Walkure
Apr 22, 2013, 06:33 PM
Conceptual
Discussing the the basic theory being used for a general weapon reinforcing/grinding system:
[SPOILER-BOX]
Probability of a String of Trials Occurring
Let’s imagine a specific probability where you fail x amount of times before succeeding a reinforcement value. With this, there is a known P probability of success and a Q=(100%-P) probability of failure. In comparing a string of several independent trials, the overall probability is simply the product of the probability of each individual event occurring. In this situation, one success event occurred, and x amount of failure events occurred. So, in effect, the probability of that specific event can be calculated as such,
http://i.imgur.com/8qLn9B1.png
Example of usage: I have a completely fair coin, which has a probability P=50% of landing heads up and a probability Q=100%-P=50% of landing tails up. I flip the coin until it reaches heads. What is the probability of the event where I fail once before landing on heads?
http://i.imgur.com/vpIpl4I.png
Since each coin flip is an independent trial, I could potentially fail several times before I end up successfully flipping heads. There is a probability for flipping tails any number of times before flipping a single heads.
Total Probability
Each of these strings (failing heads once, twice, etc.) are mutually exclusive events; there is no overlap in probabilities between events. The sum of all these event’s probabilities is one; this is an exhaustive list of probabilities.
For example, is the sum of all events possible in Pstring(x) account for all situations? Can the probabilities of flipping a fair coin be found? The answer is yes; you can calculate the probability of every single event, adding up to a total probability of 100%.
http://i.imgur.com/ouFcZLx.png
After adding the 10th element in the infinite series, the sum total is at 99.9%. As you add more and more examples, the total probability of Pstring adds up to 100%.
Determining Average Cost With Probability
If every outcome, and their probability, is known, and a cost can be assigned to each outcome, then the average cost can be determined:
http://i.imgur.com/cthLUSE.png
So, applying this to a fair coin, with a P=50%=.5, and assigning a cost to a heads and a tails flip, the average cost would be the midpoint between the two costs.
General Formula:
Now, we can look at an item-grinding system, and calculate average cost from there. Assuming that the percentage success rates are known, each event can have a calculated probability, and a cost can be associated with that event. So, it is possible to calculate for the average upgrade cost.
http://i.imgur.com/FEwRVDB.png
This is simply taking Pstring, without assuming the probability of a fair coin (50%), and assigning a cost to each event happening!
[/SPOILER-BOX]
Applying the general formula to PSO2:
[SPOILER-BOX]
I. Examining Attempt Cost
Attempt cost in PSO2 is determined by a base cost of some (relatively small) amount of meseta, a certain amount of Grinders, and any support items you use in each grind attempt. Usually, unless a support item is used, each individual grind attempt has a total cost of under 10,000 meseta. However, this quickly adds up after repeated fails.
II. Examining Success Rate
The success rates used in these calculations are based on this collection of data for 10* weapons (http://pso2osusume.com/kyouka-custom/rare10buki-kanwamae-kanwago-hikaku/). It is relatively recent, dating to April 4, 2013, and has a decently high amount of trials, up to 3801 trials if I'm reading right from Google Translate. This is close enough for a rough estimate. As 10* weapons are usually the most relevant grind costs, as they are well-desired, decently obtainable, and extremely expensive to grind compared to lower rarities, they’ll be the focus of this calculation.
III. Examining Failure Penalty Distribution
There are four possible penalties for failing a grind attempt; your weapon drops in grind value from 0 to 3 levels. Those probabilities are shown in the table above. However, when calculating the failure penalty for usage of the formula, it helps to know the probability given that a failure occurred. In this case, the probability of a given penalty, assuming a failed grind attempt is simply the percentage chance of overall occurrence divided by the sum total of each fail rate. There is no risk of an item being destroyed; the usual penalty is solely determined by the cost of lost grind levels.
Shown below is a table showing the aggregated grind probabilities, using the collection of data as shown in the previous section, with this calculated penalty distribution in a separate segment.
http://i.imgur.com/GqpMtnk.png
As an example, with an initial grind value of 1, there is a 19% chance to fail the grind attempt with no loss in grind value, and no chance of losing one, two, or three grind levels. Thus, it takes up 100% of the failure distribution, as it is the only possible event when a failure occurs.
IV. Examining the Cost of Failure
Considering that there are multiple possible ways to suffer from a grind failure, the failure cost will be an average cost of those penalties.
At +0, and +1, there is no penalty on failure, so the “Fail Cost” of the formula for those sections will be zero, and, since their attempt rate and success rate is also known, their average cost can be calculated. However, how can the proceeding failure costs be determined?
When the weapon drops down a grind a grind level, the cost of such an event would be the cost required to take it back up a level. So the average failure rate depends on the average cost of upgrading previous, as well.
http://i.imgur.com/Judoaic.png
V. Examining Support Items
Support items throw an interesting kink into the mix. There are two kinds of support items to use when grinding: success amplifying support items and risk reducing support items. Using either of these increases the attempt cost, with hopefully positive returns on investment by reducing the average attempts needed to succeed a roll or the failure penalty cost associated with failing a roll.
V.a. Handling Success-Amplifying Items
The most likely way these items work is by raising the overall success rate by a certain percentage, while still maintaining the proportions of penalties during a failure event. So, this would increase the success rate, without changing the derived failure penalty distributions.
V.b. Handling Risk-Reducing Items
A risk penalty is determined by averaging the upgrade costs of previous levels. So, losing two levels would require, on average, the average cost of raising the weapon up again the previous two levels. With the risk reduction (+1) item, only one level would be lost. So, to calculate the penalty cost when using a risk reduction (+1) item, the quickest way would be to treat the -3 penalty like a -2 penalty, -2 like a -1, and a -1 like no penalty.
http://i.imgur.com/mS92kfZ.png
With a full protection item, there is no penalty incurred on any failure; the failure cost is zero.
VI. Optimization
In order to optimize the costs, the most efficient method of upgrading previous upgrades is needed. After all, calculating the failure penalty on a +20% ticket with (+1) protection at +9 shouldn’t necessarily rely on you using the same items on +7 or +8; it could be more effective to use other options for those other circumstances. So, the most optimal methods of grinding low grind-value items should be used for grinding higher levels. This means calculating for the most efficient way to grind a +0, then using that to find the most efficient way to grind a +1, and so on. Through this method, finding the optimal path to a +10 10* Weapon can be determined.[/SPOILER-BOX]
Analysis
Google Drive Version. (https://docs.google.com/spreadsheet/ccc?key=0Al4_gG8wPfu_dGN5ZGhZWkZwS2IwWm5tT1labVFUR Gc&usp=sharing)
Here is a link to the downloadable version. (https://docs.google.com/file/d/0B14_gG8wPfu_V3pQdFFBWHlLRlU/edit?usp=sharing)
Using the downloadable version, you can play around and see how adjusting prices of items affects upgrade costs and likelihoods.
With these assumed rates, and costs, the most effective strategy seems to be:
Start using (-1) protection when your weapon is +7 or higher.
Using a 5% ticket when your weapon is at +9 currently is about the same as not using one.
Discussing the the basic theory being used for a general weapon reinforcing/grinding system:
[SPOILER-BOX]
Probability of a String of Trials Occurring
Let’s imagine a specific probability where you fail x amount of times before succeeding a reinforcement value. With this, there is a known P probability of success and a Q=(100%-P) probability of failure. In comparing a string of several independent trials, the overall probability is simply the product of the probability of each individual event occurring. In this situation, one success event occurred, and x amount of failure events occurred. So, in effect, the probability of that specific event can be calculated as such,
http://i.imgur.com/8qLn9B1.png
Example of usage: I have a completely fair coin, which has a probability P=50% of landing heads up and a probability Q=100%-P=50% of landing tails up. I flip the coin until it reaches heads. What is the probability of the event where I fail once before landing on heads?
http://i.imgur.com/vpIpl4I.png
Since each coin flip is an independent trial, I could potentially fail several times before I end up successfully flipping heads. There is a probability for flipping tails any number of times before flipping a single heads.
Total Probability
Each of these strings (failing heads once, twice, etc.) are mutually exclusive events; there is no overlap in probabilities between events. The sum of all these event’s probabilities is one; this is an exhaustive list of probabilities.
For example, is the sum of all events possible in Pstring(x) account for all situations? Can the probabilities of flipping a fair coin be found? The answer is yes; you can calculate the probability of every single event, adding up to a total probability of 100%.
http://i.imgur.com/ouFcZLx.png
After adding the 10th element in the infinite series, the sum total is at 99.9%. As you add more and more examples, the total probability of Pstring adds up to 100%.
Determining Average Cost With Probability
If every outcome, and their probability, is known, and a cost can be assigned to each outcome, then the average cost can be determined:
http://i.imgur.com/cthLUSE.png
So, applying this to a fair coin, with a P=50%=.5, and assigning a cost to a heads and a tails flip, the average cost would be the midpoint between the two costs.
General Formula:
Now, we can look at an item-grinding system, and calculate average cost from there. Assuming that the percentage success rates are known, each event can have a calculated probability, and a cost can be associated with that event. So, it is possible to calculate for the average upgrade cost.
http://i.imgur.com/FEwRVDB.png
This is simply taking Pstring, without assuming the probability of a fair coin (50%), and assigning a cost to each event happening!
[/SPOILER-BOX]
Applying the general formula to PSO2:
[SPOILER-BOX]
I. Examining Attempt Cost
Attempt cost in PSO2 is determined by a base cost of some (relatively small) amount of meseta, a certain amount of Grinders, and any support items you use in each grind attempt. Usually, unless a support item is used, each individual grind attempt has a total cost of under 10,000 meseta. However, this quickly adds up after repeated fails.
II. Examining Success Rate
The success rates used in these calculations are based on this collection of data for 10* weapons (http://pso2osusume.com/kyouka-custom/rare10buki-kanwamae-kanwago-hikaku/). It is relatively recent, dating to April 4, 2013, and has a decently high amount of trials, up to 3801 trials if I'm reading right from Google Translate. This is close enough for a rough estimate. As 10* weapons are usually the most relevant grind costs, as they are well-desired, decently obtainable, and extremely expensive to grind compared to lower rarities, they’ll be the focus of this calculation.
III. Examining Failure Penalty Distribution
There are four possible penalties for failing a grind attempt; your weapon drops in grind value from 0 to 3 levels. Those probabilities are shown in the table above. However, when calculating the failure penalty for usage of the formula, it helps to know the probability given that a failure occurred. In this case, the probability of a given penalty, assuming a failed grind attempt is simply the percentage chance of overall occurrence divided by the sum total of each fail rate. There is no risk of an item being destroyed; the usual penalty is solely determined by the cost of lost grind levels.
Shown below is a table showing the aggregated grind probabilities, using the collection of data as shown in the previous section, with this calculated penalty distribution in a separate segment.
http://i.imgur.com/GqpMtnk.png
As an example, with an initial grind value of 1, there is a 19% chance to fail the grind attempt with no loss in grind value, and no chance of losing one, two, or three grind levels. Thus, it takes up 100% of the failure distribution, as it is the only possible event when a failure occurs.
IV. Examining the Cost of Failure
Considering that there are multiple possible ways to suffer from a grind failure, the failure cost will be an average cost of those penalties.
At +0, and +1, there is no penalty on failure, so the “Fail Cost” of the formula for those sections will be zero, and, since their attempt rate and success rate is also known, their average cost can be calculated. However, how can the proceeding failure costs be determined?
When the weapon drops down a grind a grind level, the cost of such an event would be the cost required to take it back up a level. So the average failure rate depends on the average cost of upgrading previous, as well.
http://i.imgur.com/Judoaic.png
V. Examining Support Items
Support items throw an interesting kink into the mix. There are two kinds of support items to use when grinding: success amplifying support items and risk reducing support items. Using either of these increases the attempt cost, with hopefully positive returns on investment by reducing the average attempts needed to succeed a roll or the failure penalty cost associated with failing a roll.
V.a. Handling Success-Amplifying Items
The most likely way these items work is by raising the overall success rate by a certain percentage, while still maintaining the proportions of penalties during a failure event. So, this would increase the success rate, without changing the derived failure penalty distributions.
V.b. Handling Risk-Reducing Items
A risk penalty is determined by averaging the upgrade costs of previous levels. So, losing two levels would require, on average, the average cost of raising the weapon up again the previous two levels. With the risk reduction (+1) item, only one level would be lost. So, to calculate the penalty cost when using a risk reduction (+1) item, the quickest way would be to treat the -3 penalty like a -2 penalty, -2 like a -1, and a -1 like no penalty.
http://i.imgur.com/mS92kfZ.png
With a full protection item, there is no penalty incurred on any failure; the failure cost is zero.
VI. Optimization
In order to optimize the costs, the most efficient method of upgrading previous upgrades is needed. After all, calculating the failure penalty on a +20% ticket with (+1) protection at +9 shouldn’t necessarily rely on you using the same items on +7 or +8; it could be more effective to use other options for those other circumstances. So, the most optimal methods of grinding low grind-value items should be used for grinding higher levels. This means calculating for the most efficient way to grind a +0, then using that to find the most efficient way to grind a +1, and so on. Through this method, finding the optimal path to a +10 10* Weapon can be determined.[/SPOILER-BOX]
Analysis
Google Drive Version. (https://docs.google.com/spreadsheet/ccc?key=0Al4_gG8wPfu_dGN5ZGhZWkZwS2IwWm5tT1labVFUR Gc&usp=sharing)
Here is a link to the downloadable version. (https://docs.google.com/file/d/0B14_gG8wPfu_V3pQdFFBWHlLRlU/edit?usp=sharing)
Using the downloadable version, you can play around and see how adjusting prices of items affects upgrade costs and likelihoods.
With these assumed rates, and costs, the most effective strategy seems to be:
Start using (-1) protection when your weapon is +7 or higher.
Using a 5% ticket when your weapon is at +9 currently is about the same as not using one.