Questioning the Fundamental Theorem of Poker...

[blackaces13]

I just read "The Theory of Poker" and I've been thinking about the fundamental theorem and situations that come up a lot in low limit hold 'em where I don't think it applies at all. Of course in heads-up play the FTOP is undeniably true in all situations, however in the book it says that it is also true for "almost all" multi-way pots and I'm not so sure about this. Anyway, here's an example that I think illustrate my point here:

You pick up pocket 9's in a pretty loose hold 'em game with 5 players still to act behind you. Its folded to you and you raise. The 5 players behind you hold the following hands: T8, J8, Q8, K8, A9.

Here we can see that no one here has any business calling your raise as they are all dead to 3 outs. If they could see your hand they would surely fold, according to the FTOP then calling is a MISTAKE by all of them, and when they make a mistke YOU WIN. So assumming thay all make this critcal mistake and call, this should be the BEST scenario for you. But is it? I'm not really sure, you now have to dodge 15 cards 5 times or hit your own 1-outer. I don't know if the 13.5 small bets in the pot is making up for this.

Lets now say that the flop is [2-2-7]. This is a very good flop for you and if you could see their hands here you'd have to lead off and bet instead of give the dreaded free card when there are 15 cards waiting to crush you. So you bet, If anyone behind you could see your cards would they call? Well, maybe because the pot is laying them 14.5 to 1 now and that may be good enough for a 3-outer. So I won't comment on this round too much.

Now lets say the turn is a 3. Again you bet because now you're ahead. The pot is pretty large and at this point I'd have to think that you'd LOVE to take it down RIGHT now. Also, the pot is now only giving about 10-1 and this is NOT enough to justify drawing to 3 outs. Therefore, all players behind you SHOULD fold if they knew your hand, FTOP then states that because they should fold if they knew your hand, then you want them to call, I do not think you want them to call here. All you stand to gain is 5 more bets but you risk losing the whole pot and you will not be able to collect any future bets if they miss again on the river.

I haven't checked the math on this example too extensively but if it were me I'd be fine with taking the blinds pre-flop if I knew those hands were behind me. I'd also bet and hope they folded on every round thereafter, even though it may be the correct play for all of them.

The example I used my not be the best because of the pot odds it gives on the flop. But what if you had 4 players behind you in the same situation with J8, Q8, K8, A8. Would they still be getting the right price for 3 outs after the flop? If not I'd STILL want them all to fold and thats what they'd do if they saw my hand.

I'm no Einstein but it seems like in an example like this it KILLS your chances when a lot of people make mistakes and call you. And that happens a lot in LL Hold Em. Am I right?

[Jim Easton]

[ QUOTE ]
I'm no Einstein but it seems like in an example like this it KILLS your chances when a lot of people make mistakes and call you.

[/ QUOTE ]

There 15 outs against you. Counting the flop and the other pocket cards, you've accounted for 15 cards. That leaves 22 safe cards in the deck. Your hand will hold up 22/37 * 22/36 (36%) of the time, your bet is getting 4:1 on each round, an overlay. This is a classic example of low limit holdem, you don't win as many hands, but the pots you do win will be bigger.

[chaos]

I am not sure about this statement:

[ QUOTE ]
Here we can see that no one here has any business calling your raise as they are all dead to 3 outs. If they could see your hand they would surely fold,

[/ QUOTE ]

Lets say the A9 is last and assume that the blinds both have small unsuited cards and will fold to the raise. It costs him 2 small bets and there are 11.5 bets in the pot. So the pot is offering him odds of 5.75 to 1.

We can account for 16 cards so there are 36 cards left in the deck. There are C(36,3) = 7140 possible flops. Since three Aces remain in the deck there are 33 non-Aces and C(33,3) = 5456 possible flops that do not contain an Ace. So there are 7140 – 5456 = 1684 flops that contain at least one Ace. 5456 to 1684 reduces to 3.24 to 1. So the A9 is getting proper odds to call preflop. Some of the other hands may be getting proper odds also if they knew there would not be a reraise.

This is the school of fish phenomena.

(No guarantee that my math or logic is ocrrect.)

[Henke]

I think Andrew Morton (now deceased) was the first to publicly prove this flaw with FTOP. Specifically, it has to do with the fact that in multiway pots, you sometimes want your opponents to fold correctly! In this case this is best illustrated preflop. When all those people call, the ones that made a mistake are those holding 99, T8, J8 and Q8 (according to twodimes, 99 only wins about 13.5% of these hands). However, IF T8, J8 and Q8 would fold correctly, you gain! Then you'll win 47% of the time!

But in this case, implied odds might still make it correct for 99 to raise preflop, because if a 9 flops, 99 will be a HUGE favorite to win the hand. Here FTOP once again comes into play. Your opponents would fold if they knew what you had, but they still think they have 3 outs and thus odds enough to call.

[Wake up CALL]

What part of "almost all" is unclear?

[blackaces13]

What part of "I think situations like this come up in low-limit hold em all the time" don't you understand? Obviously, my example is a strage case but that's just cause it illustrates my point more easily. I feel like things similiar to this happen all the time with a bunch of calling stations.

[blackaces13]

What is the "school of fish phenomenon"? It sounds very interesting and as though it may support my sneaking suspicion that too many calling stations can be a BAD thing.

[blackaces13]

Yeah, but I doubt it is even profitable to play this hand if you knew all those hands would call you and you have 1 out to a win. The odds you quoted assumed you made it to the river, usually you'd never get that far. And its still a pretty slim margin when you're gaining 5 Big Bets on an already 15BB pot and knowing there's a 36% chance you're about to lose the whole pot. Snatching the pot on the turn would be fine by me.

[blackaces13]

I' wrote that first post right before I went to work this morning and that made it convoluted and overly-complicated. But today I came up with a much better/easier one:

3 handed NL hold 'em. blinds are $5/10, each of the 3 players has a $500 stack. You are SB with pocket 9's again. UTG picks up AK, here you are a favorite over this hand so if he could see your hand he would fold, the opposite of folding would be going all-in. So you want him to go all-in. Lets then assume that he does because that's the opposite of what he'd do if he could see your hand. Now, since you can see his hand you call and know that you have a 13:10 edge or so. Fantastic.

But here's the killer, BB has QT suited!!! If QT could see both your hand and UTG's Big Slick, BB would muck at being such a dog to UTG right? So, according the FTOP, if BB would fold if he knew the cards out then you want him to call. WRONG. When he calls he stomps a mud hole in you and you are taking BY FAR the worst of it in the hand.

In this example you'd like nothing more than for both you and UTG with AK to flip over your cards and show the QT suited that a fold is correct.

I could probably come up with a lot of similiar examples which seem to go far beyond what is defined in the book as extremely rare instances where you'd want the best hand to raie out draws after you bet and such. Also, this example shows a case where you get hammered if the person doesn't do what they would do if they saw your cards. In a situation where you want a slightly better hand to raise out draws you lose rather marginally.

[blackaces13]

Actually, after re-reading your post it seems that I misinterpreted it. I now see what you're saying but it seems to assume that you dodged a flop, this would happen less than 1/2 the time I believe. I wouldn't even know how to go about verifying this but my gut tells me that stealing the blinds is more profitable than getting 5 calls. Yet 5 calls is what the FTOP tells us most desirable.

[Ed Miller]

It sounds very interesting and as though it may support my sneaking suspicion that too many calling stations can be a BAD thing.

While this concept (Morton's Theorem) is correct, you are drawing an inaccurate conclusion.

There are situations (usually when you have a weak made hand with little chance to improve) that you would prefer your opponents to fold EVEN if calling hurts them. That is, if they call, they lose money, and so do you (benefiting a third player... usually with a strong draw like a flush draw).

The existence of these situations does NOT mean that "too many calling stations can be a BAD thing." The reason is that the calling stations greatly improve your situation when YOU are the one with the flush draw (or set or other big hand).

If you were doomed to play only weak made hands for the rest of your poker days, you would greatly prefer that most of your opponents fold immediately. But for the range of ALL hands that you will play, calling stations increase your winrate.

Your analysis is similar to this mistaken thinking from blackjack:

"I think it's better if the dealer hits soft 17. Say I get 11 and double down against a 6. I catch a 5, giving me 16. The dealer's hole card is an ace. If the dealer stands on soft 17, I'm guaranteed to lose. If he hits, I might win. So it's better if he hits than if he stands."

The fallacy is that you are using a specific example where the rule hurts you and extrapolating that to all hands.

You make less money in your example (pocket nines) if your opponents fold than if they call. You make more money on ALL your hands considered as a group if your opponents call.

[Ed Miller]

What part of "I think situations like this come up in low-limit hold em all the time" don't you understand?

The frequency of occurrences is not the only thing that matters. It's a risk vs. reward thing.

In situations where the presence of calling stations hurts you, it only hurts you a little bit. In situations where the presence of calling stations helps you, it can help you a LOT.

If opponents call your bet taking the worst of it with a three outter, that call may cost you a small fraction of a bet.

If opponents call your bet taking the worst of it while drawing dead, that call pads the pot one full bet.

The one call your opponent makes drawing dead more than makes up for all the times he calls you taking just barely the worst of it, hurting you as well.

In general, an opponent who calls too much when the pot is small and folds too much when it is big is an ideal opponent. Unfortunately, it's basically impossible to find such an opponent. Given a choice between an opponent who calls far too much, and one who folds correctly or slightly too often, on aggregate, you are better off against the opponent who calls too much.

[Easy E]

I liked both of your answers very much, Kongs SIR!

[chaos]

The school of fish phenomena is that while any single fish is not getting proper odds to call as more of them make weak calls their odds improve. This will make their calls less bad or even correct.

In your example if the A9 was the only caller, it would cost him money to call. But since other players made bad calls ahead of him he now has the correct odds to call.

[blackaces13]

Yeah I see what you're saying. I guess that the times when I have AJ and the flop comes [J-4-2] and I know that I have a better hand than the 4 callers I'm gonna get on the flop yet end up losing the hand when a guy with 23o catches a two-pair on the river piss me off so much that I forget about all the extra bets I get when my AJ is suited and I hit a flush on the river against people who can't bring themselves to fold with bottom pair.

One important question though if you don't mind my asking seeing as how you're a soon to be author on the subject: Is it better to have a nut-flush draw against 4 calling-stations than it is to have top pair top kicker, assuming none of the CS's has a better hand than tptk but they may have a piece of the flop in someway? I've always thought that the made hand was better because its better than the flush heads-up as it will win about 65% of the time. But now I see that with all the extra callers in there, the flush draw may be the more profitable hand. Is this typically true? This happens all the time in LL as you know.

Thanks for the replies btw.

[MrDannimal]

And this, my friends, is why I

a) Love this forum
b) Am buying Mr. Miller's book as soon as humanly possible.

For a), we have a well thought out discussion that addresses an often mentioned, and fairly confusing point ("Why is it so hard to beat LL HE games?").

For b), how could you not want to buy a book that so easily and clearly describes a key LL concept?

[Ed Miller]

It isn't really possible to answer your question precisely. Compare these two instances of TPTK:

A /images/graemlins/club.gifK /images/graemlins/club.gif on a K /images/graemlins/heart.gif7 /images/graemlins/club.gif2 /images/graemlins/diamond.gif flop
A /images/graemlins/club.gifJ /images/graemlins/diamond.gif on a J /images/graemlins/spade.gifT /images/graemlins/spade.gif8 /images/graemlins/diamond.gif flop

Against 4 calling stations you are in great shape with the first hand, but real trouble with the second hand.

To try to answer your question, I would usually prefer
A /images/graemlins/club.gifK /images/graemlins/club.gif on a K /images/graemlins/heart.gif7 /images/graemlins/club.gif2 /images/graemlins/diamond.gif flop to
A /images/graemlins/club.gif6 /images/graemlins/club.gif on a J /images/graemlins/club.gif9 /images/graemlins/club.gif2 /images/graemlins/diamond.gif flop

If your opponents will call a flop bet on the first flop with a hand like queen-jack or ace-six, then you are definitely better off with that hand.

The value of a hand is determined both by the chance you will prevail AND the amount of action you expect to get. Against tight opponents, you will get little action on a K72 rainbow flop. Against calling stations, the hand is worth a lot more. As the board becomes more coordinated (and your pair more vulnerable as a result), you begin to prefer the flush draw.

[Zeno]

[ QUOTE ]
I've always thought that the made hand was better....

[/ QUOTE ]

What's a made hand? This is a little off the topic but I think it is important as a general concept of how to think about the game. You mentioned "made hand" meaning a five-card hand (your two hidden cards and the 3 community cards on the board (the flop)). But think about it this way -No had is made (completed) until the last card is dealt (let's discount opponents drawing dead).

This may seem silly and obvious to say, but I think many players do not see the game or fully understand the game of poker in this light. A small but important change in your frame of reference about this may help your overall perspective of LL Hold’em and also for poker in general. In some situations in many Poker games, for example Omaha 8 or pot-limit Omaha high, flopping a draw or draws is best – you have two more cards to make your hand (draw to numerous outs). Indeed, you can be the numerical favorite.

Another thing is that no person wins the pot until all the cards are dealt or everyone else folds. You are not entitled to the pot with a five-card ‘made hand’. Again, this seems silly to say and everyone ‘knows this’ but I think many do not ‘understand it’. This post may be more appropriate for the psychology forum but I thought it more important to say it here in this tread.

-Zeno

[blackaces13]

I just meant that tptk was already a fair hand, while the fllush draw was a potentially powerful hand but at the moment was nothing and only has a 35% chance of becoming something. It seems like you are just knit-picking over semantics. I know its a 7 card game, just wondering which is a more profitable flop, TPTK or a nut-flush draw against 4 typical calling stations. I guess I shouldn't have used the term "made hand".

[Mung]

A "made hand" is when you have AKs, and QJT of the same suit comes on the flop. :P

[Zeno]

[ QUOTE ]
It seems like you are just knit-picking over semantics.

[/ QUOTE ]

A justifiable criticism. I hesitated in writing my response but chose to error on the side of posting it. I have, over the years, observed that many poker players display an entitlement mentality to the game. It is very self-destructive in my opinion, thus my post. I hope this serves as an explanation.

-Zeno

[NaobisDad]

It might have already been argued, but I believe that S&M adress this in HEFAP. They state that in multiway pots there are exceptions to the FTOP.

I'll look it up, cuz I can't find it right now.