Poker Tournament Strategy: "True M" vs. Harrington's M
"True M" versus Harrington's M: Critical Flaws in Harrington's M Theory, And Why Tournament Structure MattersBy Arnold Snyder
(From Blackjack Forum Vol. XXVI #1, Spring 2007)
© Blackjack Forum Online 2007
Critical Flaws in the Theory and Use of "M" in Poker Tournaments
In this article, I will address critical flaws in the concept of “M” as a measure of player viability in poker tournaments. I will specifically be addressing the concept of M as put forth by Dan Harrington in Harrington on Hold’em II (HOH II). My book, The Poker Tournament Formula (PTF), has been criticized by some poker writers who contend that my strategies for fast tournaments must be wrong, since they violate strategies based on Harrington’s M.
I will show that it is instead Harrington’s theory and advice that are wrong. I will explain in this article exactly where Harrington made his errors, why Harrington’s strategies are incorrect not only for fast tournaments, but for slow blind structures as well, and why poker tournament structure, which Harrington ignores, is the key factor in devising optimal tournament strategies.
This article will also address a common error in the thinking of players who are using a combination of PTF and HOH strategies in tournaments. Specifically, some of the players who are using the strategies from my book, and acknowledge that structure is a crucial factor in any poker tournament, tell me they still calculate M at the tables because they believe it provides a “more accurate” assessment of a player’s current chip stack status than the simpler way I propose—gauging your current stack as a multiple of the big blind. But M, in fact, is a less accurate number, and this article will explain why.
There is a way to calculate what I call “True M,” that would provide the information that Harrington’s false M is purported to provide, but I do not believe there is any real strategic value in calculating this number, and I will explain the reason for that too.
The Basics of Harrington’s M Strategy
Harrington uses a zone system to categorize a player’s current chip position. In the “green zone,” a player’s chip stack is very healthy and the player can use a full range of poker skills. As a player’s chip stack diminishes, the player goes through the yellow zone, the orange zone, the red zone, and finally the dead zone. The zones are identified by a simple rating number Harrington calls “M.”
What Is “M”?
In HOH II, on page 125, Dan Harrington defines M as: “…the ratio of your stack to the current total of blinds and antes.” For example, if your chip stack totals 3000, and the blinds are 100-200 (with no ante), then you find your M by dividing 3000 / 300 = 10.
On page 126, Harrington expounds on the meaning of M to a tournament player: “What M tells you is the number of rounds of the table that you can survive before being blinded off, assuming you play no pots in the meantime.” In other words, Harrington describes M as a player’s survival indicator.
If your M = 5, then Harrington is saying you will survive for five more rounds of the table (five circuits of the blinds) if you do not play a hand. At a 10-handed table, this would mean you have about 50 hands until you would be blinded off. All of Harrington’s zone strategies are based on this understanding of how to calculate M, and what M means to your current chances of tournament survival.
Amateur tournament players tend to tighten up their play as their chip stacks diminish. They tend to become overly protective of their remaining chips. This is due to the natural survival instinct of players. They know that they cannot purchase more chips if they lose their whole stack, so they try to hold on to the precious few chips that are keeping them alive.
If they have read a few books on the subject of tournament play, they may also have been influenced by the unfortunate writings of Mason Malmuth and David Sklansky, who for many years have promulgated the misguided theory that the fewer chips you have in a tournament, the more each chip is worth. (This fallacious notion has been addressed in other articles in our online Library, including: Reverse Chip Value for Poker Tournaments: Good Math, Bad Logid (A Reply to David Sklansky.)
But in HOH II, Harrington explains that as your M diminishes, which is to say as your stack size becomes smaller in relation to the cost of the blinds and antes, “…the blinds are starting to catch you, so you have to loosen your play… you have to start making moves with hands weaker than those a conservative player would elect to play.” I agree with Harrington on this point, and I also concur with his explanation of why looser play is correct as a player’s chip stack gets shorter: “Another way of looking at M is to see it as a measure of just how likely you are to get a better hand in a better situation, with a reasonable amount of money left.” (Italics his.)
In other words, Harrington devised his looser pot-entering strategy, which begins when your M falls below 20, and goes through four zones as it continues to shrink, based on the likelihood of your being dealt better cards to make chips with than your present starting hand. For example, with an M of 15 (yellow zone according to Harrington), if a player is dealt an 8-3 offsuit in early position (a pretty awful starting hand by anyone’s definition), Harrington’s yellow zone strategy would have the player fold this hand preflop because of the likelihood that he will be dealt a better hand to play while he still has a reasonable amount of money left.
By contrast, if the player is dealt an ace-ten offsuit in early position, Harrington’s yellow zone strategy would advise the player to enter the pot with a raise. This play is not advised in Harrington’s green zone strategy (with an M > 20) because he considers ace-ten offsuit to be too weak of a hand to play from early position, since your bigger chip stack means you will be likely to catch a better pot-entering opportunity if you wait. The desperation of your reduced chip stack in the yellow zone, however, has made it necessary for you to take a risk with this hand because with the number of hands remaining before you will be blinded off, you are unlikely “…to get a better hand in a better situation, with a reasonable amount of money left.”
Again, I fully agree with the logic of loosening starting hand requirements as a player’s chip stack gets short. In fact, the strategies in The Poker Tournament Formula are based in part (but not in whole) on the same logic.
But despite the similarity of some of the logic behind our strategies, there are big differences between our specific strategies for any specific size of chip stack. For starters, my strategy for entering a pot with what I categorize as a “competitive stack” (a stack size more or less comparable to Harrington’s “green zone”) is far looser and more aggressive than his. And my short-stack strategies are downright maniacal compared to Harrington’s strategies for his yellow, orange, and red zones.
There are two major reasons why our strategies are so different, even though we agree on the logic that looser play is required as stacks get shorter. Again, the first is a fundamental difference in our overriding tournament theory, which I will deal with later in this article. The second reason, which I will deal with now, is a serious flaw in Harrington’s method of calculating and interpreting M. Again, what Harrington specifically assumes, as per HOH II, is that: “What M tells you is the number of rounds of the table that you can survive before being blinded off, assuming you play no pots in the meantime.”
But that’s simply not correct. The only way M, as defined by Harrington, could indicate the number of rounds a player could survive is by ignoring the tournament structure.
Why Tournament Structure Matters in Devising Optimal Strategy
Let’s look at some sample poker tournaments to show how structure matters, and how it affects the underlying meaning of M, or “the number of rounds of the table that you can survive before being blinded off, assuming you play no pots in the meantime.” Let’s say the blinds are 50-100, and you have 3000 in chips. What is your M, according to Harrington?
M = 3000 / 150 = 20
So, according to the explanation of M provided in HOH II, you could survive 20 more rounds of the table before being blinded off, assuming you play no pots in the meantime. This is not correct, however, because the actual number of rounds you can survive before being blinded off is entirely dependent on the tournament’s blind structure.
For example, what if this tournament has 60-minute blind levels? Would you survive 20 rounds with the blinds at 50-100 if you entered no pots? No way. Assuming this is a ten-handed table, you would go through the blinds about once every twenty minutes, which is to say, you would only play three rounds at this 50-100 level. Then the blinds would go up.
If we use the blind structure from the WSOP Circuit events recently played at Caesars Palace in Las Vegas, after 60 minutes the blinds would go from 50-100 to 100-200, then to 100-200 with a 25 ante 60 minutes after that. What is the actual number of rounds you would survive without entering a pot in this tournament from this point? Assuming you go through the blinds at each level three times,
3 x 150 = 450
3 x 300 = 900
3 x 550 = 1650
Add up the blind costs: 450 + 900 + 1650 = 3000.
That’s a total of only 9 rounds.
This measure of the true “…number of rounds of the table that you can survive before being blinded off, assuming you play no pots in the meantime,” is crucial in evaluating your likelihood of getting “…a better hand in a better situation, with a reasonable amount of money left,” and it is entirely dependent on this tournament’s blind structure. For the rest of this article, I will refer to this more accurate structure-based measure as “True M.” True M for this real-world tournament would indicate to the player that his survival time was less than half that predicted by Harrington’s miscalculation of M.
True M in Fast Poker Tournaments
To really drill home the flaw in M—as Harrington defines it—let’s look at a fast tournament structure. Let’s assume the exact same 3000 in chips, and the exact same 50-100 blind level, but with the 20-minute blind levels we find in many small buy-in tourneys. With this blind structure, the blinds will be one level higher each time we go through them. How many rounds of play will our 3000 in chips survive, assuming we play no pots? (Again, I’ll use the Caesars WSOP levels, as above, changing only the blind length.)
150 + 300 + 550 + 1100 (4 rounds) = 1950
The next round the blinds are 300-600 with a 75 ante, so the cost of a ten-handed round is 1650, and we only have 1050 remaining. That means that with this faster tournament structure, our True M at the start of that 50-100 blind level is actually about 4.6, a very far cry from the 20 that Harrington would estimate, and quite far from the 9 rounds we would survive in the 60-minute structure described above.
And, in a small buy-in tournament with 15-minute blind levels—and these fast tournaments are very common in poker rooms today—this same 3000 chip position starting at this same blind level would indicate a True M of only 3.9.
True M in Slow Poker Tournaments
But what if you were playing in the $10K main event of the WSOP, where the blind levels last 100 minutes? In this tournament, if you were at the 50-100 blind level with 3000 in chips, your True M would be 11.4. (As a matter of fact, it has only been in recent years that the blind levels of the main event of the WSOP have been reduced from their traditional 2-hour length. With 2-hour blind levels, as Harrington would have played throughout most of the years he has played the main event, his True M starting with this chip position would be 12.6.)
Unfortunately, that's still nowhere near the 20 rounds Harrington's M gives you.
True M Adjusts for Tournament Structure
Note that in each of these tournaments, 20 M means something very different as a survival indicator. True M shows that the survival equivalent of 3000 in chips at the same blind level can range from 3.9 rounds (39 hands) to 12.6 (126 hands), depending solely on the length of the blinds.
Furthermore, even within the same blind level of the same tournament, True M can have different values, depending on how deep you are into that blind level. For example, what if you have 3000 in chips but instead of being at the very start of that 50-100 blind level (assuming 60-minute levels), you are somewhere in the middle of it, so that although the blinds are currently 50-100, the blinds will go up to the 100-200 level before you go through them three more times? Does this change your True M?
It most certainly does. That True M of 9 in this tournament, as demonstrated above, only pertains to your chip position at the 50-100 blind level if you will be going through those 50-100 blinds three times before the next level. If you’ve already gone through those blinds at that level one or more times, then your True M will not be 9, but will range from 6.4 to 8.1, depending on how deep into the 50-100 blind level you are.
Most important, if you are under the mistaken impression that at any point in the 50-100 blind level in any of the tournaments described above, 3000 in chips is sufficient to go through 20 rounds of play (200 hands), you are way off the mark. What Harrington says “M tells you,” is not at all what M tells you. If you actually stopped and calculated True M, as defined above, then True M would tell you what Harrington’s M purports to tell you.
And if it really is important for you to know how many times you can go through the blinds before you are blinded off, then why not at least figure out the number accurately? M, as described in Harrington’s book, is simply woefully inadequate at performing this function.
If Harrington had actually realized that his M was not an accurate survival indicator, and he had stopped and calculated True M for a variety of tournaments, would he still be advising you to employ the same starting hand standards and playing strategies at a True M of 3.9 (with 39 hands before blind-off) that you would be employing at a True M of 12.6 (with 126 hands before blind-off)?
If he believes that a player with 20 M has 20 rounds of play to wait for a good hand before he is blinded off (and again, 20 rounds at a ten-player table would be 200 hands), then his assessment of your likelihood of getting “…a better hand in a better situation, with a reasonable amount of money left,” would be quite different than if he realized that his True M was 9 (90 hands remaining till blind-off), or in a faster blind structure, as low as 3.9 (only 39 hands remaining until blind-off).
Those radically different blind-off times would drastically alter the frequencies of occurrence of the premium starting hands, and aren’t the likelihood of getting those hands what his M theory and strategy are based on?
A Blackjack Analogy
For blackjack players—and I know a lot of my readers come from the world of blackjack card counting—Harrington’s M might best be compared to the “running count.” If I am using a traditional balanced card counting system at a casino blackjack table, and I make my playing and betting decisions according to my running count, I will often be playing incorrectly, because the structure of the game—the number of decks in play and the number of cards that have already been dealt since the last shuffle—must be taken into account in order for me to adjust my running count to a “true” count.
A +6 running count in a single-deck game means something entirely different from a +6 running count in a six-deck shoe game. And even within the same game, a +6 running count at the beginning of the deck or shoe means something different from a +6 running count toward the end of the deck or shoe.
Professional blackjack players adjust their running count to the true count to estimate their advantage accurately and make their strategy decisions accordingly. The unadjusted running count cannot do this with any accuracy. Harrington’s M could be considered a kind of Running M, which must be adjusted to a True M in order for it to have any validity as a survival gauge.
When Harrington’s Running M Is Occasionally Correct
Harrington’s Running M can “accidentally” become correct without a True M adjustment when a player is very short-stacked in a tournament with lengthy blind levels. For example, if a player has an M of 4 or 5 in a tournament with 2-hour blind levels, then in the early rounds of that blind level, since he could expect to go through the same blind costs 4 or 5 times, Harrington’s unadjusted M would be the same as True M.
This might also occur when the game is short-handed, since players will be going through the blinds more frequently. (This same thing happens in blackjack games where the running count equals the true count at specific points in the deal. For example, if a blackjack player is using a count-per-deck adjustment in a six-deck game, then when the dealer is down to the last deck in play, the running count will equal the true count.)
In rare situations like these, where Running M equals True M, Harrington’s “red zone” strategies may be correct—not because Harrington was correct in his application of M, but because of the tournament structure and the player’s poor chip position at that point.
In tournaments with 60-minute blind levels, this type of "Running M = True M” situation could only occur at a full table when a player’s M is 3 or less. And in fast tournaments with 15 or 20-minute blind levels, Harrington's M could only equal True M when a player’s M = 1 or less.
Harrington’s yellow and orange zone strategies, however, will always be pretty worthless, even in the slowest tournaments, because there are no tournaments with blind levels that last long enough to require no True M adjustments.
Why Harrington's Strategies Can't Be Said to Adjust Automatically for True M
Some Harrington supporters may wish to make a case that Dan Harrington made some kind of automatic adjustment for approximate True M in devising his yellow and orange zone strategies. But in HOH II, he clearly states that M tells you how many rounds of the table you will survive—period.
In order to select which hands a player should play in these zones, based on the likelihood of better hands occurring while the player still has a reasonable chip stack, it was necessary for Harrington to specify some number of rounds in order to develop a table of the frequencies of occurrence of the starting hands. His book tells us that he assumes an M of 20 simply means 20 rounds remaining—which we know is wrong for all real-world tournaments.
But for those who wish to make a case that Harrington made some kind of a True M adjustment that he elected not to inform us about, my answer is that it’s impossible that whatever adjustment he used would be even close to accurate for all tournaments and blind structures. If, for example, he assumed 20 M meant a True M of 12, and he developed his starting-hand frequency charts with this assumption, then his strategies would be fairly accurate for the slowest blind structures we find in major events. But they would still be very wrong for the faster blind structures we find in events with smaller buy-ins and in most online tournaments.
In HOH II, he does provide quite a few sample hands from online tournaments, with no mention whatsoever of the blind structures of these events, but 15-minute blind levels are less common online than 5-, 8-, and 12-minute blind levels. Thus, we are forced to believe that what Mason Malmuth claims is true: that Harrington considers his strategies correct for tournaments of all speeds. So it is doubtful that he made any True M adjustments, even for slower tournament structures. Simply put, Harrington is oblivious to the true mathematics of M.
Simplifying True M for Real-Life Tournament Strategy
If all poker tournaments had the same blind structure, then we could just memorize chart data that would indicate True M with any chip stack at any point in any blind level. Unfortunately, there are almost as many blind structures as there are tournaments.
There are ways, however, that Harrington’s Running M could be adjusted to an approximate True M without literally figuring out the exact cost of each blind level at every point in the tournament. With 90-minute blind levels, after dividing your chip stack by the cost of a round, simply divide your Running M by two, and you’ll have a reasonable approximation of your True M.
With 60-minute blind levels, take about 40% of the Running M. With 30-minute blind levels, divide the Running M by three. And with 15- or 20-minute blind levels, divide the Running M by five. These will be far from perfect adjustments, but they will be much closer to reality than Harrington’s unadjusted Running M numbers.
Do Tournament Players Need to Know Their “True M”?
Am I suggesting that poker tournament players should start estimating their True M, instead of the Running M that Harrington proposes? No, because I disagree with Harrington’s emphasis on survival and basing so much of your play on your cards. I just want to make it clear that M, as defined and described by Harrington in HOH II, is wrong, a bad measure of what it purports and aims to measure. It is based on an error in logic, in which a crucial factor in the formula—tournament structure—is ignored (the same error that David Sklansky and Mason Malmuth have made continually in their writings and analyses of tournaments.)
Although it would be possible for a player to correct Harrington’s mistake by estimating his True M at any point in a tournament, I don’t advise it. Admittedly, it’s a pain in the ass trying to calculate True M exactly, not something most players could do quickly and easily at the tables. But that’s not the reason I think True M should be ignored.
The reason is related to the overarching difference between Harrington’s strategies and mine, which I mentioned at the beginning of this article. That is: It’s a grave error for tournament players to focus on how long they can survive if they just sit and wait for premium cards. That’s not what tournaments are about. It’s a matter of perspective. When you look at your stack size, you shouldn’t be thinking, “How long can I survive?” but, “How much of a threat do I pose to my opponents?”
The whole concept of M is geared to the player who is tight and conservative, waiting for premium hands (or premium enough at that point). Harrington’s strategy is overly focused on cards as the primary pot entering factor, as opposed to entering pots based predominately (or purely) on position, chip stack, and opponent(s).
In The Poker Tournament Formula, I suggest that players assess their chip position by considering their chip stacks as a simple multiple of the current big blind. If you have 3000 in chips, and the big blind is 100, then you have 30 big blinds. This number, 30, tells you nothing about how many rounds you can survive if you don’t enter any pots. But frankly, that doesn’t matter. What matters in a tournament is that you have sufficient chips to employ your full range of skills, and—just as important—that you have sufficient chips to threaten your opponents with a raise, and an all-in raise if that is what you need for the threat to be successful to win you the pot.
Your ability to to be a threat is directly related to the health of your chip stack in relation to the current betting level, which is most strongly influenced by the size of the blinds. In my PTF strategy, tournaments are not so much about survival as they are about stealing pots. If you’re going to depend on surviving until you get premium cards to get you to the final table, you’re going to see very few final tables. You must outplay your opponents with the cards you are dealt, not wait and hope for cards that are superior to theirs.
I’m not suggesting that you ignore the size of the preflop pot and focus all of your attention on the size of the big blind. You should always total the chips in the pot preflop, but not because you want to know how long you can survive if you sit there waiting for your miracle cards. You simply need to know the size of the preflop pot so you can make your betting and playing decisions, both pre- and post-flop, based on all of the factors in the current hand.
What other players, if any have entered the pot? Is this a pot you can steal if you don’t have a viable hand? Is this pot worth the risk of an attempted steal? If you have a drawing hand, do you have the odds to call, or are you giving an opponent the odds to call? Are any of your opponent(s) pot-committed? Do you have sufficient chips to play a speculative hand for this pot? There are dozens of reasons why you need to know the size of a pot you are considering getting involved in, but M is not a factor in any of these decisions.
So, again, although you will always be totaling the chips in the pot in order to make betting and playing decisions, sitting there and estimating your blind-off time by dividing your chip stack by the total chips in the preflop pot is an exercise in futility. It has absolutely nothing to do with your actual chances of survival. You shouldn’t even be thinking in terms of survival, but of domination.
Harrington on Hold'em II versus The Poker Tournament Formula: A Sample Situation
Let’s say the blinds are 100-200, and you have 4000 in chips. Harrington would have you thinking that your M is 13 (yellow zone), and he advises: “…you have to switch to smallball moves: get in, win the pot, but get out when you encounter resistance.” (HOH II, p. 136)
In The Poker Tournament Formula basic strategy for fast tournaments (PTF p. 158), I categorize this chip stack equal to 20 big blinds as “very short,” and my advice is: “…you must face the fact that you are not all that far from the exit door. But you still have enough chips to scare any player who does not have a really big chip stack and/or a really strong hand. Two things are important when you are this short on chips. One is that unless you have an all-in raising hand as defined below, do not enter any pot unless you are the first in. And second, any bet when you are this short will always be all-in.”
The fact is, you don't have enough chips for “smallball” when you’re this short on chips in a fast tournament, and one of the most profitable moves you can make is picking on Harrington-type players who think it's time for smallball.
Harrington sees this yellow zone player as still having 13 rounds of play (130 hands, which is a big overestimation resulting from his failure to adjust to True M) to look for a pretty decent hand to get involved with. My thinking in a fast tournament, by contrast, would be: “The blinds are now 100-200. By the time they get around to me fifteen minutes from now, they will be 200-400. If I don’t make a move before the blinds get around to me, and I have to go through those blinds, my 4000 will become 3400, and the chip position I’m in right now, which is having a stack equal to 20 times the big blind, will be reduced to a stack of only 8.5 times the big blind. Right now, my chip stack is scary. Ten to fifteen minutes from now (in 7-8 hands), any legitimate hand will call me down.”
So, my advice to players this short on chips in a fast tournament is to raise all-in with any two cards from any late position seat in an unopened pot. My raising hands from earlier positions include all pairs higher than 66, and pretty much any two high cards. And my advice with these hands is to raise or reraise all-in, including calling any all-ins. You need a double-up so badly here that you simply must take big risks. As per The Poker Tournament Formula (p. 159): “When you’re this short on chips you must take risks, because the risk of tournament death is greater if you don’t play than if you do.”
There is also a side effect of using a loose aggressive strategy when you have enough chips to hurt your opponents, and that is that you build an image of a player who is not to be messed with, and that is always the preferred image to have in any no-limit hold’em tournament. But while Harrington sees this player surviving for another 13 rounds of play, the reality is that he will survive fewer than 4 more rounds in a fast tournament, and within two rounds he will be so short-stacked that he will be unable to scare anybody out of a pot, and even a double-up will not get him anywhere near a competitive chip stack.
The Good News for Poker Tournament Players
The good news for poker tournament players is that Harrington’s books have become so popular, and his M theory so widely accepted as valid by many players and “experts” alike, that today’s NLH tournaments are overrun with his disciples playing the same tight, conservative style through the early green zone blind levels, then predictably entering pots with more marginal hands as their M diminishes—which their early tight play almost always guarantees. And, though many of the top players know that looser, more aggressive play is what’s getting them to the final tables, I doubt that Harrington’s misguided advice will be abandoned by the masses any time soon.
In a recent issue of Card Player magazine (March 28, 2007), columnist Steve Zolotow reviewed The Poker Tournament Formula, stating: “Snyder originates a complicated formula for determining the speed of a tournament, which he calls the patience factor. Dan Harrington's discussion of M and my columns on CPR cover this same material, but much more accurately. Your strategy should be based not upon the speed of the tournament as a whole, but on your current chip position in relation to current blinds. If your M (the number of rounds you can survive without playing a hand) is 20, you should base your strategy primarily on that fact. Whether the blinds will double and reduce your M to 10 in 15 minutes or four hours should not have much influence on your strategic decisions.”
Zolotow’s “CPR” articles were simply a couple of columns he wrote last year in which he did nothing but explain Harrington’s M theory, as if it were 100% correct. He added nothing to the theory of M, and is clearly as ignorant of the math as Harrington is.
So money-making opportunities in poker tournaments continue to abound.
In any case, I want to thank SlackerInc for posting a question on our poker discussion forum, in which he pointed out many of the key differences between Harrington’s short-stack strategies and those in The Poker Tournament Formula. He wanted to know why our pot-entering strategies were so far apart.
The answer is that the strategies in my book are specifically identified as strategies for fast tournaments of a specific speed, so my assumptions, based on a player’s current chip stack, would usually be that the player is about five times more desperate than Harrington would see him (his Running M of 20 being roughly equivalent to my True M of about 4). ♠
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