Snyder, Malmuth, and Sklansky all partly right?

[BigAlK]

One of the regular posters on the forum at Arnold Snyder's site posted a response to Snyder's utility chip article. In it he makes a case that Mason's chip valuation theory and Snyder's utility value theory aren't incompatible. He argues that neither one is without flaws, but both also have value. Thoughts?

[George Rice]

[ QUOTE ]
Suppose that you are in a small multi-table tournament, with 10,000 chips in play, in which the prize pool is $1,000, the top five places pay, and fifth place pays 10%. With five players remaining, you have 2,000 chips. You are not involved in the current hand. On this hand, one of the other players busts out. You had 2,000 chips at the start of the hand, and you have 2,000 at the end. Under Mason's and David's reasoning, your chips were worth $200 at the start of the hand, because you had one-fifth of the chips, and $200 was one-fifth of the prize money remaining available. At the end of the hand, those same 2,000 chips are worth less--$180, which is one-fifth of the $900 of the prize money remaining available to active players. However, it should be obvious that the EV of your finish has increased, not declined, because you are now guaranteed at least a fourth-place finish. So how can it be that your chips have, in an instant, without you even being involved in the hand, lost value?

[/ QUOTE ]

What? Is he kidding?

If the EV was based on percentage of chips the EV after the player was eliminated would be $280, not $180. And if he is trying to show what chips decreasing in value would make the EV of the 2000 stack, it would be over $280. And that would be for equal skilled players.

[Mason Malmuth]

Hi George:

It's almost funny reading some of this stuff. If first place pays $250 and second pays $200 (for zero chips), I suppose they'll say his chips are now worth $50 but that there is an additional $200 for finishing at least second.

It's some of the most convoluted reasoning I have ever seen.

best wishes,
Mason

[WRX]

[ QUOTE ]
It's almost funny reading some of this stuff. If first place pays $250 and second pays $200 (for zero chips), I suppose they'll say his chips are now worth $50 but that there is an additional $200 for finishing at least second.

It's some of the most convoluted reasoning I have ever seen.

[/ QUOTE ]

Mason:

I can't speak for what "they" would say. There's only one of me.

Again, I don't say that a certain number of chips are "worth" anything in particular. Among other points, analysis using the ICM or another random decision model will imply that a player's stack, of constant size, can result in the player having widely varying $EV in the tournament, depending on how the remaining chips are distributed among the other active players.

I think that these misconceptions of what I'm trying to say are largely the result of tenaciously holding onto the idea that "chip value" is a meaningful concept, the very idea I am calling into question.

I would be glad to see any criticism of my reasoning (just so you know, I'm getting it from both sides) when you've truly taken the time to consider and absorb it. There's not much I can say in response to discourteous, dismissive comments.

[CityFan]

Can I take you up on the idea that "chip value" is a meaningful concept?

In any tournament scenario S (including size of the blinds, position of players at the tables, time until next level etc.), there is a function F_r which gives player r's equity in the tournament as a function of every player's chip stack.

F_r = F_r(x_1,...,x_r,...,x_n,S)

Move the chips around, leaving all other conditions S unchanged, and F_r will change for each player.

Such a function exists whatever assumptions you make about how the players play. You don't have to assume that they play optimally, merely that each will play according to SOME strategy.

Now suppose you artificially increase player r's chip stack by an amount h, drawing the chips equally from each of the other stacks.

F_r[new] = F_r(x_1*(1-h/T),...,x_r + h,...,x_n*(1-h/T),S)

Where T = total chips - x_r

Note that the total number of chips in the tourney has not changed.

Player r's increase in equity is given by

F_r[new] - F_r
= F_r(x_1*(1-h/T),...,x_r + h,...,x_n*(1-h/T),S) - F_r(x_1,...,x_r,...,x_n,S)

Now, usually we would study the derivative of player r's equity w.r.t the number of chips he receives

lim(h->0) (F_r[new]-F_r)/h

I think this is valid, but it could be argued that this will often be zero, since the addition of one chip may not change his equity at all, because all bets are in multiples of the small blind.

Even then, we can look at (F_r[new]-F_r)/h for the smallest SIGNIFICANT changes in r's chip stack (or some other way of studying the "gradient" of a step function) and we then have a workable definition for the incremental value of a chip to player r.

[CityFan]

but your point that the incremental chip value depends on the other players' stacks is of course correct - I didn't realise this was in question!

[djames]

Perfect. So most people use ICM as F_r and call F_r an equity function. What does Snyder use for his utility function? Anyone pull this out of him? I don't believe I've read anywhere that he's stating his utility function is modeled by the independent chip model's equity function. So what model is he using to value scenarios where he states the utility per chip can increase as a players' chips increase?

[WRX]

[ QUOTE ]
Can I take you up on the idea that "chip value" is a meaningful concept?

In any tournament scenario S (including size of the blinds, position of players at the tables, time until next level etc.), there is a function F_r which gives player r's equity in the tournament as a function of every player's chip stack.

F_r = F_r(x_1,...,x_r,...,x_n,S)

Move the chips around, leaving all other conditions S unchanged, and F_r will change for each player.

Such a function exists whatever assumptions you make about how the players play. You don't have to assume that they play optimally, merely that each will play according to SOME strategy.

Now suppose you artificially increase player r's chip stack by an amount h, drawing the chips equally from each of the other stacks.

F_r[new] = F_r(x_1*(1-h/T),...,x_r + h,...,x_n*(1-h/T),S)

Where T = total chips - x_r

Note that the total number of chips in the tourney has not changed.

Player r's increase in equity is given by

F_r[new] - F_r
= F_r(x_1*(1-h/T),...,x_r + h,...,x_n*(1-h/T),S) - F_r(x_1,...,x_r,...,x_n,S)

Now, usually we would study the derivative of player r's equity w.r.t the number of chips he receives

lim(h->0) (F_r[new]-F_r)/h

I think this is valid, but it could be argued that this will often be zero, since the addition of one chip may not change his equity at all, because all bets are in multiples of the small blind.

Even then, we can look at (F_r[new]-F_r)/h for the smallest SIGNIFICANT changes in r's chip stack (or some other way of studying the "gradient" of a step function) and we then have a workable definition for the incremental value of a chip to player r.

[/ QUOTE ]

Greetings, fellow insomniacs.

All right, CityFan, thank you for this. I accept the validity of your formulation. It's so great to have qualified mathematicians contributing. (And I'm glad I read earlier threads, so I know what "w.r.t." means!) I understand scenario "S" to include the strategies, skills, and idiosyncracies of all players in the tournament. Therefore, this is a completely general formulation, which does not assume equal skill levels, as does, say, the ICM.

The derivative of F_r with respect to h shows the incremental value (meaning, impact on $EV) of player r gaining or losing a chip. I certainly agree that studying how the value of this derivative changes as x_r increases or decreases is informative. If you read my full article, you'll recognize that this value (the derivative of F_r) corresponds to what I call "relative chip value." I argued that this was a legitimate way of analyzing tournament situations, although I spoke of limitations on the value or validity of results of calculations using the approximation methods that, in practice, one is forced to use. But to give an example, I believe that it is instructive to attempt, in a given tournament situation, to determine the risk to your tournament $EV of risking 100 chips in an effort to win 100 chips, and to compare that to the potential reward to your tournament $EV of winning those 100 chips. Note that this does not entail actually calculating the derivative of F_r along every point in the range from x_r-100 through x_r+100. All that is needed is a comparison of F_r for three values of x_r: the current x_r, x_r-100, and x_r+100.

What I attacked in my article was the idea that there is a meaningful concept of what I called "absolute chip value." So, if we take a static tournament situation, we can, in theory, calculate player r's current F_r, which is his tournament $EV. What I objected to was the idea that this value, equal to the current F_r, should be, in our minds, invested in the chips sitting in front of player r, and that the average per-chip value should then be either (1) carried forward into future hands, and viewed as still embodied in those same chips, or (2) compared to an average per-chip value based on a new calculation of F_r, at some future stage of the tournament. Neither of these exercises will produce meaningful insights.

F_r is a function of multiple variables. The value of F_r can and often does shift dramatically based on chips being shifted around among the other stacks x, as a result of events in which player r is not even involved, even when x_r remains constant. This demonstrates that it is a conceptual mistake to invest "value" in chips, for the purpose of analyzing anything beyond the present hand of poker. The "value" of one's place in the tournament is a function of the full scenario S, and of all the stack sizes, not just of one's stack x_r.

Now, in practice, in order to do relative chip value analysis, for the purpose of analyzing the utility of a play, one is has to use greatly simplified models like the ICM. I think that these can be informative, but they have definite limitations. First, these models usually have to assume that all players have equal skill levels. They are random decision models, in the sense that they assume that the tournament will be decided by random events that are not biased in favor of one player or another. Second, the simple random decision models do not correspond in any exact way to how a poker tournament is actually decided. An important example is that they generally fail to take account of the effect of increasing blinds. I noted in my article how this limitation can lead one to an incorrect conclusion that one should not exploit small edges in the early stages of a tournament. Third, computation of the ICM appears to become intractable as you increase the number of paying finish positions, and as you increase the number of currently-active stacks. However, even for a large number of active players, one can readily calculate ICM results as long as one specifies that the stacks of all active players other than the hero are equal to one another. This may be good enough to give valuable insights.

More later.

[Mason Malmuth]

[ QUOTE ]
Among other points, analysis using the ICM or another random decision model will imply that a player's stack, of constant size, can result in the player having widely varying $EV in the tournament, depending on how the remaining chips are distributed among the other active players.

[/ QUOTE ]

Okay. And the reason for this has something to do with the percentage payback nature of tournaments.

For example. If the leader has $1,000,000 in tournament chips, you have $10,000, and the other two remaining plaers each have $1, your $10,000 is worth second place money. (This assumes the blinds are relatively large so that skill differences are minimized.)

On the other hand, if the leader only has $980,000, you still have $10,000, and the other two players now each have $10,001, your $10,000 is now worth less than second place money (assuming third and fourth places do pay a fair amount less than second).

From your post, I know that we both understand this. But when I pointed out to Snyder that he missed this sort of thing in his analysis in his book, all of this started with him claiming that his book was written as some sort of response to twenty years of bad advice form David and me.

MM

[WRX]

[ QUOTE ]
[ QUOTE ]
Among other points, analysis using the ICM or another random decision model will imply that a player's stack, of constant size, can result in the player having widely varying $EV in the tournament, depending on how the remaining chips are distributed among the other active players.

[/ QUOTE ]

Okay. And the reason for this has something to do with the percentage payback nature of tournaments.

For example. If the leader has $1,000,000 in tournament chips, you have $10,000, and the other two remaining plaers each have $1, your $10,000 is worth second place money. (This assumes the blinds are relatively large so that skill differences are minimized.)

On the other hand, if the leader only has $980,000, you still have $10,000, and the other two players now each have $10,001, your $10,000 is now worth less than second place money (assuming third and fourth places do pay a fair amount less than second).

From your post, I know that we both understand this. But when I pointed out to Snyder that he missed this sort of thing in his analysis in his book, all of this started with him claiming that his book was written as some sort of response to twenty years of bad advice form David and me.

MM

[/ QUOTE ]

This is a great example. In analyzing gambling problems, I love comparisons of a couple of extreme examples, because they show us the range of possible answers, with the answer for more "normal" examples usually falling somewhere in between.

Obviously, your conclusions are not precisely correct, but they depart only trivially from the truth.

Discussion of "chip value" (a term I don't much like, and that I would rather call the effect of plays on one's $EV, but that's unwieldy) has come up recently, but it's not something Arnold wrote much about in The Poker Tournament Formula. He had little to say about theory. The book's approach was to give very practical advice, with less discussion of the reasoning behind it. I don't take that to be a flaw. With a lot of academic discussion added, it could have been 700 pages instead of 350. The debate was sparked by your reviews, in which you raised just one major criticism. You disagreed with Arnold's advice that the speed with which you play should be influenced by the speed of the tournament--that is to say, that one should play faster if declining "M" is approaching, but hasn't yet arrived.