Exploring how marginal chip value changes with stack size

[WRX]

I posted this message, in somewhat different form, as part of a thread in the Books and Publications forum. It didn't get much of a response. I'm hoping that more people will see it here, and will be interested in commenting. It's relevant to certain ongoing controversies.


General Form--Tournament Equity as a Function of Stack Size

In the earlier thread, CityFan proposed the following function, which appears to me a valid formulation, to address the question of the marginal value of gaining or losing a chip:

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In any tournament scenario S (including size of the blinds, position of players at the tables, time until next level[, the strategies, skills, and idiosyncracies of all players in the tournament,] 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 [with respect to] 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.

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Note that this is a completely general formulation, which does not assume equal skill levels, as does, say, the independent chip model (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. Studying how the value of this derivative changes as x_r increases or decreases shows how "relative chip value" or "marginal chip value" changes over the range of possible stack sizes.

To give an example, it would be instructive, in a given tournament situation, to be able 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. Unfortunately, just stating the form of the function F_r does not enable us to solve it, to attach any actual value to F_r for any value of x_r.

CityFan defined h as the amount by which x_r grows, and specified that h is removed in equal amounts from the other stacks x. 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 CityFan's more specialized scenario.


Implications as to Range Across Which Marginal Chip Value Can Be Increasing

Even though it is not practical to calculate exact values of F_r for a full scenario S, reflection on the general F_r function quickly leads to important implications.

There has, of course, been a debate raging 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. 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.

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.

This is a formal proof of a point that a lot of people consider intuitively apparent, and that could be adequately explained in many fewer words. But it lays the groundwork for what follows.


Further Conclusions as to Superior and Inferior Players

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.


Approximations Using Random Decision Models

In practice, in order to do marginal chip value analysis, one has to use greatly simplified models like the ICM. 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. The ICM is one such specialized model for solving the function F_r. This offers a certain amount of 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. 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 specialized form of the ICM 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 CityFan's 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.

Quite independent of the results of the ICM, I earlier stated the conclusion that for a superior player, across the initial range of x_r, between zero and the buy-in amount, here 1,000 chips, the slope of the curve plotting F_r against x_r should have a slope of $1/1 chip or greater. A curve, continuing from this initial segment, either concave or linearly, reaches the maximum value of F_r, $29,000, at the point at which x_r = 29,000, or earlier. This apparently places an outside limit on the range of values of x_r for which the curve can be concave--in other words, the range for which marginal chip value can be increasing. More generally, if z_1 is first prize, marginal chip value cannot increase past the point at which x_r = (z_1/Z)*X. In practice, there would not be a sharp break from positive to zero marginal chip value at such a point, so marginal chip value must begin to decline at some point before the stack reaches (z_1/Z)*X, or 29,000 chips in the example given.

(This will hold true at all points of the tournament until some of the prize money has been awarded. Thereafter, marginal chip value could be constant or increasing until some larger stack size was reached.)


Questions Remaining Unanswered

This still doesn't prove or disprove the principle that has been stated as, "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.

The modest conclusions just stated still leave a considerable range of stack sizes over which marginal chip value may be increasing for a superior player, due to the chip utility effect. In the example given, it would be a major achievement to increase one's stack from 1,000 to, say, 6,000--and it's entirely plausible that marginal chip value could be increasing up to that point, or much higher.

F_r is a function of multiple variables. Note that the curve plotting F_r against x_r, and hence player r's location on that curve, can and routinely does change substantially as a result of changes to the other stacks x, or the scenario S. Events leading to this include, as examples only, other players winning chips from each other, players busting out, blinds increasing, or other players wising up and starting to play better poker. All of this has the potential for placing player r in a position in which marginal chip value is now increasing, although it was previously decreasing for him. However, the range across which marginal chip value can potentially be increasing remains subject to constraints as discussed above.

As CityFan has noted, "In tournaments we all find ourselves rooting for one or other player in an all-in situation that doesn't involve us, precisely because the result will have an effect on our tournament equity."

I welcome comments on these observations. I think that it may be possible to extend these ideas, to further quantify the situations in which increasing chip utility has a potential for putting a player in a situation of overall increasing marginal chip value. Any thoughts along those lines would be helpful.