Median Best Hand Part II

[Aisthesis]

That's strange. I'm almost sure that it was in cardplayer. If UPF is archived back that far, I think Lee was the one who originally referred me to it. But it was probably something like last December or January.

[Aisthesis]

Ok, in line with what I was saying in the previous post, I started doing some figuring here.

Let's set the amount in the pot at 1 and solve for stack-size X (for any pot-size this will be the factor to multiply by)

f=average winning percent (weighted by frequency) of superior hands over the given hand

s=percentage of same hands (very small, but just to be exact--as we will see, s is almost completely irrelevant)

p=probability with the given number of players that your hand is best (note: p DEPENDS ON THE NUMBER OF PLAYERS IN THE HAND!!)

Given these numbers, you will win uncontested (since you have the best hand or a tie) on average p + .5s

And you will lose on average .5s + (1-p-s)(f-(1-f))X

So, break-even occurs when
p + .5s = .5s + (1-p-s)(2f-1)X

So, for break-even: X = p/[(1-p-s)(2f-1)]

[Aisthesis]

No, I didn't handle calling requirements for SB and BB differently, and that will definitely make a big difference when we're talking about such a small stack (hurting the all-in, since you no longer get a 100% win due to folding).

The reason I didn't do it was just because it looked to me like it would complicate things enormously, and I don't think it's going to make a huge difference once we get into the more relevant stack-sizes.

In the (hopefully rare) cases where you get down to 3 BB or less, I think the solution will still give some fairly accurate values. In that case, I think I probably would consider going all-in with A9s UTG, although the "solution" up to now seems to suggest that as long as there are no antes, it might be worth waiting out the round.

I think where the math becomes most interesting is in the (for me much more frequent) situations where you have something like 6-12 BB left in your stack. And in that case, the BB still has to call a bet of at least 5 BB. I guess that probably would allow him to call even as a 65-35 underdog, but in practice, I don't think most people will.

But obviously, some non-mathematical factors are going to come into play here: A big stack that won't be hurt by losing against your all-in is going to be much more likely to call, whereas a small stack who would go out if he called and lost is going to be very picky.

Anyhow, for the reasons mentioned, I'm hoping that it won't hurt the model severely if one just ignores the fact that SB and BB are going to have different calling criteria than the others at the table. Factoring that in does seem to me to make a big difference in the "complication level" because one would have to go through ALL the hands in which, for example, BB is less than a 2:1 underdog or whatever.

[PairTheBoard]

Yes. I agree with your thinking. Better to get SOME good results than NO perfect ones. Something to keep in mind though is that the Stack/Pot ratios will be somewhat higher than without the simplification. That makes the 1.53 you calculated for A9s even more impressive. What a difference a field of competitor hands makes.

Something all this brings to light is that when tight play is advocated for early position in a ring game, it's not just the fact that you will be in a positional disadvantage, but it is also the fact that you are up against many unseen hands that can have yours beat.

PairTheBoard

[PairTheBoard]

It was definitely in the CardPlayer. I don't know why Sklansky is not showing up in the Writers' Archive. I don't recall if he did a follow up article but in the one I remember David worked out the math for one example and discussed the principles behind the idea that you could compute a +EV for the play even assuming your opponent played perfectly against you. Since your opponent won't play perfectly the real EV will be even better.

David invited people to carry out the calculations for all hands. That discussion took place here at 2+2 where I believe K ended up doing the work. It would be nice if someone could find those threads and post a link to them that works. When I get the ambition I'll look for them myself.

PairTheBoard

[PairTheBoard]

That looks right to me. I suppose to be precise you would make p the probabilty that no one has a hand good enough for pot odds to call, and f the average win frequency for those hands that do have pot odds to call. But similiar to your treatment of the Blinds it may not be worth the complication when X is relatively large.

You can then calculate p using the same math technique as when you calcuated the Ranking for Median Best Hand.

Ideally it would be nice to have a spreadsheet with win percentages for all 2 Hand matchups. I suspect K used something like that when he did his calculations. You could then automate a lot of solutions. I myself would not want to individually enter every hand matchup in a 2-dimes type program to get the figures though. However I think you could get a good enough aproximation with a simplification by way of Hand Matchup Types. There are only a few Hand Matchup Types and all matchups of the same type have fairly close win frequencies. Sklansky had a thread going here recently which basically asked what your average win freqency would be if you could see your opponent's hand and only played when you had an edge against it. I was able to make a pretty good aproximation to the correct answer by just considering Hand Matchup Types.

For non pocket pairs you are up against:
1. 2 Undercards
2. 1 Undercard and 1 Middle card
3. 2 Middle cards
4. 1 Overcard and 1 Undercard
5. 1 Overcard and 1 Middle card
6. 2 Overcards
7. An underpair
8. A Middle Pair
9. An Over pair

With Pocket pairs you are up against the Hand Types:

1. Under cards
2. 1 undercard and 1 overcard
3. overcards
4. An underpair
5. An overpair

I think I got them all. You can pick examples of each Matchup type. I would use differing suits with minor straight chances. Then use those win freqencies for all hands of the same type. Figure all the greater and lesser flush and straight potential hands cancel each other out - aproximately.

I think such a settup would allow you to pump out a lot of good aproximate solutions without a whole lot more work.

PairTheBoard

[Aisthesis]

Another complication I just thought of. What I have been doing here is assuming that all hands lower on the list would fold. BUT, there are some hands lower on the list that are actual favorites over the given hand but lower in ranking: namely the little pairs. This will make things even worse for A9s.

This kind of situation is going to come up with every list one uses, as AK is pretty much always ranked higher than 22 or 33 but is still a small dog to those hands.

I'd be interested to hear how you think one should deal with this: Should one include on the side of callers any hand that is a favorite, however slight, to the hand in question? Or should one include only hands higher in ranking, since those lower on the list may be superior to the particular hand but will be huge dogs to others in the range of possible holdings given the all-in raise?

Somehow, including all favorite hands seems better to me and should give a more conservative estimate of maximum stack-size for the all-in.

It's not clear to me whether this would ultimately favor pairs more or not: For pair hands, it would increase the number of hands that fold against the little pairs (AK will just fold), but the small pairs will suffer enormous losses if anyone else has a bigger pair. Also, as I discovered doing some calculations in response to a recent UPF post, AKs actually is a slight favorite over 22 when neither 2 is the same suit as the AKs (not sure whether this even still holds for 33--probably not, since the margin is so slim already for 22).

I really do think we already have one significant conclusion from all this: In the outlined tourney situation, with any reasonable stack-size, you need a VERY good hand to move in UTG.

Do you see what I'm driving at here? Although inclined to go with the "any favorite" method, I'm somewhat ambivalent about which is really best.

[PairTheBoard]

Yes. This is what makes the Skalnsky idea work. You assume that any hand with an edge to call does so. If you can show you have an edge under that assumption then clearly it's a +EV play because in reality your opponents will play less than perfectly. So you essentially use a hand ranking taylored to the hand you're looking at. Every hand that's a showdown favorite should be ranked higher. You can still use the p probability of a hand being "better" on that basis. I wouldn't worry too much about borderline cases like AKs vs 22 unless you know them already. Going by the Hand Matchup Types that I listed it's easy to see which hands should be favorites. I'd suggest trying to simplify things as much as possible. I don't think anyone's going to make precise decisions with the solutions. The value is in giving us a general idea of what's going on. Like you say, the solution for A9s UTG has already opened our eyes.

PairTheBoard

[Aisthesis]

I'm guessing the complete coinflips won't influence the results much anyway. What's interesting also, after all this, is that really it can be calculated completely independently of any ranking list!!! A stack-size will be the result.

If one wanted to get really picky, it would also be worth noting that it can actually be rational to call even outside of the blinds as slight underdog at times: for example, if you're only a 49-51 underdog but you only have to call $1,000 for a shot at a pot of $2,000. But I don't think it's going to make much difference there.

Anyhow, I'm probably going to be away from my computer for a couple of days (so don't think that my lack of posts means that I have lost any interest in this idea).

I think we both have a pretty good idea as to what hands are underdogs to any given hand, so that won't even require pokerstove in all but a few cases. Only the superior hands are really relevant, and stack-size is going to depend a lot on how big a dog one actually is. Pairs will also be fairly easy down to a certain point.

Anyhow, a preliminary consideration before wrapping this up for the next few days: With 9 players, blinds at 100/200 and antes at 25, there is 525 in the pot before anyone has bet. In my experience, the most relevant stack-sizes for thinking about this problem are when you're in the range of 1,200-2,400--that is, 2.5 to 5 times the current pot. Below that, you're getting extremely desparate (also useful to figure out, but the A9s question has already given some pointers in this direction). Above that, you're in a position to actually play the hand, and, at least for me, I wouldn't be inclined to go all-in at that point with a stack any larger than 2,400 (actually even at 2,400, I'd be awfully inclined to try a limp even with a big hand and then proceed from there). The stack of 1,200 also has waited awfully long to make a move, but I do have something like that happen once in a great while if the cards are running in some bad ways. But I think a stack of 2.5 times the pot really is the low end of the spectrum.

With blinds at 100/200 and no antes, the pot is only 300 prior to all betting, and I think that tends to raise the maximum value of interest. With as much as 1,800 (possibly even more), I think the all-in is well worth considering, as it's awfully hard even to limp at that point. That's a stack of 6 times the pot, and I think the math of the all-in move might be worth investigating up to stacks as big as 7 times the pot.

In practice, I'm still a bit undecided as to when you've actually reached the "all-in or fold" point, but I think this problem should have a lot of interest for stacks from 2.5 times pot up to 7 times pot. That at least narrows down the range of hands for which one will need to go through all the necessary calculations. (a complete table matching a hand with a stack-size depending on number of players is also possible for any given hand, but I think it's really mainly worth the effort on hands where the result is in that range).

[PairTheBoard]

Yes. Absolutely. Look at the most interesting situations. I'm glad to hear you have some steam left for this. I for one am very appreciative of your work. I know I am guessing when in those situations you describe. Some hard analysis to support those decisions would be a big help imo. Looking forward to your return.

PairTheBoard