LK vs Travincal vs Cows - runefinding guide

[Gripphon1]

I had in mind p7 focusing only Vizier seal with maybe adding Infector seal though that's questionable. Overall not touching De Seis nor doing this seal killing thing. Hammer can try the seal tactics however sorc could do only Vizier plus few ghost packs.

Comparison is one ghost killed on p7 is roughly as killing 2.7 cows on p5. Good characters can aim to kill 2.5 cows per second average which would mean sorc should kill ~1 ghosts per second to be comparable. Each ghost counts as 3.56 other creep types in sanctuary. Whether or not is this possible should be tested. Hammer can do seal trick.

 

[EatSleepDiablo]

Sorry for the delay in posting this...

In terms of runs, I collected 112 of diablo's essences. A drop rate of 1: 10 (@ P7) would indicate 1,120 runs, but that factor is just a guess. As indicated previously, leveling was from 87-ish to 96.

I typically clear out all three seal bosses and any boss packs around the pentagram during my initial battle orders. I then teleport from Infector's seal, to the entrance, to Visers seal. Only monsters aggo'd from the entrance to Viser's seal drop and any misclicks can impact the total significantly. If I were to guess, total drops, including gold, potions, etc. number at least 100. At least two boss packs are caught up in the glitch.

HRs collected with seal glitch: 2 Jahs, Ber, Sur, 1 Ohm, at least one Gul and Um, and a bunch of Mals/Puls. TC87s include Gargoyle's Bite, Gris Redemption, BK CB, and Nats Mark. Notable socketables include two +3 grand matron bows, several eth sacred armors (to be socketed), and a bunch of class specific items.

 

[Carny]

I've started running CS again in the last few weeks and documented it and I have to say so far the Runes keep dropping. So far I have done exactly 675 runs and I got 6 HR and 10 Mid Runes out of it, which is 1 HR every 112,5 runs in average. Drop Order was:

15 Pul
167 Sur
235 Pul
293 Ist
299 Cham
302 Lo
403 Mal
405 Um
412 Mal
552 Ber
576 Pul
600 Ist
617 Pul
641 Cham
661 Pul
673 Jah


As you can see sometimes I had periods where I got nothing for 150 runs and other times I got 2HR within 3 runs or 3 Mid Runes within 9 runs. Runs were about ~2:20 - 2:30 on P7 and I killed all bosspacks in CS and RoF manually and also finished off Diablo at the end for xp. In these 675 runs I also got 3 TC87 unique items (Crown of Ages, Templar's Might and Ghostflame). Set Items I don't keep track, since I got them all already. If i only went for Runes, runtime would be halved to 1:20 - 1:30 due to sealglitch farming. 675 is ofc not a huge number so I will definitely do atleast 2-3k runs over the next weeks/months to see how the numbers change.
Whoever wants to finish first in the RF marathon thats coming soon, should definitely consider CS.

Edit: Edited according to progress

 

[drmalawi]

In this video, I show how to properly calculate probability for finding "effective" runes - for instance Cubing up Ber from Surs + straight drop of Ber.

In particular, I review and give improved statistics for LK p7: Effective Ber, effective Infinity, effective Enigma, effective Infinity + Enigma (that is mentioned in this guide).

Enjoy and become enlighten

* NOTE : I use the LK rune patterns mentioned in the OP guide, just for the sake of comparison.

You guys started to talk about averages and expected values...

Here is an explanation:

E(x) = Integral over the entire space x * p(x) dx ( http://mathb.in/34183 )

where p(x) is the probability mass density

(the definition of E(x) is the continuous variant of the "sum of all values" divided by "number of trials/players")

What I have in my graphs are the cumulative probability distribution which is the integral of p(x).

Due to the fundamental theorem of calculus, we obtain p(x) when we take the derivative of the cumulative distribution.

Here are the results (if you guys want expected value, which have no / weak statistical interpretation)

1 Ber or 2 Surs : 1393 runs

"Infinity" : 2683 runs

"Enigma" : 3969 runs

"Infinity" & "Enigma" : 6539 runs

All of these values are very close to the numbers given in the OP guide.
Still it does not help just to improve these numbers, we need to give people a better statistical interpretation.

 

[sir_cyclops]

@drmalawi amazing video! Happy to see you.

 

[T72on1]

Great !!! Can you also just give us the numbers, so @Gripphon can add it to this guide?

 

[sir_cyclops]

Great !!! Can you also just give us the numbers, so @Gripphon can add it to this guide?

It all depends what confidence level you want. The last quarter of the video has the run count for 3 different confidence levels for different gear (enigma, infinity, and both I believe).

 

[T72on1]

Hmm, that is chinese to me. So there isn't a simple answer as to what is the expected value?

 

[sir_cyclops]

Hmm, that is chinese to me. So there isn't a simple answer as to what is the expected value?

I’m no expert either, but I’ve always viewed confidence level as “the chance you will have it.” So if it’s the count of runs for 95% confidence level, it means you have a 95% chance of obtaining the items with that run count. So in my mind, the number that jumps out that you want is 50%. (Edit: accidentally hit reply.) But @Luhkoh normally uses 65% for the confidence level when making a general statement about average number of runs to obtain an item. @drmalawi uses 63 point something in the video so maybe that’s the best number. Probability is hard haha. The conclusion is that the way everyone normally calculates the chance is less accurate than the real chances, and the graphs at the end depict it best. I bet if you watch the video, it would make more sense

 

[art_vandelay]

@drmalawi I very much appreciate those videos. I should know better but I have used the expected value as a measure for comparing probabilities way too often myself!


I think there is yet another flaw with the numbers used in the guide. The pattern thread doesn't distinguish locked and unlocked patterns. So if for example there are 4 locked and 4 unlocked patterns for rune X and the first three locked and unlocked ones agree whereas the last two do not then to us it will look as if there are five X patterns, i.e. we would think the chance to get X out of a chest is 5/65536 when in reality it actually is 4/65536.

One example are Sur patterns with ID 144 and 314 in the pattern thread where the second one looks like the locked version of the first. Sadly it is impossible for us to tell from a screenshot wether a pattern is locked or unlocked. One would have to simulate all the patterns and count the ones containing HRs both in the locked and the unlocked case.


@sir_cyclops and anyone else who is interested in this stuff: The 'naive' approach is in fact entirely wrong if you are after the actual probability. The formula for it is basically just the definition of the expected value in this particular case.

Slightly more technical: The model underlying that formula is that there is a random variable X which has X(rune)=0 if the rune is less than Sur and X(Sur)=1/2 as well as X(Ber)=1. Then the expected value (per definition) of X is:

ExpectedValue(X)=Prob(El)*X(El)+...+Prob(Lo)*X(Lo)+Prob(Sur)*X(Sur)+Prob(Ber)*X(Ber)=0+...+0+Prob(Sur)*(1/2)+Prob(Ber)

So really the naive approach is just a mislabeling, we have sucessfully calculated the expected number of cubed Bers but it is not a probability.


I think it is easy to confuse the expected value and the probability because our random numbers here do not have a dimension and are between 0 and 1. So since the thing we calculate in the 'naive' approach is the expected value of cubed Bers in 1 run it should be called 1285th of a cubed Ber.

 

[Fruit]

For slightly more accurate results you'll want to use 65534, not 65536. Also, I believe the well-used lists have mistakes, for example I believe p7 LK has only 2 Ber patterns and 9 Sur patterns. @art_vandelay also has a point, though specifically for p7 Hell Lower Kurast I think locked and non-locked have the same amount of high rune patterns. Though for other effective player numbers, locked versus non-locked would have an impact.

 

[ffs]

@Fruit What makes you think the Ber/Sur patterns are not correct, and which ones?

 

[sir_cyclops]

@Fruit What makes you think the Ber/Sur patterns are not correct, and which ones?

@Fruit been waiting for fruit to drop this info

 

[Fruit]

@ffs

From the pattern list here on diabloii, incorrectly under p7:

Ber #3 Brandistock Great Sword Light Belt +3 others

Output from my generator thingy, including an in-game screenshot that shows the player number (3):

Spoiler

----------start_seed: 43640
Gold
Gold
Gold
Rondel
Super Mana Potion
Arrows
Great Sword
Bolts
Scroll of Town Portal
Brandistock
Light Belt
Gold
Two-Handed Sword
Ber Rune
Scroll of Town Portal
Arrows
Gold
Stamina Potion
Gold
Gold
Voulge
Gold
Key

From that same list, Sur #5 is a p5 pattern, and Sur #7 is just the locked version of Sur #11. You could argue that #7 and #11 are not really the same patterns, but now the list makes it seem like there is a 11/66534 chance for Sur, but it's really only 9/65534.

 

[Yng]

@Fruit you’re doing god’s work

 

[Luhkoh]

So much stuff to comment on here.

@drmalawi I'm glad to see you rejoined! Hope you stick around. Already told you on discord that I loved the video, but I hope work that into my runefinding sheet at some point in the future (not feeling motivated on it now).

@sir_cyclops So 63.2% confidence level (CL) is the average of a binomial distribution, which is what we have in diablo 2 for single drops. This is not a binomial distribution anymore, so if we wanted to find the "average" we would have to take the derivative of his graph and find the geometric centroid. Better thing to do would just be decide on what confidence level you want and pull the number off his graph.

@T72on1 So what that means is you decide "what chance do I want to find this item?" (I like to use 75%), and then you pull the total runs needed off of his graph. Usually (but not necessarily in this case) the average is 63.2% chance, and that's why drmalawi includes that number in his video, to compare with the old, supposed, average. Another way you can use the graphs is if you say "I've done 8000 runs and still dont have infinity", you can see how unlucky you are.

@Fruit so glad you commented on the numbers of LK patterns

@ffs I actually got this info from fruit's generator a while back and did some looking into it, as well as some calculating on discord with @drmalawi. On p7 there are 2/9 ber/sur and 2/5 Lo/ohm. On p5 there are 4/5(7) for ber/sur (with the number in parentheses being sur patterns for locked chests) and 4/2 for Lo/ohm. In LK fruit let me know there is a 19% chance for a chest to be locked. To me this meant that maybe everyone should be doing p5 runs for runes rather than p7. However drmalawi generated CL graphs for both and they are extremely similar. With p5 being better at lower CL's and p7 being better at high CL's. So not as huge a revelation as i thought. But new takeaways would be:

1. p5 is pretty much as good as p7 for runes (maybe relevant for HC players)
2. p5 is better IF you want to see a direct ber/Lo drop for some reason

@art_vandelay when the locked and unlocked pattern high rune counts are equal, we don't need to distinguish them. P7 LK for example, the locked are the same counts as unlocked. P5 though, there are 2 extra sur patterns in locked variants. I THINK we could say there are "effectively" 5(1-0.19)+7*0.19 = 5.38 sur patterns on P5. (19% chance for chest to be locked in LK specifically, its different based on area) But I still want to do some thinking to see if this is accurate since these linear approximations are how we approached rune cubing incorrectly.

 

[art_vandelay]

@Luhkoh It's still a binomial distribution if you look at a series of drops. The problem is that using this exact model is very cumbersome because of numerical reasons. drmalawi circumvents those by using the Poisson distribution as an approximation which is much easier to deal with and less prone to numerical errors. (It's almost like he's an expert )

On the matter of patterns: There are less Gul patterns on p5 than on p7 (5 vs. 11) and the same amount of Vex patterns for both settings so it takes longer to farm HotO on average.

Also much more important: I would assume that there are fewer drops per p5 chest as well, so less charms, jewels, gems on average? (Is that actually true @Fruit ?)

I doubt p5 wins against p7, if anything it might be a little bit below it because runes are only one aspect of LK in 1.13.

 

[drmalawi]

For slightly more accurate results you'll want to use 65534, not 65536. Also, I believe the well-used lists have mistakes, for example I believe p7 LK has only 2 Ber patterns and 9 Sur patterns. @art_vandelay also has a point, though specifically for p7 Hell Lower Kurast I think locked and non-locked have the same amount of high rune patterns. Though for other effective player numbers, locked versus non-locked would have an impact.

That will not even be visible/noticeable due to rounding in the calculations and the width of the graph

The point was not to discuss what patterns there are, but simply to provide a more accurate and useful description of what some of the things do claim.

Hmm, that is chinese to me. So there isn't a simple answer as to what is the expected value?

As I claim in the video, expected value is not sensible / useful (even in the limit of infinitely many runs there will be a huge spread so it will still not help anyone). I do tell in the video what the graphs mean (the guide in OP do not state what is meant by "expected" either... to me, it just seems a bit "hand-waving")

If there is any interest, I can make a text and picture version of it.

I can of course also recommend my "mathematical magic find" videos.

@Luhkoh It's still a binomial distribution if you look at a series of drops. The problem is that using this exact model is very cumbersome because of numerical reasons. drmalawi circumvents those by using the Poisson distribution as an approximation which is much easier to deal with and less prone to numerical errors. (It's almost like he's an expert )

I have phd in theoretical particle physics and probability, statistics and computations is second nature to me. I am also high school teacher in maths so I hope I can be considered "an expert".

But yes, Poisson is easier to implement, it has nice algebraic properties and in addition have less numerical / rounding errors. We are dealing with such small probabilities here that Binomial and Poisson is indistinguishable (I made a video about that too...)

However, the resulting distribution from cubings is NOT a Binomial distribution.

 

[drmalawi]

Perhaps we all should join forces?

All I need is number of patterns (i.e. drop chance per chest) of various runes, what runes can/should be cubed, what runeword should be made etc. and I am "happy" about doing CL graphs and tables etc.

 

[Luhkoh]

@art_vandelay I could be wrong, but I'm pretty sure the resulting probability distribution from including cubing is no longer a plain binomial distribution (OR a high number/low probability poisson distribution that is essentially the same thing). Been discussing this with malawi on discord a while now. If it's indeed a different shape, it's not a given that 63.2% CL is the average. Would have to derive and find the centroid like i mentioned.

It was also my assumption that charms and gems would be fewer on p5, and that's why I said p5 is approximately as good for runes specifically. That being said, if you set out to farm enigma and infinity from LK,and if p5 was substantially better for ber, then you might be satisfied with your chances at hoto and your charms/jewel count at that number of runs. But like I said it's NOT really better than p7 for ber or Lo, so P7 should still be the gold standard except in those two cases i mentioned.