Median Best Hand Part II

[PairTheBoard]

How many times playing Holdem have you asked yourself, What is the Best hand I can expect to be playing against?

We began looking at this question in the thread:

http://forumserver.twoplustwo.com/sh...fpart=1#702500

There are some who say this question is not interesting. They are free to read no further. But some of us find this question intriging. There are problems with this question such as, what do you condsider "best hand". Certainly it depends on the situation but these are some of the situations where it appears most relevant.

1. You are thinking of enterring the fray with many hands yet to act. Many pots at this table end up heads up. How does my hand fare against the stregnth of the field behind me?

2. I am in a tournament and I am short stacked. My stack will be pretty much eaten up by my next blind. I look at my cards and ask myself, is this hand better than what I can expect to get in the next n hands dealt to me?

3. I am in a tourny short stacked. I would like to take a shot at stealing the blinds. There are n players to act between me and the BB. Is my hand good enough to make an All-In move?

imo, These are the kind of questions where it is natural to ask if my hand is better than the BEST hand I can expect to be up against.

One problem is settling on a Hand Ranking System. In other words, what do you mean by "Best Hand".

Aisthesis has shown interest in solving this problem for case 3. A solution would be an expansion on the recent Sklansky article in the CardPlayer in which he anyalized the situation for the Small Blind. There is a Skalnsky Ranking system developed with K (I'm sorry I don't know K's name better) where this has been solved for the Small Blind situation. This thread will look at the same problem where you are not in the Small Blind but are thinking similiarly.

Please do not [censored] on this Thead


PairTheBoard

[Aisthesis]

Nice summary of the discussion until now!

I decided to go ahead and figure it through at a 9-player table for UTG, basically just following the calculation in the Sklansky article and using the K-S rankings, which give A9s as median best hand.

The result was actually pretty shocking to me! Even under the given assumptions (which may actually favor the all-in a little bit), you need to be VERY desperate to move in UTG. Assuming all worse hands will fold (not necessarily the case for BB) and all better hands will play, the move is plus EV only if your stack is 1.53 times the total amount in the pot up to that point (blinds plus antes). I expected it to be WAY higher than that! So, anyhow, it looks to me like from a mathematical standpoint, you're going to be much better off moving in in the small stack from LP with a lesser hand than you are in EP with a fairly decent one.

Here was my methodology: I figured that 50% of the time, you win the blinds plus antes with the all-in (I just set this value to 1 for simplicity). The other 50% of the time, you see the flop against a superior hand.

Taking all the superior hands according to the K-S ranking, weighting them according to frequency of occurrence, and then running them through pokerstove individually, I get that the superior hands will win on average 66.38% of the time. Then if you calculate a break-even in terms of stack size, that yields that your stack can only be at most 1.53 times the amount of the pot for a positive EV on this move!!!

Of course, it can still be a valid play even with bigger stack-sizes if you consider that some superior hands may fold (like ATo, 66, 77 maybe). But purely in terms of cards, you need to be pretty desperate to move in with A9s UTG.

In light of Sklansky's results on the SB version of the problem, where the stack-sizes allowing an all-in are surprizingly large, the curve as you move toward the later positions is going to have to be very steep indeed.

[cnfuzzd]

Is the above mentioned article available online? If so, could someone please provide a handy-dandy link. Thanks.

peace

john nickle

[PairTheBoard]

Very Interesting. So that uses the fact that if you are UTG in a 9 handed game the Median Best Hand of the remaining 8 players is A9s. That's what allows the simplification of assuming you'll get called 50% of the time.

UTG the stack/pot ratio is 1.53. What was the K-S ratio for A9s when in the SB?

PairTheBoard

[magic_man]

According to this page, the K-S ratio for the SB with A9s is 104.12.

~Magic_Man

[Aisthesis]

It was in cardplayer. So, if you go there and look under Sklansky, you should be able to find it.

[Aisthesis]

Quite a difference, huh?!! That value is actually somewhat too high, since Sklansky is figuring stack-size as a ratio to BB rather than total pot, so you can knock off 1/3 right there, and more if there are antes. Also there's a small boost due to the fact that the SB part of it is "pre-paid." But still, the difference is HUGE!

I'm going to try and calculate it through for 5 hands (just rather tedious running so many hands through pokerstove) and see what I come up with.

Part of the problem with A9s is that it's such a big dog to lots of the hands that beat it--all of the aces with better kickers. It only holds up reasonably well against the pairs 66-88, which are still rated as superior hands in K-S. So, the "how big of an underdog" aspect definitely is going to play a big role on this type of problem.

Another consideration: With the formula, it's also easy to run this through for the hand that has, say, a 70% (or pick any %) chance of being the best among n hands. So, for the question of when it is plus EV to go all-in from a tournament shortstack, a higher probability might be desirable--certainly is from EP, but I'll be interested in the results at 50% for 5 players as a starting point.

Actually, an "equation" for all this would involve 2 variables (given the number of players), it seems to me, in order to solve for stack-size: "f" as the percent favorite the aggregate of all superior hands is over the given hand; and "p" the percentile ranking of the given hand--p would determine how often the whole table would fold with inferior hands, hence increasing the probability of winning the pot uncontested. For example, if your hand is in the top 30%, the assumption would be that you win the pot uncontested 70% of the time...

The interesting thing is that both f and p are determined in the list for each given hand. So, it's possible (easy if you could devise a program to calculate f--with pokerstove it requires a lot of legwork) for every position to determine the maximum stacksize with which it is plus EV to move in with any given hand.

So, that's another interesting thing to try: Figure out the maximum stack-sizes for 9 players on all hands superior to A9s. I think I could actually do that by hand (although I think I'd start at the top of the list and work my way down, perhaps avoiding some of the marginal ones close to A9s in rank; anyhow, the further down on the list, the more tedious the calculation becomes).

[PairTheBoard]

That's an astounding difference. It really gives you a sense of how the Monster on your Left Grows with the number of players.

Do you handle the calling requirements for the SB and BB differently Aisthesis since they will have better pot odds for the smaller amount they would have to call the raise with?

PairTheBoard

[PairTheBoard]

I'm afraid that link is not working for me magic_man. Didn't K-S post their solutions here at 2+2 a while back?

PairTheBoard

[PairTheBoard]

You know I'm not finding Sklansky in the Writers Section. I wonder what's up with that.

PairTheBoard