Drop Units -- An Objective Trade Values Guide

[pairadoxen]

Drop Units -- An Objective Trade Values Guide


The drop unit (DU) is a unit I invented to try to answer the endless queries of the value of an item. The drop unit is similar to pH, if you remember your chemistry. The definition of the drop unit is:

DU = - Log10 ( SUM(P) )

Where P is the probability of a particular monster dropping a particular item, Log10 is the base10 logarithm function, and DU is the drop unit. I used ATMA Drop Calculator (1.10) with players 1, 0% MF, Monster class regular, Difficulty Any. I search for a Dol Rune and got:

<PRE>
1 : 22642 Ghost - (N) Tower Cellar Level 1 - {mlvl 38}
1 : 22642 Ghost - (N) Tower Cellar Level 2 - {mlvl 39}
1 : 22642 Ghost - (N) Tower Cellar Level 3 - {mlvl 40}
1 : 22642 Ghost - (N) Tower Cellar Level 4 - {mlvl 41}
1 : 22642 Wraith - (N) Jail Level 1 - {mlvl 41}
1 : 22642 Wraith - (N) Jail Level 2 - {mlvl 41}
1 : 22642 Wraith - (N) Jail Level 3 - {mlvl 41}
1 : 22799 Ghost - (N) Tower Cellar Level 5 - {mlvl 42}
1 : 22799 Wraith - (N) Cathedral - {mlvl 42}
[... cut ...]
1 : 47058 Hell Bovine - (N) The Secret Cow Level - {mlvl 64}
1 : 47115 Hell Bovine - [H] The Secret Cow Level - {mlvl 81}
1 : 50320 Hell Bovine The Secret Cow Level - {mlvl 28}
[... cut ...]
1 : 69668 Black Archer - (N) Inner Cloister - {mlvl 41}
</PRE>

Now to analyze this information, I pasted it into a text file and imported it into Excel. (Use delimited format, both Tab and Space as delimiters.) If you did it right the third column should start with 22642 in this example. Overwrite the first column with the formula "=1/C1" (ie calculate the reciprocal of the third column). Make an entry at the bottom of the worksheet that takes the log of the sum of the first column. In this case, cell A111 contains the formula "=LOG(SUM(A1:A110))" and I get 2.38 (rounded). I'm going to write a perl script so I don't have to do this by hand anymore.

Therefore a Dol Rune has 2.38 drop units (DU) because killing all the regular monsters in all difficulties has a total probability of 0.00417 of dropping one Dol Rune. I repeated this for some other runes to get:

<PRE>
Dol Rune 2.38
Ko Rune 2.31
Gul Rune 3.43
Vex Rune 3.61
Zod Rune 5.18
</PRE>

From this we can see that the drop unit is consistent with our understanding about frequency of dropping because these are in the right order (except the Ko rune, not sure what happened there). Note that 1 DU makes a huge difference because this is a logarithmic scale. Also, the transition from 1 DU to 2 DU represents a much smaller interval than 4 DU to 5 DU.



The effect of monster class


Still using 0% MF and players 1, I compared the effect of Monster Class.

<PRE>
Monster Class Unique item DU
==================================================
Regular monsters Baranar's Star (7mt) 3.51
Unique monsters Baranar's Star (7mt) 1.64

Minion monsters Tyrael's Might (uar) 5.52
Unique monsters Tyrael's Might (uar) 3.84
</PRE>

Unique monsters include act bosses (both first drop and further drops), super uniques (eg Lister), and plain uniques (Bishbosh). This calculation does not factor in the effect of the multiple drops from act bosses.



Frequently found items


Since these drop so frequently, it's good to know their DU as a reference. 0% MF and players 1, Regular monsters.

<PRE>
Really common items DU
====================================
Chipped Amethyst (gcv) 0.43
Flawless Amethyst (gzv) 2.79
Cathan's Seal (rin) 1.65
</PRE>

It takes 27 Chipped Amethysts to make a Flawless Amethyst with the Horadric Cube recipe. Since the probability of a Chipped Amethyst drop is 0.37 (invert the formula), the probability of getting 27 should be (0.37)^27, which is 11.61 DU by doing the calculation and taking DU again. From this we can see that cubing gems is really unfavorable. An alternative way to calculate this value is to observe that logarithms transform exponentiation into multiplication, so if you know that a Chipped Amethyst is 0.43 DU, and you want to get 27 of them, just multiply by 27 to get the total DU. This is the basis of the Objective Trade Values Guide.

Multiplication is transformed into addition in DU, so another important implication is that the value of getting one Baranar's Star from Regular monsters (3.51 DU) is about equal to 8 Chipped Amethysts (8*0.43 = 3.44 DU). Therefore the true value of finding one Baranar's Star from Regular monsters is about the same as finding 8 Chipped Amethysts.

Now here you should be skeptical because nobody would ever trade away their Baranar's Star for 8 Chipped Amethysts. The problem is that we assumed that we killed every Regular monster which could possibly drop either of those, and each of those monsters dropped exactly one item. We need to be more selective about which monsters we include in the calculation.



Effect of how many monsters


<PRE>
Monsters used Common item DU
==================================================
Top 3 Isenhart's Case (brs) 4.23
All 1425 Isenhart's Case (brs) 1.84

Top 3 El Rune (r01) 2.33
All 1642 El Rune (r01) 0.19
</PRE>

I cite these two because they very frequently drop and it's very easy to kill the monsters which are most likely to drop them. In fact, here's a list for Isenhart's Case:

<PRE>
1 : 51213 Infidel Kurast Causeway - {mlvl 24}
1 : 51539 Bog Creature Great Marsh - {mlvl 22}
1 : 51539 Bramble Hulk Great Marsh - {mlvl 22}
[...]
1 : 442213 Horror Archer Palace Cellar Level 1 - {mlvl 18}
1 : 442213 Horror Archer Palace Cellar Level 2 - {mlvl 18}
1 : 442213 Horror Archer Palace Cellar Level 3 - {mlvl 18}
</PRE>

Among the regular monsters, the top 3 most likely to drop Isenhart's Case (players 1, 0% MF) aren't really much harder to find or kill than the bottom 3 least likely. Also, nobody cares about Isenhart's Case because it's not very useful. I calculated Tyrael's Might at 5.52 DU and a Zod Rune at 5.18 DU, so Isenhart's Case shouldn't be anywhere close to them. 4.23 DU is far too close to 5.18 DU even though it's logarithmic.



TODO List


- Figure out which monsters should be included when calculating DU.

- Do some math to decide if the trade value of items is equal to their sum in DU.

- Account for rares and items with variable properties.

- Check the Horadric Cube upgrade and rerolling recipes to see if the value of a Perfect Gem can be used as an objective price baseline.

- Get some other guides and evaluate each of their items in DU, then see if the usefulness of an item can be correlated to DU.

- Show how bad the SOJ economy is. Go on eBay and get a guess of DU to dollars.

- Try to find some way to account for duping. It might be as simple as subtracting 1 DU from all high runes.


Stat forum folks: Please help me! I think this has a lot of promise for giving objective answers to item prices.

[Myrakh-2]

- Do some math to decide if the trade value of items is equal to their sum in DU.

Trade value has almost nothing to do with drop frequency (just look at how many great items you can get for a few pgems).

[Lyrs]

Your scenario of value = drop ratio will only work in the absence of dupes.

On East sccl, it's now possible to trade an ist for a mal. I don't know exactly why the mal rune went up in value, but that's what the supply and demand is showing. Furthermore, hr = ist + mal, and is dropping to ist um in several instances. I believe that if this rate continues, in a week or two, 1 hr will be equivalent to 1 ist, even though the probability of a hr dropping is much lower then an ist.

Lastly, a boss in the pits dropped a Sur rune! :clap:

[Quickdeath]

First, I don't understand why you take the logarithm (base 10) of the drop probability. That means that every unit of drop probability corresponds to a factor of 10.

So if a chipped amethyst has a DU=0,43 and a Baranar's Star has a DU of about 3.4, this means that the Baranar's Star is equal to 1,000 chipped amethysts, not 8 chipped amathysts. That's because the 2 drop units have a difference of 3 and ten to the third power is 1,000.

Another things you've ignored is the minimum qlvl of an item. An item with a DU=1 that can only be dropped by lvl 85 monsters is harder to get than an item with a DU=1 that can be dropped by lower level monsters.

[purplelocust]

Does this sum of drop probabilities account for the number of monsters occurring in the levels? That is, there is a big difference between:

Hell Bovine: 1 in 100 chance of dropping item A

and

The Summoner: 1 in 100 chance of dropping item A

since if you did full clears of all monsters, you would have several hundred cows but only one Summoner. If DU is based on "one of each monster on a difficulty" that would be quite different than a DU based on "a full clear of all monsters on a difficulty."

The "full clear" upon which this Drop Unit is based isn't the way that most people look for items- instead, the amount of time that people look for items is heavily skewed towards Meph and boss runs, so perhaps the sum of probabilities should be weighted by what people actually spend their time doing. Easy-to-get-to bosses that drop well should be weighted heavily, and more obscure areas that don't yield good drops should be weighted lightly.

This seems like an interesting project but as was pointed out above, rarity is different than value. There are plenty of extremely unlikely drops that would be totally useless, particularly when talking about rares, where some comically useless combinations of prefixes and affixes could be astronomically unlikely and completely valueless.

[pairadoxen]

There are plenty of extremely unlikely drops that would be totally useless,


I am aware of this problem, and DU was only intended as an approximation of value. However, duping and the uselessness of many high level uniques makes this project pointless. Perhaps I will pick it up again later.

[Cormallon]

Hmm, your DU seems to state that a DOL rune (2.38) is more frequent than a flawless Amethyst (2.79)? I don't think so, there must be some flaw in your calculation.

Besides that, items are worth what people are willing to pay for them. Value depends on usefulness and only sometimes on rarity.