This will be my last post until at least September 28. I'm heading to Paris for my honeymoon.

Concept No. 12: Be wary of overcallers.

Yes! Good idea. Overcallers are often on a draw, but they are also often slowplaying a big hand. This is especially true if there are no draws on the board, such as the JJ4 rainbow flop in the example in the book.

That's all I have to say for this one. See you in a few weeks!

## Friday, September 11, 2009

## Monday, September 07, 2009

### Analyzing NLHE:TAP Concept 11

I'm commenting on the sixty concepts at the end of No Limit Hold 'em: Theory and Practice by David Sklansky and Ed Miller. Let me apologize in advance: this one is a little more mathematical and a little less conclusive than most.

Concept No. 11: A big bet is the most relevant and accurate information at the table.

Once again, I think Sklansky and Miller's advice here is probably bad, but this one is particularly difficult to analyze because it's unclear what is meant by "relevant and accurate information." In any case, this concept's claim is far too broadly stated, and only holds in some circumstances.

If we take the claim in the most literal sense, this concept is not always true. Infinite counterexamples exist. Perhaps you know that your opponent always makes big bets; in such a case, the big bet provides essentially no information at all. Or, sometimes you may have the nuts, which is probably the most "relevant" factor if you're trying to figure out how to proceed with the hand. As far as "accurate" information goes, usually preflop play is most accurate, since you will have observed your opponent in many more preflop situations than postflop situations. Of course, all of this varies from player to player. It seems overly bold for S+M to make a claim that a big bet is (presumably, always) the most relevant and accurate.

If we're not to take this concept literally, let's consider whether it is at least good advice in a figurative sense. This concept seems suggest that players should respect the big bet more than they do, and thus fold to it more. Do players tend to underestimate the importance of a big bet when trying to determine their opponent's possible hand? Human nature actually suggests the exact opposite; there is a phenomenon called the "base rate fallacy" that seems like it could apply here. It has been established that people tend to discount base rates (which could correspond to your read of your opponent before the big bet), and focus only on the most immediate and salient information (such as the big bet) when estimating likelihoods. Are poker players similarly likely to focus to much on the most immediate information? I think sometimes yes, sometimes no. There are certainly a lot of players who are not sensitive enough to bet sizes, and for these players, taking this concept to heart is probably a good idea. I'm not sure that most players fall into this category, though.

In their explanation of this concept, Sklansky and Miller say "all information from the past takes a back seat to the fact that they've made a big bet." I think a fair way to interpret this is "the probability your opponent holds the nuts, or nearly the nuts, given that he just made a big bet, is greater than the likelihood that he did NOT hold the nuts before he made that big bet." Mathematically, we can write this as:

Suppose we are facing a big bet and we think our opponent is either bluffing or holding the nuts. Before he made the big bet, suppose we thought that the way he played the hand meant he had only a 0.05% chance of holding the nuts. This seems reasonable in S+M's example where your opponent is very tight, raised pre-flop, and the nuts is 74 (actually, .05% seems a bit too high, but let's give S+M the benefit of the doubt). Now let's say we think our opponent would make a big bet 80% of the time if he held 74. Otherwise, he would make the same big bet 2% of the time as a bluff. The base rate fallacy would be to think that your opponent would now be holding the nuts 80/82 times, or about 97.6% of the time. Of course, this is way too high because it ignores base rates. In this example, we can use Bayes' theorem to find the actual probability that you are up against the nuts. Bayes gives us the following equation:

P(big bet|nuts) = .8

P(nuts) = .0005

P(big bet|nuts') = .02

P(nuts') = .9999

and so:

P(big bet|nuts)*P(nuts) = .0004

P(big bet|nuts')*P(nuts') = .019998

Then,

If you don't like the rates I chose, I guess it's most likely because you think that even very tight players tend to hold 74 on the river more than .05% of the time in this situation (the situation: he raised preflop, and 74 is the nuts on the river). I admit, this is a tricky one to estimate, but we can do another Bayesian inference to examine my .05% claim. That is, we want to know if .05% is a reasonable approximation for how often a tight, pre-flop raiser holds 74 on the river in this situation. That is, we want to find P(74|saw river).

First of all, let's assume our "tight" player raises preflop with 74 only 1% of the time when dealt to him. Also, assuming we don't have a 7 or a 4 in our hand, our opponent will have been dealt 74 only 1.3% of the time. So, P(74) = .013*.01 = .00013.

Second of all, we need to approximate how often 74 will see the river in this situation. Remember, the situation is: 74 has already raised preflop and seen a flop that gives him a draw to the nuts. IN order to be conservative and to simplify the math, let's assume he will never fold before the river in this situation. So, P(saw river|74) = 1.

Third of all, we need to approximate how often a player would have seen the river in this situation given that he raised preflop holding anything. This is another tough one. I think this is likely over 50%, but let's be conservative and choose 26%, which will also help out with the arithmetic. So, assume P(saw river) = .26. Bayes theorem then gives us:

Concept No. 11: A big bet is the most relevant and accurate information at the table.

Once again, I think Sklansky and Miller's advice here is probably bad, but this one is particularly difficult to analyze because it's unclear what is meant by "relevant and accurate information." In any case, this concept's claim is far too broadly stated, and only holds in some circumstances.

If we take the claim in the most literal sense, this concept is not always true. Infinite counterexamples exist. Perhaps you know that your opponent always makes big bets; in such a case, the big bet provides essentially no information at all. Or, sometimes you may have the nuts, which is probably the most "relevant" factor if you're trying to figure out how to proceed with the hand. As far as "accurate" information goes, usually preflop play is most accurate, since you will have observed your opponent in many more preflop situations than postflop situations. Of course, all of this varies from player to player. It seems overly bold for S+M to make a claim that a big bet is (presumably, always) the most relevant and accurate.

If we're not to take this concept literally, let's consider whether it is at least good advice in a figurative sense. This concept seems suggest that players should respect the big bet more than they do, and thus fold to it more. Do players tend to underestimate the importance of a big bet when trying to determine their opponent's possible hand? Human nature actually suggests the exact opposite; there is a phenomenon called the "base rate fallacy" that seems like it could apply here. It has been established that people tend to discount base rates (which could correspond to your read of your opponent before the big bet), and focus only on the most immediate and salient information (such as the big bet) when estimating likelihoods. Are poker players similarly likely to focus to much on the most immediate information? I think sometimes yes, sometimes no. There are certainly a lot of players who are not sensitive enough to bet sizes, and for these players, taking this concept to heart is probably a good idea. I'm not sure that most players fall into this category, though.

In their explanation of this concept, Sklansky and Miller say "all information from the past takes a back seat to the fact that they've made a big bet." I think a fair way to interpret this is "the probability your opponent holds the nuts, or nearly the nuts, given that he just made a big bet, is greater than the likelihood that he did NOT hold the nuts before he made that big bet." Mathematically, we can write this as:

P(nuts|big bet) > p(nuts' before the big bet). That is: P(nuts|big bet) > p(nuts').I'm fairly confident that this is not always true, although it is might be true if your opponent is a very good player. I think that Sklansky and Miller might have succumbed to the base rate fallacy and are underestimating the importance of their initial read of their opponent's preflop tendencies. Let's do some Bayesian analysis of the example they give in their explanation of this concept.

(Note that nuts' means "not the nuts.")

Suppose we are facing a big bet and we think our opponent is either bluffing or holding the nuts. Before he made the big bet, suppose we thought that the way he played the hand meant he had only a 0.05% chance of holding the nuts. This seems reasonable in S+M's example where your opponent is very tight, raised pre-flop, and the nuts is 74 (actually, .05% seems a bit too high, but let's give S+M the benefit of the doubt). Now let's say we think our opponent would make a big bet 80% of the time if he held 74. Otherwise, he would make the same big bet 2% of the time as a bluff. The base rate fallacy would be to think that your opponent would now be holding the nuts 80/82 times, or about 97.6% of the time. Of course, this is way too high because it ignores base rates. In this example, we can use Bayes' theorem to find the actual probability that you are up against the nuts. Bayes gives us the following equation:

P(nuts|big bet) = P(big bet|nuts)*P(nuts) / (P(big bet|nuts)*P(nuts)From our assumptions above, we have:

+ P(big bet|nuts')*P(nuts'))

P(big bet|nuts) = .8

P(nuts) = .0005

P(big bet|nuts') = .02

P(nuts') = .9999

and so:

P(big bet|nuts)*P(nuts) = .0004

P(big bet|nuts')*P(nuts') = .019998

Then,

P(nuts|big bet) = .0004 / (.0004 + .019998) = .0004/.020398So, under these assumptions, our opponent holds the nuts less than 2% of the time when he makes a big bet! You may quibble with the rates I chose, but even if our calculations are off by a factor of 10, our opponent will still only be holding a winning hand 20% of the time, and bluffing 80%, making it correct to call any sized bet.

= .0196

If you don't like the rates I chose, I guess it's most likely because you think that even very tight players tend to hold 74 on the river more than .05% of the time in this situation (the situation: he raised preflop, and 74 is the nuts on the river). I admit, this is a tricky one to estimate, but we can do another Bayesian inference to examine my .05% claim. That is, we want to know if .05% is a reasonable approximation for how often a tight, pre-flop raiser holds 74 on the river in this situation. That is, we want to find P(74|saw river).

First of all, let's assume our "tight" player raises preflop with 74 only 1% of the time when dealt to him. Also, assuming we don't have a 7 or a 4 in our hand, our opponent will have been dealt 74 only 1.3% of the time. So, P(74) = .013*.01 = .00013.

Second of all, we need to approximate how often 74 will see the river in this situation. Remember, the situation is: 74 has already raised preflop and seen a flop that gives him a draw to the nuts. IN order to be conservative and to simplify the math, let's assume he will never fold before the river in this situation. So, P(saw river|74) = 1.

Third of all, we need to approximate how often a player would have seen the river in this situation given that he raised preflop holding anything. This is another tough one. I think this is likely over 50%, but let's be conservative and choose 26%, which will also help out with the arithmetic. So, assume P(saw river) = .26. Bayes theorem then gives us:

P(74|saw river) = P(saw river|74)*P(74) / P(saw river)Well, this is precisely the number I used above, and I was using conservative numbers in this example. Of course, you could still contend that my numbers or other assumptions are unreasonable, but I have at least convinced myself that in Sklansky and Miller's example, our opponent's big bet usually will not indicate that he has the nuts.

= 1*.00013/.26 = .0005 = .05%

## Friday, September 04, 2009

### Analyzing NLHE:TAP Concept 10

I'm working through all 60 concepts at the end of Sklansky and Miller's No Limit Hold 'em: Theory and Practice.

Concept No. 10: Sometimes you should go for a check-raise bluff on the river when a bluff bet would be unprofitable.

Sklansky and Miller are technically correct that you should sometimes check-raise as a bluff, but the advice is still bad. It's confusing for them to state this advice without noting the unusual circumstances under which it holds. They say, "sometimes you should..." Well, yes, but rarely. Readers who take Concept 10 at face value are in danger of trying check-raise bluffs indiscriminately, which would probably be a big mistake. My problem with this advice is actually very similar to Concept No. 4: "Sometimes you should bluff to stop a bluff." In both cases, Sklansky and Miller are missing the main point, an idea that I've attributed to Mike Caro (though I can't find it anywhere - maybe I thought of it myself!): If your opponent bluffs either way too much or way too little, be more willing to check to him. Or, more generally: If you've identified any weakness in your opponent's play (such as bluffing too much), try to put him in situations that will allow you to exploit this (for example, by checking and letting him bluff).

In Concept 4, Sklansky and Miller refer to an opponent who folds way too often to a small bet. You should bluff often against this opponent. Here in Concept 10, they use an example where your opponent bets 5/9 of the pot and is bluffing "the majority of the time." This is too often. (When betting 5/9 of the pot, game theory suggests you should be bluffing only 5/14 = 35.7% of the time.) Against an opponent who bluffs too much, checking is commonly the best play. This is the real lesson of the example they give: If you know your opponent bluffs too much, exploit it!

I know Sklansky and Miller are not talking about game theory here, but I just want to acknowledge that theoretically, it is indeed sometimes profitable to check-raise bluff even if your opponent is playing perfectly. For an optimal strategy, these bluffs are needed to balance the times you check-raise with a strong hand. In this theoretical case, the hands that are best to check-raise bluff with are actually not the very worst hands in your range (which are the best hands to try regular bluff bets with). Instead, the best hands to check-raise bluff with are the very best of the hands that you would otherwise fold. For example, if 66 is the worst hand you would call with in a given situation, 55 is actually the best hand to try a check-raise bluff with. In practice, if you want to try a check-raise bluff, the decision should depend a lot more on your read of your opponent than on your own hand, but I always think it's worth considering what game theory has to say about a situation.

Concept No. 10: Sometimes you should go for a check-raise bluff on the river when a bluff bet would be unprofitable.

Sklansky and Miller are technically correct that you should sometimes check-raise as a bluff, but the advice is still bad. It's confusing for them to state this advice without noting the unusual circumstances under which it holds. They say, "sometimes you should..." Well, yes, but rarely. Readers who take Concept 10 at face value are in danger of trying check-raise bluffs indiscriminately, which would probably be a big mistake. My problem with this advice is actually very similar to Concept No. 4: "Sometimes you should bluff to stop a bluff." In both cases, Sklansky and Miller are missing the main point, an idea that I've attributed to Mike Caro (though I can't find it anywhere - maybe I thought of it myself!): If your opponent bluffs either way too much or way too little, be more willing to check to him. Or, more generally: If you've identified any weakness in your opponent's play (such as bluffing too much), try to put him in situations that will allow you to exploit this (for example, by checking and letting him bluff).

In Concept 4, Sklansky and Miller refer to an opponent who folds way too often to a small bet. You should bluff often against this opponent. Here in Concept 10, they use an example where your opponent bets 5/9 of the pot and is bluffing "the majority of the time." This is too often. (When betting 5/9 of the pot, game theory suggests you should be bluffing only 5/14 = 35.7% of the time.) Against an opponent who bluffs too much, checking is commonly the best play. This is the real lesson of the example they give: If you know your opponent bluffs too much, exploit it!

I know Sklansky and Miller are not talking about game theory here, but I just want to acknowledge that theoretically, it is indeed sometimes profitable to check-raise bluff even if your opponent is playing perfectly. For an optimal strategy, these bluffs are needed to balance the times you check-raise with a strong hand. In this theoretical case, the hands that are best to check-raise bluff with are actually not the very worst hands in your range (which are the best hands to try regular bluff bets with). Instead, the best hands to check-raise bluff with are the very best of the hands that you would otherwise fold. For example, if 66 is the worst hand you would call with in a given situation, 55 is actually the best hand to try a check-raise bluff with. In practice, if you want to try a check-raise bluff, the decision should depend a lot more on your read of your opponent than on your own hand, but I always think it's worth considering what game theory has to say about a situation.

## Tuesday, September 01, 2009

### Analyzing NLHE:TAP Concepts 8-9

Discussing the concepts at the end of Sklansky and Miller's No Limit Hold 'em: Theory and Practice.

Concept No. 8: Other things being equal, when you're in one of the blinds your preflop raises should generally be a little larger than normal.

I often follow this advice in my own play, but I'm not convinced that it is generally correct. Maybe it is, but I'm not sure. This is rather difficult to analyze objectively, and I think it will be a subject of discussion among poker theorists and enthusiasts for a long time. Anyway, Sklansky and Malmuth list three reasons why they think this advice is correct. Let's look briefly at each.

1. S+M say:

You'll be out of position, and raising bigger increases the chances you'll win right away.

Me: Yes, but you could also argue that you'd rather play a small pot if you're out of position, and so you should either raise smaller or only with very strong hands like JJ+.

2. S+M say: Cuts down your opponents' implied odds.

Me: Well, yes, but it also costs you more. Is it worth it?

3. S+M say: In early position, it's more important that you narrow down your opponents' possible hands, which you can do by raising more.

Me: Good point. Again, I wonder: is it worth the extra investment?

Finding the right strategy for raising in the blinds is something I'm still working on. It probably deserves a lot more consideration than I'm giving it in this post, but I don't feel that I have anything insightful to say about it beyond what I'm writing here. My current strategy is this: In the blinds, I like to raise only with a small number of strong hands. The actual range will depend on who has limped into the pot so far, but sometimes I will check with a hand as strong as pocket tens (although I will usually raise this hand). Occasionally, I will bluff with weaker hands (sometimes even extremely weak hands such as 72o if I think my image is very tight and I will likely make everyone fold). Since I seldom bluff from the blinds, I don't usually feel the need to make my raises particularly large. If my opponents know me well, they should fold to a normal-sized raise of about 2 BB larger than the current pot size, which is about what I consider normal-sized from other positions, as well. However, I will often raise larger than this if my opponents don't know me or if they seem to play too loosely. Since this is often the case, I actually do raise larger than normal from the blinds pretty often.

A related question is how much to raise in early position versus late position. I've seen all three alternatives argued: larger raises in early position; larger raises in late position; keep raises about the same size. The argument that you should raise bigger from early position is the most popular: you don't want to get called and then have to play the pot out of position, which is the same as S+M's 1st point above. I've also seen it argued that you should raise the same amount from each position. One argument for this is that it will always give the blinds the same pot odds if nobody else joins the pot. I've rarely seen it argued that you should raise bigger from late position than from early position, but this is what Chen and Ankenman argue in The Mathematics of Poker. They say that since your raising range is much smaller from early position, it's not necessary to raise as much. Also, by raising small, you save money if one of your opponents reraises, which is much more likely if you are raising from early position with 7-9 players yet to act than if you raise from the button with just 2 players left after you. Personally, I've decided to take the middle road and raise about the same regardless of my position. For me, this philosophy extends to how I play from the blinds, where I raise about the same as from anywhere else.

I think this topic deserves a much more objective, EV-based analysis, but this seems like a big project. I'm not up to the challenge just yet. I hope to return to this question at some point after I'm done going through the rest of these concepts, but that will not be for quite a while.

Concept No. 9: Bets are usually more important than pots.

I'm tacking this one on the end here because I don't have much to say about it. The concept's claim is not literally true because even in no-limit holdem, bets are usually smaller than the pot, but S+M's point is well-taken: in contrast to limit poker, winning or saving extra bets is extremely important in no-limit poker.

No-limit players should be much more willing to give up a pot in order to save a bet. In limit poker, on the other hand, the pots tend to be big compared to the bets, and so calling is much more commonly the right decision.

For limit players, it can be difficult to adjust to this difference. I had the opposite problem when I started playing more limit holdem last year: I was folding way too often for a limit game. My experience with no-limit poker had gotten me used to the idea that saving bets was important, and giving up pots was okay. In limit, the exact opposite is true!

Concept No. 8: Other things being equal, when you're in one of the blinds your preflop raises should generally be a little larger than normal.

I often follow this advice in my own play, but I'm not convinced that it is generally correct. Maybe it is, but I'm not sure. This is rather difficult to analyze objectively, and I think it will be a subject of discussion among poker theorists and enthusiasts for a long time. Anyway, Sklansky and Malmuth list three reasons why they think this advice is correct. Let's look briefly at each.

1. S+M say:

You'll be out of position, and raising bigger increases the chances you'll win right away.

Me: Yes, but you could also argue that you'd rather play a small pot if you're out of position, and so you should either raise smaller or only with very strong hands like JJ+.

2. S+M say: Cuts down your opponents' implied odds.

Me: Well, yes, but it also costs you more. Is it worth it?

3. S+M say: In early position, it's more important that you narrow down your opponents' possible hands, which you can do by raising more.

Me: Good point. Again, I wonder: is it worth the extra investment?

Finding the right strategy for raising in the blinds is something I'm still working on. It probably deserves a lot more consideration than I'm giving it in this post, but I don't feel that I have anything insightful to say about it beyond what I'm writing here. My current strategy is this: In the blinds, I like to raise only with a small number of strong hands. The actual range will depend on who has limped into the pot so far, but sometimes I will check with a hand as strong as pocket tens (although I will usually raise this hand). Occasionally, I will bluff with weaker hands (sometimes even extremely weak hands such as 72o if I think my image is very tight and I will likely make everyone fold). Since I seldom bluff from the blinds, I don't usually feel the need to make my raises particularly large. If my opponents know me well, they should fold to a normal-sized raise of about 2 BB larger than the current pot size, which is about what I consider normal-sized from other positions, as well. However, I will often raise larger than this if my opponents don't know me or if they seem to play too loosely. Since this is often the case, I actually do raise larger than normal from the blinds pretty often.

A related question is how much to raise in early position versus late position. I've seen all three alternatives argued: larger raises in early position; larger raises in late position; keep raises about the same size. The argument that you should raise bigger from early position is the most popular: you don't want to get called and then have to play the pot out of position, which is the same as S+M's 1st point above. I've also seen it argued that you should raise the same amount from each position. One argument for this is that it will always give the blinds the same pot odds if nobody else joins the pot. I've rarely seen it argued that you should raise bigger from late position than from early position, but this is what Chen and Ankenman argue in The Mathematics of Poker. They say that since your raising range is much smaller from early position, it's not necessary to raise as much. Also, by raising small, you save money if one of your opponents reraises, which is much more likely if you are raising from early position with 7-9 players yet to act than if you raise from the button with just 2 players left after you. Personally, I've decided to take the middle road and raise about the same regardless of my position. For me, this philosophy extends to how I play from the blinds, where I raise about the same as from anywhere else.

I think this topic deserves a much more objective, EV-based analysis, but this seems like a big project. I'm not up to the challenge just yet. I hope to return to this question at some point after I'm done going through the rest of these concepts, but that will not be for quite a while.

Concept No. 9: Bets are usually more important than pots.

I'm tacking this one on the end here because I don't have much to say about it. The concept's claim is not literally true because even in no-limit holdem, bets are usually smaller than the pot, but S+M's point is well-taken: in contrast to limit poker, winning or saving extra bets is extremely important in no-limit poker.

No-limit players should be much more willing to give up a pot in order to save a bet. In limit poker, on the other hand, the pots tend to be big compared to the bets, and so calling is much more commonly the right decision.

For limit players, it can be difficult to adjust to this difference. I had the opposite problem when I started playing more limit holdem last year: I was folding way too often for a limit game. My experience with no-limit poker had gotten me used to the idea that saving bets was important, and giving up pots was okay. In limit, the exact opposite is true!

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