I was going to post this under discussion of the LV Hilton tournament or with the calendar thread for this tournament, but thought that perhaps other tournaments might have this "feature" as well. I intend to attend an upcoming Las Vegas Hilton tournament, so I began studying the use of the mulligan rule for various situations. I added mulligan capability to my simulation software and used it to generate strategies for 3 common end-game situations: Must win 1 bet (push is as bad as a loss) Must win 2 bets Must not lose (push is as good as a win) The tables were generated using what I understand to be the LVH tournament conditions: 6 Decks, BJ 3/2, DOA, DAS, DL, DBJ, SP4 and mulligan on the first two cards or on the last card dealt. Unfortunately, I am unable to attach the tables to my post because: We are not able to attach .html files at all. The html versions of the files are colour coded and provide the percentages for each play when you hover your mouse over a given cell. The limits on the size of other attachments (.txt, .pdf, etc.) are too small for my tables. With .pdf, I could still provide colour coding. I, unfortunately, do not currently have a place on the internet where I could post the tables and provide a link. So, if you would like to have the tables, PM me with your email address and I'll send them to you. What I can do is post the results for success in each of these situations using my mulligan strategies vs optimal non-mulligan strategies vs basic strategy. This will at least give you some idea of the value of the mulligan in these situations when used to its full potential. Code: Situation/Strategy | Optimal With Mulligan Optimal Without Mulligan Basic Strategy ------------------------------------------------------------------------------------ Must Win | 55.65% 43.80% 43.39% Must Win 2 | 43.52% 32.33% 6.25% Must Not Lose | 66.69% 52.67% 52.12% You will want to keep these percentages in mind when betting if you or your opponents have mulligans available in any of these situations. Hopefully those of you going to the tournament this week will have time to benefit from this information.

Very Nice Work Very nice work, Gronbog. You can see from these figures that the mulligan significantly increases your chances of winning a bet or two on a hand. This is why you want to save it for the last hand if at all possible. If you're acting early on the last hand you'll want to double down even more aggressively than usual because you can anticipate that your opponents with mulligans will do that too. In fact, doubling down and then replacing a bad double down card if needed will almost always be preferable to replacing one of your starting two cards. Armed with a mulligan, your probability of making a hand when you double hard 17 is better than half (52%). The main situation in which you'd replace one of your starting cards is a low soft hand, i.e. A-6, AND 1-1/2 bets is enough to cover your opponents' doubles.

very bad luck I too, appreciate gronbog's work here and per usual, Monkey's input is spot on. Having said that, the last time I used a mulligan at the LVH I was dealt a hard 12 on the last hand with max bet out, doubled it down, caught a 10 value card, used mulligan and caught another 10 value card. This happened with a count of something like - 6TC at the time. With that count, I think I would have played it the same way had I been dealt a hard 13,14 or 15, too. Sitting on 12 with 2 "draw" chances, I certainly was shocked to get two "tens" in a row with the count being what it was. As G.I. Joseph used to say "such is the life of a tournament junkie"!!!!! Billy C

Dang Dang Billy, that's some bad luck there... The greatest strategizing in the world won't help you when you're handed a klunker. Just ask Green Bay Packers head coach Mike McCarthy these days about that...

Tables Now on the Web I now have a place on the web from which to share these kinds of results. You can find the strategy tables that I had offered via PM at: http://gronbog.org/results/blackjack/strategy/bjt.com/mustWinWithMulligan.html http://gronbog.org/results/blackjack/strategy/bjt.com/mustWin2WithMulligan.html http://gronbog.org/results/blackjack/strategy/bjt.com/mustNotLoseWithMulligan.html Some notes about the tables: The tables are indexed by hand+card where 'card' is the final card dealt and 'hand' is the hand you had before the card was dealt. For example, 11+2 refers to an 11 which was hit with a 2. We want to know whether to mulligan the 2 or not and, if not, what the next best play is. The tables are divided into 5 sections (if applicable) Hard+Card: hard hands + card Soft+Card: soft hands (including black jack) + card Pairs Hard+Double: hard hands doubled down Soft+Double: soft hands doubled down Within each section, the tables are organized by the overall hand total and, within each total, the various combinations if initial hand + final card which can make up that total. In the interest of making the tables smaller: B refers to any card which will bust the original hand. For example the B in 13+B could be a 9 or a card valued as 10 For doubled hands, S refers to any card which leaves the hand stiff. For example the S in 13+S could be an ace, a 2 or a 3. The S in A,3+S could be ace, 2, 8, 9 or a card valued as ten. More Notation: 7+7 does not refer to a pair of 7s. It is a hand of 7 completed with a final card of 7. 7,7 does refer to a pair of 7s. T+A does not refer to black jack. It is a hand of 10 completed by an ace. BJ does refer to blackjack A,T+4 refers to a soft 21 (including blackjack) which was completed by the card 4. Some hands are "invertible". For example, 8+7 and 7+8 are really the same hand, but either the 7 or the 8 could be the original hand and either (or both) could be the mulligan candidate card. In the tables, for clarity, these hands are listed in the table twice; once in each inversion. I can produce tables without the duplication, if necessary. The recommended actions should be familiar (but in lower case) except for the new ones for mulligans: m: means to mulligan the card to the left of the + in the table row M: means to mulligan the card to the right of the + in the table row. You may see both in the same table cell for some invertible hands. For example, the entry for 5+9 vs a dealer 6 in the mustWinWithMulligan.html table is mM, which means to mulligan the 5 if you can (5 is the final card or the first card of a 2 card hand), otherwise mulligan the 9 (9 is the final card or the first card of a 2 card hand). If you hover your mouse over the text in a cell in the table, you will see the simulated percentages for each available action. This works best with Internet Explorer, which respects line breaks in the hover text. Firefox ignores the line breaks and generates the hover text as one large block of text. I have not tested any other browsers. The mustWinWithMulligan.html and mustNotLoseWithMulligan.html tables do not include recommendations for doubling down, since hitting is always superior when the goal is not 2 units. I know that the tables are large, but there are a few patterns that will take you a long way toward making the mulligan plays correctly. Hopefully the colour coding will help you to see them.

More Comments on Mulligan Strategy Monkeysystem is correct here. There are only a few situations in which replacing cards in your initial hand is preferable when doubling: Your initial hand totals 5, 6 or 7 In these cases you can not hope for much of a hand when doubling and doing it twice will not help. Hard 17, 18, 19 where one of your cards is a ten In these cases replacing the ten and then doubling reduces the risk of busting over attempting to double twice. Note that my results (the percentages you see when you hover your mouse over the table) are the percentages for successfully reaching the stated goal, not just of making a hand. The generation of a table for "Must Win 1.5 Bets" is currently underway. Preliminary results support this. I'll post the table once it stabilizes. The cases above also seem to apply to this situation. Other situations in which you would replace a card from your starting hand: Must Win 1 Bet or Must not Lose: Most stiff hands - replace the smaller card Most Soft Hands <= 19 - replace the non-Ace

Excellent work A great kudos to gronbog for his work on simulating Mulligan strategy and putting the results in tables. This, along with bluelight’s 4 players’ correlation tables, is one of the best projects published here. There had to be a lot of work put into getting this well thought out project. Thanks, S. Yama

Thanks for your kind words! The software changes were not too bad. The longest amount of time was in running the sims. It took 3 weeks each for the sims required to generate the strategies to converge!

Using the Mulligan to Win 1.5 Bets Sorry this took so long. I've actually had these results for a while, but got side tracked by a couple of other projects. Still these results may be in time for anyone playing at LVH this weekend. The table for using your mulligan to win 1.5 bets is now available at: http://gronbog.org/results/blackjack/strategy/bjt.com/mustWin1.5WithMulligan.html The main difference between this table and the one for needing to win 2 bets http://gronbog.org/results/blackjack/strategy/bjt.com/mustWin2WithMulligan.html is that you mulligan the non-ace in your initial low-soft hands if possible, hoping to get a blackjack. This is as Monkeysystem asserted. Another difference is that it recommends doubling all low hard hands (7 or less) vs Ace where the "must WIn 2" table recommends using the mulligan. I'm not quite sure what the explanation for this is since using the mulligan after doubling these hands would not be very helpful, especially for 5, for which it would not be helpful in any way. All hands must be doubled or split at some point (except blackjack) and I expected the results for hands which cannot be mulliganed into a blackjack to be the same. It should be noted that this strategy conisders doubling for 1/2 as well as for the full amount, since a loss of a double for 1/2 in one hand can be rescued by winning the remaining 3 hands of a 4-way split (among other possibilities). For single hands, the expectation is always the same, and the full double is recommended for simplicity.

Mulligan thoughts that may be moot. I have no idea if a 4 hwin would ever be needed. Would you ever need to win 4? I witnessed a guy win 3 if memory serves me. No mulligan involved. He split TT and caught a third T. Won 1st place. If you need to win 4, assuming Aces cannot be resplit, you have to mulligan the Ace and hope to pair the non-ace, split it, and then double anything on both hands or re-split and double at least 1 of the three hands. I've never had enough chips left to do this but I could see a chip leader possibly running into it. I imagine it would be tempting to keep an Ace out of habit or lack of knowledge of the resplit. A second thought; If you catch a natural on your first two and need to toss the Ace and attempt a 4 win, I'm curious if there would be confusion. I know that some tournaments pay 2 to 1 on a natural. This negates doubling a natural. But, I do not know if you'd be allowed to mulligan one of the cards. The mulligan rules I've seen state you can mulligan either of your 1st two. My concern would be a misinterpretation to the affect: you can't double a natural so you can't make any changes at all to a natural. I don't recall reading a rule that specifically forbids changing a natural with a mulligan (and being "stuck" with a natural and your 3-win). I think it's worth reading the rules carefully and maybe asking so the TD can't weasel out of it in the heat of the battle. No offense, but if an Asian player is involved and he has friends on the rail, the TD will be catching an earful of pressure and could make a dumb decision. If it's an invited player, a rule decision could go that way as well. These scenarios have to be far left and right of the norm but they came to mind.