Walkure
Jun 24, 2013, 01:30 AM
Since more 11* and 12* weapons are being released, there's a good chance that people are going to be looking to grind these weapons at some point. However, there isn't a large amount of data flowing around for the actual rates behind success rates, or the penalty distributions, so it's hard to find which support items to use to find the minimum average cost to upgrade such weapons. There's also a good chance, considering how much risk there is with a -4 penalty, that people won't be exactly thrilled to grind at +7 and above with no protection tickets for science, so I doubt there's going to be accurate information flowing around for a good amount of time.
This makes my previous method for minimizing costs for 10* weapons (http://www.pso-world.com/forums/showthread.php?t=206991), which requires known or approximate rates for every outcome, unlikely to be directly usable for 11*+ weapons for a very long time.
Currently, on Ship 10, the prices for support items are listed below:
http://i.imgur.com/PCe53vF.png
So, for starters, let's assume:
Risk is high enough at a certain point to always warrant full protection
The base meseta cost of grinding a weapon is around 13.5k
This weapon requires four grinders to attempt a grind
Exact success rate is low, and unknown, but greater than or equal to 0%.
Prices listed above are constant
The first assumption isn't necessarily true, depending on the failure penalty rates. For example, the failure penalty distributions on a 10* weapons are heavily slanted towards -1 being the penalty for failure that, on average, it isn't worth using a full protect ticket, even at +9. The same COULD be true for a 11* or 12* weapon, although it is unlikely.
The other assumptions are based on some lurking and seeing people trying to grind such weapons, or simplifying the math for later usage. This puts the minimum attempt cost at 16,750 meseta, which is about twice the cost for any 10* weapon.
Let's go through some basic concepts real quick.
Full Protect-Assumed Methods
With my previous model, there was this concept of a "failure penalty", which was a cost incurred as a direct result of failing a grind. With a full protection ticket, there is no direct penalty. All of the cost, when working with a full protect item, is in the "attempt cost", which is now going to be fairly high.
Let's say we know the exact rate of a certain step has a 10% success rate, and the attempt cost is 100k. This would take 10 attempts on average, as we're looking for a mean (average) of 1 with a given rate of 10%. In probability terms, where E[X] is the expected (mean, or average) value, n is the number of trials, and P is the probability:
http://i.imgur.com/osocMza.png
Which, multiplied by the cost per attempt, would be 1 million meseta. So, given that we know the rate of a grind attempt, and we're working with a known attempt cost, and zero failure penalty, the average cost for upgrading, using that step, would be:
http://i.imgur.com/iDqcK6n.png
And, expanding that for every variable listed, would be:
http://i.imgur.com/VEYN9Eg.png
The problem, of course, is that the rate isn't even known. So, let's make that into a function of the base success rate.
http://i.imgur.com/BrNVhwP.png
Through doing a sweep of the variable for base probability, we'll be able to find out ranges that a certain percent booster is most effective.
Finding the Right Success Rate Booster
Because there is a minimal success rate of zero percent, and the rate of success is low (below 30% probably) let's have the variable sweep between 0% and 30%. The sweep will go in steps of .5% of the base success rate.
Chart Form:
http://i.imgur.com/mS677Hw.png
The prices without any success rate booster are not included; they throw off the chart and never reach a point of being the most effective choice throughout this sweep. Chances are, by the time that success rate is high enough for using NO success-enhancing support item is true, the risk penalty is also low enough that using a full protection ticket is unwise.
At any point past 1% base success rate, the +20% ticket has a higher average cost than the +10% ticket. At a rate of ~6%, the +5% ticket becomes more effective than the 20% ticket. At a rate of 20% base success, the +5% ticket becomes more cost-effective than the +10% ticket.
Raw results:
[SPOILER-BOX]
http://i.imgur.com/ftQ3bR0.png
http://i.imgur.com/Wv1HvaM.png
[/SPOILER-BOX]
Practically speaking, what does this mean?
In any situation where the full protect ticket is cost-effective to use, chances are the +10% ticket will be the most cost-effective ticket to use in tandem with it. While most support items are dropping in price, items other than the +20% boosting ticket are dropping faster.
The +30% ticket requires 20 Excubes. While it may not have a direct cost to the user, it does have a potential cost; if you were to instead use those Excubes to make grinders, and sell those for as low as 1k a pop, you could make 1.2m. It's pretty much out of the question, as far as minimizing cost and/or effort goes.
This makes my previous method for minimizing costs for 10* weapons (http://www.pso-world.com/forums/showthread.php?t=206991), which requires known or approximate rates for every outcome, unlikely to be directly usable for 11*+ weapons for a very long time.
Currently, on Ship 10, the prices for support items are listed below:
http://i.imgur.com/PCe53vF.png
So, for starters, let's assume:
Risk is high enough at a certain point to always warrant full protection
The base meseta cost of grinding a weapon is around 13.5k
This weapon requires four grinders to attempt a grind
Exact success rate is low, and unknown, but greater than or equal to 0%.
Prices listed above are constant
The first assumption isn't necessarily true, depending on the failure penalty rates. For example, the failure penalty distributions on a 10* weapons are heavily slanted towards -1 being the penalty for failure that, on average, it isn't worth using a full protect ticket, even at +9. The same COULD be true for a 11* or 12* weapon, although it is unlikely.
The other assumptions are based on some lurking and seeing people trying to grind such weapons, or simplifying the math for later usage. This puts the minimum attempt cost at 16,750 meseta, which is about twice the cost for any 10* weapon.
Let's go through some basic concepts real quick.
Full Protect-Assumed Methods
With my previous model, there was this concept of a "failure penalty", which was a cost incurred as a direct result of failing a grind. With a full protection ticket, there is no direct penalty. All of the cost, when working with a full protect item, is in the "attempt cost", which is now going to be fairly high.
Let's say we know the exact rate of a certain step has a 10% success rate, and the attempt cost is 100k. This would take 10 attempts on average, as we're looking for a mean (average) of 1 with a given rate of 10%. In probability terms, where E[X] is the expected (mean, or average) value, n is the number of trials, and P is the probability:
http://i.imgur.com/osocMza.png
Which, multiplied by the cost per attempt, would be 1 million meseta. So, given that we know the rate of a grind attempt, and we're working with a known attempt cost, and zero failure penalty, the average cost for upgrading, using that step, would be:
http://i.imgur.com/iDqcK6n.png
And, expanding that for every variable listed, would be:
http://i.imgur.com/VEYN9Eg.png
The problem, of course, is that the rate isn't even known. So, let's make that into a function of the base success rate.
http://i.imgur.com/BrNVhwP.png
Through doing a sweep of the variable for base probability, we'll be able to find out ranges that a certain percent booster is most effective.
Finding the Right Success Rate Booster
Because there is a minimal success rate of zero percent, and the rate of success is low (below 30% probably) let's have the variable sweep between 0% and 30%. The sweep will go in steps of .5% of the base success rate.
Chart Form:
http://i.imgur.com/mS677Hw.png
The prices without any success rate booster are not included; they throw off the chart and never reach a point of being the most effective choice throughout this sweep. Chances are, by the time that success rate is high enough for using NO success-enhancing support item is true, the risk penalty is also low enough that using a full protection ticket is unwise.
At any point past 1% base success rate, the +20% ticket has a higher average cost than the +10% ticket. At a rate of ~6%, the +5% ticket becomes more effective than the 20% ticket. At a rate of 20% base success, the +5% ticket becomes more cost-effective than the +10% ticket.
Raw results:
[SPOILER-BOX]
http://i.imgur.com/ftQ3bR0.png
http://i.imgur.com/Wv1HvaM.png
[/SPOILER-BOX]
Practically speaking, what does this mean?
In any situation where the full protect ticket is cost-effective to use, chances are the +10% ticket will be the most cost-effective ticket to use in tandem with it. While most support items are dropping in price, items other than the +20% boosting ticket are dropping faster.
The +30% ticket requires 20 Excubes. While it may not have a direct cost to the user, it does have a potential cost; if you were to instead use those Excubes to make grinders, and sell those for as low as 1k a pop, you could make 1.2m. It's pretty much out of the question, as far as minimizing cost and/or effort goes.