What's the correct play here?

The term 'optimal' has a context. The information you have to work with is not just the situation within the game, but also what you know about your opponent's skill level.

$225 is optimal against an expert opponent; $5000 is optimal against a lesser opponent, if the probability that they will make the mistake of matching your $5000 bet is sufficiently high.

Here's a link to one of the previous threads - Betting First final hand

 

The optimal bet depends on what your opponent bets. This is why position is so important. When you bet after your opponent, or when you know what he will bet when he bets after you, you can make the optimal bet in response. In this case, you are betting first and, assuming you don't have a read on him, the best you can do is to take the best initial position which, in this case, is to take the low. The situation gets more complex with more than one opponent involved.

I like to think of it as a series of tradeoffs. You can cover A, B and C and the cost of giving up X, Y and Z. Some combination of A, B, C, X, Y and Z gives you the best chance to achieve your goal.

Ken Smith's signature used to say "There's always a better bet". Sometimes it feels that way at the table!

 

Did someone say simulation?

I ran the sim for this situation and the results are a little surprising to me. The sim was run for 8 decks, S17, DOA, SP4, RSA and assuming that player 2 will respond optimally, which is to say that he will play for a swing when appropriate and play for the high otherwise.

For the case of A,A vs 13 vs T, the best play for player 1 is to hit for a 74.58% chance of success vs 73.18% for splitting and 69.52% for doubling. A pretty close call between hitting and splitting. While splitting/doubling does allow player 1 to take the high, it does expose him to a potential half swing should he lose two bets.

For A,3+A,A vs 13 vs T, the best play for the A,A is to resplit for a 72.48% chance of advancing vs 68.63% for standing. I previously said that I could see no value in resplitting, because the extra bet does not provide any additional coverage relative to player 2's bet of 500. I see now that the value comes in the form of protection against a 2 bet loss which exposes him to a potential swing. In this case, if player 1 stands with A,3+A,A, he suffers such a loss if the dealer makes any hand. He wins 2 bets if the dealer busts. There is no possibility of a net push. However if he resplits, then he still enjoys a net win when the dealer busts, but also has the potential to lose less than 2 bets should the dealer make a hand.

 

I knew you wouldn't be able to resist!

That is quite surprising.

I think the underlying reason that hitting comes out on top must be that the penalty player 2 incurs when obliged to draw only one card by doubling on 13 v T is quite small (they would always wish to draw again after an A or 2; after a 3 it depends on what player 1 did - following a split/double drawing to 16 for a possible push comes into the equation; after a 4,5,6,7,8,9,T,J,Q or K doubling is no different to hitting). So the benefit to player 1 of imposing this penalty by getting two bets out is apparently outweighed by the cost of the potential half swing.

 

Ken's response makes the most sense to me, although some of the others did too. Let me give a little background on myself.... I play flawless BS both in tournament & non-tournament play, feel I have a good understanding of tournament fundamentals and proper bet sizing, tournament strategy, etc. as I read and own Wong's awesome book "Casino Tournament Strategy" with tons of great real-life examples. I've also made it to the final table of Biloxi's Hard Rock Casino weekly BJ thrice (never won it though since finishing 2nd is same as finishing 7th), made final table in Palace Casino's weekly, and finished 2nd in one of Beau Rivage's $50K tournament for a cool $15,000. Most recently I made it to the semi's of Horseshoe's Winner take All 25K last weekend in a weird elimination format.... anyway, I really enjoy BJ tournaments and feel us tournament players really have an edge over the majority of players although one must still have LUCK of course to advance!

In the example in this post though, since player 1 bet 225 and even with a split + double opportunity win by him (225 x 3) he can ONLY end up with 7925 so betting 950 makes sense to me. I'm not sure where 975 comes in though... Having said this, and even though it's quite rare to be able to have more money out than 3x your bet, a SAFER bet would simply be 2000 since you're just hoping that both players beat the dealer so player 2 can advance. (Aside from him winning and player 1 losing, that's the easiest way for player 2 advance) Betting 2000 would still leave you room to double down on a proper double too, so 2000 is probably what I'd bet to protect myself even such a rare occurrence. Since player 1 has more chips to start, you'd need the dealer to win as both players losing leaves player 2 eliminated anyway, right? Just my 2 cents....

 

But why limit player 1's possible gain to one split with one double? If he doubles the first hand, he could just as easily double the second for a potential total of 8150. Also don't discount the possibility of multiple splits with doubles. This happens more often than you might think, especially in the case of splitting 10s (T, J, Q, K). Ken's suggestion to cover an 8 bet win (split to 4 hands and double all) with a single bet of 2075 comes at no cost to player 2.

I know I posted that player 2 should at least cover player 1's 2 bet win. That's because the chances of winning a planned double/split before the cards are dealt are quite significant at around 30%. However, if you can cover more outcomes at no additional cost (in terms of the effect of losing your hand), then there is no reason not to. I then posted that, in the heat of battle, I would probably just make a max bet, but Ken's suggestion of limiting the bet to 2325 has merit, since sometimes the best way to get a net win from a pair is to split it, perhaps more than once.

 

I see your point. Did not think about how many times (or ways!) a 10-10 hand could come up and I have seen this actually myself in tournament play do it in my table! But the chances of having a split + double is lower than simply doubling the money out on the table so in effect, a bet of 3500 would be good too to maximize the 7K in chips. Betting the max would run into the problem of a splitting opportunity that you wouldn't be able to do so that's why I like 3500. Besides, you could also split too and instead of doubling a proper hand, just hit.

 

I think we're basically agreeing now. Ken's bet of 2325 allows for the possibility of a second split or a double which squeezes out an additional very small percentage. Once again, since there is no cost to this limit, there is no reason not to do it. The rule of thumb would be to cover as much as you can if there is no tradeoff.