Snyder, Malmuth, and Sklansky all partly right?

[Mason Malmuth]

Hi WRX:

I only skimmed through your post because I'm getting tired of reading this stuff and Snyder is now someone who in my eyes has lost all credibility. But here's a quote from page 213 of my Gambling Theory book which I suspect you're unaware of.

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Percentage-payback mathematics also shows the opposite if you are against a small stack. Now the penalty for you to lose the pot will be less than it will be for your opponent.

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And from page 217:

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In addition, keep in mind that the penalty to your opponent for losing the pot (since he is short stacked) will be greater than your penalty for losing the pot.

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From page 218:

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Another reason for not making steal plays late in a tournament is that the chips have changed value. Specifically, when you are low on chips the penalty for having your steal fail may be greater than the penalty to your opponent, depending on his chip position.

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And finally from page 221:

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This means that if you have a lot of chips and your opponent is on a short stack, the penalty to him for calling and losing will be greater than the penalty to you if he calls and wins, even though he will be calling you with the same number of chips.

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Best wishes,
Mason

[Mason Malmuth]

Hi WRX:

I just read a little more, and perhaps I didn't read enough, but it seems to me you are just confused. Towards the end of a tournament, the payoff structure should be what influences your play, not so much the value of your chips. Here are two examples.

Suppose you are in a Satellite tournament where the top nine finishers get the same prize, there are ten players left, and you are the chip leader with a lot of chips. I guess you can now argue that your chips are worth the least, but so what. Your best strategy might be not to play another hand.

Now suppose it's down to ten players and the chip distribution is just as before but there are big differences between the payouts for the different places, making it correct for your opponents to play very conservatively in an effort to move up the payout ladder. Your best strategy is now to be loose-aggressive, especially if first in.

So there you have it. In each of these two examples the chip distributions are exactly the same, yet your strategy is as opposite as possible.

Best wishes,
Mason

[WRX]

Hello, Mason.

You quote yourself, "Percentage-payback mathematics also shows the opposite if you are against a small stack. Now the penalty for you to lose the pot will be less than it will be for your opponent."

Thank you, I'll keep that in mind.

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I just read a little more, and perhaps I didn't read enough, but it seems to me you are just confused. Towards the end of a tournament, the payoff structure should be what influences your play, not so much the value of your chips. Here are two examples.

Suppose you are in a Satellite tournament where the top nine finishers get the same prize, there are ten players left, and you are the chip leader with a lot of chips. I guess you can now argue that your chips are worth the least, but so what. Your best strategy might be not to play another hand.

Now suppose it's down to ten players and the chip distribution is just as before but there are big differences between the payouts for the different places, making it correct for your opponents to play very conservatively in an effort to move up the payout ladder. Your best strategy is now to be loose-aggressive, especially if first in.

So there you have it. In each of these two examples the chip distributions are exactly the same, yet your strategy is as opposite as possible.

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I don't know what you think I'm confused about. I certainly agree with your point about payoff structure drastically affecting the $EV of a potential play, and thus one's best strategy, and I comprehend your examples. In fact, coincidentally, your example of a tournament in which the top nine finishers get the same prize is essentially the same as an example I gave in the article at BJFO that prompted this thread. (It was in Section VIII, if you're interested.)

Of course you don't have to respond to my comments at all--although I appreciate that you have taken the time to respond--but I think any dialogue will be better if, when you choose to respond, you first read my full message. :-)

The conclusions I stated in my response to George Rice et al. were:

(1) The statement, "Chips decline in value as a tournament progresses," is not true in any meaningful sense.

(2) The statement, "The value of each successive chip you acquire is less than the value of the previous chip," is a very different proposition, and may well be true. (I did not make an effort to address the issue one way or the other. I'm not writing my magnum opus in one Internet forum message.)

(3) Analysis of marginal chip value in discrete tournament situations is appropriate and useful. The form of function stated by CityFan in an earlier post is a true general expression of how an increase or decrease in one's stack affects one's tournament $EV. Thus, if solvable (which in practice will have to be by approximation methods), it is a valid way of deriving the marginal values of acquiring or losing chips. Even without a solution to this function, elementary observations will lead one to realize that in a percentage-payback tournament, the marginal value of acquired chips must begin to decline at some point. (I haven't elaborated on this, but I did note that it will be true after taking into account distortions caused by tournament prize structure, chip utility effect, and any other effects that might exist.)

Maybe this isn't earthshaking. I never said it was. Baby steps.

Even though in your eyes, Arnold has lost credibility, I hope I haven't lost credibility. Not that I ever had a reputation to begin with. You'll note that I haven't accepted everything Arnold writes uncritically, and am perfectly willing to question what he says, when I think it might be wrong.

You know how I feel about gratuitous disparaging remarks from either side.

[Mason Malmuth]

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You know how I feel about gratuitous disparaging remarks from either side.

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I don't know how you feel about this. In fact, I suspect that you are probably very suspect in this area. Remember, both David and I have been personally attacked with statements such as that we have been putting out bad information for twenty years (in all our books) and I have read numerous misquotes of my (and David's) material claiming that we say things and advocate strategies which we don't come close to. Yet I have not seen you speak out here.

To be specific, I suggest you read this post.

MM

[George Rice]

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So it is possible that in attacking the syllogism stated above, I am attacking a straw man. I am sure that there are people who will set me straight as to whether it actually represents David and Mason's views, or is an inaccurate spin that others have put on it.

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Exactly. David and Mason haven't stated that the reason chips lose value is because prize money has been awarded (that will actually increase value, at least in most cases (no time to think this through now)).

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(1) Chips decline in value as a tournament progresses.

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No, the opposite. It's more a factor of stack size. But they can go up if your stack stays the same size and players are eliminated. How much is one chip worth if it gets you into the money?

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Again, don't accuse me of adopting the logic I'm critiquing.

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Fine. But you were critiquing logic that no one was stating. You misunderstand their poistion. That was my point:

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Most 2+2'ers, especially S&M, will tell you it's worth $315 (give or take, probably less

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And the 315 number came from your example of equal distribution of the remaining prize money. I don't agree with it, just that if you were applying your logic to an example (whether or not it represents your views) you should have used that method. And you were right to divide by 5 instead of 4. I accepted MRBAA's point prematurely.

The actual value of that stack, if you used an actual prize distribution in your example, would be too complicated for me to figure out. You would also need the chip counts of the other players. The articles are in GTOT which explain how to do this. The solution would require using determinants, which I long ago have forgotten how to do.

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George Rice and MRBAA accuse me of "mistakes," but the calculations they criticize are the calculations called for by the very theory I am questioning.

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The calculations were wrong, base on the supposed theory you were questioning. And I see your point about the 180 (which I originally accepted), but you should have taken 1/5 of the remaining prize after 5th an 4th place had been awarded and added that to 4th place. So if 120 for forth, then 276; if 100 (same as 5th), then 260. The three of us messed that up. [img]/images/graemlins/blush.gif[/img] But this is the wrong method anyway, as you misunderstood S&M.

[WRX]

Hi, Mason. I had previously read the material you point me to, and now I've read it again, because it's good and important.

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Yet I have not seen you speak out here.

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And I won't, because I'm not interested in cutting anyone down. I'm interested in good poker theory, and good practical strategies. The cream will rise to the top.

[Mason Malmuth]

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Hi, Mason. I had previously read the material you point me to, and now I've read it again, because it's good and important.

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Yet I have not seen you speak out here.

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And I won't, because I'm not interested in cutting anyone down. I'm interested in good poker theory, and good practical strategies. The cream will rise to the top.

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That's fine. But given that's the case, you shouldn't bring this subject up.

MM

[WRX]

Hi, CityFan. I'd like to follow up with observations on certain implications of your general "F_r" function. There has, of course, been a debate raging on the question of how changes in one's stack affect one's $EV in the tournament--and in particular on the question of under what circumstances the marginal value of acquiring chips is increasing as one's stack grows, and under what circumstances it is decreasing. This question is equivalent to the question of how the value of F_r changes as the value of h changes. You defined h as the amount by which x_r grows, and specified that h is removed in equal amounts from the other stacks x. (A slight extension of this principle is to posit that for negative values of h, the loss of chips from x_r is added to all other stacks x in equal amounts.)

Of course removing h from the other stacks x is not the only way that h could be added to stack x_r in an actual poker tournament. In practice, chips will usually be removed from just one or a few of the other stacks. One could devise functions to model these other scenarios. However, in its broad outlines, I think the analysis of these other scenarios would parallel the analysis of your more specialized scenario.

The rules of a tournament impose constraints on the value that F_r can take:

(1) Define Z as the total prize pool, and z_m as the prize for finishing in position m. Then Z = z_1+z_2+...+z_m. F_r can never be greater than z_1.

(2) Define X = x_1+x_2+...+x_n (the total of all chips in play). Then F_r = z_1 when x_r = X. In all other cases, F_r < z_1, necessarily, because there will always be at least a slight chance of another surviving player winning first place.

(3) F_r can never be less than zero.

(4) When x_r = 0, F_r will always be zero, unless the number of surviving players has previously been reduced to the number of payout positions, or less. (This factor would be included in the scenario S.)

(5) When x_r > 0, F_r will always be positive, because player r will retain at least a slight chance of finishing in the money.

Assuming that no rebuys or add-ons are allowed, and assuming that player r's skills are equal to the skills of the field, or are superior, we would ordinarily expect the following to be true at the start of the tournament: F_r >= Z/n (that is to say, the amount that player r paid for his stack), because player r's prospects should be at least as good as those of the average player. Under any plausible assumptions as to the characteristics of the players and the entire scenario S (and unless the tournament field is very small in comparison to the number of paying finish positions), this leads to the following conclusion: in the neighborhood of the starting value of r_1, the slope of the curve plotting F_r against x_r is close to Z/X or is greater. In other words, the value of the first derivative, F'_r, is close to Z/X or is greater.

The reason for this is that we know that the starting point of the curve is the point 0,0, and we do not believe the rate of change in F_r/x_r to be huge in the range between zero and the initial value of x_r = Z/n. This suggests that the curve in that range plotting F_r against r_1 is close to a straight line segment. Since we are discussing an average or superior player, whose initial F_r >= Z/n, we know that the average initial slope of the curve over the range of values of r_1 from 0 to X/n is greater than or equal to (Z/n - 0)/(X/n - 0) = Z/X.

If the curve were to continue in a straight line with the same slope, when the value of x_r reached X, the value of F_r would be equal to or greater than Z. However, this is impossible, because F_r cannot exceed z_1, which is less than Z. Therefore, the slope must decline at some point, and must decline very substantially.

If on first entering the tournament, player r has an overall positive tournament EV of 200% of his entry fee, this translates to an initial F_r = 3*Z/n. This implies an initial slope of 3*Z/X, which extrapolates to F_r = 3*Z when x_r reaches X. Again this is impossible, by an even greater margin than would be the case for the player with no initial positive tournament EV. So the ultimate decline in the slope of the curve will be even sharper. What goes up, must come down, and the higher you fly....

In terms David Sklansky has previously used, for an average or superior player, the curve must at some point become convex upward. For a greatly inferior player, this is not necessarily true. It is conceivable that the curve could start at such a low angle that it would be concave through its full path. "Convex" translates to declining marginal chip value, while "concave" translates to increasing marginal chip value. Whether marginal chip value is increasing or decreasing answers the question of the utility of a "coin flip."

This is by no means a full solution to marginal chip value problems, because the curve could conceivably follow many paths to its ultimate destination.

One approach to trying to attach real numbers to the function F_r is to apply the independent chip model (ICM). As we have discussed, the ICM makes very specialized assumptions, and in other ways does not exactly correspond to how a poker tournament is actually decided. Significantly, it assumes no skill advantage for any one player. It also makes no allowance for the increasing blind structure of a tournament. Nevertheless, it offers insight into the impact of the payout structure on the utility of various stack sizes.

Others have created calculators for applying the ICM to final-table problems with only three prizes. As we have previously discussed, it appears that solving the ICM in its general form becomes intractable with increasing numbers of prizes and increasing numbers of players remaining active. However, I have created a spreadsheet for calculating one flavor of the ICM in situations involving large numbers of active players. This assumes that the stacks of all remaining players, except the player being analyzed (player "r"), are equal to one another. Note that this is consistent with your version of F_r, which assumes that chips acquired by player r are taken from the stacks of all other player in equal shares.

Using this model, I ran a calculation of F_r given the situation at the start of a tournament with the following payout structure:

Finish rank: % of pool awarded:
1 29.00%
2 18.50%
3 12.00%
4 10.00%
5 8.00%
6 6.50%
7 5.50%
8 4.50%
9 3.50%
10 2.50%

100.00%

The tournament has 100 players, the buy-in is $1,000, and each player receives 1,000 tournament chips. This results in a $100,000 prize pool, and a $29,000 first prize.

I then took the results, and plotted them as a graph. The results look like this:



Again, this is a rather specialized model, one of many in the universe of models that could be true for a particular tournament structure, and just one of many that assume no skill advantage for any one player. Nevertheless, it may be observed:

(1) For a player with a skill advantage over the field, we would expect the value of F_r to be above the ICM curve, for any value of x_r. This is just another way of saying that, for any stack size, skilled player r's tournament expected value should be greater than that of an average player holding the same stack. From this, one can theorize as to where the curve begins to be convex for a skilled player. (The observation that the skilled player's curve is above the average player's curve at all points might not be true if the generally-skilled player were extremely deficient in the play of stacks of a certain size--very small, very large, or even medium.)

(2) For a player with skills below those of the field, we would expect the value of F_r to be below the ICM curve at all points. Note that this could result in a curve that was still convex at all points, was flat, was concave at all points, or had a wavy form, partially concave and partially convex.

This still doesn't prove or disprove your stated principle, "Chips gained will usually increase your equity less than chips lost will decrease it." That would depend a great deal not only on the shape of the curve, but also on where on the curve the player usually spends the most time.

More later.

[George Rice]

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The calculations were wrong, base on the supposed theory you were questioning. And I see your point about the 180 (which I originally accepted), but you should have taken 1/5 of the remaining prize after 5th an 4th place had been awarded and added that to 4th place. So if 120 for forth, then 276; if 100 (same as 5th), then 260. The three of us messed that up. But this is the wrong method anyway, as you misunderstood S&M.

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Oh good, nobody noticed. I can correct my own mistake. [img]/images/graemlins/smile.gif[/img]

The 120 should be subtracted for all four remaining players, and also the 100 for fifth. So $420 remains and that should have been divided by 5. So if 5th pays 100 and 4th pays 120, the 2000 stack would be worth $204 if chips didn't change value. If chips decrease in value as S&M claim, then the 2000 stack is probably worth more, depending on chip distribution (figured out as in GTOT). And this is more than 200, so his EV does go up when a player is eliminated.

[Radar_O'Reilly]

Again, the mistake you guys are making in this thread is in assigning chip value as an overly simplified function of prize. You are completely failing to deal with the real-life complexity of chip value, and there is no reason for this, because it is addressed in Arnold Snyder's article. To be blunt, the math in this thread is pretty, but it has nothing whatsoever to do with real-life tournament strategy and success.
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I realize that none of you are actual tournament players, or that, if you do play, you are playing at a very very low level.
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However, gambling advice is properly seen as investment advice, and you are responsible for giving players very bad advice on what to do with their money. In addition, you are failing to identify yourselves as non-practitioners, as well as failing to alert players that you have not performed statistically significant tests of your theories, while Arnold Snyder has gathered statistically significant win results on his theories.