Failstacking with armor

A popular strategy used by players enhancing endgame gear is deliberately downgrading armor to build stacks for more important components, such as weapons. Today we will examine the soundness of this strategy from a financial point of view.

We will only examine downgrading DUO and TRI fodder tier armor today. In other words, the mathematics will be done with Heve/Grunil equivalent armor at DUO/TRI.

The concept of what we’re doing here is best illustrated with a tree diagram. Tree diagrams allow us to examine possible outcomes based on the odds, and make inferences based on them.

Capture.JPG

In mathematical terms:

  • The failstack cost analysis for armor enhance can be calculated as
    • ( CostArmorFail + CostEnhanceAttempt) * FailChance + (CostXFailstack + CostEnhanceAttempt – ExtraArmorValue) * SuccessChance
    • That equation calculates the expected outcome per action, but of course to reach your failstack goal you need to retry the outcome multiple times until you fail the enhancement.
    • The entire thing has to be divided by the probability of enhancement fail to give you the final equation:
      • ((CostArmorFail + CostEnhanceAttempt) * FailChance + (CostXFailstack + CostEnhanceAttempt – ExtraArmorValue) * SuccessChance) / FailChance
  • The failstack cost analysis for blackstone fodder can be calculated as
    • (CostXFailstacks + AverageStonesLost) * SuccessChance + AverageStonesLost * FailChance
    • That equation calculates the expected outcome per action, but of course to reach your failstack goal you need to retry the outcome multiple times until you fail the enhancement.
    • The entire thing has to be divided by the probability of enhancement fail to give you the final equation:
      • ((CostXFailstacks + AverageStonesLost) * SuccessChance + AverageStonesLost * FailChance) / FailChance

 

Next we need to figure out the multiple variables we need to complete the equations. We will examine this at 30 and 40 failstacks using DUO/TRI armor for the first strategy.

The cost of 30 failstacks and 40 failstacks can be worked out using combinatorics. This is too complex to summarize in this article, so I’ll just write the answers instead.

  • 30 Failstacks – 274.5 blackstones with Reblath+14
    • At 250k per stone and repair, this comes out to 68.6mil
  • 40 Failstacks – 1417.5 blackstones with Reblath+14
    • At 250k per stone and repair, this comes out to 354mil

The cost of DUO/TRI/TET Grunil as dictated by the market is:

  • PRI->DUO – 40m
  • DUO->TRI – 80m
  • TRI->TET – 120m

Using this and the KR enhancement table, it is now possible to work out all 8 scenarios and the cost of building additional failstacks using these.

DUO Grunil 30 stacks

((40 + 4) * 0.8 + (68.6 + 4 – 80) * 0.2) / 0.8 = 42.15m for 4 more stacks

TRI Grunil 30 stacks

((80 + 4) * 0.905 + (68.6 + 4 – 120) * 0.095) / 0.905 = 79m for 5 more stacks

DUO Grunil 40 stacks

((40 + 4) * 0.75 + (354 + 4 – 80) * 0.25) / 0.75 = 136.67m for 4 more stacks

TRI Grunil 40 stacks

((80 + 4) * 0.88 + (354 + 4 – 120) * 0.12) / 0.88 = 116.5m for 5 more stacks

 

And of course the cost using Reblath method.

  • Average stones used for 4 stacks = 1 + 0.85 + 0.85^2 + 0.85^3 = 3.19
  • Average stones used for 5 stacks = 1 + 0.85 + 0.85^2 + 0.85^3 + 0.85^4 = 3.71

How the above is derived can be found in an earlier article: https://bdomaths.wordpress.com/2016/07/15/demythifying-failstacks/

Reblath 30 stacks -> 34

((68.6 + 3.19 * 0.25) * (1-0.85^4) + 3.19 * 0.25 * 0.85^4) / 0.85^4 = 64.3m for 4 more stacks

Reblath 30 stacks -> 35

((68.6 + 3.71 * 0.25) * (1-0.85^5) + 3.71 * 0.25 * 0.85^5) / 0.85^5 = 88.1m for 5 more stacks

Reblath 40 stacks -> 44

((354+ 3.19 * 0.25) * (1-0.85^4) + 3.19 * 0.25 * 0.85^4) / 0.85^4 = 325.7m for 4 more stacks

Reblath 40 stacks -> 45

((354+ 3.71 * 0.25) * (1-0.85^5) + 3.71 * 0.25 * 0.85^5) / 0.85^5 = 445.9m for 5 more stacks

Conclusion:

The numbers are self-explanatory. The values used in the market prices of various Grunil pieces are before tax, so they slightly understate the cost of building stacks using the high enhance armor fail method.

Because of how simple these equations are, I’m not going to elaborate on these findings in more detail. Feel free to play around with them by substituting different armor costs and come up with your own conclusions.

Additional thoughts:

Given these numbers, it’s pretty obvious anyone with their hands on the enhancement equations can easily manipulate this mechanic with Value Pack. This is also a proof of concept on how to directly accumulate a Valk’s Cry effect on your enhancement through buying and selling gear with Value Pack. In order to prevent the market from being ruined more than it is already, I have no intention of releasing the specifics of this strategy at this point in time.

On RNG – Lies, Deceptions and Misinterpretations

Warning: This blog post is extremely heavy on the mathematical side and does not have much direct relevance to BDO gameplay. There were feedback that asked for more maths-orientated posts, so this is an experiment on how much demand there is on this type of content. Read this if you want to understand RNG. Do not read this if you don’t care about maths.

  • RNG is by far the most misunderstood popular concept in the history of everything.
  • RNG is frustrating, it makes people want to ragequit.
  • RNG is without skill, and your outcome rests solely on a dice roll.
  • RNG is psychologically rewarding through Operant conditioning, no matter how much you want to deny it.
  • RNG is part of almost everything in BDO.
  • But luckily, RNG does not actually change your fate in BDO.

I’ve wanted to write an article on RNG since I started this blog, however I never found a good way to discuss this topic in a relevant, succinct and straightforward manner. RNG is a complex issue that had been constantly misunderstood by people since the beginning of time. In fact, interpreting RNG is so hard that even scientists cannot easily explain the foundation statistical concept known as the p-value.

Today after seeing a reddit thread, I was inspired to make this blog post. Of course I was inspired because of how misleading that thread was, otherwise I’d just copy it here with credits. Nevertheless, it showed me the exact angle in which this topic should be best explained.

Look at these two pictures:

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  1. Which one was definitely generated by a RNG generator and which one was deterministic?
  2. If I gave you magical sand and asked you to spread it randomly and continuously gather and spread it again, which pattern would I more likely make?

Before we start, let’s assume both pictures have the exact same number of dots.

And here are the answers:

  1. We don’t know for sure.
  2. Both are equally likely.

The first answer should be easy to comprehend. An RNG mechanism of spreading dots on a sheet can produce literally any pattern. In fact, the dots can simply create a smiley face shape or all be confined to the top left quadrant of the canvas, albeit both extremely improbable.

The answer to the second question? The picture is just as likely to spread itself into a smiley face as it is to copy the left picture exactly.

Too deep down the rabbit hole? Don’t worry, this article ends with a very simple conclusion that everyone should take home.

This can be difficult to comprehend, so let us explain this by first cutting the canvas into discrete blocks. A canvas of X pixels height and Y pixels width can be:

  1. Cut into a canvas of X times Y number of boxes.
  2. Each box has a chance to contain either a dot or no dot.
  3. There are n dots in the canvas.

Therefore, the number of combinations the dots can take is finite. Because this concept applies to any number of dots, I’ll simplify it by demonstrating what I’m talking about in a 3 by 3 tic-tac-toe tiny canvas with n = 3 dots.

Mathematically, we know that 3 dots on a 3×3 canvas has a total of 9 Combination 3 solutions, thus 84 possible arrangements.

Here are examples of three arrangements.

Capture

Which one is truly RNG? Which one is more likely to appear?

The answer is that all 3 can be generated by an RNG generator, and that all 3 have an equal probability to appear.

The probability to appear for each combination of black and white boxes can be calculated by the equation:

p(black total) * p(black total -1) * p(black total -2)

Which in mathematical form is:

3/9 * 2/8 * 1/7 = 1/84

Each combination has a 1/84 chance to be produced. There are 84 combinations.

To conclude partially, the above explanation shows that:

  1. Any pattern can be generated by a true RNG generator.
  2. It does NOT show that every pattern is generated by a true RNG generator.
  3. It does NOT show that there are no RNG generators that are inherently shit.

 

Now consider two sequences:

  1. 0101010101010101010101010101010101010101010101010101010101010101
  2. 1101000101011011100010101010110000011000001011110101000001000001

Which one is more likely to be generated by a true RNG generator? Still the same. But let’s ask a different question, which is more likely to be generated by a fake RNG generator?

The first sequence can be eyeballed and a pattern is already apparent. It is 01 repeated 32 times. The second sequence wasn’t generated with a pattern; I generated it with random.org.

Knowing the probability of both is not important if you want to know whether the pattern is surely random, but it is important if you want to know the odds of the RNG generator behind the sequence being a good RNG generator (i.e. closer to random).

We can now map the odds into a grid:

Capture

This is not an advanced mathematics class so I won’t go into the several methods one can use to calculate the bottom right box, all you need to know is that it’s pretty much as low as the probability of it being truly random.

In other words, the odds of:

  • The first sequence is 18,400,000,000,000,000,000 to 1 in favor of fake RNG.
  • The second sequence is about 1 to 1 in favor of nothing.

We can therefore make a probabilistic claim that the first sequence is not actually RNG. We cannot make that claim for the second sequence.

Let’s go back to this picture:

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The same method can be applied to these pictures but because this is not an advanced maths class I will not do it step by step. The conclusion using a similar randomness approach to the above is that:

  • The left side image is probably generated by RNG.
  • The right side image is probably generated by a mixed RNG algorithm. It is rigged by a formula which decreases the probability of a dot being close to another preexisting dot.
    • MOBAs such as DotA and LoL use something similar to the image on the right side to reduce the snowball effect of RNG streaks.

The correct conclusions to draw from the reddit thread is:

  • The left side image in its exact form is more probable than the right side image to appear in a true RNG system.
    • [False]
  • The left side image is RNG, the right side isn’t.
    • [Impossible to determine with certainty]
  • The left side image is more likely than the right side image to be generated with a true RNG generator compared to a partial/fake one.
    • [True]

The most beautiful part of all of this? None of this is relevant to BDO at all. Suppose a square X unit tall and width represents 1 hour of grinding loot, now consider these two images:

ecdw5pmfz9bx3

If I place the same red box on the image randomly (simulating 1 hour of grinding), which side is more likely to give me a higher number of drops?

ecdw5pmfz9bx2

If I place the same red box on the image randomly (simulating 1 hour of grinding), which side is more likely to give me a higher number of drops?

The answer to both is it makes no difference as long as the algorithms do not use the red box as a variable. In other words, as long as BDO RNG isn’t controlled by a temperamental gnome that hates you, it honestly makes zero difference which formula it is in the long run. However, the smaller red box is likely to give larger fluctuations in the number of loot you get. This applies no matter whether you look at the left side or right side.

In fact, we can magnify both images to simulate events with low probability of occurrence, such as obtaining an Ogre Ring…. If you’re smart you’d realize that it has the same effect as making the red box smaller.

In other words:

  1. If an occurrence is more common, randomness becomes lesser of a factor.
  2. If the RNG dice is rolled more times, randomness becomes lesser of a factor.

To summarize, I will show you two graphs.

This is the cumulative probability graph of an event tested to be 10% success chance in game over 1000 tries using a real RNG generator.

Capture.JPG

This is the cumulative probability graph of an event tested to be 10% success chance in game over 1000 tries using a broken RNG generator that has double the chance of rolling 9.

Capture

 

So the final TL;DR:

  1. A true RNG generator has an equal likelihood of producing both images.
  2. RNG generators can be tested for randomness using many methods including statistics, transformations, string complexities, etc.
  3. The left image is more likely to be generated by a true RNG generator than a deterministic or partial RNG one.
  4. The right image is more likely to be generated by a partial RNG generator than a true RNG one.
  5. None of the above actually matters. In other words, the reddit thread posting that image was a red herring.
  6. What does matter is the frequency of RNG dice rolls and the frequency of success in each dice roll.
  7. Although I have to admit if BDO uses an algorithm like the one on the right, it will smooth the progression curve at least in the early stages.
  8. Point 7 shows why it can actually be beneficial to have rigged RNG systems, as long as you’re not the variable being rigged against. And you aren’t, so tinfoil hats off please.

The entire point of this long blog post? The effect of RNG gets lower the longer you play. Simple right?

 

Cost vs Scaling – Mark of Shadows vs Crescent Guardians

A Bares necklace gives 6 AP 4 AP (credits to reddit user Caspersaur for noticing this error) and costs 100,000 silver (16.7k 25k per AP). An Ogre ring gives 10 AP and costs 100,000,000 silver (10million per AP). The nature of gear cost scaling in BDO is not proportional to the base stats of items. This is often a point of confusion, so in today’s short blog post, we will examine Mark of Shadows and Crescent Guardian’s Ring.

Crescent Guardian vs Mark of Shadows is the only gearing dilemma introduced in Valencia that I consider purely mathematical in nature. Blue Whale Earring gives the same AP as Witch Earring but also an additional 100 HP. How much is 100 HP worth? I don’t know. Basilisk belt gives 3 AP and Tree Spirit Belt gives 2 AP and 1 accuracy. Again, how much is 1 AP worth compared to 1 accuracy? I don’t know. The dilemmas in earring and belt choices are not mathematical in nature, hence I cannot analyze them objectively with simple numbers. If you want my opinion, Blue Whale Earrings are not worth it because of the exponentially higher cost for 100 HP at higher enhancements. Tree vs Basilisk is highly dependent on the rest of your build.

Using the strategies outlined in the previous blog posts, the cost of optimal enhancement for these two pieces of gear can easily be mapped out, so I will not go into details about the mathematics today. For more information about failstacking, read this post.

In short the cost of jewelry can be modeled by the equation:

(CostOfJewelry + CostOfBaseJewelry) * ExpectedTries + FailstackCost

The total consumption of rings can be modeled by the equation:

(PreviousEnhanceRingNumber + 1) * ExpectedTries

Here is a table of MoS and Crescent AP, cost, cost per AP and total rings required using the optimal tactic for enhancement. The failstacks are obtained with Reblath+14 and 200k blackstones. The failstacks at extremely high values are obtained with Reblath+19 and 4mil concentrated stones as after a certain stack breakpoint, this becomes cheaper to stack with than +14 Reblath.

Capture

And there you have it. The reason it costs more MoS than Crescent is because you would stack higher on Crescents, thus increasing success chance resulting in less raw rings burnt. Note that the cost of jewelry quoted in the table are the minimum based on perfect enhancement strategies, if you use a sub-optimal strategy, you will likely spend more.

In conclusion:

  1. Be realistic about your jewelry enhancement goals. Most people vastly underestimate how time consuming and costly it is at the higher enhancements.
  2. A DUO Crescent Ring is better than a TRI MoS in terms of cost effectiveness. They offer the same AP, but DUO Crescent is cheaper to make.
  3. The above means you should not invest in a MoS beyond DUO.
  4. Most jewelry enhancements follow similar patterns, thus can be approximately extrapolated using the tables above.
  5. As with all RNG based enhancements, this is subjected to fluctuations. If you go for TET/PEN enhancements, don’t expect that you’ll get it with certainty.
  6. Expect a jewelry enhancement guide for Base->TRI for a few common jewelry types in the near future.

 

As with likely all posts in the future, I won’t be responding directly to reddit threads unless I really like the question. To contact me, send me a PM or leave a message here.

Revisiting Liverto vs Yuria

Two primary concerns brought up for the previous article was the cost of materials and the value of Yuria as a cheaper substitute. Let’s examine those two propositions in this short blog post.

Liverto – Material Cost

I redid the maths on Liverto with 1.2m repairs (480k memory fragments), 300k armor blackstone, 300k weapon blackstone. As you can see, these new numbers are extremely unrealistic and biased. Not only can I list memory fragments in the market and make more than 480k even with the 35% cut, but why is memory fragment far lower than market value when blackstone isn’t, yet most people are actually bottlenecked by memory fragments?

I don’t know the answer to that question, all I know is some people think this is a good idea. To them, if you farm both memory fragments and blackstones with relic scrolls, memory fragments are cheaper but the stones aren’t. #logic?

Regardless here they are:

  1. +7->+8 force
  2. +8->+9 3 stacks
  3. +9->+10 3 stacks
  4. +10->+11 4 stacks
  5. +11->+12 4 stacks
  6. +12->+13 5 stacks
  7. +13->+14 7 stacks
  8. +14->+15 9 stacks

I strongly recommend not following the above values because obviously they underestimate the cost of memory fragments significantly. However, it is your decision.

If you’re a more reasonable person but still think the memory fragments are overpriced, at 1.8m repairs and 300k stones, take the failstacks in the guide and subtract each one by 2.

Liverto vs Yuria

If you still believe at equal enhancement Yuria is better than Liverto, you are in denial. However the assertion that Yuria is cheaper to enhance hence better is one worth investigating.

Here are the PRI, DUO, TRI, TET cost for Yuria and Liverto using the optimal strategy, 10mil sharp fragments, 300k weapon blackstone, 300k armor blackstone and 3.5mil repair Liverto, 200k repair Yuria. Increased gain of failstacks on fail, delevel, etc, are all accounted for.

Liverto:

  1. PRI: 37.5mil
  2. DUO: 60.5mil
  3. TRI: 306mil
  4. TET: 1.9bil

Yuria:

  1. PRI: 32.5mil
  2. DUO: 50mil
  3. TRI: 257mil
  4. TET: 1.6bil

As you can see, the cost of PRI->PEN weapon enhancement is predominantly gated by the sharp crystal cost. This is pretty obvious because it dwarfs the repair cost regardless of what weapon you choose to enhance.

What does this mean?

  1. We already know Yuria is weaker than Liverto at the same enhancement level. Slightly higher raw damage on human targets, but much lower accuracy.
  2. We now know that enhancing Yuria is also incredibly expensive, this means the whole “I can upgrade a TET Yuria instead of a DUO/TRI Liverto” argument is false.

The above two inferences direct prove that Yuria is terrible even for the purposes of conserving money and resources. If this doesn’t seal the debate for you, I’m afraid nothing will change your mind.

Conclusion

Hopefully this clears up a few queries about the Liverto upgrade guide. Due to the sheer stupidity of some of the comments I’ve been seeing, I will very likely no longer be responding to reddit threads. To reach me, you can contact me through PM.

Your New Liverto – A Daunting Experience

When a new player starts the game, I always tell them to skip Yuria and go straight to Liverto. The reason for this is simple, Liverto gives 15% more accuracy than Yuria, making it comparable to +15 Yuria when it’s only +12 or so. The cost of getting Liverto +11~+13 is lower than the cost of getting +15 Ultimate Yuria, yet functions identically and have longevity.

To me, investing in a Yuria is immediately setting yourself back 25million silver, because you quickly replace it with a Liverto, which is now easily accessible. A new player could simply cash in their free leveling gold bars for a shiny Liverto, and never have to worry about changing their weapon months or even years down the track.

[Edit] If you’re level 45 and you need a weapon ASAP, you can borrow a Kaia weapon from Kanobas in Calpheon for 50 contribution points. This weapon can be used while you get your Liverto to an acceptable enhancement level. Credits to jimcdiver on BDO Discord for bringing this to my attention.

If this is your experience, you’re in luck. This blog post will teach you how to get to +15 Liverto in the cheapest way possible. Note: as with all RNG, this is subject to fluctuation. The strategy here maximizes your odds, but does not guarantee them.

In order to work out the optimal strategy, you need to know:

  1. The amount of tries on average. I covered that topic here previously.
  2. The cost of each enhance attempt, blackstone + memory fragments. Straightforward.
  3. The cost of building X number of failstacks. I will cover this topic eventually.

For every value of failstack, the price of the enhancement can be calculated by:

(BlackstoneCost)*(1/p(success))+(MemFragCost)*odds(failure)+FailstacksCost

To save everyone the trouble of doing thousands of rows on Excel, I did this for you already.

Capture

 

Focus only on the blue cells. Some of the rest do not apply to enhancements before +16. The costs used in this spreadsheet:

  1. Memory Fragment: 800k each.
  2. Blackstone (Weapon): 300k each.
  3. Blackstone (Armor): 200k each.

How to build failstacks:

  1. Make a level 10 alt. Kill 1-2 horses so your karma is negative.
  2. Enhance Reblath pieces until desired failstacks reached. Reblath can be purchased from the Velia Blacksmith.
  3. If your Reblath hits +15 at any point, equip it on your negative karma alt and suicide in PvE a few times until it delevels back to +14.
  4. Continue to build stacks on your +14 Reblath until desired stacks reached.

[Edit] If you still have armor pieces not +15 yet, you should build failstacks on those instead of buying a Reblath. Thanks to reddit user asdf2100asd for bringing this to my attention.

And here is the optimal strategy.

  1. Buy Liverto+15 premade if it’s on the market and you can afford it. If not, follow step 2 to 10.
  2. Enhance your Liverto to +7. This has a 100% success chance.
  3. Force enhance your Liverto to +8.
  4. Build 7 failstacks and attempt to upgrade your Liverto to +9. This will cost you 6.3mil on average.
  5. Build 8 failstacks and attempt to upgrade your Liverto to +10. This will cost you 7.7mil on average.
  6. Build 9 failstacks and attempt to upgrade your Liverto to +11. This will cost you 9mil on average.
  7. Build 9 failstacks and attempt to upgrade your Liverto to +12. This will cost you 12.3mil on average.
  8. Build 10 failstacks and attempt to upgrade your Liverto to +13. This will cost you 14.8mil on average.
  9. Build 12 failstacks and attempt to upgrade your Liverto to +14. This will cost you 18.7mil on average.
  10. Build 13 failstacks and attempt to upgrade your Liverto to +15. This will cost you 21.8mil on average.

I listed the prices above so you can have realistic expectations about how far you go. You could be luckier, you could be cursed. To those who are slightly more maths savvy, the numbers quoted are means. Because the distribution of enhancement is right skewed, more than half the players who attempt this will be able to do it with cheaper than the stated prices (median).

That’s all for today. Remember:

  1. Yuria is meme. Never use that junk.
  2. Enhancement is all about maths. Delete “it’s all RNG” from your vocabulary. Your IQ will go up 20 points instantaneously, trust me.

[Edit] Response to some queries.