math people - help me better understand pokerstars tlb formula

Hey math guys, I'm bad at math.

Here is the page on PokerStars.com that explains it: The PokerStars Tournament Ranking System

The formula they use is:

Quote:

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Points = 10 * [sqrt(n)/sqrt(k)] * [1+log(b+0.25)]

Where:

n is the number of entrants
k is the place of finish (k=1 for the first-place finisher, and so on)
b is the buy-in amount in US Dollars (excluding entry fee). For freerolls the buy-in is $0, and for FPP tournaments 1 FPP is counted as $0.0161. If the tournament is not a US Dollar tournament, the buy-in will be converted to US Dollars using the current exchange rate at the time the tournament begins.

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The main questions I have are related to the square roots and logs. Why use them? Is this a fair ranking system considering the buy-in level, amount of entrants, etc? Could it be improved?

If anyone has the time to break it down and explain the weights and such, I'll be forever grateful, and give you a custom title if ya dig.

Points = 10 * [sqrt(n)/sqrt(k)] * [1+log(b+0.25)]

Where:

n is the number of entrants
k is the place of finish (k=1 for the first-place finisher, and so on)
b is the buy-in amount in US Dollars (excluding entry fee). For freerolls the buy-in is $0, and for FPP tournaments 1 FPP is counted as $0.0161. If the tournament is not a US Dollar tournament, the buy-in will be converted to US Dollars using the current exchange rate at the time the tournament begins.

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First off, the 10 is a factor of proportionality. It scales the outcome as you'd expect happens when you multiply by 10.

Next, the [sqrt(n)/sqrt(k)]. This is equal to sqrt(n/k). It makes more sense to me in this form. The n/k bit means that the more people entered, the higher the value, and the better you place in that event, the higher the value. Taking the square root of this means that the n/k term has a diminishing return. In order to double your points, you need to quadruple your n/k value.
The same holds for the n in the numerator. Given the same k and b, you need to quadruple n to double your points.
The k in the denominator means that there will be a significantly greater effect when its value is small than when it is large. The difference between n/100 and n/99 is much less than the difference between n/2 and n/1. This holds up to our valuation norm that first is worth much more than 2nd which is worth much more than...

Finally, the [1+log(b+0.25)].
log(0.1) = -1 ; log(1) = 0 ; log(10) = 1 ; log(100) = 2 ; log(1000) = 3
See the pattern?
log(0) is undefined. It gets negative very quickly as the input gets close to 0. log of negative numbers requires the use of i=sqrt(-1).
So the +0.25 inside the log is to force the result above some threshhold to avoid hugely negative numbers. The 1 + is because the log(b+0.25) term can still be negative, but definitely not less than -1.
So it's roughly a measure of how much the buy-in was by counting the decimal places of the amount. Except without rounding to the nearest whole number.

Why did they choose a sqrt? Why did they choose a log? Is it fair?
Well, it's an arbitrary points system. It's fair if it applies equally to all. It will favor some strategy, as all points structures do.
This one favors playing in (and doing well in) large events. It rewards playing in more expensive events, but at an increasing lower weight as the buy-in increases.

Now I have to dig, eh?

http://cdn.meme.am/instances/58470240.jpg

thanks for breaking that down for me! i was actually hoping you would see the thread after i wrote it!

and now you will forever be known as the monkey that won't dig. that is unless you want it to say something else.

[DELETED]

EDIT: WP, sir.

That was an interesting read. I must say though that every time I saw sqrt I thought squirt and not square root. It was hard to remember that this was a serious maths discussion.