Arnold Snyder's Response to Sklansky's Article is Up

[George Rice]

Snyder's Response

It's lengthy and some of it is a re-hashing of what he said in his earlier articles. But he addressed David's 2+2 Magazine article. Here's a point he made:

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But there are two major errors in logic in Sklansky’s “proof.” First, when we say that a player has a $150 expected value (or EV) on his $100 buy-in, we are saying that this player, because of his skill and overall strategy, expects an average return of $150 per $100 buy-in for every tournament like this that he enters. You can’t assign that overall expectation on the tournament as the chip utility value of the starting stack, nor can you automatically multiply this overall EV as you multiply your stack.

The reason you can’t do this is because your overall $150 EV is derived from your aggressive stack-building strategy; it’s because the player is consistently building chip leads that he has that $150 EV. The $150 does not represent the chips’ value, it represents the strategy’s value, and the value of an overall strategy doesn’t double every time a player’s chip stack does.

In incorrectly assigning the player’s overall EV as the utility value of his starting chips, Sklansky errs in his tournament logic.

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First, if we were to agree with Snyder's assertion that more chips mean more utility, and this increase in utility increase a player's chance of winning, then it must increase's his EV when he has those extra chips. Since a skilled player, I would assume, has a greater likelihood of obtaining those utility ridden chips, and those utility ridden chips add EV to his stack, it's perfectly acceptable to assign a hypothetical value to it. Perhaps this skilled player is X% likely to get to a point where the utility of the chips gained add $Y to the value of his stack and 1-X% likely to lose $Z value and overall this adds $50 to his starting value. Wouldn't a skilled player be more likely to get to a position where the "utility" of his chips increase his chances of winning than to a point a "lack of utility" decreases his chances of winning? If not, then isn't Snyder saying that a skilled player is not more likely to win than a non-skilled player? Obviously he doesn't think that. So why can't David assign an EV to the stack?

Second, why can't David multiply this overall EV as he multiplies the stacks? David was demonstrating that the chips can't be gaining value. His example proved that by doubling stack size you can't be doubling EV. The reason he did that was, in part, because Arnold Snyder claimed the following in his "Implied Discount" article:

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

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David's example disproves that statement (A claim re-asserted in Snyder's current article).

Could it be that Snyder is claiming that each additional chip is gaining worth (and EV), but the remaining chips haven’t gained EV? That could be a way of looking at it (but obviously each chip gained EV on average). Each successive chip in this stack is gaining EV but the average EV per chip in the stack is rising at a much slower rate. I don't know if Snyder agrees with that. But I hope everyone sees that if the average EV per chip is increasing, the doubling of a stack will more than double it's EV.

Snyder does go on to claim that the $150 does not represent the "chip's value", it represents the "strategy's value", . . Ah hah, now I get it, the chips only have strategy value. But does the SV of the stack grow faster that the size of the stack? If so, the individual chips in the stack are gaining SV the larger the stack gets. If not, then they are either remaining the same or losing SV the larger the stack gets.

So Sklansky was wrong. He should have anticipated the new "SV" value of chips and addressed that instead of EV. This is all getting so very entertaining.

In addressing Sklansky's "second major error," Snyder gives an example where a player's stack has doubled up, but the other player's stacks have increased more. I'm not sure of the specifics of the entire tournament, as he addresses only the table he's playing at. But he does cite consideration for blind and ante costs.

First, I'm sure (hope) everyone realizes that if a player's stack has doubled and he has an average stack for the tournament, that his EV will more than double (on average, in Sklansky's example based on Arnold's claim of increasing chip value) because that means half of the players have been eliminated from the tournament (A fact Snyder seemed to miss). Second, if for some reason he was up against a lot of big stacks at his table, then the other tables have more players on short stacks would be even more effected by the blinds and antes. This will effect overall EV, UV, SV or whatever. So while he may be up against some big stacks, the other table will have players dropping like flies (and increasing his EV). This would be an example hard to model. Seems to be a better idea to use an example when only one table left.

If it's Snyder's purpose to claim that an average stack sitting at a table where is average stack is well below the average of the table it's at, that is an interesting theory that may have merit. But if hero has 200, and the rest of the table has 800, then where is Arnold's utility value for the big stacks? Against the one lone 200 stack? Obviously the big stacks also derive some value from the other tables as players are dropping out, and the 200 stack would too. If Snyder wants to use an example like this, he can't ignore what's going on at the other tables. He should give an example when it's down to one table.

The point of David's example was to show that the chips aren't gaining value. And even if hero's chips did gain value at some point, they would have to reverse that trend and lose value at another point, because his stack can never be worth more than 3200.

Snyder has asserted that even when heads up the more chips you have the more each chip is worth. This fact is easily disproved. And unlike multi-table and multi-player situations, is not complicated to show dollar values for the chips.

For example, in David's 32 player tournament, it gets down to two players. And we'll assume it's winner take all for this example. Hero has 3199 chips and Goat has 1 chip. How much is this 1 chip worth? How much is the 3199 stack worth? We assign a value of X for one chip, and 3200-X for the value Hero's stack (because the total value must equal 3200). In order for Arnold's claim that chips gain value to be correct, the last chip won by Hero must be worth the most. Therefore it must be worth more than $1. If it was worth exactly $1, then Hero's stack would be worth 3199 (3200-1), the same as the number of chips in his stack, meaning his chips were worth $1 to begin with (in violation of Snyder's assertion). If the last chip won is worth less than $1, then Hero's stack is worth more than 3199, and the average value of each chip in that stack was worth more than $1 (again, in violation of Snyder's assertion). So to keep with Snyder's assertion, the last chip must be worth more than $1 (and hence, each chip in Hero's stack is worth less than $1)

But that means X>1. So Goat's 1 lone chip must also be worth more than $1. So Goat's lone chip is worth more than each chip in Hero's stack! And if each chip he gains is worth more than the previous chip, all Goats individual chips are worth more than Hero's individual chips (they can never be worth less than $1). You see where this is going. This is also in violation of Snyder's assertion. Therefore, Snyder's assertion that chips gain value in a head-up situation in not valid.

And this isn't a phenomenon that occurs only at the extreme ranges. Because when heads-up any EV or "SV" gained by a large stack has to be lost by the small stack. Any EV or "SV" lost by the large stack has to be gained by the small stack. EV doesn't appear from or vanish into thin air.

The conclusion you can draw from this is that when heads-up, individual chips neither gain nor lose value. Snyder proved this himself in his second article "Chip Value in Poker Tournaments." here

But he abandoned the idea because it was based on all-in "coin-flip" situations. He favored his utility value theory when the blinds are small compared to stack sizes. He came to the wrong conclusion because of that. And if this is his basis for his theory in multi-player and multi-table situations, his whole argument is based on a flawed foundation.

Snyder claimed that Malmuth's chips losing value theory doesn't hold water because it doesn't hold water in heads-up situations. Snyder's utility value theory doesn't hold water in heads-up situations (even though he claims it does). So how can Snyder claim that his chip gaining value theory hold water outside of heads-up situations? What's good for the goose . . . Seems to me that he's neither proven his theory correct, nor Malmuth's theory incorrect.

[BigAlK]

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Snyder has asserted that even when heads up the more chips you have the more each chip is worth.

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George,

I understand the logic behind your contention that this can't be true when heads up, but I've gone through the latest article twice and can't find anywhere that makes this claim specifically for heads up situations. (I've gone so far as doing a search for every time the string 'head' occurs in the latest article and can't find it.) You do point out that this would disagree with what Snyder said in his last article. What am I missing?

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Snyder claimed that Malmuth's chips losing value theory doesn't hold water because it doesn't hold water in heads-up situations.

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I also can't find where Snyder says this although I haven't searched for it as hard. The only thing I've found that seems to come close to this is that he thinks Mason is incorrect in assigning the value of 2nd place plus some value for the possibility of finishing in first to the chips of the short stack when heads up (instead assigning only some value based on a possible first place finish to these chips due to the fact that he's guaranteed to finish 2nd even with zero chips). Again, am I missing something?

Al

[George Rice]

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I understand the logic behind your contention that this can't be true when heads up, but I've gone through the latest article twice and can't find anywhere that makes this claim specifically for heads up situations. (I've gone so far as doing a search for every time the string 'head' occurs in the latest article and can't find it.) You do point out that this would disagree with what Snyder said in his last article. What am I missing?

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What he claimed again in his latest article was that chips gain value. The heads-up situation was in his second article.

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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.

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He's claiming that the player with the 10% stack has almost no chance of winning. I think he has a 10% chance of winning. The larger stack certainly didn't gain more value in each subsequent chip he won.

Your second point, from first article.

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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.

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In his second article he disagrees with the chips-losing-value premise, and proves it (a detail I had forgotten, remembering only that he disproved it in Article 2, forgetting that he accepted it in Article 1).

Snyder proved that Malmuth's chips losing value didn't hold water heads-up (which is the only thing I remember him actually proving). He claims what is quoted above from the first article, (questioning whether the since disproved heads-up-chip-losing-value theory can be applied throught a tournament and saying that it's a pecular idea). His explanation of why this is so is because of his utility factor theory.

So he was saying that because of his chip utility theory, Malmuth is wrong to make the jump from heads-up to whole tournament. Then he is saying (in Article 2) that Malmuth is wrong even in the heads-up situation. Then he is saying (later in atticle 2) that the utility theory now applies to heads-up too (more to the point of your first question).

[Mason Malmuth]

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Snyder proved that Malmuth's chips losing value didn't hold water heads-up (which is the only thing I remember him actually proving).

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Hi George:

I just want to make a small point. Once there are two plyers left, each player is guaranteed a second place prize. Thus they are only playing for the difference between first and second which means that the chips no longer change value as long as both players are equal in skill. So Snyder proving this was not something that no one knew, and it was first pointed out by Mark Weitzman in his guest essays about settling up in tournaments that appear in my book GTOT, and this was originally written almost twenty years ago.

Best wishes,
Mason

[George Rice]

[ QUOTE ]
I just want to make a small point. Once there are two plyers left, each player is guaranteed a second place prize. Thus they are only playing for the difference between first and second which means that the chips no longer change value. So Snyder proving this was not something that no one knew, and it was first pointed out by Mark Weitzman in his guest essays about settling up in tournaments that appear in my book GTOT, and this was originally written almost twenty years ago.


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I haven't read GTOT in years, I just assumed if Snyder was referring to it, and going through all that trouble, that he at least had that much right.

In his second article after proving the chips don't change value he writes in part:

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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:

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Why in the world would he be using "Weitzman's erroneous method" to calculate a heads-up situation again if Weitzman pointed this out 20 years ago?

Appoligies for not double checking with the original source and assuming that you were advocating a decreasing chip value theory in heads-up play.

Now doing so, on page 180 of the 1990 edition, I find the following paragraph:

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The second method of solving this problem is to note that they are only fighting for the remaining $600. . .

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Mark then goes on to solve the problem to arrive at the same figure he arrived at using the first method (which simply took a percentage of first place money and a percentage of second place money, the percentage based on percentage of stack size).

By the way, I tried to look at Snyder's articles with an open mind and I haven't dismissed the notion of a utility value for chips. I'm just having a hard time getting past the errors he's making in the simple cases. He's asking the reader to accept his ideas in more complex situations without providing the same type of examples he did for the simple cases.

And he advocates using zero-chip-value to help explain his theory. But never adds that back in later, or describes how that varies based on stack size, etc. And until a tournament get's heads-up, he has to account for it (but to be fair, I haven't finished reading his third article, if we're lucky, he addressed it).

[Mason Malmuth]

Hi George:

My post wasn't meant to be critical of you. I just wanted to make the small point that once a percentage payback topurnament gets down to two people the idea that the chips no longer change value is a trivial point even though Snyder has presented it like it's not.

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Why in the world would he be using "Weitzman's erroneous method" to calculate a heads-up situation again if Weitzman pointed this out 20 years ago?

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That's a good question.

Best wishes,
Mason

[George Rice]

I didn't interpret your comment as being critical. You were pointing out something I didn't realize, having gotten the oppisite impression from Snyder's Article.

[PairTheBoard]

I read Sklansky's article and quite a bit of Snyder's. I suspect Snyder misjudges the implications S/M's chip theory has on optimum play. In fact, a lot of what David is doing in his article is clarifying some of those implications - especially in the small stack case.

However, here's a thought that might be a little out of the box. Suppose certain suboptimum play wrt $EV gives you better chances at making the final table of a high prestige tournament. Might that play actually provide better Overall EV when you figure in the value of the Fame you win by getting to the final table? With of course some extra big Percs if you can actually Win the High Prestige Tourny.

PairTheBoard

[mornelth]

[ QUOTE ]
I read Sklansky's article and quite a bit of Snyder's. I suspect Snyder misjudges the implications S/M's chip theory has on optimum play. In fact, a lot of what David is doing in his article is clarifying some of those implications - especially in the small stack case.

However, here's a thought that might be a little out of the box. Suppose certain suboptimum play wrt $EV gives you better chances at making the final table of a high prestige tournament. Might that play actually provide better Overall EV when you figure in the value of the Fame you win by getting to the final table? With of course some extra big Percs if you can actually Win the High Prestige Tourny.

PairTheBoard

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Believe it or not, DS addresses it in TPFAP...

It's a one-sentense semi-humorous blurb, but it's there.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
I read Sklansky's article and quite a bit of Snyder's. I suspect Snyder misjudges the implications S/M's chip theory has on optimum play. In fact, a lot of what David is doing in his article is clarifying some of those implications - especially in the small stack case.

However, here's a thought that might be a little out of the box. Suppose certain suboptimum play wrt $EV gives you better chances at making the final table of a high prestige tournament. Might that play actually provide better Overall EV when you figure in the value of the Fame you win by getting to the final table? With of course some extra big Percs if you can actually Win the High Prestige Tourny.

PairTheBoard

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Believe it or not, DS addresses it in TPFAP...

It's a one-sentense semi-humorous blurb, but it's there.

[/ QUOTE ]

What does he say?

PairTheBoard