NLHE Tournament Hand Ranking for All-In

[Aisthesis]

The following are selected results matching a hand to a maximum stack-size given a number of opponents. The main application I’m thinking of on this is with a short-stack in tournament situations, where one is often faced with the alternative of moving in or folding.

The calculation does not include any special status for the blinds, who might have odds to call with inferior hands. I assume here that your opponents know exactly what you have and simply fold every inferior hand (hands that are inferior but would have odds to call are not considered). The result is a MAXIMUM stack size for which it is exactly 0 EV to move in in the given situation.

The stack-size is expressed as a multiple of the total pot prior to seeing any cards. This seemed to make more sense to me (rather than using BB as starting point) because it also provides a way to figure antes into the equation. For example, with blinds at 100/200 with no antes. The pot-size is 300 before any cards are seen, so a value of “6” would mean that moving in with the given hand is plus EV as long as your stack is smaller than 1,800. But with 100/200 blinds and an ante of 25 at a 9 player table, the pot is 525 before any cards are seen. So, a plus EV here is possible with the same value of “6” up to a stack size of 3,150—large enough that moving in is probably not the best play even though it should be plus EV for a stack of 3,000.

For this reason, I’ve tried in my calculations up until now to focus on those hands where the stack-size is between 3 and 9 times the pot before cards are seen. Those should provide some insight into the hands that one can consider in short-stack situations where this move is often a viable way to play the hand.

Anyhow, here are some selected results up to now (sorry about the format--I'm copying it from Word and the tabs don't seem to translate here. The first "column" is the hand in question, then the stack-sizes for 9 down to 3 players):

9 players 8 7 6 5 4 3
99 7.91 9.15 10.80 13.11 16.59 22.37 33.95
AQo 6.60 7.82 9.46 11.77 15.24 21.05 32.69
88 6.50 7.54 8.92 10.85 13.76 18.61 28.30
AJs 6.43 7.62 9.21 11.44 14.80 20.42 31.68
77 5.39 6.27 7.45 9.11 11.59 15.74 24.03
AJo 4.51 5.38 6.55 8.19 10.66 14.81 23.12
66 4.48 5.24 6.27 7.70 9.86 13.47 20.69
A9s 3.14 3.80 4.67 5.91 7.79 10.94 17.26
A8o 2.07 2.52 3.12 3.97 5.26 7.42 11.78
22 1.97 2.40 2.98 3.80 5.05 7.15 11.37
A6o 1.51 1.85 2.32 3.00 4.02 5.74 9.21
A2o 0.99 1.25 1.62 2.14 2.95 4.31 7.09
KJo 0.85 1.12 1.51 2.08 2.97 4.51 7.68
K7s 0.52 0.71 0.98 1.39 2.04 3.17 5.51
JTs 0.13 0.21 0.36 0.63 1.12 2.09 4.28
K2o 0.20 0.29 0.42 0.63 0.98 1.59 2.90

Those are the most important hands I’ve run up to now. For background as to how all this is calculated and how we arrived at this idea, see PairTheBoard’s thread “Median Best Starting Hand Part II.”

[Aisthesis]

Well, the table in my last post has gotten a fair number of views but no responses yet...

So, I'll try to make it a little more readable. Also, I should note that it is basically designed as a generalization of the Karlsson-Sklansky rankings. These should turn out identical to the stack sizes listed in my table with the exception that my table does NOT give any special calling criteria to the blinds. If a blind hand is an underdog, it is assumed here that it is folded. This shouldn't influence the actual ranking of hands at all, but it will have a small influence on the actual stack-size values (I think I'll actually run the same calculations on a 2-handed situation using my formulae just to see how much difference there is in comparison to K-S).

Anyhow, here's my attempt at putting the table in a more readable form. The column headings refer to number of players at the table (or, potentially in the hand, as the case may be--in which case the column headings correspond to a position with everyone folding to you in that position).

------9-------8------7-------6------5------4-------3
99---7.91----9.15---10.80--13.11--16.59--22.37---33.95
AQo--6.60----7.82---9.46---11.77--15.24--21.05---32.69
88---6.50----7.54---8.92---10.85--13.76--18.61---28.30
AJs--6.43----7.62---9.21---11.44--14.80--20.42---31.68
77---5.39----6.27---7.45---9.11---11.59--15.74---24.03
AJo--4.51----5.38---6.55---8.19---10.66--14.81---23.12
66---4.48----5.24---6.27---7.70---9.86---13.47---20.69
A9s--3.14----3.80---4.67---5.91---7.79---10.94---17.26
A8o--2.07----2.52---3.12---3.97---5.26---7.42----11.78
22---1.97----2.40---2.98---3.80---5.05---7.15----11.37
A6o--1.51----1.85---2.32---3.00---4.02---5.74----9.21
A2o--0.99----1.25---1.62---2.14---2.95---4.31----7.09
KJo--0.85----1.12---1.51---2.08---2.97---4.51----7.68
K7s--0.52----0.71---0.98---1.39---2.04---3.17----5.51
JTs--0.13----0.21---0.36---0.63---1.12---2.09----4.28
K2o--0.20----0.29---0.42---0.63---0.98---1.59----2.90

[Dynasty]

I don't know where you're going with this but I'll provide this link just in case it may help you.

http://gocee.com/poker/HE_Val_Sort.htm

[Aisthesis]

Thanks for the link! That's a very different table that can perhaps help me to better explain what I'm trying to do here.

The table you give shows how a hand holds up against random hands assuming the full table stays all the way to the river. I'm making almost diametrically opposite assumptions with my table: I'm assuming no one calls unless they have a hand that is favored over yours, and that they always call if they do have a better hand than you do.

The basic scenario is this: You are somewhat shortstacked in a tournament and have reached the point where you feel you have only two choices pre-flop: all-in or fold. The question is, what hands are good choices for the all-in?

My table here attempts to answer this question given two additional parameters: the number of people at the table and the size of your stack relative to the initial pot (before any cards are seen).

Some examples: Blinds are at 100/200 with no antes, and you have a stack of 1600 holding AJo UTG with 9 players (including yourself) at the table--a pretty good hand. Is this a hand where you might want to move all-in in the hope of a favorable coinflip?

According to the table, the answer is "no." The initial pot is 300, so your stack is 5.33 times this pot. Given the possibility of callers after you with superior hands, your stack-size would have to be at most 4.51 times the pot (1,353) to make moving in a +EV play on that hand.

But if you had AJs in the same position, moving in would have +EV up to a stack-size of 6.43 times the pot, so AJs WOULD be sufficient for the all-in move. If you do get a caller with a superior hand, AJs will hold up well enough that you are justified in taking the risk for the sake of winning the blinds (or doubling up).

Basically, with AJo in that situation, you will lose more total tournament dollars than you win if you move in, but with AJs, you will win more tournament dollars than you lose.

However, let's change the situation just a little: Same blind structure and same stack-size. But now you have AJo at UTG+2, and it's folded to you. So, there are now only 7 players, including yourself, in the hand. In this position, the table shows that moving in is +EV up to a stack-size of 6.55 (1,965 in this case). So, given the size of your stack, AJo is a good bet in this situation. You will now win more tournament dollars than you lose if you move in with AJo in this position.

Does this explanation help in explaining why I assume only a caller with a superior hand rather than everyone playing to the river? I felt that it was a more accurate reflection of practice to assume that either (1) you steal the blinds without a contest by moving in with these hands or (2) you get a caller with a superior hand.

For assumption (2) one of the main issues is just how often a superior hand to yours is going to occur at a table with n players. Actually, with AJo, more often than not you will have the best hand among 9 players, but when you don't, you lose so frequently that it is apparently not worth the risk.

Does this help at all in explaining where I'm trying to go with this?

[PairTheBoard]

Brilliant work Aisthesis. Anyone interested in the Karlsson-Sklansky work shoud be interested in this - including Karlsson and Sklansky.

We should compare the ratios you get for 3 players to the Karlsson-Sklansky ratios for two players. If your work is correct the ratios should be somewhat close. Your ratios will be higher because of the extra player and lower because of measuring agaisnt the pot rather then the BB. Your ratio will also tend lower because of not giving the SB and BB pot odds consideration for their calls. Taking all that into account, if your work is correct there should be noticable correlation between your 3 player case and their 2 player case.

imo This is an important attempt to extend the Karlsson-Sklansky Concept to all positions at the table.

PairTheBoard

[Aisthesis]

Thanks very much for the encouragement--as well as the initial idea that got this whole thing going!!

Anyhow, here's the comparison to Karlsson-Sklansky. I'm going to abbreviate my all-in value for 2 players as "AV2." "DF" stands for difference factor and is simply the K-S result divided by my AV2.

Hand---------AV2--------K/S--------DF
99-----------68.68------191.41-----2.79
AQo----------67.67------192.67-----2.85
88-----------57.39------159.30-----2.78
AJs----------65.50------183.22-----2.80
77-----------48.93------134.85-----2.76
AJo----------48.09------136.31-----2.83
66-----------42.35------115.35-----2.72
A9s----------36.29------104.12-----2.87
...
22-----------24.08------48.06------2.00
...
JTs----------11.46------36.11------3.15

Well, I left some out, but did want to include the atypical cases of 22 and JTs. Everything else seems to hover pretty much around a difference factor of 2.8.

The main difference is that K/S does have hands not folding that are slight underdogs--they only fold if they don't have odds to call. On my list, all underdogs just fold (even in the blinds).

Obviously, the K-S method is more accurate, but I thought the simplification was necessary just to make it doable for more than 2 players. The comparison would suggest that my stack-sizes are generally conservative.

The big discrepancies seem to occur with little pairs (22 having the smallest difference factor) and suited connectors (JTs, which is the only suited connector I've calculated, has the biggest difference factor). I'm not sure exactly why that is.

Clearly, 22 is going to get lots of underdog callers on the K/S scenario because almost any hand that doesn't include a 2 is going to have odds to call from BB. I'm just not sure why that would drive the difference factor down. It would in any case suggest that I may not be giving the little pairs quite enough credit, although even 22 comes off pretty well (marginal with 6 players, decent with 5, and downright good with only 4 players).

JTs will no doubt be typical of the other suited connectors, and the comparison suggests that I may be ranking it too high--although my calculations show it as marginal in LP. QJs will be slightly better, but I think the rest of the suited connectors will turn out completely untenable except possibly from SB, as in K-S. So, while my simplification appears to have the worst consequences here, these hands aren't going to be particularly relevant for this problem anyway.

[PairTheBoard]

I'm afraid I'm troubled by such large DF's. Your AV2 is your calculation of the same SB situation as K-S did if I'm reading it right. The fact that you're using a Stack/Pot ratio and K-S uses a Stack/BB ratio should account for a DF on the order of 1.5. For example, if the K-S ratio was 12-2, all other things being equal your ratio would be 12-3. ie. K-S would be 6, yours would be 4 and the DF would be 1.5. The fact that you assume calls are not made by inferior hands that have pot odds to call would INFLATE your ratio. Besides, that would be a very small factor for such large ratios as you show. I can't see why the DF's should be so much larger than 1.5 unless there is some other fundamental difference in the calculations that we are not seeing.

I would suggest making a pure K-S calculation for one of the hands and see if you can duplicate the K-S ratio. Then compare the two methods to see why the DF is so much larger than 1.5. Of course it's possible the K-S calculations were in error. I don't know if they were ever double checked by anybody.

PairTheBoard

[karlson]

The BB in the SK rankings was 2, not 1. So the right ratio should be around 3.

The fact that you're not allowing worse hands to call should swing it in favor of moving in, so your numbers seem reasonable.

[Aisthesis]

That explains a lot! Thanks. I was wondering why it was closer to 3 here rather than 1.5. But 3 is the number one would logically expect if the assumptions were completely identical.

I'm assuming that the fluctuations have to do with the specifics of how many underdog hands there are with odds to call in your scenario, and exactly how well these hands do against the given hand.

[cnfuzzd]

This is wonderfull work, thank you very much for supplying the effort on this.

peace

john nickle