Not As Simple As You Might Think

Here's the situation that came up for me tonight:

Last hand. One advance. Bet range 5-200 in increments of 5. No surrender. BJ's pay 3:2. Only two players left. I am first to act.

=> Me, bankroll 277.50, bet ?
Nice Lady, bankroll 162.50

Nice Lady is not a reckless, aggressive bettor. The starting bankroll was 500; the dealer was so hot he was sizzlin' the whole round. I think Nice Lady is likely to go all in, but she had played for full swings a couple times in the hands leading up to this hand.

This is a good example of a situation in which you can systematically think your way to the best bet you can come up with. Come up with a range of bets that would maximize my chances of advancing and explain your logic.

 

Tx has a couple of typos in his hand, though his result numbers look correct. I think he means she could bet up to $160 of her original $162.50.

It's worth looking here at what a blackjack for you can do. Bet at least $85, but not more than $110. Your blackjack now beats her blackjack without you having to double down.

 

Let me clarify for hopinglarry:

You have bet the $85, but Nice Lady does get a blackjack. Now you need to win two bets. You split Aces.

Hand One: Ace-Ten
Hand Two: Ace-Ace

His question is do you resplit the Aces, or just hit the second hand out? Which is the best way to win at least 2 bets?

However, there is a problem with hopinglarry's question. When splitting aces you aren't allowed to hit the hand out. Remember, it's one card only on split Aces. If you are allowed to resplit, you definitely should do so because you might end up with three strong hands instead of solely relying on the dealer to bust.

Now, if you were somehow allowed to hit multiple times on the pair of Aces, larry's question remains.

 

Heh, don't feel bad. I remember spending a lot of effort to work out some details on a similar question before I remembered the one card on split aces rule. It's such a special case it is easy to forget.

On a similar note, I get a lot of questions from people who are developing their own blackjack simulation software and wondering why their answers aren't matching widely published numbers. More often than not, they have forgotten one of the two oddities about splitting Aces: 1) You get only one card on each. 2) Ace-Ten is no longer a blackjack.

 

To backtrack a few posts, I wanted to respond to this one. Larry is right. The best bet appears to me to be either 85 or 90. In the very slight possible chance of Nice Lady getting a blackjack and Monkey needing to win 2 bets, there are certainly occasions where splitting to make three hands improves your chances of winning at least 2 bets.

The most obvious one is this:
Consider if the dealer has a small card showing, and you are dealt a pair of 8s. You make a 21 on the first hand. If you now get another 8 on your second hand, you are obviously better off splitting again than standing on the 16. You might make two more non-stiff hands.

 

Another Consideration

I didn't really emphasize the following idea too strongly in my original post: The thought burning through my mind when I was considering my bet was that Nice Lady had made bets once or twice in the late hands that seemed to be attempts to get full swings. I believe I read this player right - she considered doing it on the final hand. After I placed my bet she paused, starting counting out a smallish swing bet, paused again, thought better of it, and shoved all in.

All the ideas mentioned in this thread occurred to me when I was considering my bet except for Ken's point that I'd want to bet big enough to cover a BJ with my own BJ. That point is even more important at Ho Chunk because they don't allow you to double on BJ's or totals of 21 in splits. But this idea didn't come to me until I was in the car on my way home.

I'll let you know exactly how things turned out, but I'll let you guys and others make more comments first.

 

I picked up on that idea from your initial post, but I didn't see a way that it could reasonably impact my bet size decision. I look forward to your thoughts on the matter. And, thanks for starting the thread!

 

How It Went Down

What I bet was 75. My first impulse was to bet my whole lead minus a chip, which would be 110. But my gut told me Nice Lady may very well make a small bet that can only full-swing me if I make a smaller bet. That would be a mistake for her that would increase my chances.

So I figured the smallest bet I could make that would allow me to double it to cover her all in BJ as well as cover her simple all in. That was 65. I started counting out a bet of 65 but decided that since I was trying to induce her to make a conservative mistake I'd better not put a rainbow out in my betting circle. It's been my experience that rainbows can befuddle casual players and can trigger them to shove. So I settled on a bet of 75 - three green chips.

As I stated in the previous post she hesitated, started counting out her smallish bet, then thought better of it and shoved.

I got 11, she got hard 16, dealer showed 2. I hit, got a five and waved off. She thought about it for a few seconds, then waved off to guarantee me a seat at the final table. The dealer made a hand - not that it mattered.

For me, a bet of 90 would've been better. It was probably still small enough to induce the mistake I was hoping for, while covering Nice Lady's BJ with my own BJ. But, as I stated earlier in this thread, covering the BJ-BJ didn't occur to me until I was in the car driving home.

I've been considering ways to improve your results against weaker players. The most important is anticipating how a weaker player will respond to your bet when you must act first on the last hand. You can estimate the possibility of an opponent's suboptimal response to your bet then treat this fraction as a probability in an EV calculation. You can modify your bet decision to maximize your EV given this additional factor.

I've never figured this particular scenario before, in which you trade off the BJ-BJ in exchange for the added benefits obtained from the possibility of inducing a tight opponent to bet too small. This isn't a complete analysis, because it doesn't take other factors into consideration such as different combinations of players' and dealer's BJ. It's meant to be an example of the kind of analysis you could use to factor in weaker opponents' suboptimal play.

The probability of both you and your opponent getting BJ is ~1/440. So for you to benefit from trading that off the increased chance of advancing times the possibility of the opponent making this mistake must be greater than 1/440. If I bet 110 and Nice Lady shoves, my probability of advancing is ~86%. If I bet 75 and Nice Lady bets, say, 75 my probability of advancing is ~88%.

Let M = the possibility of Nice Lady making a Mistake and betting too small. Then for me to benefit from this smaller bet M(88% - 86%) > 1/440. M = ~1/9. In other words, this was a profitable decision for me if the possibility of Nice Lady making the small bet mistake were >1/9.

9:1 is a small chance. It seemed more likely than that she would make this mistake.

Don't get me wrong. I didn't sit there at the blackjack table at Ho Chunk doing these mental EV calculations. I'm doing all this after the fact.

Just because an opponent didn't actually do what you were hoping for doesn't necessarily mean you misread that player. You can't predict the actions of another human being with 100% accuracy. All you can do is estimate the possibility of an action they'll take. In many cases you're estimating the chances of behavior that will happen less than half the time, then modifying your own actions based on that estimate.

An example of this concept I've written about in the past is the situation in which you're acting first, heads up, with a small lead against an aggressive opponent. Against a strong opponent you'd bet small. But against an uninformed aggressive player you'd bet the max. The max bet gives you better EV if the possibility of the opponent matching your max bet is >1/3.