Response to Sklansky's article "Chips Changing Value in Tournaments"

[Piers]

I think the basic shape is not too far off; the exact gradient of the main liner body seems to be contentious.

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As you get more and more chips their value increases (and along the steepest portions of the curve - it increases MORE than just the straight-up dollar value).

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Why do you believe this? Are you guessing blind, do you have stats to back up the claim, or are you just blindly reproducing Snyder’s statements?

Clearly the gradient will vary depending on context, in theory it is conceivable that for some people in some situations it might exceed one. However I do not see why this should necessarily be true.

The $EV of the chips stack is dependant on the amalgamation of numerous factors. Two of which are

A) Reduction in chip value due to prize structure. So that winning all the chips does not win you all the money.

B) As your stack gets bigger, you have more playing options, Good enough players will be able to leverage those extra options effectively giving a larger than linear boost to $EV.

Snyder appears to be saying that (A) is not just too small to counter (B) in some circumstances, but that (A) is completely wrong. His articles suggest he is not thinking clearly here.

Sklansky appears to be ignoring (B), despite the fact he clearly understands the point. I presume this is because he thinks it would weaken his argument if he acknowledged that (B) is a factor?

Personally I do not think shouting match, while amusing to watch, will help clarify things. What would be needed is some model that can be used to quantify the effects. Basically draw your graph to scale with real numbers; the gradient can then be measured with a protractor. Giving the Poker Tracker database of some of the clearly best tournament players in the world, to a database expert might prove instructive.

[mornelth]

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I think the basic shape is not too far off; the exact gradient of the main liner body seems to be contentious.

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As you get more and more chips their value increases (and along the steepest portions of the curve - it increases MORE than just the straight-up dollar value).

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Why do you believe this? Are you guessing blind, do you have stats to back up the claim, or are you just blindly reproducing Snyder’s statements?

Clearly the gradient will vary depending on context, in theory it is conceivable that for some people in some situations it might exceed one. However I do not see why this should necessarily be true.

The $EV of the chips stack is dependant on the amalgamation of numerous factors. Two of which are

A) Reduction in chip value due to prize structure. So that winning all the chips does not win you all the money.

B) As your stack gets bigger, you have more playing options, Good enough players will be able to leverage those extra options effectively giving a larger than linear boost to $EV.

Snyder appears to be saying that (A) is not just too small to counter (B) in some circumstances, but that (A) is completely wrong. His articles suggest he is not thinking clearly here.

Sklansky appears to be ignoring (B), despite the fact he clearly understands the point. I presume this is because he thinks it would weaken his argument if he acknowledged that (B) is a factor?

Personally I do not think shouting match, while amusing to watch, will help clarify things. What would be needed is some model that can be used to quantify the effects. Basically draw your graph to scale with real numbers; the gradient can then be measured with a protractor. Giving the Poker Tracker database of some of the clearly best tournament players in the world, to a database expert might prove instructive.

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I think the waters of this discussion are SEVEREKY muddied by the fact that Sklansky/Mason assertion applies to the $EV gain/loss of ACTUAL CHIPS with regards to te payout structure, whereas Snyder asserts that talking about the value of actual chip is essentially meaningless and the $EV of a chip is dependent a lot more on the UTILITY value of the said chip.

Both points are VALID within their own context, however Snyder's point(s) are set in the much more realistic, real-world-tournament context and therefore (at least to me) are a lot more relevant.

Data modeling - The only guy I know of around here who MAY have close to sufficient data to model is 2p2er "shaundeeb" who made a run at the Stars TLB last month (I think he ended being #8 or something like that), I think he was 30-tabling MTTs for a month and just may have a big enough sample size over a short enough period where we can say that his skillset did not change MUCH. I do not have either practical knowledge nor time to learn the appropriate methodology for statistical analysis and modeling of what we're discussing here, so I'll leave it up to whoever is qualified/interested in doing the work.

Anoter point I do not see mentioned JUST ABOUT ANYWHERE is that when you take a coinflip and double-up - your chip's EV may go down - but what about the increase of your $EV due to the fact that there are now LESS people in the tourney that you need to go thru to get to the 1st?... Obviously, the closer we get to the money - the stronger this effect becomes (and it's obv negligible on day 1 of WSOP ME)...

[BigBuffet]

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Another point I do not see mentioned JUST ABOUT ANYWHERE is that when you take a coinflip and double-up - your chip's EV may go down - but what about the increase of your $EV due to the fact that there are now LESS people in the tourney that you need to go thru to get to the 1st?...

Obviously, the closer we get to the money - the stronger this effect becomes...

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I agree. However, I see Mike Caro walking to his keyboard to argue that we should worry about knocking people out.

[David Sklansky]

B) As your stack gets bigger, you have more playing options, Good enough players will be able to leverage those extra options effectively giving a larger than linear boost to $EV.

"Snyder appears to be saying that (A) is not just too small to counter (B) in some circumstances, but that (A) is completely wrong. His articles suggest he is not thinking clearly here.

Sklansky appears to be ignoring (B), despite the fact he clearly understands the point. I presume this is because he thinks it would weaken his argument if he acknowledged that (B) is a factor?"

You skimmed over this in my article:

"Meanwhile there is a more common situation where a good player is not favored to double up before going broke. I speak of those times where his stack is very short.

When this occurs he may well have an EV greater than the amount of his chips. But that is only because if he does double up he is in a lot better shape."

In other words I specify that there are stack sizes where your EV is positive where it is nevertheless correct to go for a coin flip because your positive EV comes from the edges you get if you obtain a larger stack. For example it is possible that $100 in chips gives you an EV of $110 while $200 gives you an EV of $225. But that syndrome disappears approximately at the point, stacksizewise, where you are favored to double up vs. going broke. Which for good players is about 30 big blinds or so. (Unless the stakes will soon rise.)

[Optisizer]

You're almost there Mornelth. Take a look at these:



It is better to use the term utility instead of value, as using value by itself implies we are measuring in dollars (or other currency units) which we are not. Instead we are simply trying to quantify how usful the chips are to us.

With Delta ChipStack we mean the difference between our stack and our opponent's stack (or the median stack at the table), measured in big blind increments, which equals the current min bet size. Note, we are NOT measuring in dollars, not even in chip units, but in BIG BLINDS!!!

This give us:



Both charts assume a skilful player. Although, for these charts never mind the X and Y scales, as they will vary with the skill of the player depending on both his playing skill relative the blinds and the stack of his opponents.

Oh, and as for measuring your chip stack in dollars, it's totally meaningless. It's your final placement that are worth something in monies, not the maximum number of chips you manage to gather at any stage during the tourney. No matter if we measure in dollars or chip units, which we are not. This is no different than how it works in any other competition, like a 100 meter dash or a tennis tourney. You win money according to how you place, not according to number of points, games or sets you win, nor how many 1/100 of a second you beat your opponents with.

And that is all I am going to say about that...

[George Rice]

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Oh, and as for measuring your chip stack in dollars, it's totally meaningless. It's your final placement that are worth something in monies, not the maximum number of chips you manage to gather at any stage during the tourney. No matter if we measure in dollars or chip units, which we are not. This is no different than how it works in any other competition, like a 100 meter dash or a tennis tourney. You win money according to how you place, not according to number of points, games or sets you win, nor how many 1/100 of a second you beat your opponents with.


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It does matter how many chips you've managed to gather. That effects the chances that you will win. And in practice, it's the measurment used to make deals.

[BigAlK]

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Both charts assume a skilful player. Although, for these charts never mind the X and Y scales, as they will vary with the skill of the player depending on both his playing skill relative the blinds and the stack of his opponents.

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Optisizer,

If I'm interpreting them correctly I think the shape of your graphs is getting closer. In the first one (delta chipstack/delta utility) I'm interpreting delta utility to be the marginal value of each additional big blind added at specific delta chip stacks, correct?

I think your disclaimer (that the actual numbers on the x and y scales will vary) is an important point. While I suspect there are situations where the exact numbers you show might be right (delta utility starts climbing when you've got a standard raise more than your average opponent and decreases when you've got about 3 times that - a standard re-raise) I think the actual numbers in most situations on the BB scale would be more stretched out than this.

I don't know exactly when the first graph would start trending downward, but don't think that the delta utility would approach zero until your chipstack is at least double the median stack. Also in most cases I believe there would be increased delta utility at most stack sizes other than the very smallest even if you've got less than the average stack. I think to accurately reflect what we're trying to illustrate the first graph would have the line shifted upward, flattened out, and shifted to the left to start showing an increase even when your delta chipstack is negative.

Also I wonder if delta chipstack is really the best measure for that axis (utility and delta utility seem right for the other one). I think it needs to be something that reflects both the difference from the average stack as well as takes into account the various stack sizes. Maybe standard deviation? I'll try to explain my thinking.

If we just consider the players at our table and we've got 2 players (us and one other) who both have roughly the same size stack, but it is significantly larger than the others (say 8 stacks of around 10BBs and 2 stacks of about 30BBs) then we're probably to the point where any utility gained by getting extra chips isn't going to increace our overall utility by very much. If these same chips are distributed differently with the mean continuing to be 14BBs but with stack sizes of the 10 players varying from 10 to 20BBs then I believe the increased utility of picking up 5 or 10 additional big blinds would be significant for any of the players (although less for a 20bb stack than a 10bb stack).

Al

[BigAlK]

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I'm not going to wade through a long article when only a few dozen people are paying attention to it and I know that my words in the 2+2 magazine article is correct. It's sort of like wading through a complex dice system where its tough finding the flaw, but simple logical principles tell you there must be one.

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David,

Obviously you can respond or not as you see fit. What I get from your response is "I don't know what Snyder is saying (since I haven't really read it) but if it disagrees with me then it must be wrong." I think you're severely underestimating the number of people who are paying attention to this discussion. A couple dozen people have posted in this thread, but there are a lot more watching from the sidelines and previous discussions regarding The Poker Tournament Formula have had thousands of views.

Al

[mornelth]

The only caveat to your (and, to some extent, mine) graph is the VERY skillful player who has the knowledge and ability to deploy his chips strategically, making certified -cEV moves with certain strategic implications in mind that will result in a long-term +$EV. In order to be able to make such moves one needs a BIG stack, and it may be hard to express all that in a graph.

I also do not think the Utility EVER becomes a fully horisontal line, but it will approach that as we get more chips. There's ALWAYS some value in being able to survive coinflips and when you get down to HU we'll have to agree that chip counts represent each player's equity in the (remaining prize pool - (2nd Place$$ x 2)), so the more chips you have - the better off you are.

[Piers]

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In other words I specify that there are stack sizes where your EV is positive where it is nevertheless correct to go for a coin flip because your positive EV comes from the edges you get if you obtain a larger stack. For example it is possible that $100 in chips gives you an EV of $110 while $200 gives you an EV of $225. But that syndrome disappears approximately at the point, stacksizewise, where you are favored to double up vs. going broke. Which for good players is about 30 big blinds or so. (Unless the stakes will soon rise.)

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Ok, fair enough.

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. But that syndrome disappears approximately at the point, stacksizewise, where you are favoured to double up vs. going broke

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A winning player must at some stack size be a favourite to double up vs. going broke. If he is always favourite to go broke he can’t be a winning payer

If at a particular stack size (A say) such a winning player is a favourite to go broke vs. double up for all smaller stack sizes, it follows that there must be an increase in stack size (to B say) that will increase the players chance of doubling up vs. going broke to over 50%.

If it was a winner take all event then it seems reasonable to assert that the average chip value for a stack of size A will be less than the average chip value for the stack of size B, Flattening the payout structure will reduce the difference between the average chip value for a stack of size A and a stack of size B, until eventually they are equal. Further flattening will then result in B having a lower average chip value than A.

So where did you dig up the number 30 from?