Arnold Snyder discusses poker tournament rebuy strategy and new insights into the value of fast play in poker tournaments.  This article on poker tournament strategy was written partly in response to mistakes in the logic, theory, and practical advise by Mason Malmuth and David Sklansky on poker tournament rebuys and overall strategy. Arnold Snyder shows that Malmuth and Sklansky have a fundamental lack of understanding of poker tournament skill and the types of skill that are most profitable in No-Limit Hold'Em poker tournaments. This discussion is an expansion of the poker tournament strategy and rebuy advice in Arnold Snyder's book, The Poker Tournament Formula.
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The Implied Discount and Poker Tournament Strategy

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The Implied Discount: New Insights Into Optimal Poker Tournament Strategy

By Arnold Snyder
(From Blackjack Forum , Fall 2006)
© Arnold Snyder 2006

This article will dispel a number of unsound theories about poker tournaments that have been around for decades. These theories have led tournament authors to promote weak and even losing strategies that are the reason why so many smart and experienced poker players have found it impossible to make money in tournaments. Arguments occurring on poker discussion boards at various web sites, including this one, with regards to the rebuy strategy I propose in The Poker Tournament Formula were the impetus for this article. But the implications of this article will go beyond rebuy strategies and deal with the fundamental realities of how you make money in poker tournaments.

Among the topics I will address will be the logic of sound rebuy strategy, and specifically why rebuys are almost always the correct strategy for a skilled player, even when the rebuy chips must be purchased at the full price of the initial buy-in. I will also show the reason why a player should add-on even when he has a lot of chips, and why it is often wrong for a player to purchase the add-on when he is short-stacked.

Most importantly, this article will address the theory that the fewer chips you have the more each chip is worth, and the more chips you have the less each chip is worth, and show that this relationship is true only in very particular instances at some final tables, and is completely inadequate for understanding true chip value throughout a tournament, or devising overall tournament strategy. The peculiar idea that this theory can be applied throughout a poker tournament can be traced back to Mason Malmuth’s 1987 book, Gambling Theory and Other Topics. It has been embraced by other prominent poker authors as well as players, and has been used to guide players toward bad tournament decisions and strategies for many years. The concept is wrong because, as I will show, chip value is primarily based on chip utility (the various ways in which chips can be used), and, in the hands of a skilled player using optimal strategy, chip utility goes up with stack size.

The First Great Misconception

Much of the advice in The Poker Tournament Formula was written specifically to address errors in the existing poker tournament literature, especially a number of serious errors put forward over roughly a twenty-year period by Mason Malmuth and David Sklansky. One of the most widely held and erroneous concepts of poker tournament logic is the concept that the fewer chips you have, the more each of your chips are worth, and the more chips you have the less each of your chips are worth, and that as a tournament progresses, all chips lose value. David Sklansky presents this idea in his Tournament Poker for Advanced Players (p. 44-5), though he credits Mason Malmuth’s 1987 Gambling Theory and Other Topics as the original source of this notion. I am not aware of any reputable poker authority having ever disputed this claim, and it has come to be accepted as the common wisdom.

In a nutshell, the theory says that at the start of a $10,000 buy-in tournament, all chips are worth their face value. At the final table, however, since many of the smaller prizes have already been distributed to players who busted out in the money, the total payout to those at the final table will be less than the initial cost of the chips in play. As a recent example, the winner of the WSOP main event this year received a $12 million prize. When he beat his last competitor, however, he was holding more than $87 million in chips, for which $87 million had been paid by the 8,700+ competitors. So, the individual chips in his stack were worth less per chip to him than their initial cost.

Malmuth carries this idea further to conclude that the individual chips in a small stack are worth more per chip than the individual chips in a big stack. Let’s say a small buy-in tournament is down to the last two players, who are heads up at the final table. The remaining prizes are $3,500 for the winner, and $1,800 for second place. The player who is the chip leader at this point has $70,000 in chips, while the player in second place has only $10,000. Clearly, the value of each of the chips of the second place player is much greater than the value of each of the chips of the chip leader. Assuming the blinds are so high at this point that both players are simply all-in before every flop regardless of cards (making this a coin-flip situation for all intents and purposes), the player in second place has an 87.5% chance of taking the second place prize, and a 12.5% chance of doubling up enough times to take first place. This makes his $10,000 in chips worth more than $2,000 in prize money. So, the short-stacked player’s chips are worth more than 20 cents each. The chip leader, however, will find that his $70,000 in chips have a prize value of less than $3,300. The chip leader’s chips are worth less than 5 cents each. And ultimately, the more chips the second place player loses to the first place player, the greater the value of each of his individual chips. In fact, if he gets’ down to a single chip, that chip will have a value of more than $1,800, while the chip leader’s chips ain’t worth even a nickel each.

This is a true fact, and I don’t dispute it. But the problem is that Sklansky and Malmuth have gone on from this premise to devise rebuy strategies, and in Malmuth’s case, even entire tournament strategies, based on the idea that the chips in big stacks are worth less than the chips in small stacks. In Gambling Theory and Other Topics, p. 232, Mason Malmuth states: “As the poker tournament essays have stressed, in a percentage payback tournament, the less chips you have, the more each individual chip is worth, and the more chips you have, the less each individual chip is worth. This idea is the major force that should govern many of your strategy decisions in a poker tournament.”

I will show that extending this chip value idea to a dominant factor in devising overall poker tournament strategy is a humongous error in logic. And, because it has led to more bad tournament strategies than just about any other “truth” about tournaments ever revealed to the public, it has been a tremendously costly error in logic to players. It is the basis of the whole conservative, sit-and-wait-for-a-hand approach to tournaments, as well as bad advice on just about every aspect of tournament play from how to play a short stack to final table play to optimal rebuy strategy to satellite strategies, and more.

The reason it’s a humongous error in logic is because it starts from the assumption that all players have equal skill, and it ignores the value of a bigger stack in the hands of a skilled player. In the hands of a skilled player, all chips essentially have greater value. Indeed, as I will show, all chip purchases made by a skilled player using optimal tournament strategy are essentially made at an implied discount. This implied discount has a substantial impact on rebuy and add-on decisions.

Why More Chips Equals More Value per Chip: The Implied Discount

 In order to make money at gambling, you have to actually gamble. That is, you must place money at risk on wagers on which you have an edge. The more money you can afford to wager with an edge, which is to say the more money you can put in action, the more money you will make. Sitting on your chips like a hen on an egg is not the way to make money gambling, especially in a tournament, which is, essentially, a race for all the chips.

It is incorrect to convert chips to dollar values with no consideration for how individual players might use those chips. It may seem logical that we could assign dollar values to chips since chips are initially purchased with dollars. But once the tournament begins, they cease to be dollars or even to represent dollars. You can’t cash them out. You can’t sell them, trade them, or buy anything with them. They are simply tools that are provided to players for competing in a contest. When the tournament director says, “Shuffle up and deal!” a battle has begun, and chips are nothing more nor less than ammunition.

The ground rules of this battle are pretty simple. If you run out of ammo, you’re dead. But if you outlast enough of your enemies, you will get some portion of the prize money depending on exactly how many of your enemies you bested. The player who survives the longest gets the biggest prize.

But you can’t survive by just hoarding your ammo. Your position will be attacked at regular intervals (the blinds), forcing you to spend some of your ammo even if you are attempting to avoid confrontations at all costs. These attacks on your position will start small, but they will escalate, costing you larger amounts of your ammo as the battle progresses. The only way for you to survive is to acquire more ammo, and the only way for you to do this is to engage in confrontations with your enemies and continually capture some (or all) of their ammo to add to your stockpile.

If a chip is a bullet, and I have 500 bullets, and you have 4500 bullets, you can utilize your ammo in many ways that I cannot. You can fire test shots to see if you can pick up a small pile of ammo that none of your enemies are all that interested in defending. You can engage in small speculative battles to try and pick up more ammo, and you can back out of these little skirmishes if necessary without much damage to your stockpile. Most importantly, because all of your enemies can see your huge stockpile, you can get them to surrender ammo to you without fighting, even in battles they would have won, were it not for their fear of losing everything.

So, intrinsically, each of your bullets has a greater value than each of mine purely as a function of its greater utility. This is due directly to the fact that you have so much more ammo than me.

The more chips you have, the more each chip is worth.

The only time chips do not have more value in a bigger stack is when the bigger stack is in the hands of a player who does not know how to use them. For example, any player who plays according to Harrington’s M strategy will not gain the full available advantage from having a bigger stack of chips. When you are waiting for hands, primarily playing your cards, and taking so little advantage of the edges available from other types of poker moves, your bigger stack will not be in action enough to earn you this greater value.

So, when I say that the more chips you have the more each chip is worth, that assumes that the player will be deploying his chips in such a way as to extract their full potential earning value.

Even Sklansky, despite his mistaken advice on rebuys, clearly recognizes that the dollar value of chips can be greatly increased by skill. On page 44 of Tournament Poker for Advanced Players he provides an example of a bystander, at the start of a $10,000 event, who is a highly-skilled player but who has arrived at the event too late to buy-in. Sklansky comments: “It might be worth it for him to buy your original $10,000 in chips for $30,000 because of his great skill.”

With these words, Sklansky demonstrates that he understands the guiding principle of all of the strategy in my book, The Poker Tournament Formula. This guiding principle is the theory of the implied discount. What is an implied discount? An implied discount comes from the fact that a skilled player can earn more with his chips than an unskilled player can earn. And how do I arrive at an implied discount from this fact?

If the player in Sklansky’s example had arrived on time to buy-in for this tournament, then (according to what Sklansky is telling us) this player would have essentially purchased chips valued at $30,000 to him, based on his skill, for only $10,000. When a player is getting $30,000 in value for $10,000 in cash, it is the same as getting his chips for one-third the price of the unskilled player.

Another way of saying this is to say that the unskilled player will have to buy in three times as often to get the same amount of winnings (not profits, but prize money) as the skilled player. (In fact, an unskilled player will never be able to profit, and may never even be able to make the same amount of prize money as a skilled player, no matter how many times he buys in. The point is that the skilled player is essentially getting his chips at a discount.)

And what are the implications of this implied discount? In The Poker Tournament Formula, I show that there is a huge value to any player in making a rebuy or add-on at a discount in an even playing field. Sklansky agrees with this concept (see Tournament Poker for Advanced Players, p. 94.) The implied discount would mean that, for a skilled player, a full-price rebuy or add-on is never really full price, and is thus a great value, with some exceptions that I cover in The Poker Tournament Formula. In fact, the only exception for a skilled player would be when the skilled player is so short-stacked that the extra chip purchase would not provide enough chips to sufficiently utilize his skill advantage. Again, when skill cannot be used, the implied discount doesn’t exist.

Unfortunately Sklansky, in his rebuy advice, fails to take into account both the extra value of chips to a skilled player and the limits on how skill can be deployed when chips are too few. He says, “I think a decent rule of thumb would be to add-on if you have less than the average number of chips at that point, and not otherwise.”

In effect, Sklansky is saying it’s fine for a skilled player to pay $30,000 for $10,000 in chips, but not to pay $20,000 for $20,000 in chips (in a $10,000 rebuy event). Sklansky has failed to realize that extra chips add to the amount of skill a player can use, and the amount of action he can generate with an edge.

More Implications for Rebuy Strategy: When Opponents Rebuy

Which brings us back to Mason Malmuth’s error in using his evaluation of chips based on stack size to generate a rebuy strategy. As I have just shown above, the fewer the chips a skilled player has, the more likely it is that he will not be able to acquire enough chips to fully utilize his skill, and the less he should be inclined to rebuy or add-on. (Again, this is exactly the opposite of Sklansky’s and Malmuth’s advice.)

On p. 196 of Malmuth’s Gambling Theory rebuy chapter, he says: “If you are leading in a tournament and someone rebuys, the pot is not being ‘sweetened’ for you… Discouraging your opponents from rebuying when they are broke should be an important part of your overall tournament strategy.”

Malmuth bases this advice on the math in a model in which all players have equal skill, making the model inappropriate for poker tournaments, and making his advice absolutely terrible for skilled players. In any real-world tournament, assuming that any players at your table actually give a damn what you recommend regarding their rebuy decisions, you should encourage the poor (read “conservative”) players to rebuy and discourage the more skillful (read “skillfully fast and aggressive”) players from rebuying. If you believe you’ve got an edge on the whole table, encourage them all to rebuy and add-on as much as they possibly can. If they don’t know how to use chips when they have them, believe me, they will indeed ultimately be “sweetening the pot” for you.

Again, this is because the skilled player’s chips come at an implied discount. Whenever players of lesser skill are in effect paying more for their chips than you, as a function of your implied discount, you will profit. They are, in effect, buying chips for you.

This is not something that mathematical analyses based on players of equal skill will reveal.

In this section, we have dealt with the implications of the implied discount for optimal rebuy strategy. Now let’s look more closely at the implications of the implied discount for the validity of various authors’ overall poker tournament strategies.

Broomcorn’s Uncle

In SuperSystem, Doyle Brunson described a South Texas player, Broomcorn, whose uncle occasionally joined the game but never played a hand. He simply sat there until his chips were dwindled away by the antes. Whenever Doyle’s fellow poker players encountered a tight player in a poker game, they would needle him by saying, “You’re gonna go like Broomcorn’s uncle.”

To this day, poker tournament pros use this saying to make fun of tight players, and for good reason.

Mason Malmuth may be the foremost advocate of a tight style of play in poker tournaments. On page 210 of Gambling Theory and Other Topics, Malmuth advises players that it is incorrect for a player who is short-stacked to raise with a “marginal hand” or push all-in with a “calling hand” because, since “the less chips you have, the more (relatively speaking) each individual chip is worth… This means that going out with a bang is wrong. You should try to go out with a whimper. That is, try to make those few remaining chips last as long as possible.”

And on page 204 of Gambling Theory and Other Topics, Malmuth provides an example of two players who enter a small buy-in tournament where each receives $100 in chips. Player A plays very conservatively, so that he always has exactly $100 in chips at the end of the first hour. Player B, on the other hand, plays a very aggressive style such that he busts out in the first hour three out of four times, but one out of four times he finishes the first hour with $400 in chips. Malmuth asks the question, “Who is better off?”

He answers his question on the next page: “…because of the mathematics that govern percentage-payback tournaments, we know that the less chips a player has, the more each individual chip is worth, and the more chips a player has, the less each individual chip is worth. This means that it is better to have $100 in chips all the time than to have $400 in chips one-fourth of the time and zero three-fourths of the time. Consequently, A’s approach of following survival tactics is clearly superior.”

Malmuth tells us little about Player B’s strategy other than that it is “aggressive and reckless.” I have described in detail in The Poker Tournament Formula how fast strategy earns more chips than conservative strategy while actually tending to lead to fewer confrontations and less risk of bouncing out of a tournament early. So there is no reason to equate the higher earnings of fast strategy with recklessness and increased bust-outs. (Despite this, even if Player B was a poker neophyte who had very little understanding of the game, but just liked to get his chips into action and gamble, I’d put my money on Player B’s overall earning power in tournaments before I’d bet a nickel on Player A. That’s how strong the value of aggression is in tournaments, as opposed to conservatism.)

But there is reason to equate Malmuth’s approach of “following survival tactics” with players being worse off, not better off as Malmuth claims. Although we can see that a situation can exist at a final table in which individual chips have less value in a big stack than a small stack, assuming all players have equal skill, Malmuth’s advice assumes that this situation exists throughout a tournament, when it does not.

The reason it does not exist for skilled players is that, as I have shown, chips have increased earning power when they are put into action with an edge more frequently, thus creating an implied discount on the chips. Malmuth’s tight “survival tactics”, by failing to use chips for their full earning power, are in effect leading to a higher chip cost for his tight player.

Again, until the point in a tournament when the remaining players get into the money, at which time, in some but not all tournaments, equal-skill models may start to make sense, the real truth of tournaments is that the value of chips is based on what you can do with them. Malmuth’s theory about individual chip values is dead wrong through most of a tournament. Throughout 90+% of a tournament, any individual chips in a short stack aren’t worth squat. They simply represent a last desperate shot at survival for players who will almost certainly not make it into the money.

The fast strategies prescribed in The Poker Tournament Formula are essentially a blueprint for earning more chips by getting more chips into action with an edge than you can get into action with a conservative strategy.

In the PTF’s fast strategy, for example, there are three positions from which a player would raise if first in with any two cards. On the button, players are advised to call any number of limpers and even to call any standard raise with any two cards. Postflop, players are advised to always bet if they were the pre-flop aggressor, even when out of position, even when the flop does not hit them, even when the flop looks dangerous, and even when their preflop raise was just a position or chip shot with a trash hand. Likewise, if a player has position on an opponent postflop, and he checks, the player is advised to bet—regardless of his hand. Other more complex and dangerous plays—like pretending to slow-play a hand in order to steal even more chips from an opponent when you have nothing yourself—are also described. But the standard positional preflop and postflop bets and raises with any two cards are presented as a “basic strategy” that a player in a fast tournament should almost always follow. (There are discussions in PTF on proper violations of the basic strategy based on the type of opponent you are facing, your chip position, your opponent’s chip position, poker tournament structure and other factors, but the book’s general approach to tournament strategy is overall much looser and more aggressive than the conservative approaches that have been written about in the past.)

The fast play strategy in The Poker Tournament Formula will consistently out-earn conservative play because it keeps your money in action while your conservative competitors are sitting there waiting for stronger cards, and hoping to make back, with trapping hands, what you have taken from them. But, in fast tournaments, a trapping hand is unlikely to arrive frequently enough to help you, and in all tournaments, trapping hands are unlikely to get paid off by skilled players even if they do arrive.

The fast play strategy in The Poker Tournament Formula is specifically designed to earn you an implied chip discount. It is designed to take chips from conservative opponents and penalize them for their weaker strategy.

The Weakness of Using Equal-Skill Models for Devising Overall Tournament Strategy

Although I use coin-flip examples in the rebuy chapter of The Poker Tournament Formula to explain specific points of tournament logic, it would be incorrect for any player to think that coin-flipping contests (or Malmuth’s “equal-skill tournaments”) are analogous to poker tournaments. It is always dangerous to use a simple example to solve a complex problem, because it may tempt a less astute researcher to use a seemingly similar analogy to answer a question for which that analogy does not apply.

For example, in The Theory of Blackjack, Peter Griffin created a game he called “Woolworth Blackjack,” which consisted of a blackjack game where the only cards in the deck were fives and tens. He used this hypothetical and vastly over-simplified version of the game to test how well a statistical estimate of expectation based on approximations might correspond to a player’s actual expectation. For the problem he was attempting to solve, his analogy works well.

Some years back I received a letter from a blackjack player who had used Griffin’s Woolworth Blackjack deck to devise a unique betting strategy for a real-world casino blackjack game. His strategy was terribly misguided, although it would work fine if the player ever found a casino that would deal actual Woolworth Blackjack. Casino blackjack is not Woolworth Blackjack. There are many complexities to the game as dealt in the casino that do not apply in the five-and-ten version. Griffin’s analogy worked well for the point of logic he was testing, but he never meant to imply that all blackjack problems could be solved by Woolworth.

On p. 196-197 of Gambling Theory and Other Topics, Malmuth demonstrates that in an “equal-skill” tournament with a percentage payout structure (as opposed to winner-take-all), there will be situations in which it would be mathematically incorrect to make a rebuy. (In fact, this particular example is incorrectly used by Malmuth as “proof” that players should not rebuy when they have “a lot of chips”, meaning as many as or more chips than their opponents.)

In The Poker Tournament Formula, I too show that, in a coin flip tournament where every player is of equal skill, there is no mathematical advantage in making a rebuy. I have no argument with the mathematics presented by Malmuth for this specific “equal skill” situation. The math is the math. My problem is with his assumption that an equal-skill event provides an adequate model upon which to devise a valid strategy for rebuying. Although his math in this example is internally correct, it is irrelevant to rebuy decisions in real-life tournaments.

In an excellent post on the Poker Board at this Web site, Pikachu provides much better models in a post titled “A Closer Look at the MTT Format and Playing With an Edge”. To summarize, he finds “A player [with] an edge of 50% should… take the rebuy if his stack is less than 4.5 times greater than his initial stack. A 100% edge player should rebuy with a stack less than 7.5 times his initial stack. For very skilled players rebuys should still be made except in extreme cases… The more skill a player has, the more often he should rebuy.”

Pikachu also shows that players with small advantages should rebuy much less often. Pikachu’s findings, by the way, validate the rebuy advice I give in The Poker Tournament Formula. (In Appendix A, I advise players that it is futile to enter multi-table poker tournaments without a very big edge, and say that with an edge as low as even 10% you should not enter multi-table poker tournaments.)

Pikachu’s models are superior to Malmuth’s because they take into account the fact that poker tournaments are not “equal skill” events. They are especially useful because they compare a wide range of realistic player skill levels (or edges) in poker tournaments.

In Closing

The Poker Tournament Formula specifically addresses winning strategies for fast tournaments. Many of the PTF concepts and strategies, however, will also have value in slow tournaments. These strategies will require adjustments for the speed of the tournament, but the concept of the advantage of fast-play over conservative strategy is still valid. Specifically, the PTF strategies will have to be adjusted to take advantage of the increased opportunities for profitable fast play action pre- and postflop, created by the slower blind structures and bigger starting chip stacks. Optimal slow tournament strategies will be very different from the conservative strategies recommended in many of the popular books today, as many of these books were written from the same erroneous perspectives on chip values as Malmuth’s book.

A number of the best professional poker tournament players believe that tight conservative play is already nearing obsolescence even in slow tournaments. Daniel Negreanu, interviewed by Peter Thomas Fornatale in the just-published Winning Secrets of Poker, says: “Basically, the math is behind an aggressive style of play. Books that have been written in the past simply didn’t have it all right as far as what hands to be playing and what hands to be folding… So without playing that [aggressive] style, you’re basically depending on getting really good cards. And when you’re depending on getting really good cards on a regular basis, you’re going to be disappointed because they don’t come often enough.”

Many of the most successful tournament pros do not follow tight, conservative strategies, even in the slowest major events. Malmuth remarks in the “Afterthoughts” on his tournament section in Gambling Theory and Other Topics, “…some people with what appears to be excellent tournament records have very little idea of what is correct… In addition, if it were possible to estimate the standard deviation for tournaments and a good tournament record was compared to a poor one, I suspect that a seemingly large difference would often not be significant. This means that some of the current superstars are probably not very good, just fortunate.”

I agree that there is a large standard deviation in poker tournaments. However, the combined win records of the aggressive fast players have moved well beyond the realm of normal fluctuations. I suspect that the fast-playing pros who keep making it into the money, keep making final tables, and keep racking up wins, know a lot about “correct” strategy.

Part II of this article deals with correcting the idea that chip value is based on stack size at a final table or non-skill tournament. See Chip Value in Poker Tournaments, With Implications for Winning Strategy.  ♠

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