Median Best Holdem Starting Hand
 

I believe this is an open question as far as Poker Literature is concerned, yet it should be basic knowledge for every profesional Holdem Player.

Pick a Hand Ranking System. Or better yet, do this for several of the best recognized hand ranking systems.

What is the Median BEST 2 card starting hand dealt to 9,8,7,6,5,4,3,2,1 players?

It has been solved and published for one player but not 2-9 as far as I know. I challenge the 2+2 analysts, including David and Mason, to answer this.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

I know i am misunderstanding the question. Please help me.

wouldnt it be aces? what about the rankings of hands listed here hot naked chicks

peace

john nickle

Re: Median Best Holdem Starting Hand
 

I believe the question is: of the ten starting hands dealt, what will be, on average, the best of those ten hands?

No idea how to work it out without generating a dataset, but I'd guess it's going to be pretty similiar to the recommened hands for an UTG raise at a full table.

Re: Median Best Holdem Starting Hand
 

You seem like a pretty sophisticated poster, so maybe I'm not understanding your question, but WTH:

The rankings against 1-9 random hands are well known:

http://www.gocee.com/poker/HE_Value.htm

The median hand for any number of players is easy to pick out from the list. E.G. against 9 opponents, the median hand is J4s, since half the 169 possible hands have more equity and half have less (assuming I picked the right row).

Of course "random" is not much of a system, so I guess you mean a ranking of hands within a preselected group of hands? I think this is pretty easy to generate assuming all hands go to showdown. I haven't seen it done, but offhand I don't see how the relative rankings would end up different from the random case.

FWIW, eastbay did some interesting things in this thread. But he started with random hands and only considered the HU case.

Edit: I see now cnfuzzed posted the gocee link too. The naked chicks thing threw me off. [img]/images/graemlins/grin.gif[/img]

Re: Median Best Holdem Starting Hand
 

He's asking about the median "best" hand. For instance, say ten random hands are dealt, and out of those hands, the "best" one is KJs. Then ten more random hands are dealt, and the "best" one is KK. It you did this like 100,000 times, what would the median "best" hand be?

That's what he's asking about. Unfortunately, I am highly skeptical that this process could possibly identify any new and useful information. One major issue is the value of hands varies significantly with how many players enter the pot. There is no relaible or meaningful way to identify the "best" hand out of 10 random hands.

Even if you could reliably identify the best hand, how would knowing the median best hand help you?

Re: Median Best Holdem Starting Hand
 

[ QUOTE ]
What is the Median BEST 2 card starting hand dealt to 9,8,7,6,5,4,3,2,1 players?

[/ QUOTE ]

Suppose the hands are represented by independent numbers drawn uniformly from [0,1]. The probability that n hands will all be between 0 and x is x^n. You want the median best, x so that x^n = 1/2; x = 1/nthroot(2). For n=10, that is 0.933, so look at the hands that are in the 93rd percentile of your ranking system.

The problem with this simplification is that the hands are not independent, e.g., hands in the top 0.4% tend to repel each other. Nevertheless, I would guess this approximation is reasonably accurate.

Re: Median Best Holdem Starting Hand
 

Good, i hate it when my simplistic responses turn out to be correct. Makes me feel dirty inside.

peace

john nickle

Re: Median Best Holdem Starting Hand
 

Yes, that's the idea Ed. However you miss the point of the Hand Ranking System when you say,

Ed: "There is no relaible or meaningful way to identify the "best" hand out of 10 random hands."

Once you pick a hand ranking system THAT is how you judge whether one hand ranks higher than the other. You also point out that some hands play differently according to the situation. That's why I suggest doing this for several different Hand Ranking Systems.

For example, you are in a tournament and thinking about going All In with 5 players left to act including the blinds. The most applicable Hand Ranking System here is probably just the one that determines which hands are favorites in a heads up show down matchup. In that situation do you or does anyone here know the Median Best hand you are up against with the 5 players left to act?

I think such knowledge would be very useful.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

I refer everyone to Ed Miller's post and my reply.

Typical of a Novel idea. The first responses are usually, "it's already been done" when it hasn't, "it wouldn't be useful" when it would, and "it can't be done" when it can. Don't worry guys, when it finally gets accomplished I'll let you say it was your idea all along.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

[ QUOTE ]
Typical of a Novel idea. The first responses are usually, "it's already been done" when it hasn't, "it wouldn't be useful" when it would, and "it can't be done" when it can. Don't worry guys, when it finally gets accomplished I'll let you say it was your idea all along.


[/ QUOTE ]
PairTheBoard,

I suggest patience.

You have almost 400 views on this thread. You got a couple of confused responses and a couple of clarifying ones. You got one helpful suggestion on how to approach the problem. Most people reading this are going to skeptical about your idea, but that's not (quite) the same as dismissing it.

The following are still possible:

The right person hasn't seen it yet.
Someone is thinking about it or actually working it out.
Someone has an idea/suggestion but is too busy to respond at the moment.
People are too bashful to post a possibly helpful but incomplete response.
People think the idea has merit but don't want to invest time in a speculative project.

--uuDevil

Re: Median Best Holdem Starting Hand
 

Good observations uuDevil. What I'd really like to see is David or Mason express interest in the idea. If they were to do so I'm sure it would stimulate some work on the problem. A solution would require a moderate amount of programming for simulations. I'm convinced it would be a valuable area for research and the results worthy of a CardPlayer Article or Chapter in a good Poker Book. Plus, I would like to know the answers. I think it would be especially interesting using various Card Ranking Systems taylored for different situations, like flops seen by 2,3,4... players - both Limit and No Limit.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

here's my answer: it doesn't matter

Re: Median Best HE Starting Hand (solution)
 

Ok, here goes. First, I'm going to skirt the ranking issue for the moment and just assume that we have one (Karlsson-Sklansky might be a good start, but it depends on what we're doing with the results).

I'm going to define "median best starting hand" in the following way: GIVEN a particular hand, it is "median best" if the probability at a table of n players is 50% that yours is the best hand among the random hands dealt.

So, assume we have a way of ranking starting hands, of which there are 1326. There are obviously lots of equivalencies, but it won't hurt the solution if we just assume that we can rank them all 1, 2, 3, 4, ...

You are dealt the hand ranked r at a table of n players. Then the probability that no other hand beats you is: [(1326-r)/1325]~(n-1). (Sorry, but "~" is the best I can find on my keyboard to represent an exponent).

So, at a table of 9 players we want .50 = [(1326-r)/1325]~8 or 2 = [1325/(1326-r)]~8. Solving this, my computer calculator puts r at 111.

Hence, we want the hand ranked 111.

Now, as to particulars (will depend on how to rank these hands, obviously): Pairs all have 6 instances. Unsuited non-pairs 12 instances; suited non-pairs 4 instances.

Pairs 88-AA cover 48 cases. ATs-AKs cover 16 instances. ATo-AKo cover 48 cases. That puts the total at 112.

So, assuming ATo is "better" than any hand other than the aforementioned and the worst of those, ATo should be the median best starting hand at a 9-player table.

Side note to PairTheBoard: This means my initial attempt on UPF, where I came up with AJo on the basis of a slightly faulty attempt at assigning probabilties, was indeed a little off.

As to the practical relevance: One fairly obvious case is shortstacked in a tournament looking for an opportunity to double up. What do you need in EP to go all-in? For me, anyway, it would depend also on just how short my stack is relative to the blinds. But it at least provides some orientation in making this decision, I think.

Re: Median Best Holdem Starting Hand
 

Here's why it doesn't matter. You have no criteria for "best hand". Secondly, it is not transitive. You cannot stick hand ranks on a number line and expect since A > B, B > C, A > C. This greatly increases the complexity of the problem. It's very hard to produce a mathematically precise answer of a poorly defined problem. So there is no way to find a "median" of them. Suppose you did three samples, and the best hands were 4s 4c, Ad Kc, Jh Th. Which of these three is the median? Each lose to one and beat another heads up.

If you actually put up a mathematical percise definition of the problem, I'm willing to bet someone will simulate it.

Re: Median Best Holdem Starting Hand
 

True, but you can put an exact percentage on the frequency with which a given hand wins against a random hand heads-up (or, for that matter, against any number of random hands, but the results will be different). I think that's probably the only meaningful way to define the problem (at least the only way I can think of).

Shouldn't the order of the Karlsson-Sklanksy rankings reflect this (even though they're expressed as stack sizes in SB)?

Re: Median Best Holdem Starting Hand
 

For a table of ten players who play showdown poker with any hand (no betting or folding), the hands are ranked according to showdown percentage against 9 random hands, and the median best is (from pzhon, etc.) best ~7%. There are 89 hands in this set, and for showdown against 9 random hands these are 99-AA,AK,XTs for 88 hands with AQo on the bubble.

For a more reallife answer, you would need to specify a complete ranking of hands. As others have pointed out, this is difficult or not too meaningful. In addition, knowing the median doesn't help much because much of the coolness of poker comes in the variations (like Ak and qq being dealt together).

Craig

Re: Median Best Holdem Starting Hand
 

Heads up if you have hand 50% then you are even money to have the best hand. That is, the median "best" hand for one player is hand 50%.

Against 2 players your hand h% is "best" when it beats the first player AND the 2nd player. Chances it beats the first player is h. Chances it beats both is h**2. .5 = h**2. h = 2ndRoot of .5; =70.1%. So if you have hand 70.1% you are 50:50 to have the best hand against 2 random hands.

** Average best hand for n opponents is nthRoot of .5. **

- Louie

Be advised that the actual rank of hands changes somewhat as N changes: trouble hands go down in relative value as N increases (A2 ranks higher heads-up than at a full table); suited hands go up.

This is interesting if you are adopting a starting strategy that says play if you are 50:50 to have the best hand or better.

Re: Median Best HE Starting Hand (solution)
 

By golly I think your on it Aisthesis. I think looking at the whole 1326 possible hands is the way to go since all are equally likely.

When I suggested finding the median best hand among 9 dealt I was thinking of sitting at a 10 player table, getting dealt my 2 cards and asking myself, what is the median best hand of the 9 remaining players? But looking at it your way, when sitting at a 10 player table, ask what hand you need for a 50% chance of being best at the table. Using your solution my calculator comes up with the hand ranked 99 out of the 1326 possible. Where do you figure that falls in the Sklansky ranking system?

btw, here's a link to a Heads UP Showdown Ranking system.

http://www.gocee.com/poker/he_ev_pe.html

How does the change in Ranking Systems affect the results?

PairTheBoard

Re: Median Best Holdem Starting Hand
 

You refernce "pzhon" Bozeman. What is that?

PairTheBoard

Re: Median Best Holdem Starting Hand
 

I'm thinking Aisthesis has the right approach below Louie. He uses your same reasoning but considers all equally likly 1326 possible starting hands. Here is a link to Hand Rankings accourding to Heads Up matchup.

http://www.gocee.com/poker/he_ev_pe.html

Interesting that J5s is on the bubble heads up.

I guess I agree that in a 3 player game the Bubble would be the hand ranked in the top 70% like you say. I'm not suggesting you think this but for others reading here, that's not the same as the hand that wins 70% of the time against a random hand when taken to showdown.

Thanks,

PairTheBoard

Re: Median Best HE Starting Hand (solution)
 

99 sounds about right for a 10 player table. I think the difficulty is going to be selecting the ranking system, because that's very much going to depend on the context. On the Landrum system, just counting through, I get AJo as including hand #99.

I notice the ranking system you cite definitely differs from my more intuitive rankings, which gave me the ATo for 9 players. I notice 77 is put higher than AJs there, fairly large suited bonus, etc.

I actually already find it useful just to keep in the back of my mind in various situations. But I think that in order to nail it down further, one would need to enumerate precisely the situation we're talking about, then do the ranking of hands on that basis.

In light of stack-sizes in tournament play, one might also consider other percentages than just the 50% mark (and this method makes that fairly straightforward despite the fact that we're dealing with large numbers).

Do you know how well the Landrum list correlates with Sklansky-Karlsson? At the moment, I'm in a rush and don't have time to call them up and compare.

Hopefully some discussion will get started on where one can go with these things. I'm thinking there are several (particularly, late) tournament situations to which it's likely applicable, but I'm a little hazy still as to how to define them precisely.

Re: Median Best Holdem Starting Hand
 

Interesting that J5s is on the bubble heads up.

I guess I agree that in a 3 player game the Bubble would be the hand ranked in the top 70% like you say. I'm not suggesting you think this but for others reading here, that's not the same as the hand that wins 70% of the time against a random hand when taken to showdown.

Ok, PairTheBoard, since you are going martyr on me with your, "The best ideas always ridiculed by the man," stuff...

What exactly is this "bubble" that you are talking about? For instance, if J5s is on the bubble, surely J2s is below the bubble. What conclusions about HOW TO PLAY J2s do you draw from that?

Re: Median Best Holdem Starting Hand
 

You refernce "pzhon" Bozeman. What is that?

"pzhon" is a dude who posted in your thread.

Re: Median Best Holdem Starting Hand
 

why is anybody responding to this?

Re: Median Best Holdem Starting Hand
 

Some of us are just trying to understand something that I'm sure is already obvious to you astroglide.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

why is anybody responding to this?

Because he isn't trolling. He got close with his melodramatic "You guys can pretend it was your idea all along," speech, though.

Re: Median Best Holdem Starting Hand
 

Most people reading this are going to skeptical about your idea, but that's not (quite) the same as dismissing it.

I'm dismissing it. Poker is played with MONEY. The Sklansky-Karlsson rankings, you will notice, are squarely focused around MONEY. His "idea" makes no mention of it. Thus, any results he might come up with are meaningless.

I planned to be diplomatic and walk him through it, but lost patience.

Re: Median Best Holdem Starting Hand
 

I was repeating the "on the bubble" phrase introduced by a previous poster. In this context it's just another way of saying Median Best Hand.

Suppose you are heads up in a tournament and first to act. The blinds are getting big tending to be pot committing if you play. You are first to act. You look at your hand. Isn't this the first most natural question you're going to ask yourself. "Is my hand better than what I can expect his to be for Showdown purposes?" Now you know. If your hand is better than J5s in the rankings at that link, the answer is yes. How you use that information is up to you.

In fact, isn't this a natural question to ask every time you play a hand? "Is my hand better than what I can expect my opponents to have for purposes of the kind of tactics likely to be involved in playing it?"


Playing at a Full Table there are many instances where all the early position players Fold around to you. You know that the more latter position you are in, the lower your raising standards are when this happens. Most players go by feel in this situation. Wouldn't it be helpful to see exactly how the Median Best Hand of your remaining opponents weakens in this situation as your position varies? There are many opinions about what your raising requirments should be but by looking at a Median Best Hand Measure you can see something that is quantifiable and free from opinions. Of course there are other factors involved, but many of them can also be considered by looking at various hand ranking systems taylored for the extra factors.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

Isn't this the first most natural question you're going to ask yourself. "Is my hand better than what I can expect his to be for Showdown purposes?"

NO, IT ISN'T! That's my point. The question I ask is, "Will this hand show a profit if I play it?" That is a 100% completely different question than, "Is this hand better than average?"

Re: Median Best Holdem Starting Hand
 

You need to find a better poker game. In the games I play, as long as your hand is in the top 3, you make loads of loot. Better than average rules!

Re: Median Best Holdem Starting Hand
 

Yes, of course you are right. But isn't a fundamental factor for whether your hand will show a profit a consideration of whether it is likely to be the best hand at the moment?

PairTheBoard

Re: Median Best Holdem Starting Hand
 

Yes, of course you are right. But isn't a fundamental factor for whether your hand will show a profit a consideration of whether it is likely to be the best hand at the moment?

No, not really.

Re: Median Best Holdem Starting Hand
 

I disagree.

PairTheBoard

Re: Median Best HE Starting Hand (solution)
 

[ QUOTE ]
By golly I think your on it Aisthesis. I think looking at the whole 1326 possible hands is the way to go since all are equally likely.



[/ QUOTE ]

Not all hands are equally likely, another nice little wrench in the problem. AKs happens less frequently than AKs, etc etc. I honestly doubt this information is as practical as you had hoped, I tend to side wth Ed on this one.

Re: Median Best Holdem Starting Hand
 

You are obviously wrong. There are many hands that are "money favorites" even though they will win infrequently. This is the whole point of implied odds and reverse implied odds. I play hands that will not be the best the vast majority of the time, but that will win a lot more than a mediocre hand when it comes in. A hand that is "better than average" but a huge money dog is ATo in hold em. Pocket pairs aren't likely to be the best hand (doubly so once you take into the flop) against a field of multiple players, but you can easily release it if you miss your set while punishing them when you hit your set.

Also, which is the better hand in this situation:

As2s or KdKc, if the board is 4s5s8h? KK is in the lead, but the A2 is a favorite.

Also, your point about tournaments is also incorrect, since you neglect the blind money that's in the pot.

If you're playing headsup limit hold em, there are some players whom I will play every hand on the button against (in fact, most players that I'll play headsup will fit into this category). I will save more than enough bets when I'm not winning while extracting extra bets when I do win. Plus, there's the fact that the blind money is laying me a price. Thus, having a hand that's above or below the median is pointless.

Perhaps the best example of this comes from NL holdem. If you and your opponent are both playing deep stacks and you have a player in front of you willing to commit his entire stack on the flop with TPTK or any overpair, you could play a ton of hands, looking to outflop him and get the money in when he's in deep kimchee.

You should spend your time thinking about other things, and let Noted Poker Authority Ed Miller finish his book!

Re: Median Best HE Starting Hand (solution)
 

No. That's the beauty of Aisthesis's approach. He's looking at all 52c2 = 52*51/2 = 1326 possible starting hands - each of which IS equally likely. Then when it comes time to look at a hand ranked say 110, he counts all combinations of cards that make up the top ranked hands in the Skalansky Hand Ranking system. His method is simple, sound, and I think gives a pretty accurate result.

As far as practicality I think this is like a lot of innovations. Until the knowledge is available people can't think of much use for it. But once the information is available and you have it right in front of you, people start getting ideas.

Look, there's been all sorts of computer generated results for how a hand fares going to showdown against 1,2,3,...9 random hands. How practical is that information? When does that actually happen? Yet people still LOOK at that information because it gives them a better idea of how a hand is likely to perform in actual Realistic situations. It sheds LIGHT on things. It's part of Poker Theory - which is the topic of this Forum.

Having an idea how your hand compares to the best hand it is likely to face depending on the remaining players yet to act seems like something poker players would be interested in. But like I said before, there is natural resistance to any new idea.

PairTheBoard

Possible next step
 

I think exactly the kind of situation you're talking about is where this consideration is most relevant. Basically, you're sufficiently shortstacked in a tournament situation that you're not going to be able to play the hand out, so you're all-in or fold.

Just to get the further consideration started, let's assume the following situation: The blinds have escalated to 100/200 with antes of 25(I like Cloutier's suggestion of viewing the ante stage as a time to move, since the antes really do noticeably increase total pot-size at this point).

So, the first question is: At what stack-size "x" (expressed as number of big blinds in your stack) is it enough for the all-in to have a 50% probability of having the best hand? With a larger stack, you'll presumably want the probability of having the best hand to be higher.

For simplicity's sake, it's probably also best to assume that it's folded to you in the given position. UTG, then, you're automatically talking about the ranking for 9-players, and it goes down from there. One will also need to assume that the "bunching factor" is irrelevant if it's folded to you in LP or in SB.

The number of callers expected is also going to be a relevant factor in choosing the hand-ranking system. But I think as first assumption, 1 caller ought to work and seems fairly typical--at least if my rough guess as to stack-size is accurate.

Then there's the further question of the probability that the all-in will fold the remainder of the field. Actually, here, the assumption made in the discussion leading to the Sklansky-Karlsson rankings seems like a good start: Just assume that only hands superior to yours will call (I think that was the way it was done in the relevant article).

I'm not sure exactly how to do this, but with the given assumptions (which seem fairly reasonable), one can assign a 50% probability to the remainder of the field folding. Then, you have to figure out what the equity is on your "median hand" vs. the superior hands. I think it should be possible to come up with an actual maximum stack-size for which it would be the best play to go all-in with the given hand (although this stack-size may vary from one position to the next).

While these assumptions restrict the scope of the question a little more than necessary (e.g., how about if you have callers before you?), I think working through the stack-size issue under these restrictive assumption would be the next step in the analysis.

Re: Median Best Holdem Starting Hand
 

I think the whole idea is only relevant in cases where you're basically playing the hand all or nothing pre-flop. As soon as you are capable of betting the hand all the way through, everything changes completely and can't be capture in this type of model.

But I think there are plenty of tournament situations (at least I run into them more often than I'd like) where my stack just isn't going to allow me even to bet the flop adequately. Those are the ones where this idea has a great deal of relevance imo.

Re: Possible next step
 

Yes. One of the things I had in mind was the recent Sklansky CardPlayer article in which he anlayzed going all in heads up. The idea was that if your stack size was small enough in relation to the BB then it could be computed to be a positive expectation play even if your opponent could see your cards and made perfect decisions when calling you. I believe Sklansky assigned a "Power Number" for each hand which determined the over-under for your Stack/Blind ratio.

I don't think such a thing has been done for when there are several players left to act behind you. While knowing the Median Best Hand for the players to act behind you may not solve the problem It may be that combining the Median Best Hand knowledge with the Sklansky Power Number concept might at least give you a good feel for the situation you're in. It may also be a jumping off point for an explicit analysis as you indicate Aisthesis.

Certainly food for thought for those interested in Poker Theory imo.

PairTheBoard

Re: Median Best Holdem Starting Hand
 

You should spend your time thinking about other things, and let Noted Poker Authority Ed Miller finish his book!

Just sent the manuscript to Mason ten minutes ago. [img]/images/graemlins/smile.gif[/img] There will still be a few odds and ends to clean up, no doubt, but essentially we have a book. [img]/images/graemlins/smile.gif[/img]