Leveling up skill : 15% or 90% ?

Leveling up skill : 15% or 90% ?

Postby captaincave » Fri Oct 31, 2014 12:48 am

This investigation is stimulated by this post
viewtopic.php?f=36&t=13837#p29443
It is to answer which strategy on leveling up skill is better, 1 material card but smallest chance per strengthen or 6 material card highest chance?

I have write a script to simulate the situation and a sample results is attached at message end.

The (theoretical) answer is , they are the same.
Statistically, about 6.67 cards (note: 6.67 * 15% = 100%) need to level up one skill level, no matter using 1 or 6 cards per strengthen.
The 6 cards 90% chance strategy may be only psychologically fell better, until you miss the level up at 90% chance ;p

The calculation is purely theoretical, under the assumption the randomness follow the numerical value presented from the game.

It may not be the case in reality.

My personal believe is that the developer may 'cheating' in favor of the players ;p
In this case, 15% is better than 90%.

bye~

Code: Select all
from random import *
def draw(n):
    return random() < n * 0.15
def reach(m,n):
    c = 0
    t = 0
    while c < m:
        t = t + 1
        if draw(n):
            c = c + 1
    return t
def sample(m,n,r = 10000):
    c = 0
    for i in range(r):   
        c = c + reach(m,n)
    return c/r

# start ------
seed()

for j in [1,2,3,4,5,10,16]:   
    print("=== reach lvl ",j)
    for i in range(1,7):
        s = sample(j,i)
        print("Each use {:1d}, total strengthen {:3d}, total used {:3d}, per level used {:5.2f}".format(i,int(s), int(s *i), s * i/j ))

# sample result
"""
=== reach lvl  1
Each use 1, total strengthen   6, total used   6, per level used  6.62
Each use 2, total strengthen   3, total used   6, per level used  6.72
Each use 3, total strengthen   2, total used   6, per level used  6.73
Each use 4, total strengthen   1, total used   6, per level used  6.68
Each use 5, total strengthen   1, total used   6, per level used  6.65
Each use 6, total strengthen   1, total used   6, per level used  6.67
=== reach lvl  2
Each use 1, total strengthen  13, total used  13, per level used  6.70
Each use 2, total strengthen   6, total used  13, per level used  6.69
Each use 3, total strengthen   4, total used  13, per level used  6.67
Each use 4, total strengthen   3, total used  13, per level used  6.74
Each use 5, total strengthen   2, total used  13, per level used  6.69
Each use 6, total strengthen   2, total used  13, per level used  6.67
=== reach lvl  3
Each use 1, total strengthen  20, total used  20, per level used  6.69
Each use 2, total strengthen   9, total used  19, per level used  6.66
Each use 3, total strengthen   6, total used  20, per level used  6.70
Each use 4, total strengthen   4, total used  19, per level used  6.67
Each use 5, total strengthen   3, total used  19, per level used  6.66
Each use 6, total strengthen   3, total used  20, per level used  6.68
=== reach lvl  4
Each use 1, total strengthen  26, total used  26, per level used  6.66
Each use 2, total strengthen  13, total used  26, per level used  6.71
Each use 3, total strengthen   8, total used  26, per level used  6.69
Each use 4, total strengthen   6, total used  26, per level used  6.64
Each use 5, total strengthen   5, total used  26, per level used  6.68
Each use 6, total strengthen   4, total used  26, per level used  6.66
=== reach lvl  5
Each use 1, total strengthen  33, total used  33, per level used  6.65
Each use 2, total strengthen  16, total used  33, per level used  6.62
Each use 3, total strengthen  11, total used  33, per level used  6.69
Each use 4, total strengthen   8, total used  33, per level used  6.63
Each use 5, total strengthen   6, total used  33, per level used  6.67
Each use 6, total strengthen   5, total used  33, per level used  6.66
=== reach lvl  10
Each use 1, total strengthen  66, total used  66, per level used  6.66
Each use 2, total strengthen  33, total used  66, per level used  6.67
Each use 3, total strengthen  22, total used  66, per level used  6.65
Each use 4, total strengthen  16, total used  66, per level used  6.66
Each use 5, total strengthen  13, total used  66, per level used  6.67
Each use 6, total strengthen  11, total used  66, per level used  6.66
=== reach lvl  16
Each use 1, total strengthen 106, total used 106, per level used  6.66
Each use 2, total strengthen  53, total used 106, per level used  6.67
Each use 3, total strengthen  35, total used 107, per level used  6.70
Each use 4, total strengthen  26, total used 106, per level used  6.66
Each use 5, total strengthen  21, total used 106, per level used  6.67
Each use 6, total strengthen  17, total used 106, per level used  6.67
"""
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Re: Leveling up skill : 15% or 90% ?

Postby Boop » Fri Oct 31, 2014 2:10 am

As someone who had maxed out both some panzer, horsemen and other cards, I feel like your codeythingguy is failing the 90% method more than it should be.

I'm not saying it isn't possible, because you know, rule of infinite probability and such. But it isn't that bad on average.

That also isn't to say that I always do 6 for skill up either. Just sharing my thoughts.

I've only ever had 3 90% skill ups fail out of a heap.

But if you roll the dice poorly, yeah I guess you could fail 10 in a row. But is PM 'cheating' just because the ability to fail exists? Odd selection of words.
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Re: Leveling up skill : 15% or 90% ?

Postby captaincave » Fri Oct 31, 2014 3:48 am

Ha, sorry, my native language is not English, but I choose the word 'cheating' intentionally.
Don't you think the phrase "the developer cheating" used mostly in situation like players' ranting of low drop rate?
So I guess the phrase "the developer cheating in favor of players" has some rhetorical contradicting sense I intended :)

I have the impression the developer in this game is generous, not like other games.
The impression is built up from personal experience on farming the 4th slot card for stage complete reward.
It is agonizing when there are seldom gold chest appeared, or the gold chest only roll to 3rd slot for so many runs of the same map stage.
But there are several instances I encountered that suddenly 2 or 3 gold chest appeared in same run and the 4th slot is rolled, sometime 2 4th slot cards at the same time.
This phenomena suggest me the developer may switching the drop rate algorithm under some condition, in favor of the players.

Of course, I recapitulate again, this is my personal speculation only.

But the experience that you based on, and said that 90% strategy "isn't that bad on average", seems support my impression and speculation (that the rate algorithm switching exist ).
Yet [90% strategy "isn't that bad on average"] does not contradict to ''15% is better than 90%" :p
ie. 15% has higher chance hitting the rate algorithm switching, if there is really such switching.

bye~
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Re: Leveling up skill : 15% or 90% ?

Postby pmsupport » Fri Oct 31, 2014 4:32 am

Hi captaincave, thank you for calling us generous and describing us as "cheating in favor of players", but to avoid confusion, I have to clarify that we actually never do such things. We can't switch the drop rates for one account and we never do. The drop rates of all the existing stages haven't been changed for a long time. Everybody has the equal chance to get the same cards in the same stage.
Enjoy the game!
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Re: Leveling up skill : 15% or 90% ?

Postby captaincave » Fri Oct 31, 2014 4:51 am

pmsupport wrote:Hi captaincave, thank you for calling us generous and describing us as "cheating in favor of players", but to avoid confusion, I have to clarify that we actually never do such things. We can't switch the drop rates for one account and we never do. The drop rates of all the existing stages haven't been changed for a long time. Everybody has the equal chance to get the same cards in the same stage.


I agree the chance is fair to every player :)

I'm ok to delete this thread if this topic is not appropriate .

Thank you~
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Re: Leveling up skill : 15% or 90% ?

Postby Boop » Fri Oct 31, 2014 5:23 am

But we have to agree with captaincave's statement that the developers are somewhat generous.

Things like energy pots don't exist in most games, and that itself is generous.

Generous cheaters. :p
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Re: Leveling up skill : 15% or 90% ?

Postby BlackFro » Fri Oct 31, 2014 9:13 am

Your calculations are wrong.

When dealing with probability, like 15% success rate for skill up for 1 and 90% success rate for 6, you actually have to look at the failure % when determining the better probably chance of something.

For example:
When sacrificing 1 at a time, you have a 85% chance of failure each time.
Now when determining the chance of failing all 6 times, you have the following % failure .85^6
This means you will fail all 6 attempts about 37.715% of the time.
Compare that to 10% failure chance of 1 attempt at 90% success rate, and you will find doing 6 at once is statistically better than doing 6 separately.

That being said, you could in theory get lucky and hit more than 1 success in that batch of 6, which would give you a better result than max of 1 for the 90% chance.

Just for giggles and grins, if you wanted the know the chance of getting 6 skill ups from 6 separate 1 sac each, that will occur .00114% of the time, or about 114 out of every 100,000 sets of 6.

The odds are never in your favor with only 1, but if you want to gamble... I won't stop you.

Just for completeness:
6 sets of 1 : 62,3% of at least one success
3 sets of 2 : 65.7% of at least one success
2 sets of 3 : 69.75% of at least one success
1 set of 6: 90% of at least one success
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Re: Leveling up skill : 15% or 90% ?

Postby captaincave » Fri Oct 31, 2014 10:10 am

BlackFro wrote:Your calculations are wrong.

When dealing with probability, like 15% success rate for skill up for 1 and 90% success rate for 6, you actually have to look at the failure % when determining the better probably chance of something.
.... snip ....

Just for completeness:
6 sets of 1 : 62,3% of at least one success
3 sets of 2 : 65.7% of at least one success
2 sets of 3 : 69.75% of at least one success
1 set of 6: 90% of at least one success


Could you please calculate the exact probability of reaching lv 5 with less than 5*6 = 30 cards under the strategy 1 card 1 strengthen?
One of the benefit of simulation is to present the mathematics truth to mathematics non-professional neatly and simply :)
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Re: Leveling up skill : 15% or 90% ?

Postby BlackFro » Fri Oct 31, 2014 1:57 pm

Well if using 1 only, we know that we have a 62.3% chance of success for getting a single upgrade in 6 attempts.

So if we are shooting for 5 levels of upgrading (taking you from level 1 skill to level 6 skill), and using 30 as our target, we can use the previous percentage to help us.

With a 62.3% chance of succeeding on 1 of 6 attempts, we can then just use the same formula as above for the success scenario.

This time because we need all instances to be successful, rather than just 1, we actually have to take the success percentage.

Final% = S% ^ N ; where N = number of sets we can have in our restriction, in this case 30 "cards", or 5 sets of 6 (the basis I am using).

F% = 62.3% ^ 5 = 9.374% chance of getting 5 successes in 30 cards or less.

Compared to using the 90% success rate which is still only 59.05% chance of getting 5 successes in 5 attempts.
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Re: Leveling up skill : 15% or 90% ?

Postby captaincave » Fri Oct 31, 2014 6:53 pm

The previous 'with less than 30 cards' condition is not right in comparison, since it is impossible to success 5 times with 'less than' 30 cards under the 6 cards strategy.
It should read 'with not more than 30 cards'.

My calculation is different.
Let r = 0.15, q = 1-r = 0.85
p(5 success with not more than 30 strengthen with 1 card strategy) = p(not the case 'success less than 5 in exactly 30 strengthen'')
= 1 - p(success less than 5 in exactly 30 strengthen')
= 1 - p(only success 0,1,2,3 or 4 in exactly 30 strengthen)
= 1 - p(success 0 in exactly 30 strengthen) - p(success 1 in exactly 30 strengthen) .. - p(success 4 in exactly 30 strengthen) ; since they are mutually exclusive
= 1 - q^30 - 30*q^29.r^1 - 30*29/2 * q^28.r^2 - 30*29*28/2/3 * q^27.r^3 - 30*29*28*27/2/3/4 * q^26.r^4
= ...
= 0.475

It is quite different to your answer, one of us must be wrong ;)
also,
p(5 success with not more than 5 strengthen with 6 card strategy) = 0.590 , strangely this match your result ;)

btw, in case of dragon skill lv16, I have these numbers:
p(success 16 with not more than 16*6 strength (96 cards) in 1-card strategy) = 0.365
p(success 16 with not more than 16 strength (96 cards) in 6-card strategy) = 0.185
but if use 6 more cards:
p(success 16 with not more than 17*6 strength (102 cards) in 1-card strategy) = 0.465
p(success 16 with not more than 17 strength (102 cards) in 6-card strategy) = 0.482
...just for reference :)

Reasoning with probability should be careful, the result may be misleading to people not familiar with probability, given the calculation is correct.
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