Is it ever desirable to put more money in the pot when your hand has less than 50% equity against an opponent's range, even when there is no fold equity?

Sklansky says no: "...it is clearly incorrect to put more money in the pot on a hand you know to be the underdog" ("The Theory of Poker," p. 102). (To be precise, Sklansky believes there are exceptions to this principle in seven-card stud and razz, but in this post I'll be talking about holdem.) Yet an interesting example in MOP ("The Mathematics of Poker") seems to say otherwise. Let me reproduce the example here (Example 7.2, p. 76).

Summary of the book's analysis:

• When stacks are $50, X checks, Y bets all-in, X calls
• When stacks are $400, X checks, Y checks
• When stacks are $1300, X bets, Y calls

The last case is pertinent to the topic of this post (and slightly counterintuitive); X, despite being the underdog, bets and Y, despite being ahead, does not raise but instead just calls. These decisions maximize the EV of X and Y.

According to this example, the answer to my question at the beginning is yes. On the other hand, this is a special situation in which both players know each other's hands and betting is rigid pot limit – conditions which do not hold in any real poker game. Yet the situation is not so far removed from actual poker play that anything similar seems impossible (and in my question I changed the condition of knowing an opponent's hand to estimating his range – something we are actually capable of doing consistently).

(MOP also takes up an example (Example 7.3, p. 80) in which the situation is the same except that betting is pot limit (not rigid pot limit), and the stack size is $400. In this case, X maximizes his EV by betting $23.26; Y is made indifferent to calling or making a pot-sized raise. Apart from being able to see each other's cards, this type of situation actually exists in poker.)

What do you think?

The amount of allin equity you have is not the same as the amount of the pot you can expect to capture when there is future action.

Yeah. I don't think I said that that is the case, but if you think I did perhaps you could point out where.

Oh, I meant that as an answer to the question.

I mean, I guess it's true that we'd rather not put money in the pot when we can't expect to win it all back on average (i.e. we capture less than 50% of the pot in the future, assuming one other player is matching our bets) but being able to capture 50% of the pot is not the same as having 50% all-in equity -- often far from it.

But isn't that precisely what happens in the MOP example when X bets with his aces despite capturing less than 50% of the pot in the future (and Y does indeed match X's bet)?

Despite being an underdog in all, X will be a favourite on the turn more often than not. So with significant money behind and complete info on the turn he can make the pot a little bigger

I don't have MOP, but I think it's important to note that both players are profiting because of the money already in the pot. When out of position, the bigger the pot is, the more you can profit by betting with <50% equity. When in position, I think you need >50% equity to value bet.

Right. So my question is, in the absence of the type of definite information about opponents' hands that both X and Y had in the example, are there any real yet similar poker situations in which it is correct to make a play like X when all we can put our opponent on is a range.

http://forumserver.twoplustwo.com/17...p-oop-1374540/

http://forumserver.twoplustwo.com/22...r-bet-1374868/

I don't think this is generally the case. For example,

X has: 109
Y has: AA
Board is: 1053
Both players know each other's cards.

Here, X has <50% out of position but would not want to bet no matter what size the pot is (although he might want to call).

I think it happened to be the case in the example because Y, who had more showdown equity than X, had a strong draw.

My statement was incomplete:

The bigger the pot is, the more you can profit by betting with between bet/(pot+bet+call) equity and 50% equity.

Let's say the pot is $60, then you bet $20 and your opponent calls. This makes a $100 pot in which you have 20% invested. If you have >20% equity then you're profiting.

That might be true in certain situations, but only if we assume our opponent is going to merely call our bet (i.e. not raise), which is usually not necessarily the case (and is not the case in the MOP example). And it may not be true even with that assumption; the previous example I gave serves as a counterexample.

But even this example doesn't work. Say we have 25% equity. (The following EV calculations assume we go to showdown immediately after the action.) Our EV from checking (assuming opponent checks back, a fair assumption since we're also assuming she will merely call our bet) is 60*(0.25)=$15. Our EV from betting and getting called is (60+20+20)*(0.25)-20=$5. In other words, instead of betting we would rather check down. (I think what you were thinking of was that if our opponent bets into us, it would be correct to call, to which I would agree.)

Your opponent isn't going to check back 100% though. You face reverse implied odds because your opponent will value bet hands that beat you and check back the hands you beat.

You're right about the possibility of a raise decreasing the profits from this play.

No, X does in fact expect to win his whole bet back and more in the future here. Despite having <50% equity on the flop, there's a lot of money left behind, and he'll be able play very well on the turn. (No bets will go in on the river, ofc.) In particular, he can give up when the draw gets there, and when he doesn't, he can make a very profitable potsized bet. And the fact that he bet flop inflates the pot so that he can make a larger turn bet in this case and thus get more value.

Again, the main point is that when future betting is possible, all-in equity is not a good measure of how much of the pot you can expect to capture after taking various actions.

A couple of practical examples, one on either side of the artificial draw–"made hand" divide:

(a) Draws (or made hands) that have <50% all-in equity but end up "capturing" >50% of the resultant pot due to future betting--what you know as an "ex-sd advantage"--often prefer to bet even without the benefits of fold equity.

Example: 97 on T82 with a high-enough SPR (NLHE).

(b) Made hands (or draws) that have st% <= x < 50% all-in equity and a significant ex-sd disadvantage, meaning they end up "capturing" less than x% equity if they play future streets, often prefer to get all the money in ASAP. "St%" is the "stack-off threshold," or the amount of equity needed to profitably stack off with 0% fold equity.

Example: KTK6 on T85 against a range and at a SPR where those conditions are met (PLO).

In both cases, the "draw" and "made hand" specifications are unnecessary, but I used them since I was focusing on practical examples.

Yaqh, I wonder if you can model PLO.

I haven't written any PLO software, if that's what you mean. Hypothetically speaking, though, it seems like a pain. How do you even input/output/visualize ranges?

If the pot is giving you 2:1 odds you only need 33% equity. I'm confused by the title premise.

It all depends how you play the flop and go to the turn, if you have around 20% equity you are making a good profit, but it all depends how the other player plays his hand

The answer is yes. Not only in NL. There are cases where you don't need to get paid more than a pot-sized bet on the next street in order for a bet/raise to be correct now for the underdog.

I showed some math on it here and at some point will factor in fold equity and the possibility of villain raising.

My point still stands, that raising in that spot isn't necessarily profitable.

Ah I see; when we factor in the money already in the pot, even if he loses more on average, the money he expects to win exceeds the amount of his bet.

If I haven't misunderstood you, you still haven't attempted to answer my original question. Do you have any ideas for a good measure of how much of the pot we can expect to capture (given an opponent's range)?

But in NLHE the dynamic changes completely. In this example, say that stacks are $100 and the pot is $4. (Just as in the MOP examples, assume players can see each other's hands.) Hero (with the draw) makes a pot-sized raise, villain (with, say, an overpair) goes all-in. Now hero does not have the pot odds to call this bet and must fold, and he would have been better off not betting at all (notice that this eliminates any future betting). This problem didn't occur in the MOP examples precisely because they were PL.

To call, sure. But not to bet.

Rei Ayanami gave a NL example above, but I think I showed why it doesn't work (namely because you can get raised all-in and not have the pot odds to call, thus making betting a losing play). I can't think of any situation that would work. I think the MOP examples worked precisely because they were PL. Could you give a NL example?