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 ]
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.

[/ QUOTE ]

The example in my article was part of a critique of the reasoning that has been given for the theory that chips lose value during the course of a tournament. Again, the reasoning that has been advanced is that 1,000 chips you buy at the start of the tournament for $1,000 are then "worth" $1,000, because leaving aside skill differences, you have a shot at the prize pool determined by the ratio between the number of chips you hold and the total chips in play. However, the reasoning goes, when a number of the prizes have already been awarded to players who have busted out, all of the chips will of course remain in play, so applying that same ratio to the reduced prize pool still available calculates the same 1,000 chips to now be "worth" less than $1,000.

The point of my exercise was to show the fallacy of this reasoning. My exercise gave an example of a situation in which this reasoning would calculate the "worth" of a player's stack, in the course of one hand, to have declined from $200 to $180, when in fact the expected value of the player's finish in the tournament had plainly increased.

I was not trying to show that the stack of 2,000 chips had any particular "value." On the contrary, I was demonstrating that an absolute chip value theory will not produce sensible or useful results or insights.

If you have a consistent, logical, useful theory of absolute chip value, please explain it.

[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.

[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

[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!

[ECDub]

Mason, that was a good example. I am actually starting to understand this stuff more. I see your point now.

[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?