Chip Value in Poker Tournaments
Correct Chip Value Theory and Implications for Winning (and Losing) Poker Tournament StrategyBy Arnold Snyder
(From Blackjack Forum , Fall 2006)
© Arnold Snyder 2006
In my recent article “The Implied Discount”, I addressed a theory (first proposed by Mason Malmuth in his 1987 Gambling Theory and Other Topics) that linked the dollar value of chips in a tournament to the size of a player’s chip stack. Specifically, the theory states that individual chips in a small stack have greater value than individual chips in a larger stack. For convenience, I am going to refer to this theory as the "reverse chip value theory" throughout this article.
The reverse chip value theory, which has also been embraced by David Sklansky in his 2002 Tournament Poker for Advanced Players, is important because, as I will show in this article, it has led to tournament strategies that are potentially very costly to players. In “The Implied Discount,” I contested this theory based on chip utility value during the portion of tournaments where skill determines results. In this article, I will show why the reverse chip value theory is invalid even later in a tournament, including at a final table, and even when skill and chip utility are not a factor in results.
At the time I wrote “The Implied Discount,” my intuition told me that this reverse chip value theory was incorrect, even in a tournament where skill is not a factor, but I didn’t pin down the source of the error in this theory until shortly after that article had been posted. In this article, I’ll explain the errors behind the theory, show the correct way to calculate the value of chips in a tournament, and discuss the implications of correct chip value theory for winning poker tournament strategy.
The Erroneous Chip Value Theory
To refresh your memories on the reverse chip value theory itself, let’s return briefly to an example I used in “The Implied Discount” to explain it:
The example above does appear to illustrate that the chips in a larger stack have less dollar value than the chips in a smaller stack, assuming that two players are at a final table where no skill advantage is possible for either player. But this logic is based on a false assumption about how chips are awarded in percentage-payout tournaments.
The false logic can be found on pages 220 and 221 of Gambling Theory and Other Topic, in an essay written by Mark Weitzman. In this essay, Weitzman provides a method for players to chop up a prize pool, based on their respective chip stacks, in a percentage payout tournament where the players decide to quit the tournament before the finish. In his example, one player has 70% of the chips while his heads-up opponent has 30% of the chips. Essentially, Weitzman assigns 70% of the first place prize, plus 30% of the second place prize, to the player with 70% of the chips.
Weitzman’s method of chopping the prize pool does provide a fair distribution based on each player’s expected value with his respective chip stack. Unfortunately, Weitzman then leaps to a very wrong conclusion about tournament chip values. He says: “This example also illustrates a general point: If you are behind in a percentage payback tournament, your chips are worth more than their face value, and if you are ahead, your chips are worth less than their face value. For similar reasons, Mason Malmuth has pointed out that the winner of a percentage-payback tournament has in a sense suffered a ‘bad beat’—he has just won all of the chips, but he gets to keep only a percentage of his winnings.”
If you look at the chip distributions Weitzman comes up with for the players in his example, it is obvious that he is saying that you can determine the value of individual chips by dividing a player’s expected win by the number of chips in his stack (which is exactly the method I used in my example above).
But it is not true that you can determine of value of individual chips in this way. In a percentage payout tournament, there are always one or more payouts that players receive for having zero chips, meaning for having busted out before first place (the payout for the player who has won all the chips). In other words, because there is a specific dollar value to having zero chips, this value must be taken into account when calculating tournament chip value. Specifically, to correctly calculate the value of the chips in your stack, you must calculate their value based on what they add to the value of having zero chips.
What Weitzman is doing wrong is assuming that all payout value is derived from your chip stack. He is failing to take into account the portion of each payout’s value that is due to having zero chips. To return to the example above, if a player expects a payout of $3500 for having all the chips (or taking first place), and $1800 for zero chips (or going out in second place), Weitzman would be incorrectly dividing the full $3500 of the first place prize by the number of chips in the first place player’s stack in order to come up with the value of an individual chip. What he should be doing to come up with the value of a chip is dividing $1700 (the amount added to the $1800 payout for zero chips) by the number of chips in the first place player’s stack.
I can see how Weitzman made this error, and why Malmuth and Sklansky accepted the error. I accepted the error myself in my initial analysis in "The Implied Discount". Unfortunately, Weitzman’s mistake causes the individual chips in a big stack to appear to be worth less than the individual chips in a small stack. In reality, as I will show, all chips have the same value at any given point in a tournament, no matter what the size of the stack they are in.
The Correct Way to Analyze Tournament Chip Value
Properly understood, all prizes in a percentage payout tournament, other than first prize, are awarded for running out of chips, or having zero chips (after having survived longer than a specific number of opponents).
Let’s return to the example above, in which two players remain heads up at the final table. Again, the total prize pool remaining consists of a $3500 prize for first place, and an $1800 prize for second place. We are also going to assume that neither player has any skill advantage, and that the blinds at this point are so high that both players must go all in on every hand. In other words, the tournament results might as well be based on coin-flips. There is a total of $80,000 in chips on the table.
Let’s look at what each player can expect to win based on the size of his chip stack (this expected win amount is called “Stack Value” in the chart below), as well as the dollar value of each $100 chip in his stack at this point. (The math for figuring out the payout value based on stack size is explained below the chart.)
|TOTAL STACK||STACK VALUE||VALUE ADDED BY $10K CHIPS||$100 CHIP VALUE|
Explanation of Chart
1. In this example, and in other examples throughout this article, the house fee is ignored.
2. The first column is the player’s stack size. There are $80,000 total chips in play. It is assumed that if the player has a $50,000 stack, his opponent will have a $30,000 stack.
3. To calculate the “Stack Value” for an equal skill event, you simply analyze it as if the players were flipping a coin to determine the winner. If one player has 90% of the chips in play, he will win first prize 90% of the time and second prize 10% of the time. For example, with a total stack of $70,000, as above, this stack represents 7/8 or 87.5% of the $80,000 total chips in play. So, we take 7/8 of the first place payout, and add this to 1/8 of the second place payout, and we have the EV, or expected value, in dollars, of that size stack when heads up in a coin-flip against the player with the remaining chips.
4. The “Value Added by $10K Chips” shows the exact dollar value added to the stack if $10K in chips are added to it, by having won those chips from the opponent. Note that in every case, adding $10K in chips adds the same amount of dollar value ($212.50) to the player’s chip stack, regardless of the stack size.
5. The “$100 Chip Value” shows the dollar value of each $100 chip in the stack, and can be obtained by dividing the $10K value by 100.
The chart shows a number of interesting things. If the theory that chips in a bigger stack are worth less than chips in a smaller stack was correct, then we would expect to see the dollar value of adding $10K in chips to a $60K stack to be less than the dollar value of adding $10K in chips to a $20K stack. But this does not occur—every $10k added is worth $212.50, regardless of the size of the stack to which it is added. Nor does subtracting $10K in chips from a stack ever diminish a stack’s dollar value by anything other than that same $212.50. Chips a player wins have the exact same dollar value as chips a player loses, and stack size is irrelevant.
For example, the dollar value of a $70,000 stack is $3287.50, while the dollar value of a $60,000 stack is $3075.00, a difference of $212.50. And this same difference can be found between any two stacks of any size that are exactly $10,000 apart.
The percentage payout tournament structure I used above is not an isolated example of individual chips always having the same value in a non-skill tournament, regardless of stack size. If you are good with spreadsheets, you can set one up to do the simple math, and you’ll find that regardless of the number of chips in play, and regardless of the percentage of the chips you assign to a player as his chip stack, adding or subtracting chips to or from his stack always adds or subtracts the same dollar value per chip, regardless of his stack size.
By contrast, if I use Weitzman’s erroneous method to calculate the value of a $100 chip in the sample tournament I described above, here is what the chip values would look like:
|TOTAL STACK||STACK VALUE||VALUE ADDED BY $10K CHIPS||$100 CHIP VALUE|
[To come up with the value of a $100 chip using Weitzman’s invalid assumption, we simply divide the stack value by the total number of chips in the stack, and multiply by 100.]
The value of a single $100 chip, based on this incorrect method of analyzing chip value in a percentage payout tournament, rises steadily from well under five bucks (with an $80,000 stack) to more than twenty bucks (with a $10,000 stack). But the data in the Stack Value column contradicts the data in the $100 Chip Value column! We can see in the Stack Value column that every addition or subtraction of $10,000 in chips is worth exactly $212.50, regardless of stack size.
So how can the individual chips within the stack differ in value when $10K in chips always has the same value? The answer is: They can’t. And, unfortunately, this incorrect chip valuation has lead to many errors in poker tournament strategy advice.
It is important to note, for example, that if you are heads up with the last remaining opponent, and you are down to your last $100 chip, it is a mistake to think that your last chip is worth $1800+. That chip is worth just a little more than two bucks, like every other $100 chip on the table at this point in the tournament (all but one of which are in your opponent’s stack).
It is zero chips that are worth $1800 at this point in the tournament, and when you bust out, you will be paid solely because you outlasted so many of your opponents. The only reason your last remaining chip has even a two-buck value is because you still have a hair of a chance to come back from the brink of death and win.
The important point, however, is that individual chips in a small stack are not worth more than individual chips in a big stack. In fact, as we’ll see, the shorter the stack, the closer to worthless the chips in that stack become. (In other words, in real-world tournaments that are almost never pure coin-flips, it’s highly unlikely that your last chip is worth even two bucks at this point.)
The erroneous concept that individual chips in a short stack are worth more than individual chips in a large stack pervades the thinking and advice on overall tournament strategy in many popular books. However, it is mainly works by Mason Malmuth and David Sklansky in which chip value theory is cited as the basis of the strategy advice. To a large degree, this is because of the care Malmuth and Sklansky take to try to justify their advice with underlying poker math and logic.
Malmuth’s advice on how this concept relates to overall tournament strategy is somewhat contradictory. For example, on page 198 of Gambling Theory and Other Topics, he says: “It needs to be pointed out that this force becomes significant only late in a tournament. Early in a tournament, it is not that crucial.”
But, in this same chapter, two pages later, Malmuth uses the reverse chip value theory to advise that progressive rebuys may not be correct. He says: “Progressive payback mathematics clearly shows that the more chips you buy, the less profitable the rebuy is.” And on page 196, Malmuth again bases rebuy advice on reverse chip value theory where he writes: “Don’t rebuy when you have a lot of chips.” Since rebuy decisions are made in the early portion of tournaments, Malmuth is clearly basing an important early tournament strategy decision on the reverse chip value theory.
Sklansky too uses reverse chip value theory when he advises on page 94 of Tournament Poker for Advanced Players that players ought not to rebuy unless they have less than the average amount of chips. More importantly, Sklansky uses reverse chip value theory in his advice on how to adjust the “gap” concept for poker tournaments versus cash games.
The gap concept essentially says that it takes a stronger hand to call a raise than it does to make a raise. In The Poker Tournament Formula I and II I challenge the gap concept in poker tournaments, stating that because of the increased value of position in tournaments, it does not take a stronger hand to call a raise, but only a stronger position. (Position beats cards.)
Sklansky, by contrast, believing that individual chips are worth less in a big stack, comes to the opposite conclusion. He writes that tournaments, because they “…increase the ‘Gap’, …force you to give up on many small edges, and frequently make overall play tighter.”
And in his chapter titled "The Gap Concept", Sklansky writes that the success of the top tournament players against players who do well in cash games is a result of the top players’ widening their “Gap” requirements. He finishes by telling us: “As important as the Gap concept is anytime in a tournament, it becomes even more important still with the Gap usually widening even more during the last stages of the tourney. The reason is related to the way prizes are paid.”
To look further at the degree to which reverse chip value theory has influenced poker tournament strategies, let’s consider another concept of Sklansky’s—the idea that a chip you lose is worth more than a chip you win (see Tournament Poker for Advanced Players, page 45). Such advice necessarily leads to overly conservative play because a player would need a bigger advantage in a hand, or better odds, to risk a chip for a chip that is worth less.
I could produce a lot more examples of the flawed and costly strategies to which their erroneous reverse chip value theory has lead.
Again, in an “equal skill” tournament (assuming such an event could exist), every chip you add to your stack adds the exact same dollar value that it adds to any other sized stack. And every chip you lose, likewise, subtracts the same value.
As I mentioned in "The Implied Discount," I wrote The Poker Tournament Formula to address conservative poker tournament strategies that I consider deeply flawed, especially in fast tournaments. And perhaps the most important flaws in these strategies, for all tournaments, are due to this erroneous and longstanding reverse chip value theory.
Some players may be wondering about the validity of my own chip value theory in calculating the value of chips when more than two players remain in a tournament. To address that question, let’s consider a ten-player sit’n’go tournament (SNG) with two cash prizes—$3500 for first place and $1800 for second place. Will the value of each chip be the same at the start of the tournament when all ten players are at the table as when only two players remain?
This is of interest because although we know that zero chips pay $1800 to second place in this tournament, zero chips pay zero dollars to third through tenth place. Does this fact affect chip value, and if so, how?
To answer this question, let’s start with a simpler example. Let’s say that this SNG is down to the last three players, so that two players will finish in the money, but the third place finisher will get paid nothing. Does this bubble player change the dollar value of the chips from what they are with only two players remaining?
Here’s an easy way to think about this question. Let’s assume that one of these three players wins all of the chips from his two competitors when all three are all-in on the same hand. His first place payout will be $3,500. One of the other two players (the player with the bigger chip stack going into the hand) will get $1800, but the other gets nothing. Because we are making an assumption that these players have “equal skill,” then the average payout for having zero chips is $900, which is the $1800 second place payout divided by the two players who have an equal shot at it.
If the average value of zero chips has gone from $1800 with only one player in contention for second place, to $900 with two players in contention, then the value of the $80,000 in chips is now $3500 minus $900, or $2600. Or, at least, that’s what logic tells me. But since logical errors are easy to make when trying to solve gambling problems, I need to test this theory.
It appears that with three players remaining, the value of the chips on the table has gone up from what it was with only two players remaining. If $80,000 in chips is worth $2600 when three players remain, then each $100 chip is worth $3.25, instead of the $2.125 each $100 chip was worth when there were only two players.
It’s easy to test if this is true by setting up an example where we first distribute chips to each of the three players, then calculate what the chips are worth. An easy example, which requires no difficult math, is to divide the $80,000 in chips exactly evenly among the three players.
If we divide $80,000 by three we'll see that each player has approximately $26,666.67 in chips. (Okay, so they now have penny chips in play. Not very practical, but it works fine in a spreadsheet for our analysis.) We also know that since we are assuming all players have equal skill in addition to equal chip stacks, each player has a one-in-three chance of winning first place, a one-in-three chance of winning second place, and a one-in-three chance of finishing on the bubble with no payout. This gives each player’s chip stack a payout value of $1766.67, or exactly one third of the total $5300 prize pool.
Now, let’s say that all three of these remaining players go all-in on the first round and Player A wins all of the chips of Players B and C. Since Players B and C had exactly equal chip stacks before this hand, they will split the $1800 second prize, earning $900 each for their remaining zero chips. Player A wins all of their chips, adding about $53,333.33 to his chip stack, giving him the full $80,000 in chips on the table.
Since we know that Players B and C had chip stacks with a payout value of $1766.67 before this final hand, and now they have zero chips, how much dollar value did each of them lose when they lost their chips? Well, at the end of the hand, their zero chips were worth exactly $900 each. And since 1766.67 minus $900.00 equals $866.67, then it follows that with this chip distribution $26,666.67 in chips are worth $866.67.
If we divide the $866.67 payout value of the chips by the $26,666.67 in chips each player lost, we get $.0325, or exactly $3.25 for each $100 chip, the exact same dollar value that we had when all $80,000 in chips were in a single stack.
So, with three players at the table, assuming equal skill and the same payout structure we have been using, all chips have the same dollar value, in this case $3.25 per $100 chip, regardless of the size of a chip stack.
We do find something interesting in this three-player example, however. The dollar value of the chips goes up as the number of players contending for the prize pool goes up. This is because the relative value of zero chips is smaller earlier in the tournament, due to the fact that busting out now will not lead to a payout. Implied actual-chip value is higher early in a tournament, and implied zero-chip value is lower.
But it would be wrong to assume that the higher dollar value of chips should lead a skilled player to more conservative play early in a tournament. That’s because the dollar value of your chips comes primarily from being able to use them to fight your way to the higher payouts.
If chips are worth more in a ten-player SNG when three players remain than when only two players remain, it follows that at the start of this ten-player SNG, the chips will have even greater value. Let’s assume that the ten initial players buy-in for $530 each, creating the $5300 prize pool—a $3500 first place payout, and an $1800 second place payout. Eight of the ten players in this SNG will get no payout. Each of the ten entrants, however, is provided with $8,000 in tournament chips to start, creating the $80,000 total chips in play. (In a real-world tournament, that extra $30 each player pays to buy-in would go to the house fee. But to keep the math simple, since we’re primarily interested in the logic, we’re ignoring the house fee.)
So, how do we calculate the dollar value of the chips at the start of this SNG? We must first subtract from the prize pool all payouts that will be paid to players who have zero chips.
In this case, we must subtract the second place prize of $1800 from the $5300 prize pool (because $1800 will be paid to a player—the second place finisher—who has zero chips), and we get $3500. Then we divide the second place prize—which is paid to a player with zero chips—by the number of players who will share it. (They won’t really share it, as we know that one player will get it all, but they will share its value.) In this case, we divide the $1800 by the nine players who will not get the first place prize, and we get a $200 average value per player.
We then subtract $200 from the $3500 first prize, and we get $3300. The $80,000 in chips at the start of this equal skill tournament have a total dollar value of $3300. And as long as there are ten players still remaining at the table, the value of the individual chips in each player’s stack—regardless of stack size—is identical. In this case, with ten payers, the value comes to $4.125 per $100 chip.
Here’s an interesting thought: When you initially buy into the above SNG for $530, the chips you receive are only worth $330, because of the average value of zero chips to each of the ten starting players. So, why would any intelligent poker player pay $530 for only $330 worth of chips? In a cash game, his $530 would purchase $530 worth of chips. Doesn’t this make a tournament a bad investment?
Before we answer this question, let’s consider what the chips are worth in a major tournament, rather than a hypothetical ten-player SNG. What are the $10,000 in chips worth that you purchase when you enter the main event of the WSOP? Let’s use the recent 2006 event as a model, but we’ll round off the numbers for convenience, and again, ignore the house fee.
In the 2006 WSOP main event, there were 8700 entrants who paid a total of $87 million to purchase chips. Of the $87 million in chips purchased, $12 million went to the winner. That means $75 million was paid out to players who finished with zero chips!
The average dollar value of a zero-chip finish at the start of the tournament was therefore $75,000,000 divided by the 8699 non-first-place finishers, or about $8,600. Which is to say, assuming this was entirely a non-skill event (yeah, right!), the chips any individual player purchased for $10,000 had a dollar value of less than $1400, because if he ultimately managed to collect all $87 million in chips in play, he would be paid less than 14 cents on the dollar of the chips’ initial purchase cost.
Now, let’s ask again why any intelligent poker player would pay $10,000 for only $1400 worth of chips? Wouldn’t this purchase be a bad investment?
Not at all. Tournaments are not cash games. In a cash game, you can never be paid for losing all of your chips. If you lose your $10,000 buy-in in a cash game, you go home broke. In a tournament, if you have the skill to outlast enough of your competitors, you can bust out—literally leave the table with zero chips—and cash out for substantially more than you bought in for.
Tournaments with percentage payout structures are designed to pay players for exhibiting exceptional skill, even if they do not beat all competitors. The chips’ dollar value is a minor factor. Let me repeat:
If you read the history of the WSOP, you’ll find that when it started in 1970, and for the first eight years, the main event was played with a winner-take-all format. During those early tournaments, every chip purchased really did have an exact dollar value based on its cost (or it would have had an equal value if the tournament players all had equal skill, which they probably did not).
In the 1970 event, there were only six players, so no one even considered paying second place. By the fifth year (1974), the number of entrants had grown to 16, and by the eighth year (1977), there were 34 players who paid their $10,000 entry fee, creating a $340,000 prize for the winner. That was the last year that the winner-take-all format was used.
In 1978, because the field of players had grown so large—there were 42 players that year, a humongous field!—it was decided to pay the top five finishers based on their finishing positions, with first place getting only 50% of the prize money instead of the whole enchilada. The WSOP main event has remained a percentage payout tournament ever since.
In a tournament like the sample SNG that we looked at earlier, you are willing to pay $530 for $330 worth of chips because—just as in the WSOP main event—you are betting that even if you do not win the whole thing, you will play with sufficient skill to outlast enough of your opponents that you will be paid well for your finishing position, even with zero chips.
Generally speaking, the bigger the event, the more the prize pool is spread out, and the smaller the percentage of the prize pool that goes to the winner. But professional players are willing to pay $10,000 for $1400 worth of chips because they see value in their skill at using those chips. Also, the dollar value of the chips without skill is negligible, and the pros know this. Without skill, chips purchased are dead money. As I pointed out in “The Implied Discount”, most of the value of tournament chips is in their value as ammunition in the battle for survival and dominance.
But many less knowledgeable poker tournament players have a tendency to confuse tournaments with cash games. This, in fact, has been true in all types of gambling tournaments. When casinos first started offering blackjack tournaments about twenty years ago, the first blackjack tournaments—like the first poker tournaments—were “real money” events. Players who entered purchased chips that they could cash out (if they had any left) when the tournament ended, and in some tournaments, they could keep any winnings they acquired in the course of the tournament.
(Unlike poker tournaments, blackjack tournaments, for reasons based on the game structure, cannot be played until one player has all the chips. They are generally played in rounds with a specified number of hands, with an elimination format that removes players based on their chip counts in relation to their opponents.)
“Play money” blackjack tournaments soon started appearing, however, in which the chips purchased for playing the tournament had no cash-out value, and any winnings the players accumulated during the tournament likewise had no value.
Which kind of blackjack tournaments do you think the pros preferred? Real money? Or play money? The answer is real money. Why? Because many of the players who entered the real money blackjack tournaments thought of the tournament chips they purchased as real money, which led them to play their chips too conservatively, and tended to give the pros a substantial advantage over them. Players in these tournaments who thought of their chips as real money were more afraid to bet aggressively, and they often could not bring themselves to violate the basic strategy blackjack plays that would be correct in cash games, but that were incorrect in many tournament situations.
Blackjack tournament pros know (as poker tournament pros know today) that the dollar value of chips in a tournament is negligible because players must beat their competitors within a game structure that makes normal “smart” cash game betting and playing decisions wrong. In fact, in the first book ever written that described winning blackjack tournament strategies—Tournament Blackjack by Stanford Wong (Pi Yee Press, 1987)—Wong explained in no uncertain terms that players who bet with extreme aggression (even when no advantage existed in the blackjack game itself due to, for example, a high proportion of tens and aces in the remaining shoe), and who were willing to violate many of the basic strategy plays that would be correct in cash games, would obtain a huge advantage over their more conservative opponents.
When Jamie Gold won $12 million in the 2006 WSOP main event, some players pointed to this result as evidence that the tournament was a “poor value,” since the winner only got paid for a fraction of the chips he won. Or, as Mason Malmuth put it in Gambling Theory, the winner suffered a “bad beat.”
But it is short-sighted to believe that Jamie Gold was in any sense “short-changed” by the payout structure. Is there a player anywhere who believes that anyone could enter a cash game with a $10,000 buy-in, and emerge some sixty hours of play later with $12 million? To make these numbers a bit more real to those who play poker at smaller levels, it would be like buying in for $100 and leaving with $120,000. Has any cash game player in history ever done this?
The fact is, no cash game player can play poker that skillfully, or find opponents who play that poorly. No opportunity to win this kind of money in such a short time with such a small buy-in will ever exist in any real-world cash game.
It remains my primary contention that tournaments are never “equal skill” events, even at a final table. Even if a tournament has become a coin-flip-with-a-prayer for the shortest stack(s) at the table, players with bigger chip stacks can almost always still play their hands more selectively, giving them a skill advantage. And tournament strategies based on the dollar value of the chips in play, as opposed to their value as ammunition, can only lead to poor results.
If you make your strategy decisions based on the idea that the fewer chips in your stack, the more each chip is worth, you will be too inclined to use low-risk, conservative playing strategies designed to “protect” your precious few chips. If I believed one of my short-stack chips was worth more than one of your big-stack chips, I would too often see it as foolish to risk one of my precious chips against one of your lower-value chips.
I would also fail to see the huge value of building a big chip stack early—or at any point in a tournament—since I would believe these extra chips to have lower value than the chips in the smaller stacks around me. Again, the reverse chip value theory is not only untrue from the start to the finish of a tournament, it represents the exact opposite of the true value of tournament chips, most of which is based on their utility.
What I am hoping to do with this article is to help struggling tournament players who have been playing according to ideas derived from an incorrect reverse chip value theory, and who can’t figure out why they’re not winning.
An equal-skill tournament analysis would lead us to believe that if only two players remained at a final table, with one player holding 90% of the chips and the other holding 10%, the player with 10% would still have a 10% chance of winning the event. In fact, if these players were flipping coins to determine the winner, that would be true. But it’s not true if the player with the bigger chip stack is a skilled tournament player who understands how to use his chips. In this case, the player with only 10% of the chips has almost no chance of winning, even if that player with 10% of the chips matches the skill level of his opponent.
We saw a perfect example of this in a televised final table a few years back, when Paul Phillips and Dewey Tomko were heads up at the end, with Phillips holding most of the chips and Tomko extremely short-stacked. Hand after hand, Dewey Tomko pushed all-in, doing his best to give himself at least a coin-flip chance of winning.
Unfortunately for him, Paul Phillips had so many chips that he didn’t have to flip coins. He just let Dewey take the blinds until he found a premium starting hand, then he called Dewey’s all-in bet. It’s not a coin-flip situation if one player is betting everything on any two random cards and the other player is playing selective strong hands.
Dewey had almost no chance of winning this tournament against Phillips’ strategy. And I’m not criticizing Tomko’s all-in-on-every-hand strategy. In fact, with his desperately short chip stack, this was his most intelligent strategy. He knew he had second-place prize money locked up if he ended up with zero chips, and he knew his short stack chips weren’t worth squat. But he had no intention of waiting for premium cards—a strategy that would almost invariably lead him to going out with a whimper.
He could not afford to give up the blinds, nor could he afford to play poker. His chips had no utility value at all, and the poker skills he possessed were crippled without the chips to play.
My analyses of chip values in this article, based on coin-flip/equal-skill assumptions, are not meant to imply that these chip values ever exist in real world tournaments. They don’t. I only present the analyses in this article to correct misinformation about chip values, based on faulty logic, that has been accepted as correct among many tournament players for decades.
Here are the two chip-value formulas that should guide your strategy decisions from the start to the finish of every tournament:
1. The more skill you have, the more your chips are worth.
2. The more chips you have, the more your skill is worth.
If you want to make money in poker tournaments, the first thing you have to do is stop thinking of your chips as money. Your chips are ammunition, and no gunfighter ever came out on top by hoarding his ammo.
You must think in terms of building your chip stack right from the start of the tournament. Nothing is more valuable to a skilled player than a chip lead on his opponents, and this is true right from the start of a tournament to the finish. You must always keep in mind not only the cash value of chips, but their more important utility value and intimidation value.
See "The Implied Discount” for a further discussion of chip utility and the value of chips in poker tournaments when combined with skill. ♠
Player Question: First, the conclusions you draw from the heads-up situation aren't really based on a percentage payback format. When the tournament becomes heads up, in effect $1800 is paid to each remaining player, and they are now playing for the remaining $1700. It is effectively now winner takes all. You alluded to this in your "Response to Mason Malmuth ..." article.
Response from Arnold Snyder: The situation I describe is very much a percentage payback situation, identical to the percentage payback situation described in Gambling Theory. All percentage payback tournaments have a prize that is awarded to the first place finisher for winning all the chips, and other prizes awarded for finishing with zero chips in various position(s) other than first, provided you outlast enough of your opponents.
The percentage payback example that Malmuth and Weitzman used in Gambling Theory was just as bad as the example that Malmuth used in his 2+2 posts where he tried to defend his position on why add-ons were bad for players who had big stacks. (For more information, see my article, The Implied Discount.)
What I show in my chart is that the two remaining players are only fighting over $1700. There was a "base prize" of $1800 already guaranteed to both. The example I give in my article shows how the percentage payback structure can lead to the erroneous thinking that chips in a big stack are worth less than chips in a small stack when you come down to the portion of the tournament where the remaining players are already guaranteed a base prize for zero chips, if you fail to account for this base prize.
If you read Weitzman's example in Gambling Theory on how to chop up a prize pool, you'll find that he uses the exact same method that I use in my article to determine a player's dollar EV based on chip stack. In fact, instead of the two-player example I used in my article, I could have just used the example that Mark Weitzman used. It is a mathematically identical situation with slightly different numbers. He specifies that there are only two players remaining. He has the chip leader with 70% of the chips and his opponent with 30%. The prize pool awards $800 to first place and $200 to second place. He then assigns 70% of $800 and 30% of $200 to the chip leader and the rest to the player with fewer chips.
There are actually two ways to do the math on this, and Weitzman shows both methods. One way is exactly as descibed above. The other way is to award each player the base prize first (in Weitzman's example $200), then give the player with 70% of the chips 70% of the $600 that is being fought over, and the player with 30% of the chips the other 30%. If you work it out both ways, you'll discover that both methods come to the same answer. In my article, I explained the first method, but I could have used either and I would have come to the same conclusion.
Weitzman correctly includes the base payout for zero chips in his calculation of EV for the prize pool chop, because both players have already earned this base payout, even if they finish with zero chips. But for calculating chip value, his statement implies that he erroneously assumed that this base payout was part of what the chips were worth rather than what zero chips were worth.
But, again, the base payout of $200 has nothing to do with the chips in players' stacks at this point. Either player would get that $200 for finishing with zero chips. If Weitzman had done the math correctly on determining the value of each player's chips--that is, realizing that the players were only fighting over the $600 that was not guaranteed--he would have seen that the individual chips in both players' stacks had the exact same dollar value.
But instead, Weitzman erroneously concludes from his analysis that "This example illustrates a general point: If you are behind in a percentage-payback tournament, your chips are worth more than their face value, and if you are ahead, your chips are worth less than their face value."
Malmuth and Sklansky then took Weitzman's erroneous notion of "what your chips are worth" (based on the size of your chip stack) to mean that individual chips in a big stack are worth less than individual chips in a small stack, which they clearly are not. They also concluded that a chip you win is worth less than a chip you lose, which would only be true if the Weitzman's conclusion about chip values were true, but it's not. The actual dollar value that any chip adds or subtracts is always identical, regardless of the size of the chip stack.
To clarify this player's question for others who may not understand, he is asking if I am making the same mistake in my percentage payback analysis that Malmuth made in his analysis in his 2+2 posts, where he based bad rebuy advice on a model that gave all players an immediate rebate on their buy-in. Essentially, this rebate may be considered a payout for zero chips.
My answer to this player is that yes, both Malmuth's model and mine award prizes for zero chips. But Malmuth failed to subtract the zero-chip payouts from his buy-in amounts, which made his rebuys cost five times more than a buy-in and made his evaluation of the profit potential of rebuys mathematically incorrect. And, in Gambling Theory, Malmuth makes a very similar mistake in failing to subtract the value of zero-chip payouts before making calculations of chip value.
My examples, by contrast, correctly subtract the payouts for zero chips.
Player Question: Second, the way you describe the value of chips seems strange to me. At the beginning of a tournament, if you were to sum up the value of all the chips you will have a total far less than the prize pool. That is because each player will also have some additional EV based on their chance of finishing in the money, but ending with 0 chips. This seems very awkward.
The additional value is tied to the chip stacks, so can't that value be attributed to the chips? It seems to me that the player's entire EV should be the same as his total chip value.
Response from Arnold Snyder: I agree with you that the correct method of calculating "chip value" will always come up with a value for all the chips in play that is equal to less than the total prize pool in a percentage payback tournament. And the reason for this is exactly as you state: There is a large value to the zero-chip finishes. The only real importance of the chip value number is in refuting the false notion that the value of individual chips goes down with stack size.
I personally think it's a waste of time for real-life players to calculate chip value in this way. In my opinion, if the prize pool is $5300, then you are shooting for as much of that prize pool as you can get, and your skill determines the value of your chips, while your chips determine the value of your skill.
But in order to determine chip value after the players have arrived at the portion of an "equal-skill" tournament where there is a base payout guaranteed for those who remain, we must isolate the zero-chip value from the first-place finish value. Again, since I don't believe that equal-skill tournaments even exist, I primarily see this as an exercise in futility. I'm only doing these analyses to refute erroneous notions of chip value that are leading to very bad strategy advice.
My example of the chip value at the start of the WSOP was included primarily to show players the absurdity of valuing chips as money. There can be a huge value to a zero-chip finish, and how much of that value you can extract will be a function of how well you can use your chips to outlast your opponents.
Players need to know that chips in a larger stack are not worth less than chips in a smaller stack. They must know that a chip you win is not worth less than a chip you lose. The analyses from which these theories came are deeply flawed.
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