calculating the value of fl

This is confusing because of the times when a player stays in fl.


Let's say we have the following:
hand 4: fl + 13
hand 13 fl - 3
hand 16 fl +20
hand 17 fl +20
hand 18 fl +10
-------------------------
sum points = 60
sum count (FROM REGULAR HANDS) = 3 (17 and 18 don't count)
avg FL value = 20 points per hand


Because we're calculating the value of getting into fl FROM A REGULAR HAND, we could say that we got in 3 times and the 2 times we stayed in were not from regular hands so they don't increase the count. Using that logic we could say the value is 20.


However, we didn't play 3 hands, we played 5 hands. This means the argument can be made that our average is inflated because the count is wrong.


What do you guys think?

This is what Carl and Tyson said about it:

[Carl]
let's say we calculate avg net points scored in POFC FL is 12, and the chance to stay in FL is 15%. Couldn't we come up with a value that includes the 15% chance of staying in FL each time for another 12 avg points?


[Tyson]
This is the same method I suggested.


If it's 12 avg points, then the calc would be 12 + (15% * 12) + 15%(15% * 12)...


Really, this is just 12 + 1.8 + 0.27...


The drop off is quick, we wouldn't need to go 10 deep, or even 4 deep.


But there is a math calculation to come up w/ infinity value - but after that 2nd iteration I don't think it'll even add up to 1.

Originally Posted by Eric

Because we're calculating the value of getting into fl FROM A REGULAR HAND, we could say that we got in 3 times and the 2 times we stayed in were not from regular hands so they don't increase the count. Using that logic we could say the value is 20.


However, we didn't play 3 hands, we played 5 hands. This means the argument can be made that our average is inflated because the count is wrong.?

Why does the previous hand affect the value of FL?

The deck is shuffled and you get 14 cards to play... I don't see how the previous hand affects the value of a FL hand.

You have to know that a certain %-age of FL hands will get to stay in FL. So the average value of FL is increased by that %-age.

E.g. Let's say FL is worth an average of 10 points in royalties, and 5% of the time, you will get to stay in FL... and 5% of those will get to stay in FL... and so on.

So the value of FL is (1 + 0.05 + 0.0025 + 0.000125 + ...) * 10 = 10.5263158


***
I'm not 100% sure I understand what you're trying to solve, though.

But there is a math calculation to come up w/ infinity value

The sum of an infinite geometric series is
a/(1 - r)
where a is the initial value and r is the geometric factor.

EDIT: strictly speaking, the value of r must be between -1 and 1 or the series will grow infinitely. This is of no concern to us, since you can't stay in FL more than 100% of the time, or less than 0%. [/EDIT]

In our example, a is the average value of FL, and r is the %-age of hands which remain in FL.

***

let's say we calculate avg net points scored in POFC FL is 12, and the chance to stay in FL is 15%. Couldn't we come up with a value that includes the 15% chance of staying in FL each time for another 12 avg points?

12/(1 - .15) = 12/.85 = 14.11765

MMM,


I'm not sure I understand what you're saying.


---


Here's one way which seems low:


When we stay in then our hands tend to be monsters worth more points. Suppose we were to calculate the average based on getting there from regular hands and then add value based on a multiplier.


hand 4: fl + 13
hand 13 fl - 3
hand 16 fl +20


This would give us an average of 30/3 or 10 and then we multiply 10 by 1.x where x is the % chance of staying in. The problem is that our initial total for the base average is low because it doesn't include one of the 20 point monsters from staying in. In other words, if we have a 15% chance of staying in then our average is greater than 1.15*10 because we're not factoring in the monster totals when we stay repeatedly in our initial average.


10*1.15 seems low.


---


Here's another way which seems high:


What about using the average of 12 (which looks at all 5 hands including monsters) and multiply that times 1.15. It seems like we're double counting here by using the monster hands that led to 12 and then multiplying on top of that.


12*1.15 seems high.

can't you just analyse all the hands that have been played on FTR and work out the actual average value of FL from regular hands and from staying in FL. is the sample size big enough for the actual average values to be tending towards the theoretical average values

*grumble* arghh

A major post just got deleted.

In summary.

You're right, I was perhaps over-simplifying.

The value you're looking for is:

A_n + A_y*r/(1-r)

A_n is the average value of hands that do not stay in FL.
A_y is the average value of hands that stay in FL.
r is the %-age of hands that stay in FL.


Derived:

EV of first hand in FL

A_n * (1 - r) + A_y * r

EV of subsequent hand

r * {above}

The sum of this infinite geometric series is:

( A_n * (1 - r) + A_y * r ) / (1 - r)

A_n + A_y*r/(1 - r)

Ok, cool. So what is the value of fl based on the data from these 5 hands?

hand 4: fl + 13
hand 13 fl - 3
hand 16 fl +20
hand 17 fl +20
hand 18 fl +10
-------------------------
sum points = 60
sum count (FROM REGULAR HANDS) = 30 (17 and 18 don't count)
sum count (FROM STAY HANDS) = 30 (17 and 18)
number of regular hands = 3
number of stay hands = 2
total number of hands = 5

A_n + A_y*r/(1 - r)

A_n = 30/3 = 10
A_y = 30/2 = 15
r = 2/5 = 0.4 = 40%

A_n + A_y*r/(1 - r)

10 + 15 * .4/.6 = 20

The EV from the above example is 20... it's absurdly high compared to the average values because of the 40% rate of staying in FL.

r is the %-age of hands that stay in FL.
r = 2/5 = 0.4 = 40%

I'm confused about that 40%. A fuller picture looks like this:
hand 3: regular +/- ?
hand 4: fl + 13
hand 12: regular +/- ?
hand 13 fl - 3
hand 15: regular +/- ?
hand 16 fl +20
hand 17 fl +20
hand 18 fl +10
We only qualified for fl 3 times from regular hands above.


Suppose we had this:
hand 101: regular
hand 102: fl
hand 103: fl
hand 110: regular
hand 111: fl
hand 112: fl
In this case one could say I stayed in at least once 100% of the time I got there from regular hands (I was 2/2).

r is the %-age of hands that stay in FL. To "stay" means you were there and you will be there... so if you weren't there to begin with, then it can't count toward this stat.

If I am not in FL, and I get to FL, this has no effect on r.

If I am in FL, then the outcome has an effect on r. Either I stay in, and it counts as a success, and r goes up - or I do not stay in, it counts as a failure, and r goes down.

***
To summarize, consider the mechanic between 2 consecutive hands:

non-FL -> non-FL :: the first hand is non-FL, so it doesn't count toward this stat.
non-FL -> FL :: the first hand is non-FL, so it doesn't count toward this stat.
FL -> non-FL :: this hand counts as a failure to stay in FL
FL -> FL :: his hand counts as a successful stay in FL

***

I'm confused about that 40%. A fuller picture looks like this:
hand 3: regular +/- ?
hand 4: fl + 13
hand 12: regular +/- ?
hand 13 fl - 3
hand 15: regular +/- ?
hand 16 fl +20
hand 17 fl +20
hand 18 fl +10
We only qualified for fl 3 times from regular hands above.

Qualifying for FL from a regular hand has no effect on the %-age of hands in which you stay in FL. A hand which doesn't start in FL can not, by definition, stay in FL.

In this example (assuming all non-listed hands are non-FL), there are only 2 hands which are in FL, which were ALSO in FL on the prior hand. There are 5 total hands in FL... 5 total hands which could have stayed in FL... of which 2 did actually stay.

2/5 = 40%

***

Suppose we had this:
hand 101: regular
hand 102: fl
hand 103: fl
hand 110: regular
hand 111: fl
hand 112: fl
In this case one could say I stayed in at least once 100% of the time I got there from regular hands (I was 2/2).

Assuming hand 104 - hand 109 are regular hands.... hand 113 is also assumed to be regular.

There are 4 opportunities to stay... of those 4, 2 were achieved.

You could say that this player has a 100% rate of getting to FL from a non-FL hand, and a 50% rate of staying in FL from a FL hand. Yes, you can say that the player stayed in at least once every time he got to FL, but that's not really a statistic, so much as an interesting fact.

***
This is all about the confusion involved when trying to extrapolate data from small sample sizes. This data is making you uncomfortable, because you have an intuitive feeling that these %-ages are too high. You're probably right. But we're looking at very (ridiculously) small sample sizes in this thread... so the results will be all over the board and indicative of nothing, really.

What we achieve through these small examples is to show how the math works in an example that can be calculated by hand. Once we add thousands of data points, the results will be much more accurate.

MMM,

It's a coincidence that your formula and my averaging method of not including continuation hands as real hands in the count both came up with 20, right?

Carl said this in email:

Also, I'm not sure if that is the best way to calculate the value, by excluding consecutive hands from the count. Wouldn't that inflate the value? In your example, you actually only netted 12 points per hand, so why is the value of FL 20 points per hand?

Again, I would counter that by saying the continuation hands aren't normal hands. They would not exist had we not stayed in fl. Their value wouldn't have been captured had we not gotten into fl in the first place. It is easier to get my point across by studying some home games. For example, I think ej said they don't even move the button in fl because it is treated as an extension of the regular hand instead of as a new hand.

Originally Posted by Eric

It's a coincidence that your formula and my averaging method of not including continuation hands as real hands in the count both came up with 20, right?

I was all primed to say yes. I'm not so sure, though. We both looked at the same situation and wanted to find the value of entering FL, and we came up with the same number. It is SUPER common in probability and statistical calculations to have methods that look extremely different that are actually equivalent.

Originally Posted by Eric

Carl said this in email:

Also, I'm not sure if that is the best way to calculate the value, by excluding consecutive hands from the count. Wouldn't that inflate the value? In your example, you actually only netted 12 points per hand, so why is the value of FL 20 points per hand?

It's 20 and not 12 because of the continuations. Does this help:

If FL scored exactly 12 every time, and there was an X% chance that you got to stay in FL, then the EV of FL is definitely greater than 12.

***
Maybe it's a misunderstanding about EV. EV is not necessarily a viable outcome. For instance, a fair 6 sided die has an EV of 3.5, however, the die contains no faces with a value of 3.5. You'll never roll a 3.5... but that's the Expected Value.

Originally Posted by Eric

Again, I would counter that by saying the continuation hands aren't normal hands. They would not exist had we not stayed in fl. Their value wouldn't have been captured had we not gotten into fl in the first place. It is easier to get my point across by studying some home games. For example, I think ej said they don't even move the button in fl because it is treated as an extension of the regular hand instead of as a new hand.

Let's make sure we all understand the population of data in the same way.

Consider this image:

The outer, grey area is ALL hands in POFC. It represents 100% of the data. It is our Population.

Within that population, we've chosen to divide it into 2 groups. Hands that are played regular, and hands that are played FL.

The inner, white area is just the FL hands.

Inside the first white rectangle is another rectangle, representing hands that stayed in Pineapple 1 time. Inside that rectangle, there is another, representing hands that stayed in Pineapple 2 times... etc... all the way down.

We acknowledge that the rate of getting into FL is different than the rate of staying in FL, as there are different conditions for each. The first white rectangle covers X% of the grey. The 2nd white rectangle covers r% of the first, and all the subsequent rectangles cover that same r% of the last rectangle.

Here, X + r = 1, as X is the %-age of non-FL hands and r is the %-age of FL hands.

The grey area is the non-FL hands.
The white area is the FL hand that ends a FL streak
The blue areas are the FL hands that stay

Let
A = the average value of non-FL hands
B = the average value of FL hands that do not stay in FL
C = the average value of FL hands that stay in FL
X = the %-age of all hands that are non-FL
r = the %-age of all hands that stay in FL

The total EV would be

A(X) + {EV of white and blue bits}(1 - X)


{EV of white and blue bits} = B(1 - r) + C(r) + C(r)(r) + C(r)(r)(r) + ...
{EV of white and blue bits} = B(1 - r) + C(r + r^2 + r^3 + ...)
{EV of white and blue bits} = B(1 - r) + C/(1 - r)


A(X) + ( B(1 - r) + C/(1 - r) )(1 - X)

Wow, this is deep.

MMM,

You asked these questions via email:

You can't score negative royalties, and we're not interested in total score. Total score will vary by opponent's skill... Don't we want to know the value in royalties only?

You were able to calculate a fl value based on the 5 sample hands I put at the top of this thread. The +- was for total score, not just royalties. That's why the point value for hand #13 was negative.

Looking at the 5 hands at the top of this thread, they can be summed like this:
regular fl hand count: 3
regular fl hand points: 30
continuation fl hand count: 2
continuation fl hand points: 30

Alex and I will get this same summary data for the 3 players with the most fl hands.

OK. I guess either stat is valid, and you are right... we are looking at the EV of FL before the hand begins... so we want to know the score value, not just the royalties.

I see the merit in both stats, but you're right. Ignore my email.

Here's what I get for Phi's pine hands:
regular fl hand count: 1,803
regular fl hand points: 22,892
continuation fl hand count: 274
continuation fl hand points: 6,032


Pine FL value of around 16. This includes hands against Wazzup, I think the value will drop when we exclude hands against Wazzup.

My numbers are lower.

reg fl:count(*)
sum(win)
463
4714


cont fl:
count(*)
sum(win)
69
1480



13.4 avg

Originally Posted by Eric

Here's what I get for Phi's pine hands:
regular fl hand count: 1,803
regular fl hand points: 22,892
continuation fl hand count: 274
continuation fl hand points: 6,032


Pine FL value of around 16. This includes hands against Wazzup, I think the value will drop when we exclude hands against Wazzup.

Average value for reg FL: 22892/1803 = 12.7
Average value for stay FL: 6032/274 = 22.0
rate of stay: 274/(1803 + 274) = 13.2%

EV of FL from these data:
A_n = 12.7
A_y = 22.0
r = 13.2%

1 - r = 86.8%

A_n*(1 - r) + A_y/(1 - r)
12.7(86.8%) + 22.0/(86.8%) = 36.38

Wait, that's not right... researching...

Oops... the A_y/(1 - r) bit counts a score of A_y at 100%. We need to subtract that bit off of the infinite geometric series, so that it counts the first A_y at r%.

A_n*(1 - r) + A_y/(1 - r) - A_y
12.7(86.8%) + 22.0/(86.8%) - 22.0= 14.37

Wait, now that's too low... working...

A hand calculation gives a minimum of 16.8
12.7*(86.8%) + (12.7 + 22.0)(13.2%) + (12.7 + 2*22.0)(13.2%)^2 + {6 more terms} =
11.02 + 4.58 + 0.99 + {less and less} = 16.805

I'm going to have to sleep on it.

This is tripping me up.

OK, I think I sussed the problem. My picture is wrong... well... it's the one we need, but the math is different.

The math lets each new rectangle exist outside the previous, but we're nesting them.

EV of FL from these data:
A_n = 12.7
A_y = 22.0
r = 13.2%

1 - r = 86.8%

A_n + A_y*( 1/(1 - r) - 1 )

12.7 + 22.0*(1/86.8% - 1)
12.7 + 22.0*(1.15 - 1)
12.7 + 22.0*(0.15) = 16.04

This is verified by hand calculating each EV up to 9 steps.

Use this picture as a reference.


PROOF:

The white part of the picture has an EV of
A_n*(1 - r)

The first blue bit (and none of the other blue bits) has EV of
(A_n + A_y)*(r - r^2)

The 2nd blue bit (and none of the other blue bits) has EV of
(A_n + 2A_y)*(r^2 - r^3)

The 3rd blue bit has EV of
(A_n + 3A_y)*(r^3 - r^4)

... and so on and so forth...


Taking the sum and collecting the terms to factor out powers of r:

A_n + A_y*(r) + A_y*(r^2) + A_y*(r^3) + ...

A_n + A_y*(r + r^2 + r^3 + ... )

Slight sidetrack
1/(1 - r) = 1 + r + r^2 + r^3 + ... = 1 + (r + r^2 + r^3 + ... )
So then
( 1/(1 - r) - 1 ) = (r + r^2 + r^3 + ... )

Finally
A_n + A_y*(r + r^2 + r^3 + ... ) = A_n + A_y*( 1/(1 - r) - 1 )

Also... aside from that... in a ridiculously rigorous fashion...

I found the following %-ages for FL draws in POFC

drawing 14 from a deck of 52, and taking only suits into account (completely ignoring card value):

no flush - 20.6%
1 flush - 66.2%
2 flushes - 13.2%

no flush = no suit in your hand has more than 4 cards
1 flush = exactly 1 suit in your hand has at least 5 cards
2 flushes = exactly 2 suits in your hand have at least 5 cards

It doesn't tell you whether you should play them as flushes.

Originally Posted by MadMojoMonkey

EV of FL from these data:
A_n = 12.7
A_y = 22.0
r = 13.2%

1 - r = 86.8%

A_n + A_y*( 1/(1 - r) - 1 )

12.7 + 22.0*(1/86.8% - 1)
12.7 + 22.0*(1.15 - 1)
12.7 + 22.0*(0.15) = 16.04

This is what I get too: 28,924/1,803 = 16.04

MMM,


Per that other correspondance, I gave you the wrong data (fland is the stamp and its timing is different than fl).


Here are the new values for Phi and they exclude bot hands:
regular fl hand count: 298
regular fl hand points: 2,692
continuation fl hand count: 47
continuation fl hand points: 394


Phi vs Human Pine FL Value = 10.4

EV of FL from these data:
A_n = 9.03
A_y = 8.38
r = 13.6%

1 - r = 86.4%

A_n + A_y*( 1/(1 - r) - 1 )
9.03 + 8.38*( 1/86.4% - 1 )
9.03 + 8.38*( 1.158 - 1 )
9.03 + 8.38*(0.158)
10.4

-.-


*sigh*

So my formula is equivalent to just taking the average of all the data.
The only advantage is that it shows you the rate and EV of various lengths of FL run... if you bother to work out the terms long hand, that is.

The upshot of all of this is that we've proven that the value of FL is just the average value of FL hands... and the fact that some of them are stays and some of them aren't is already reflected in the average, so we don't have to do anything fancy at all to get the EV of FL.

That is something.

5 players with the most regular fl hands in pine vs humans:

uid	count(*)	sum(win)
444	509	4763
449	386	4327
75	298	2692
30	215	1572
518	194	1903

players with the most continuation fl hands in pine vs humans:

uid	count(*)	sum(win)
449	69	807
444	64	452
75	47	394
518	41	401
270	36	270
359	35	363
30	33	324

Pine FL Values:
444 (ben.neimqn): 10.25
449 (ubirman): 13.30
75 (OneByPhi): 10.36
30 (Eric): 8.82
518 (bandrei17): 11.88