NLHE Tournament Hand Ranking for All-In

[Aisthesis]

A question for the board:

At what (maximum) stack-size (if any) relative to the blinds does "all-in or fold" generally become the best strategy to pursue?

I usually put it somewhere around 10 times the BB, but I have the impression that I start adopting this shortstack strategy earlier than a lot of people do. (I just really don't like getting into a position where even doubling up isn't going to get me back into the game)

I also see players who will even limp when they're all the way down to stacks of around only 3 times the BB. I personally would prefer to just risk going out than getting down quite that far, but some people may have success with refraining from shortstack all-ins altogether.

Anyhow, my question is obviously directed toward trying to figure out the most important hands to run and post in my "table for short-stack all-in decisions." So, I'd be most interested in hearing others' views on this.

[PairTheBoard]

karlson, Great to have you join the discussion. Is there a link anywhere to your work? I've been going on memory as we haven't been able to find the 2+2 Thread where you posted it in the Archives, and David's CardPlayer Articles don't seem to be showing up in the archives there either.

Are you saying that the K-S number is actually a Stack/SB ratio? Aisthesis is giving a Stack/Pot ratio so that would account for a DF of 3. But I thought your K-S number was a Stack/BB ratio.

Also, I believe Aisthesis' excluding lesser hands that have pot odds to call should move his ratio slightly HIGHER. Under his conditions the opponent is playing somewhat LESS than perfectly. That makes the move MORE advantageous and therefore doable with a LARGER stack. Yet his DF's are mostly smaller than the 3.0 - Assuming K-S is a Stack/SB ratio that would give the 3.0.

Thanks,

PairTheBoard

[Aisthesis]

I was thinking the same thing.

Actually, with the K-S list being a stack-size in dollars with blinds at $1 and $2 (if I'm understanding this correctly), it will end up being a stack to SB ratio, as you say.

But then why does my ratio to K-S hover somewhere around 2.8 rather than a little over 3? The underdog callers with pot odds should indeed mean that you win a little LESS than the BB rather than more... ??

I think the reason for that is probably that you already have $1 in the pot before cards are seen on the K-S calculation. Roughly speaking, that should just add 1/3 of the pot to your initial stack.

While this figure probably also involves a little fudging due to odds complexities, we have: 2.8 + 0.3 = 3.1

If you just add 1/3 to my figures for 2 players (where I also ignore the fact that you have money in the pot already--mainly because I already viewed that problem as solved and was more concerned about cases outside of the blinds), that puts it very very close to the expected correlation with K-S.

[PairTheBoard]

Yes, the DF should be slightly over 3 rather than under.

I'm not sure about what you're saying concerning the SB having already been invested. Say the Blinds are $1 and $2. You've already put in the SB. You look at your remaining Stack Size, say S. The pot is $3. You make your All-In move. The BB must call an additional S-1. There is a value for S where you break even under assumed calling criteria. At this point Your's and karlson's calculations should be identical except for the calling criteria. Your breakeven S should be slightly higher than his. He forms the ratio S/1 while you form the ratio S/3. I don't see where the SB bet makes a difference in the two calculations.

PairTheBoard

[Aisthesis]

Just to give an impression of the math underlying all this, I thought I'd post an experiment I did with 22.

First, the "correct" method according to the assumptions I've made was that ALL superior hands would call and all inferior hands fold (exception: a favorite of less than 50.49% was taken as a fold). This is listed as 22A in the table below.

But for the case of 22 (and all smallish pairs), this results in some calls that are rather unusual in practice: Lots of suited connectors and one-gappers end up calling whereas AKo has to fold. Of course, this isn't going to happen in practice given the range of hands that you're likely to move in with. But all of these hands are very close to coinflips against 22.

So, anyhow, I thought I'd run 22 only against the pairs and see how much that result differed from the "correct" result. This is listed as 22B.

Third, I wanted to see how much it influenced the results if you just had a lot of callers that were minimal favorites. This was mainly to see if I was still ok eliminating the 50.49% favorites from the "call" category. So, I just decided to run 22 under the assumption that there were 30 additional hands as 50.4% favorites. I added these to the pairs under 22B (suited connectors all fold on this one). The results here are listed as 22C and should be compared with 22B since the suited connectors are out.

Here are the results:

---------9------8------7------6------5------4------3
22A------1.97---2.40---2.98---3.80---5.05---7.15---11.37
22B------2.79---3.29---3.95---4.88---6.28---8.62---13.31
22C------2.47---2.95---3.59---4.50---5.86---8.16---12.75

So, as expected, excluding the superior callers does drive the value of the hand up somewhat, and throwing in 30 marginally superior callers drives the value down.

While the extent of change on these various scenarios is pretty noticeable from a purely mathematical standpoint, it really hardly changes the value of the hand at all from a more practical point of view. All 3 ways of calculating the hand put it above A6o and below A9s. However, the "incorrect" methods (22B and 22C) do put the hand as better than A8o, while the "correct" method ranks 22 below A8o.

Also, we're actually talking about pretty much the same positions and the same degrees of desperation (stack-sizes) where 22 becomes a candidate for moving in.

So, I basically concluded that the "correct" method isn't going to distort the realistic results very much even though it's unlikely that one will get an all-in call from a hand like 76s. Realistically, it is the superior pairs, to which 22 is going to be a big dog consistently (actually it will fare better than is given if a hand like 44 folds) that are the main factor in determining the maximum stack-size and position for an all-in on 22.

In practice, 22 will also get callers from some inferior hands (like AKo for sure, but presumably a lot of other aces and some decent kings), but these coinflip hands are not going to influence its total value much one way or the other.

[PairTheBoard]

Wait a minute. Maybe I'm getting it. When you did the AV2 case you just extended the same method as you were using for the other positions. In the other positions you bet your Stack S and the BB has to call S-2 rather than S-1 as I say above. If you still have the BB calling S-2 in the AV2 case then you are looking at S before the SB is taken out of it while karlson is looking at S after the SB has been taken out. But wouldn't that make your break even S LARGER than his. For example, If your Break Even S was 4 his would be 3.

PairTheBoard

[Aisthesis]

Maybe you're right... ?? You definitely are from the standpoint of the BB and his calling criteria.

But I was thinking that (whether or not Karlson's stack-size is before or after posting the blind), the fact that SB has already put in $1 should make the all-in a little more favorable for the SB than it is in the case where you have nothing invested in the pot prior to seeing any cards.

Just to construct an extreme example: Your stack is $4 in the K-S scenario, but there is no SB at all in this scenario (let's call it scenario A.

Then if the normal SB scenario is scenario B, but you already have to post $1 out of your $4 for the SB, you'll have different all-in criteria, I would think.

In scenario A, you risk $4 to take down a $2 pot (if BB folds). In scenario B, you risk $3 to take down a $3 pot (if BB folds), so you're much more inclined to move in in scenario B, I would think--although BB has exactly the same situation either way. BB in both cases has to call $2 for a shot at a $6 pot.

So, it does seem to me that BB faces the same decision either way, but I think the extra dollar you've put in pre-flop is going to prejudice you more toward an all-in from the SB than in cases where you haven't invested anything yet.

Simply adding 1/3 of the pot to your stack is presumably overly simplistic as an approximation of the difference, but it should be a little bit like that.

[Aisthesis]

Hmmmm... I'm getting a bit confused here, but I think you're right.

Ok, taking out the factor of 3, my results give a stack-size pretty consistently HIGHER than K-S. That means, on my calculation that you will go all-in MORE frequently.

So, actually my SB consideration would be a reason why my results should have a difference factor greater than 3 in comparison to K-S.

On the other hand, the special calling criteria of the BB is going to detract from the EV I assume because BB is going to win part of his BB back by calling on some marginally inferior hands. That should tend to make K-S (again, taking out the factor of 3) more conservative.

Ok, so assuming my math is correct, I think the conclusion would have to be that the marginal calls by the BB tend normally to drive stack-size down more than the fact of putting in the SB drives the stack-size up.

Does that make sense? It does mean that my SB consideration was completely misleading... (sorry)

[karlson]

I think you guys basically have it down, but I thought I'd clarify what I could.

The DF, I thought, was KS stack/A stack. So if it's less than 3, that means A is larger, so it means you go in more often.

The stack in the KS rankings includes the SB, as far as I remember. I'll check on this. So this means that the BB has to call S-2, and the SB has to put in S-1. If you take s = S-1 and p = pot, then the SB risks s, and the BB has to call s-1 to win s+p. So if you want to duplicate it exactly, give the BB some extra pot odds (he only calls s-1).

So I was thinking about the extension to 9 players or whatnot, and I realized there might be a really bizarre dynamic with a hand like 22. Right now it won't occur, because you don't allow underdogs to call. But let's say you change that to allow slight underdogs to call. As far as I understand, all hands call independently if they can. So let's say I go all in with 22 for like 5 times the pot. Virtually no hands, unless they have a 2 in them are enough of an underdog to fold. So they all call! About half the time (very rough estimate) no one else will have a pair. All of a sudden 22 is in a big time +EV spot, so it'll be willing to do this with quite a high stack. I'm not saying this will definitely occur, it depends on exactly how +EV it is and how often it's against a bigger pair. But maybe the other hands shouldn't call independently. You could do it so that the caller flips his hand up, and the next player has to consider beating both of them.

PS. If you guys want to check the SK rankings or whatever else, I'd probably be willing to share my code.

[Aisthesis]

Thanks very much for the clarification! I was initially confusing the question, I think, of whether more or less than 3 makes my "AV" more conservative or more aggressive than your calculation.

I also agree fully on the weirdness of 22 from a mathematical standpoint. I'm still inclined to think that my table will do pretty well in terms of practical evaluation of the hand--even if one were to throw in all kinds of callers, since, as you point out, when it does hold up, it will win an enormous pot.

The offer on the code is also EXTREMELY generous, but I suspect it wouldn't do me much good, since I really doubt that I'll understand the language. I'm really only versed in working with certain aspects of Excel and Access, but PairTheBoard might have some interest in that, as I think he has a lot more computer knowledge than I do.

I do have a question for you on that note: Basically, what would it take to run the truly precise answer for various numbers of players?

I've run all this just through moderately simple formulae in Excel and gotten my relative hand strengths by running everything individually through pokerstove. Even keeping all of my other assumptions but just giving special status to SB and BB, I think the project of figuring the problem for n players (I probably should also include 10, which would be really easy) would only be doable by letting the computer do a lot more of the legwork and hence writing a lot of code (which I'm probably incapable of doing without at least spending a few months immersing myself in the language). Also, particularly for small stack-sizes, you have the possibility of underdog callers with odds even outside of the blinds.

So, what I'm driving at is whether the additional work involved would be worth it in terms of polishing up some inaccuracies. It doesn't look to me like the results up to now are off very much (obviously, complete accuracy would be better), and I have no clue how much work would be involved in fine-tuning it to factor in the additional complexities for n players.