the #s for 5

EDIT: It is the same for 5, goes from negative to positive

So at minimum, if I did this right, you should hit all 12s

 

This was my first instinct when I saw the promotion, but then I started thinking about why I WASN'T hitting 12 against a bust card. Which is how I started down this rabbit-hole.

Now I'm wondering not only as to strategy but overall expectation. I mean, if your expected value still holding the coupon is negative, its not worth using unless you were going to play there anyway.

 

The PLOT THICKENS

By now I am assuming you always hit 12. So the next question becomes do you ever hit more than 12?

The most borderline case for that would be 13 against a dealer 2, so I took a look.

Normally, hitting 13 against a 2 is a -0.307791 expectation, and standing is a -0.292784 expectation, so you stand.

But with the push on 22? Now hitting is a -.230868 expectation.

Which is a minor conundrum in itself. Suddenly, hitting is the better mathematical play. But is .076923 enough of a gain to risk the coupon? And should we risk the coupon for a hand that will not be in our favour regardless?

 

will they do the push for any size bet? for instance might not be worth it to "waste" your coupon getting you out of a pickle when your bet is 10 bucks. where as if you vary your betting a bit and you have a bigger 50 dollar bet out.

 

It is to a max of $25.

I tend to flat bet and am assuming that, but variable bets would be another interesting wrinkle.


EDIT: and below we get my first dumb logic error. I was assuming you are only using this on hard 12's, but that is not true. You are also using it on ANY basic strategy hit that adds up to 22 without resulting from a split. That makes it far more likely that you will use the coupon far sooner. Makes the coupon an even higher player edge while you hold it, but you won't have it as long. Given that, I am assuming the edge cases I was looking at above (like hard 13 vs 2) are never worth it. If I get really dedicated, I may recalculate the below based on total number of hands until you get a 22.



I am working on tackling the overall issue (does it give you a temporary overall edge, and what is the expected value?) without simulation.

Near as I can tell, looking at 2 and 3 card combinations adding to 12, you'll get a hard 12 to hit roughly 1 in every 9.94 hands.

You'll end up using the coupon one in every 3.25 hard 12s you get, assuming that you only hit hard 12s.

So on average, your coupon will last 32.305 hands or so. In 32 hands of $25 bj , (this is assuming 3:2, s17, no das simply because that is how i figured the above), the house expected win would normally be around $4.75 (# of hands * bet * house edge of 0.5884%). So even if we don't factor in better expectation on hands which you don't bust, I think this is an advantage game to play while you have the coupon. Your net expected win would become just under $20.25 in those hands, meaning an edge of about 2.5% per hand.

Still TBD on whether there are other hands you should hit. And this is depending on my assumptions and calculations (yikes).

 

EDIT: whoops, had this backwards. On 13 vs a 2, if you DISCOUNT the 9 (pretend no 9s in the deck), hitting is actually very slightly correct. So having 9s exist and be a push only help the matter further.

Still hoping to do a simulation, but I think an optimal 'basic strategy' for this coupon would include some counter-intuitive hits.

 

I played basic strategy last year on these coupons figuring I would be playing for a while anyway. This year is different as it will be more of a hit and run type venture. I'm glad your math is backing my gut instinct. I had already decided that hitting any hard 12 is the way to go. I'll play everything else by the book.