I've been playing against the 'bots at GameAccount recently. An interesting final hand came up in one of my robotic encounters - Min bet: 100, Max bet 500. Me to act first. My lead was insufficient to cover high and low, so I bet 'lead minus a chip'. I'll make up some representative figures - Me : 1191. Bet 190. Bot: 1000. Bet 500. Dealer 9: Me: Don't actually recall the cards, but it was a bustable hand and a BS hit. Bot: 7,7. I chose to double down my hand in order to go for the high and low. This does mean opening up the possibility of losing the hand while the opponent pushes; I generally only let that stop me if the opponent has 17v7, 18v8, etc., but I don't know if that is an entirely optimal strategy. In this instance I busted, so the 'bot needed a push or a win to beat me. What surprised me was that it then split the 7s. So it now needed to either win at least one of the split hands, or push them both. (It duly did the former .) So my questions are - Was the bot's play the correct move? And does there exist a table of which splits to make when a push is as good as a win? (Should all pairs be split against any upcard?) And should I avoid the double-down move when my opponent has a pair (or a particular pair-versus-upcard which they ought to split, if any such exist) ?
Taking S. Yama's advice, I'll take a shot at this before running to the sim results (although the sim is already underway ). I like your bet of 190. It's Wong 101. Betting between 207 and 290 would allow you to win with a 3/2 blackjack (~4%), but you would give up the push off the top (~8%). Betting between 292 and 308 would give up the low without gaining you anything. Betting 310 or more gives you the high (~43%) at the expense of the low (~49%). As for the play of the hands: My gut says that your double to take the high was the right move. Even if you lose, the bot still has to either hit its stiff hand for a push or a win, or split its 7's vs the dealer 9 for a push or a win. You were likely to take a card anyway to avoid being swung, so why not take the high as well? Once you doubled, I also think that the bot made the right move. It went from having to win or push a hard 14 vs a 9 to having two shots at winning from 7, plus the small chance of pushing both. My gut tells me that the latter would be better due to the reduced chances of busting. I'll leave it to the math wizards to come up with the numbers for these plays. Since I myself am a programming wizard , I'll wait for my sim results!
Thanks gronbog I should add that this was a no-surrender game. Otherwise I would have bet 380, with the option to surrender back. Well there was no issue with being swung prior to doubling, since BR2 had the high. Without doubling it really doesn't matter whether I hit or stand; all BR2 has to do is win the hand. The issue is whether the risk of being beaten by a push is outweighed by the potential benefit of gaining the high and low (assuming I do not bust, as happened in this particular case). What hadn't really occurred to me is that by splitting a pair BR2 presumably has quite a large chance of a net push.
Results You guys are right on the money. A quick spreadsheet shows me that in the above case hitting pair of 7s would advance robot 40.6% of the times. Splitting and playing basic strategy improves the chances to 45.4%. Splitting and hitting first hand to 18 and if busted playing second hand “push as bad as loss” raises it to 47.6%. Comments later, S. Yama PS I originally wrote "push as bad as loss" was a bs play after busting first hand but we stand on 16.
Simulation Results My sim results for this are ready and, as is often the case, they contradict my original gut feeling but, as always, looking into the results has given me new insight. For the bot, once Colin busted, the percentages are: Split: 48.61% Hit: 31.93% Double: 30.96% Stand: 22.85% The 'split' result is resounding and is close to what S.Yama computed (47.6%), although my software computed the optimal strategy for the bot as: - Hit hand 1 to 15 or better - If Hand 1 busts, hit hand 2 to 16 or better otherwise hit it to 18 or better The 'hit' result disagrees with S.Yama by a wide margin (31.93% vs 40.6%). Yama, if you could recheck your number, I would appreciate it. You are almost always correct and, if there is a bug in my software, I would like to know it! For completeness, in my simulation, the bot hit to 17 or better. Here is the optimal strategy for for the bot, once you busted, by my software: Code: 2 3 4 5 6 7 8 9 10 A 2,2: +p +p +p +p +p +p +p +p +p +p 3,3: +p +p +p +p +p +p +p +p +p +p 4,4: +p +p +p +p +p +p +p +p +p +p 5,5: +h +h +h +h +h +h +h +h +h +h 6,6: +p +p +p +p +p +p +p +p +p +p 7,7: +p +p +p +p +p +p +p +p +p +p 8,8: +p +p +p +p +p +p +p +p +p +p 9,9: +p +p +p +p +p +s +s +p +p +p T,T: +s +s +s +s +s +s +s +s +s +s A,A: +p +p +p +p +p +p +p +p +p +p The strategy says to split all pairs against all up cards except: 5,5: hit against all up cards T,T: stand against all up cards 9,9: stand against 7 and 8 This strategy assumes optimal play of the split hands. The answer to this is, in general yes. With the exception of vs 10,10, my software computed only a few instances where doubling was advantageous. They were a few non-stiff hands vs 2,2, 4,4 and 9,9. These cases were very close in percentage to simply hitting. Vs 10,10, it is recommended to double or split almost every hand because BR2's best play is to NOT split his 10s since he has a high probability of winning with his 20. Let's look at your actual situation (stiff vs 7,7 vs 9). The sim result disagrees with our original assessment. You couldn't recall the exact hand you had, but luckily the results are close for all stiff hands from 12 through 16: For 12: Stand: 74.08% Hit: 74.00% Double: 61.88% For 16: Stand: 73.90% Hit: 73.85% Double: 59.25% The results for 13, 14 and 15 were all somewhere in between, as expected. As you pointed out, hitting vs standing is irrelevant, however, contrary to what we originally believed, doubling is much worse for this situation. I began to wonder why that might be and I think I may have at least part of the answer. What doubling does is cover one situation (a win by the bot) at the expense of exposing you to another (a push by the bot). I looked at how often the BR2 won from 7,7 when playing for a 1 bet or better win vs 9 and how often BR pushed when playing for a push or better from 7,7 vs 9 during the simulation. The results were: Won when playing for a win or better: 26.02% Pushed when playing for a push or better: 35.81% I think that by doubling, you exposed yourself to a situation which was much more likely than the one you were covering. I was surprised myself by the rate of pushes vs wins possible from 7,7 vs 9. Now these numbers must be multiplied by the odds that you win/lose your double respectively from 12-16 (S. Yama?) to come up with the actual value/exposure of the move, but the difference in this case could explain the result. In summary, exposing yourself to a push via split by your opponent appears to be a bad idea.
very nice simulaton Wow! This is a great simulation Gronbog. The data you provided covers the specific hand as well general strategies that are so multifaceted and thought provoking. I don’t even know where to start, especially that some of the issues are rather more related to general nature of blackjack game. There is no discrepancy in the hit numbers, just my clumsiness describing what it represents. 40.6% (30.4% win, 10.2% push) represents wins and pushes hitting a single hand starting with a single card 7 after the split, which I needed to evaluate additional wins if first split hands doesn’t win. In the next post I will describe my most likely “train of thoughts” and calculation I would attempt at the table. Doubling a stiff hand vs. opponent’s pairs, to protect against single win, may seem to be a good play. Not so - it is counterintuitive to the results of your sims. I think the main reason for it is that we habitually perceive pairs as they are played in basic strategy blackjack. By doubling we are giving up advancement when we lose and the opponent pushes. In the first impulse we may look at the push chances for the hand played accordingly to basic strategy (in our case hitting) and since splitting is often (significantly) a worse play we may not give it a second look. When working on some tournament problem years ago, I created table of chances to win/lose one/two bets or push (zero net). I, too, was surprised by the results. Splitting pairs has relatively low percentage of winning/losing just one bet and considerably increased “pushes”. Pushes occur equally or more frequently than single win and single loss combined. Yet, another aspect of splitting, not related to this case but important and not rare in bj tournaments, is very high percentage of two bets wins relying on dealer bust. It is true even for splitting high cards. For example about 90% of two bet wins when splitting eights (hit to no bust and except dealer’s 7) happens when dealer busts. When splitting nines and tens dealer’s bust contribution is in high 70’s and 80’s percentile Figuring out value of playing decision at the table can be done in two ways – by calculating (estimating) total chances of success for different plays or, if most results have similar consequences, comparing only differences between gain from one play and what we give up versus the other play. In our particular case there are too many different elements to consider. BR1 can stand/hit or double down, and then depending on each total of that double BR2 can respond by doubling, hitting or splitting and playing it differently then basic strategy. We are better off trying to estimate chances of advancing for easier to calculate situation and then try to add up as many elements of the other hoping that there will be a noticeable difference indicating to us the right decision. Mathematically optimal playing strategies sometimes are not the best option at the tournament table. We should weigh in assumed probability of our opponent making a mistake. In our specific case likely opponent’s mistakes are not very costly. If BR1 stands/hits BR2 can play to win, if we double, depending on our score BR2 right play may be to split. But even if he, instead of splitting, doubles we gain little because hitting 14 (7,7) is in most cases one card hit, and we still give up our busting and BR2 push. The most interesting (to me) part of gronbog simulation was that optimal playing strategy after split is to hit first hand to 15 and if busted hit the second one to 16, otherwise hit second to 18 or better. I think program also included to hit second hand if not busted to at least one point higher than the total of the first hand. So, if the first hand is 20 then the second should be hit to 21 or bust (perhaps with the exception when both hands are 17). Once I knew that splitting is a better play I thought that since we need to win just one hand to advance it should be more advantageous to improved one of them above total of 17. If we bust the first hand we still may have a “back up” of playing the other to less than 17, so dealer's bust still advances us. Playing first hand to 18 and than playing second hand if the first was busted to “push as bad as loss” improved overall chances by slightly over 2%. Benefits of hitting first one to more than 17 are that once we have total 18 or higher we can afford to try to improve the second hand higher than 18. Apparently, simulation shows that once the first hand is 15 chances of achieving good total with the second hand (starting with 7) are better and we keep that stiff hand as protection for dealer bust. I wonder how big is the difference between the two playing strategies (gronbog?). S. Yama
This is interesting, informative stuff. I love your simulator. I just wish my old brain could remember all the things it generates. I would have never believed that splitting was the proper play. Just think somebody else programmed their machines to play correctly, so they also have a simulator. Could be that their machine plays perfectly? I don't know what the overall demand would be for such a simulator if produced commercially. Probably only the people who really don't need it would want one just to look at scenarios after the fact for future reference. Unless you play in a lot of tournaments (where repetitions) or have a phenomenal memory, can help you remember all the different things. You will likely make percentage mistakes. Luckily or unluckily, depending how you look at it, a lot of the plays have very small percentage differences which ever way you go. Larry
Thanks to both Gronbog and S. Yama for the detailed analyses. An additional curiosity of this game is that the first card is dealt to each of the split hands before the user has to act. So that would offer a tiny extra benefit to the splitter, whose strategy can take into account the initial, two-card total of the second hand when choosing how to play the first. But I'll forgive you if you don't rush to add that option into your simulation!
Before I attempt this comparison, I want to make sure that we agree on the details of both strategies. S. Yama's instincts are, once again, good in that I neglected to specify what the generated strategy for the second hand is when the first hand ends up as 17 through 21. His instinct (hit the second hand to one better) is close to the generated recommendation. Complete generated strategy for the second hand is: - 1st hand stood stiff: Hit to 18 (very close: stand = 34.83%, hit = 34.78%) - 1st hand 17: Hit to 19 - 1st hand 18: Hit to 18 - 1st hand 19: Hit to 19 - 1st hand 20: Hit to 20 - 1st hand 21: Hit to 21 - 1st hand busts: Hit to 16 i.e., the recommendation is to hit to the same total as the first hand (as opposed to one better), except for 17. I want to clarify the the strategy that S. Yama wants me to compare this to is: - Hit first hand to 18 or better - 2nd hand: - 1st hand 18: Hit to 19 - 1st hand 19: Hit to 20 - 1st hand 20: Hit to 21 - 1st hand 21: Hit to 21 - 1st hand busts: Hit to 16 or better (Push is as bad as a loss)
Thanks for your kind words! It is quite possible that the bots at GameAccount play perfect hand strategy. It's reasonable to compute the correct play for a given situation with a given deck composition in real time. It does lead to the question of how much information is fair for the bot to use. Clearly using the exact order of the cards to come would be unethical, but would it be unethical for the bot to take the current composition of the deck into account? As has been correctly pointed out several times here, sims are great for after the fact analysis, confirmation of CA calculations, and providing the final word on opposing opinions, however, as you point out, we, as humans can never hope to remember the exact numbers and percentages that are spit out. Sometimes, however, we do learn some general things which we can use. In this case we learned to beware the power of the push via splitting!
Thanks to you as well for an interesting scenario! I actually do add everything I see to a prioritized list for inclusion in my software. After all, the more situations it can handle, the better! In this case, knowing the second card dealt to the second hand after the split would certainly affect the way that the first hand is played. It would be interesting to see the resulting strategy even though, as Larry points out, most of us could never hope to remember it for future use.
While there might be some oddities that would be a test of memory, such as what to do with soft hands, I think there is a single, general rule that covers much of what you gain from seeing the second hand in advance - If the second hand already has a total on which you would stand if you had to win it, then you have a free hit on the first hand whenever - the first hand is a stiff the first hand's total is less than the second's Essentially, some of the time we get to do the hand comparison that you've already discussed twice over. We can play the first hand in the knowledge of what the second hand is, and then play the second hand having developed the first.
A similar situation just came up, with the boot on the other foot: I was the one with the option to split, but it only occurred to me after the fact that I should have done so. BR1 (Me): 1112. Bet 300 BR2 (bot): 911. Bet 455 I bet 300 (rather than 200) because of the 100 min bet, which means I can take the high and low in this situation. Dealer 5. Me 3,3 bot T,5 The bot is going to double, which will give it the high even if I win 2 bets. There is no doubling for less allowed, so I can't split and then double one of the hands for less (e.g., 56) to go for the high without giving up the low. So I reasoned it didn't really matter what I did, and I just hit, despite splitting being the BS play. But of course, since my bet of more than 200 could be beaten by a push, it would make sense to maximise my own chance of getting a push by splitting. I wouldn't lose anything by it. Fortunately the blackjack gods smiled on me, and my mistake wasn't punished.
You and the Bot, sounds like fun. Hopefully, it is not the "Blue" of the BJT world. Is the game like a CSM or does it deal X% of the shoe? If the game is X% of the shoe, does the Bot make BS plays or does it make non-BS plays at some point, indicating it is counting the deck? Does it make, what might be considered perfect bets in the last few hands? One thing about it, if you can stand the cost of play, you can hone your skills. This is hard to do for a person like me (live in Dallas Area), since there is no gambling and I have to travel a long way in general to play in a BJT. Larry
Larry, It's not a CSM. The game uses four decks with a notional cut-card 'randomly placed towards the back of the cards'. But with 10 rounds of heads-up play we don't get deep enough to encounter a shuffle, or for card counting to be much of an issue (it seems to me). There is also a 40-round variant. I've avoided playing that, in part because I think the bot could gain a bit of a card-counting advantage from the deeper penetration. The bot does take insurance at odd times, leading me to wonder if it is varying some of its decisions according to the composition of the remaining cards. Without wishing to get too cloak-and-dagger about all this, I'd rather not say too much about what I've observed in the bot's play, in case my comments are seen by the bot's author, who might then apply some fixes. But there do seem to be some exploitable weaknesses. A while ago, Ken posted a link to an article about the software author and his bots, which you might find interesting. (Although there is not much mention of blackjack.) - https://www.blackjacktournaments.com/threads/5864 It seems to me that I currently do have enough of an edge over the bot to overcome the rake (which is only 5%) and expect to make a worthwhile profit. So I'm making a more concerted effort than I have in the past. But if Ken's article is to be believed, then if I do too well I'm likely to prompt some hasty software updates in my opponent!
