A proposal of making an EV estimation tool for scrolled gears

Hi,

After some thinking, I decide to publish what I'm working on to see if I can recuit/attract people who's capable solving this with me, to create the tool together.

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Abstract

I am calculating the expected consumption of in-game currency (meso) for the gear scrolling system. The current rules for the gear scrolling system are as follows: a certain gear has a maximum of x enhancement slots, with an initial price of X meso. The initial gear is considered as +0 with x remaining slots, indicating zero successful enhancements and x remaining slots. Each time gear is scrolled, there are several scrolling options:

1. Use a 10% scroll, each scroll valued at A meso. Each time a 10% scroll is used, there is a 10% chance of successful scrolling, changing the gear from +n to +n+1, and a 90% chance of consuming one slot.
2. Use a 30% scroll, each valued at B meso. Each time a 30% scroll is used, there is a 30% chance of +1, a 35% chance of no effect consuming a slot, and a 35% chance of the gear being destroyed.
3. Use a CSS (Clean Slate Scroll) with a 20% success rate, each valued at C meso. Each time, there is a 20% chance of restoring a failed slot, a 40% chance of no effect without reducing the slot, and a 40% chance of destruction.
4. Use a 10% Scroll + ws, each valued at D meso. Each time there is a 10% chance of +1, a 90% chance of unsuccessful enhancement without reducing the slot.
5. Use a 30% Scroll +ws, each valued at E meso. Each time there is a 30% chance of +1, a 35% nothing happen, and a 35% chance of destruction.

I want to solve this typical Markov chain problem. First, list out every possible state of each gear, including +0 remaining 0 slots, remaining 1 slot, remaining 2 slots... remaining x slots, +1 remaining 0 slots, remaining 1 slot, remaining 2 slots... remaining x-1 slots, and so on. The above five methods of scrolling will cause the gear to transition between different states. I hope to propose a model to simulate and estimate the theoretical value of the gear in each state, allowing for a comparison of the optimal scrolling solution for each state.

 

While the ultimate simulation on the way, I made a side tool.

Here is my crappy calculator for estimate between 10%/30%/10%+ws%, and find out the best strategy of making the destination gear.
The second sheet in this spreadsheet is adapting from the thread how to make a perfect weapon. The original work compares only 30%+css20, 30%+ws, 10%+ws. I add 10% and 30% into comparison.

I personally do not quite agree compare all 5 scrolling strategies together, bacause, 10% and 30% has a chance generating byproduct, which should cause the EV cost drop (depending on the specific byproduct market value). So in most cases, 30% better than 30%+CSS20%

 

Very imforative and inspiring, ty for sharing!!

Especially the Scrolling Cost and Probability Analyser, great work indeed.

 

Hi! I happen to solved the algorithm to calculate this one (spent quite some time). You can find the recursive funciton for calculate it in my calculator.


The recursive function for a succesful rate x% scroll should be:


Here is my explanation for how to get the recursive function for each scroll, in case you might be interested.

Spoiler: explanation

Because of my poor English it might be a little bit hard for me to explain the algorithm, I'll give it a try.

Let's start with a simple example:
How many times on average you need to roll a die, to succeed 2 * "6" in a row? What about n * "6" in a row?

Spoiler: answer

36 is the most intuitive answer some people might try. But sadly, WRONG.
36 is the answer for "roll 2 dice at once, how many times you need to roll on avg to succeed a double 6?". But in our case, for roll, if your roll is not 6, you will start immediatly rolling the die for "first 6" to start your "consecutive double 6", instead of roll the other die to get a pair then see if you get "double 6". Or in our scrolling cases, this answer is the "EV for number of gears".

In this special scene, the avg rolling times = the EV for the rolling times ≈ the EV for scrolling times
Let's mark E(n) = the EV for rolling times to succeed n consecutive number6.
Then pretty easy that E(1) = 1/(1/6)= 6
Here comes to the tricky solution to avoid going straight for the universal math expression for E(n), but to find the relationship between E(n) and E(n-1) to get a recursive function.

The relationship between E(n) and E(n-1) is:


How comes it? Consider what's the take from succeeding (n-1) * "6" to n * "6".
for : You need to roll a more "6", then you have 1/6 chance to pass n consecutive "6". The total rolling times for this case is E(n-1)+1.
for : Same above. If you rolled a non-"6" after (n-1) * "6", you have to roll another E(n) times to reach n * "6" in a row.

Simplify above equition, we get


Then E(2) = 6 * [ E(1) + 1 ] = 42
E(n) = 6 * [ E(n-1) + 1 ]

It can be generalized that, for a successful rate x% scroll, if we mark E(n) = on average, how many scrolls I would need to pass n consecutive times.

Simplify it we get:

 

And have you ever considered to post a separate thread for Scrolling Cost and Probability Analyser?
It's great work, but while I searching relavant info myself I never found it, it deserves to be seen by more people.

 

Update - my tool inspired by kiwiz is on the way: my github repostiroy link

My aim changes a lot from OP, this is basicly a calculator works the same as kiwiz's Scrolling Cost and Probability Analyser, but contains options to compare all possible sequences for the same destination stats of gears, and generate the best cost approach.
Not done yet, will implement step by step.