I received some questions about this post the answers to which might be interesting to you, so I’m going to post them here.

If I read it correctly, it says you should bet (value + bluffs) the same percentage of your range that you offer pot odds to your opponent.

Or does it say you should BLUFF the same percentage of your range that you offer pot odds to your opponent?

If I pot the river, I offer 2-1 to my opponent.

Does this mean that I should bet 33% of my range regardless of my actual cards this time?

If my value hands only represent 20% of my range, then I should bluff with 13% of my range?

If I bet 50% of the pot, I offer 3-1 and should bet 25% of my range, meaning I should be bluffing only 5% of the time.

Or does it mean that I should be bluffing 33% of the time and value betting 67% of the time (when I choose to bet pot on the river)?

And if I bet 50% of the pot, I offer 3-1 and should be bluffing 25% of the time and value betting 75% of the time (when I choose to bet half pot)?

Or does it mean you should bet 33% and 25% of my misses respectively, exactly equal to the pot odds I offer my opponent to call?

Sklansky article says with a 40% chance to beat the bluff catcher, you should bet 60% of your hands (made hands and 1/3 of your misses).

Is his reference to 1/3 of misses the number that should be equal to the pot odds (regardless of the percentage of winning made hands)?

Can you clarify the logic or game theory for me?

BTW, I am one of those players who probably would have folded the flop on a paired board (I know this is likely too passive).

I love the way you played this hand and it is an example of the kind of pressure you put on the table that I seldom do.

It is clear that I don’t bluff enough.

I’ve never actually tried to count combos and bluff with absolutely optimal frequency at the table. It’s more just a matter of having an idea of what GTO play looks like, so that I can spot exploitable tendencies from my opponents or be aware of how I myself could be exploited and whether my opponents are actually capable of doing those things.

Just because you aren’t bluffing with game-theoretically optimal frequency doesn’t necessarily mean you aren’t bluffing enough. Calling too much is perhaps the single most common mistake people make in poker, so when in doubt, erring on the side of bluffing less or even not at all against certain players makes sense.

Villain’s Range

Ian encapsulates my thoughts on the flop: “Neither villain’s actions on the flop are particularly scary – the donk is usually weak in my experience, and the raise can therefore be quite weak, too – unpaired overs, an overpair possibly too, I suppose a 9. On the other hand, your overcall is really quite scary.” At every point where Villain has put money into the pot, he’s had a decent expectation of fold equity. I don’t have enough experience with him to say exactly how wide his range will be in a spot like this, and of course he could play strong hands like overpairs or even quads this way, but he could easily be wider than that.

This is the impetus for my wanting to play back at him in the first place. I have the rare hand that actually has some equity against the top of his range plus good reason to believe his range is wider than it “should” be.

Target

This raises the question of just ambitious we should be with our bluffing. The choice is whether to go after hands as strong as AA or simply to try to make him fold the portion of his range with which he himself was bluffing.

The latter can be done cheaply and is in my opinion not optional. Any player with half a clue is not going to expect you to be light here, and if he himself has nothing, you can expect him to give up easily. A bet of $150 – $200 offers you great odds and should drive him off the non-pair portion of his range.

The bigger question is whether to try to run Villain off of stronger hands. This is a more opponent-dependent decision, as some players simply aren’t going to fold big pairs no matter how plausibly you represent a narrow range.

Against the right player, though, you really can represent that narrow range here. There aren’t many hands that are going to cold-call a raise on this flop, and a big bet on either the turn or river should polarize you to the point where a good hand reader won’t expect you to bet worse than KK for value. At that point, he just has to decide whether you have quads or some sort of airball with which you called his flop raise. Many players won’t even call the flop raise with a straight draw, so it’s really very difficult to put Hero on a bluff here, with 66 being the most likely candidate.

Given the stacks, I think bombing the turn in order to set up a river shove is the best way to do this. Even if you don’t actually shove the river, you get the most leverage by sizing your turn bet in a way that suggests it’s coming. This should generate more fold equity per dollar than checking back and betting the river, even if you were to overbet shove the river. It’s also most consistent with how Villain probably expects you to play quads.

Balance

If you believe you know whether and when Villain will fold AA (or whatever you think the top of his folding range will be), then you can exploit that by taking the appropriate line. If you don’t know, then the best thing you can try to do is use the few bluffing hands in your range to balance the few combos of quads in your range.

We can reasonably suppose that the only hand with which you will pot the turn and shove the river for value is quads. This is an unbalanced range, since Villain could exploit it by folding all of his bluff-catchers when you pot the turn. Similarly, if you have bluffs in your range when you pot the turn but never when you shove the river, then Villain could exploit that by calling the turn with his bluff-catchers and folding if you shove the river.

Thus, you need to have some bluffs inyour range for both of these plays if you don’t know how Villain will respond to these bets with his bluff-catchers. Specifically, you need to bluff in proportion to the pot odds that your bet offers on each street.

Working this out over multiple streets is complicated, and David Sklansky’s article on the subject has been a great help to me. With a stack-to-pot ratio of roughly 4, there is room for a pot-sized bet on the turn followed by a pot-sized shove on the river.

The point I’m still mulling over is how the presence of the nuts in Villain’s range changes Hero’s optimal frequency, since Sklansky’s example assumes that Villain has a clearly-defined bluff-catcher (or “mediocre hand”, to use his terminology). I’m going to keep working on it myself, but any insight that you all may have would be appreciated. As I said, I’ll post an update when I think I have it figured out.

Results

The basic thrust of Sklansky’s article, and the fundamental value of leverage, is that a big turn bet puts tremendous pressure on Villains bluff-catching range. My plan, then, was to bet big on the turn and then give up on the river. This line could be exploited by a Villain who’s willing to call the turn with a lot of bluff-catchers, so I was gambling that my opponent was not that player. I bet $600, and he folded.

Q: I have been a regular in small to mid and occasionally high-stakes MTT’s on Stars. The ranges for open-shoving, reshoving and calling that I was taught (by a very strong and respected HSMTT crusher) are miles wider than the ranges that most regulars consider to be correct. Only recently have I discovered, through many threads and posts like this (about hands I wouldn’t have thought even a second about) how wide this disparity is. Despite what should then be a massive hole in my game, I have continued to have a strong winning record over good samples especially in turbos where this sort of leak should see me be destroyed.

It seems to me then that there may be situations where despite the concept of tournament life, there may be value to taking neutral or even -EV gambles for huge stacks. It seems pretty obvious to me that even if Hero has a skill edge over the field, a 20 BB stack doesn’t give much room to exploit that skill edge; in weak tourneys like this stealing lots of pots and chipping up gradually is tough because players call raises too wide creating spots that will become awkward with the 20 BB stack. In tough tourneys too, a 20 BB stack gives you no “skill advantage” because with that stack, there is limited room to exploit whatever few mistakes people may make.

It’s just my intuition and I don’t know how to research to verify this, but I am pretty confident Andre Coimbra with 12k chips 33% of the time would have a higher ROI from this point than him with 4k chips 100% of the time.

A: It sounds like you’re talking about two different senses of the word “correct” here, and I want to address both. First there’s correct in the sense of +EV, that perhaps many people including myself underestimate how wide jamming ranges can be. That’s very possibly true, and I don’t claim to be an expert in that area – that’s a big part of why I wanted to work out the math here.

As for whether it’s correct to take neutral or -EV gambles to get a big stack, I don’t see how it can be. A big stack is valuable primarily for its buffer effect, meaning that it makes it possible for you to take some losses without getting eliminated. Obviously risking loss/elimination in order to acquire those chips directly undercuts that value.

It’s true that certain plays become available to you as a big stack that you wouldn’t otherwise have. This may be because you can make +EV plays at other deep stacks, such as 5-betting or 3-barreling, that require a lot of stack depth. Or it may be because you can “afford” to gamble in spots you otherwise couldn’t. This latter point is a direct function of the fact that each of your chips is worth less when you have a lot of them than when you don’t, so I don’t think that helps your case any.

I’ll admit that the former of these points is theoretically possible, but in practice I doubt those situations arise much. A short stacked player actually has an intrinsic advantage when everyone else at the table is deep. This is because they are all playing deep-stacked poker against each other, which often means playing more speculative hands that don’t have great pre-flop all-in equity. A short stack can exercise the tremendously powerful pre-flop shove, precluding any possibility of a rebluff, to either pick up a good-sized pot with no showdown or get it in good with a lot of dead money in the pot. Really the only drawback of a shortstack is the heightened risk of elimination – otherwise it’s nearly always advantageous vis-a-vis several big stacks. So I don’t see a lot of merit in heightening your risk of elimination in order to acquire a big stack.

The only time I’d seriously consider knowingly making a -EV play is when I feel that if I don’t, I’ll be forced to do something even more -EV later. The best example of this, which I believe comes from David Sklansky, would involve being UTG with a slightly better than average hand and an extremely short stack. Shoving may be -EV, but it will probably be less -EV than posting 50% of your stack blind on the next hand and being forced to go with a random hand.

Do you have a question for the Thinking Poker Mailbag? Please leave it as a comment below!

I just began rereading Tommy Angelo’s Elements of Poker, and in his introduction, I came across a sentence I underlined on my first read-through, a sentence that changed the way I approached my attempts to improve as a player even though I promptly forgot where I first encountered the idea. Angelo writes, “This book is about imagining idealistic extremes and then implementing practical methods of moving toward them.”

I’d heard generic advice about visualizing success before. It always seemed like vacuous mumbo-jumbo to me. What good would it do to imagine myself sitting alone at the final table of the World Series of Poker Main Event, a fistful of hundred dollar bills in each hand? How would that make me play better or bring me any closer to actuallysitting at that table?

Turns out I had it backwards. It’s not about visualizing yourself as the winner. It’s about visualizing yourself winning, playing like a winner. It’s about knowing what it looks like and what it feels like to be playing your very best poker, and then about charting a course towards achieving that vision.

Understanding this concept is critical to reading Elements of Poker correctly, and it should be a big help to you in improving as a poker player as well. So I hope you’ll find something of use here!

“‘The guy burst into the bedroom, it was dark but I did see what appeared to be a gun,'” home invasion victim David Sklansky recalls.

Sklansky was asleep in his bedroom when two men broke into his home near Wigwam and Bermuda shortly after 3 am Thursday morning.”

Thankfully, no one was hurt. Sklansky’s cool-headed (some might say robotic) personality surely helped in that regard:

“‘During the time he was there, my main focus was to make sure that he didn’t think that I was panicking or that I would do something silly,'” Sklansky explains. “‘The advice is the same as in a poker game; you try and figure out what the other guy is thinking and then act accordingly. I can imagine if he was doing this same robbery with someone who didn’t’ behave the way I did it could have turned out terribly.'”

I think it’s particularly funny that the article doesn’t in any way explain Sklansky’s involvement with poker, so I’m sure that quote comes across as bizarre to those who don’t know who he is.

Based on his forum posts, Sklansky seems to be taking it remarkably well, though in characteristically oddball fashion:

“I have no personal animosity toward these guys. They used the absolute lowest level of force necessary to accomplish their purposes. In one of my books I wrote that there is not enough difference in the punishment of criminals who harm victims and those who don’t, using Steve Wynn’s daughter’s kidnapping as an example. I’m not going to change my mind just because I am now one of those victims. In this case I doubt they will be caught anyway since I never saw them and they let the other person in the house sleep through it!

Meanwhile if this is being read by the robbers or someone who knows them I would request that they mail me those few things (photos, etc) that are of no value to them but are to me.”

Early reports indicate that the parrot is unharmed.

There are two seminal books that address the value of tournament chips: David Sklansky’s Tournament Poker for Advanced Players and Mason Malmuth’s Gambling Theory and Other Topics. I haven’t read Malmuth’s book, but my understanding is that both make a similar argument that this value is non-linear. Another words, your last chip is worth more than your second to last chip is worth more than your third to last chip etc. Each chip you add to your stack increases the value of your stack, but by less than the preceeding chip did. So if you have on chip worth x, and you double up, your stack is worth marginally less than 2x. And if you double again, your stack will not be worth 4x. This is because survival has value in a tournament where all of the prize pool is not paid to the winner.

The countervailing principle is that chips are worth more in the hands of a skilled player than an unskilled player. This is because better players will have the opportunity to use those chips to win more chips. They contain within them the player’s expected value for the tournament. In FTOPS Event 4, the first 2000 chips, which cost $300, were probably worth $750 or so to the best player in the field. Conversely, they were probably worth about $50 to the worst player.

You can see that at this rate, it is clearly correct for the best players to rebuy immediately in a tournament that allows unlimited rebuys. Even though his second $300 buys him less than his first expenditure, it still buys him more than $300 worth of value. Conversely, it is incorrect for the worst players to rebuy. In fact, it is incorrect for them to play at all.

When only a single rebuy is permitted, this could at least theoretically change. If a skilled player uses his rebuy immediately to double his starting stack, he risks losing all 4000 of his chips at once. Although his rebuy was a good investment, he missed out on the opportunity to make an even better investment in a second tournament life. Remember, going from 0 to 2000 is worth more than going from 2000 to 4000. By opting for the latter, the player cost himself the chance to do the former a second time.

However, this must be balanced against the risk of immediately getting over 2000 chips, remaining there for the entire rebuy period, and thus losing the opportunity, to invest that second $300 at all.

The one other thing I haven’t mentioned yet is that it matters how many chips the other players at your table have and how good they are. A player cannot realize the extra value that his skill imparts to his 4000 chip stack if no one else at the table has more than 2000. If several bad players to his right all rebought immediately, then the good player should do so as well, so that he will have the opportunity to invest his additional 2000 chips well against those weak players. If several very good players to his immediate left rebought to 4000, the same player might be better off saving his rebuy as an insurance policy against elimination. Otherwise, he risks playing larger pots out of position against very good players- a scenario where those additional chips would not be invested well. If the first hour is drawing to a close and his stack is below 2000, he can always take his second rebuy then.

Unlike an unlimited rebuy tournament, where it generally makes sense to push any edge during the rebuy period since survival is never at risk, this structure makes survival during the first hour particularly important. This is because the player will have the opportunity at the end of the first hour to buy chips at a discount. $300 will buy him, not 2000 chips, as with the rebuys, but 2500 chips for an add-on. It is pretty much always profitable for a good player to add-on, and if he is eliminated in the first hour, he won’t have a chance to take this good investment.

I include this exegesis on rebuy tournament theory because I have nothing interesting to say about the tournament itself. No one at my table rebought immediately, so I didn’t either. I lost my first 2000 chips with two pair in a limped pot at the 10/20 level. My opponent seemed like a fish, so I thought he would call me down with worse, but he ended up having a better two pair. Then I rebought, added on, lasted for about an hour more despite missing tons of flops, shoved over a very weak limper with K7s, and got called by one of the blinds who had AK.

I’m not all that good at PLO, but I do want to learn, and I’m sure it will be a weak field, so I’ll probably play today’s 2:00 $500 PLO 6-Max. I’ll be playing the Stars $300 anyway. Then at 4:30 there’s a $100 rebuy FTOPS that I’ll play if I’m not sick of poker by then.