## Excel sheet to calculate Average vs Number of games.

jericho
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Poker-TH2.xlsx
This excel file is for evaluate if it is worth to play more games or you must improve your average in order to reach a desired position. The cells with red background are related to a total score greater than 150.

jericho
SLC42
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Moin!
jericho wrote: Fri May 19, 2023 10:25 am Poker-TH2.xlsx This excel file is for evaluate if it is worth to play more games or you must improve your average in order to reach a desired position. The cells with red background are related to a total score greater than 150.

As the partial derivation of the formula over "average" is a constant, you really just need the first row.

Example: n = 57, f(n, 1) = 23.72
That means: avg = 6.34, f(n, avg) = avg * f(n, 1) = 6.34 * 23.72 = 150.3848 (150.38).

Next step would be the partial derivation over "games", that's a bit more complicated.
It also depends on considering "average" or "points", because avg = pts / games.
If you use points, the term depends on games in a different way.
Other than if you hold the average constant. Want some more?

What's also missing in your sheet, are input fields for avg and n (games).
Or is this just an overview/table?

Have fun, SLC42
SLC42
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Moin!

Some more obvious facts:

f(n, 1) <= 25.00 for all n (lim(n->inf)=25 convergent).
f(n, avg) <= 15 * 25 = 375.00 (avg <= 15, so this is also the limit for winning an infinite number of games)

d/d avg f(n, avg) = f(n, 1)
f(n, avg) = avg * f(n, 1)

d/d n f(n, avg) = 3 * n² * 25 * avg * 10,000 / (10,000 + n³)² = 750,000 * avg * n² / (10,000 + n³)²

This has a local maximum at n = 10 * cubrt(5) = 17.1 of
d/d n f(17.1, avg) = 750,000 * avg * 17.1² / (10,000 + 5000)¹ = avg * 17.1² / 300 = 1.026 * avg

Isn't it funny, that the most effective number of games is already at 17.1? With approx. 1 point per game and average point...

I for now omit the case with points instead of average...

Grüße, SLC
SLC42
Posts: 5
Joined: Tue Jan 31, 2023 3:54 pm
Location: Hamburg
Contact:
Moin!

Okay, for g(n, pts) = f(n, pts/n) or avg = pts / n, the local maximum is at n = 10 * cubrt(5 * (7 - 3 * sqrt(5))) = 11.3418 with
d/d n g(11.3418, pts) = 0.0400373 * pts

Edit: Another interesting thing is the real root at n = cubrt(20,000) = 27.1442, that means
d/d n g(n, pts) < 0 for n > 27.1442! You have to increase your points by a lot to hold your result after this point.

There's also a local minimum at 40.9245 in the negative range, but as the lim(n->inf) is 0, that doesn't matter!

For your purposes, the first formula may be enough!

Grüße, SLC