Archive for the ‘Statistics’ Category

Comparison of Mounts

November 16, 2014 4 comments

Nowadays there’s a shitload of various kinds of mounts available in Age of Conan, and I was getting a bit confused about their speed and various other properties, so I decided to conduct a few measurements.

To measure the speed, I chose a section of road in the northern half of the Wild Lands of Zelata, starting at (526, 1028) and moving east from there. I used several different end points depending on how long a course I wanted: ending at (671, 1053) gives you approx. 150 meters; ending at (721, 1058) gives you 200 m; ending at (982, 1174) gives you 500 m; and ending at (992, 1314) near the Wild Plains rez pad gives you 650 m. I used the longer courses for testing the faster mounts, and the shorter ones for slower mounts. In particular, the 150 m course is useful for testing sprinting with those mounts that would run out of energy before completing the 200 m course.

(Click to enlarge.)

Of course the time measurements (using F9) are only accurate up to 1 sec, so the resulting calculations of speed are only approximate. Each speed is based on the average time of 2 or 4 runs. All the speed tests were done with a level 80 guardian with 800 skill points in the Running skill and without any other speed buffs (e.g. from armor).

Note that each mount has 3 possible speeds: walk, run, and sprint. Sprint is when you hold the shift key; otherwise, you’re walking or running, and you can use backspace to switch between walking and running mode. (If you press shift, the sprint speed is the same regardless of whether you started in walk mode or run mode.) Walking while mounted is very much not recommended, as it is even slower than walking on foot!

Another measurement that I found interesting was the turning speed of the mount. In my opinion, one of the big advantages of e.g. a tiger compared to a swift horse are that a tiger can turn more quickly; with a horse you either have to slow down or make a much bigger circle before you manage to turn around. I measured turning speed by standing in one place and holding the A key until I made 10 full turns, then dividing the time it took to do this by 10.

Before proceeding, it’s worth taking a look at the update notes for update 3.3.8, which included the sprinting revamp (introduction of the Energy resource) and a minor revamp of mounts. As the update notes say, the mounts are now divided into 5 categories and all mounts from a particular category are supposed to have largely the same stats and abilities. This categorization is still useful, although there were some exceptions to it even at the time of 3.3.8 and there are a few more exceptions to it now; more on that later.





Speed [km/h]



Exotic Mounts

−35 stagger






Level 80 Mounts

−20/35/40 stagger








Level 40 Mounts

−20 stagger






Basic Mounts









Siege Mounts

+1900 HP, −40 stagger chance






no mount








In the lists below, an asterisk * marks mounts that I haven’t tested myself.

(By the way, for another great overview of AoC mounts, see this post on Henryx’s blog: link.)

Exotic mounts

This category consists of tigers, wolves, Yothian war-mares, and Hyrkanian horses. They require the Exotic Animal Handling skill. As far as I know, all the mounts in this category have the same speeds and the same buff (−35 chance of being staggered).

But in addition to that, some of these mounts have special abilities: the faction tiger can hide; wolves have the Terrifying Howl ability (which puts a 50% hinder movement debuff for 10 sec on nearby enemies; 60 sec cooldown).


  • Vaaghasan Slaughter Steed: this is the tiger from the Tamarin’s Tigers faction. To get it, you need to be rank 4 with them, complete a quest chain and buy a saddle, which costs 800 marks of acclaim and 10 gold. As far as I know, this is the only mount that can hide. It’s available in three colors (light, dark and black) but they all have the same name.
  • White Vaaghasan Slaughter Steed*: this tiger was available with some subscription offers.
  • Purebred White Vaaghasan Slaughter Steed*: from the item shop (900 Funcom points).
  • Imperial Bronzesteel Slaughter Steed*: sold by gilding vendors (13 gilding tokens); formerly also available in the item shop [1, 2].
  • Imperial Shadowsteel Slaughter Steed*: sold by vendors in PvP armories (120 campaign badges + 5 gold).
  • Imperial Greensteel Slaughter Steed: sold by the Hand of Glory faction vendor in Ardashir (requires no faction rank; costs 500 marks of acclaim + 200 rare trophies + 5 gold).
  • Imperial Bluesteel Slaughter Steed*: drops from the Zodiac in the Jade Cidatel (T4 raid).
  • Imperial Redsteel Slaughter Steed*: was available in the past in the item shop [1, 2].

AFAIK the only difference between these various tigers (apart from the hide ability on the original faction tiger) is in the color and shape of their armor and other paraphernalia.


  • Ridden Death: this is the wolf from the Wolves of the Steppes faction. To get it, you need to be rank 4 with them, complete a quest chain and buy a saddle, which costs 800 marks of acclaim and 10 gold.
  • Xanthic Ridden Death*: from the item shop (1260 Funcom points).
  • Bluefury War Wolf: drops from the Imp in the Jade Citadel (T4 raid).
  • Greenfury War Wolf: sold by the Hand of Glory faction vendor in Ardashir (requires no faction rank; costs 500 marks of acclaim + 200 rare trophies + 5 gold).
  • Shadowfury War Wolf*: sold by vendors in PvP armories (120 campaign badges + 5 gold).
  • Bronzefury War Wolf*, Redfury War Wolf*: these two were available in the past in the item shop [1, 2].

AFAIK the only difference between these various wolves is in the color and shape of their armor and other paraphernalia. All the wolves I’ve tested (Ridden Death, Bluefury, Greenfury) have the Terrifying Howl ability.

Yothian War-Mares

These mounts are only available in the item shop, where they cost 972 Funcom points each. Three variants are available (Green Yothian War-Mare*, Purple Yothian War-Mare, Red Yothian War-Mare*).

Perhaps you remember the Curses of Travius Blacktongue quests in the Armsman’s Tavern in Tarantia Noble District. One of the insults you use during the conversation with Travius includes the following line: “Your riding skills resemble the most debased acts of love between beast and man.” Well, this is what I’m always reminded of when I see someone riding a Yothian War-Mare: it looks as if the player was molesting an enormous snail. Additionally, having now tried riding one myself, it makes your entire screen wobble left and right so badly that I almost got seasick after just a few minutes of riding. So I very much don’t recommend these mounts.

On the positive side, you might not expect an oversized snail (with hands) to be as quick as a tiger, but it is — I measured.

Hyrkanian horses

To buy these horses, you need to be rank 4 with the Hyrkanian faction; each horse costs 800 marks of acclaim and 10 gold. The Hyrkanian horses are the only horses that have the same speed or maneuverability as a tiger or wolf, so they are a great choice if you find the looks of the tiger or wolf to be too exotic. Other swift horse mounts (which we’ll see in the next category) are slower.

The following 8 horses are available; each is painted with a different pattern:

  • Hyrkanian Bitter Wind-Biter*
  • Hyrkanian Dread Ash-Tramper*
  • Hyrkanian Fleet Scream-Catcher*
  • Hyrkanian Grim Skull-Treader*
  • Hyrkanian Pale Ghost-Chaser*
  • Hyrkanian Resplendent Sun-Strider
  • Hyrkanian Sable Shadow-Dasher*
  • Hyrkanian Wicked Chaos-Clapper*

Other mounts

The Great Khan’s War Bringer (a legendary mount which can drop from the Hoard of the War Bringer, a lottery box that was introduced in the item shop in February 2015; the horse is also tradable between players) also belongs to this category. It looks like an armored horse, but has the same speed (including turning speed) as tigers, wolves etc. Unlike all the other mounts in this category, this one requires just 7 seconds to fully regenerate its energy (others require 9 seconds). Its mounted buff is −50 chance of being staggered.

The Hoard of the War Keeper can also drop another mount, Swift Ymirish Colossal Cold One, which we’ll see in the next category. (In the early days the hoard was bugged and this mount couldn’t drop, but it was fixed in a subsequent hotfix.)

Level 80 mounts

This category includes various non-exotic armored and swift mounts. In our table above, there are 2 rows for this category, because some of the mounts in it have a slightly higher run speed than the others. These will be noted below, but most of the mounts in this group belong to the slower subgroup (39 km/h run speed instead of 42 km/h). (In fact, by now a pattern is starting to emerge: it seems that all the horses are in the slower subgroup and all the non-horses are in the faster subgroup.)

Armored horses

There are 32 armored horses, all named systematically X-barded Y Horse, where X is the color of the armor (Azure, Dark, Pale, Umber) and Y gives the color of the horse itself (Buckskin, Dapple Gray, Fading Black, Mahogany Bay, Rabicano, Skewbald Pinto, Tobiano, White). The horse vendors in hub cities (Conarch Village, Old Tarantia, Khemi) sell only the Azure variants; the horse vendor in your guild city sells all variants. Each horse costs 50 gold. The buff while mounted is −20 chance of being staggered.

Swift horses

There are 8 swift horses, named Swift Y Horse, where Y is one of the horse colors, same as above. They are sold by horse vendors in hub cities (Conarch Village, Old Tarantia, Khemi), where they cost 150 gold each; in the item shop (1125 Funcom points each); and by veteran vendors (85 veteran tokens each). The buff while mounted is −20 chance of being staggered.


We’ll start with a few mounts that have recently been added to the reliquary vendors. The first two of these have the peculiar feature that their run speed is about 10% faster than that of the other mounts in this category. Perhaps this applies to the Swift Killer Rhino as well, but I haven’t tested it yet (so TBH I can’t even be completely sure that it belongs to this category at all).

  • Fleet Black Riding Camel: sold in the reliquaries (1 rare relic + 120 simple relics III + 15 gold). Buff: −20 chance of being staggered.
  • Swift Hyperborean Siege Mammoth: sold in the reliquaries (50 simple relics I + 40 gold). Buff: −40 chance of being staggered (unlike the normal siege mounts, you get no HP buff from this one).
  • Swift Killer Rhino*: sold in the reliquaries (50 simple relics II + 40 gold).

Anyway, if you ever wanted a mammoth or rhino that handles exactly like a traditional swift horse, now’s your chance to get it! 🙂

The Wolf Pict Envoys in the PvP armories also sell two swift siege mounts: Swift Vendhyan Thundering Colossus* (a kind of mammoth) and Swift Wolf Pict Rampaging Behemoth* (a kind of rhino); each costs 300 victory tokens (from open-world PvP objectives). Judging by the tooltips, their mount buff gives you −80 chance of being staggered. I don’t know how fast they are, but I imagine they most likely belong to this category, same as the other swift siege mounts mentioned above.

Another siege mount in this category is the legendary Swift Ymirish Colossal Cold One, which drops from the Hoard of the War Bringer (see above). It looks like a white mammoth with some red markings; mounting it morphs you (the player character) into an Ymirish frost giant; and it has the usual siege mount buff (1900 HP, −40 chance of being staggered). In terms of speed, this mount belongs to the faster subgroup of this category.

Next, we have a few more horse mounts. The ones that I haven’t tested yet (marked with an asterisk) are included in this list tentatively; I can’t be completely sure that they belong to this category, though I think it’s unlikely that they belong anywhere else.

  • Swift Stygian Festival Stallion: has a chance of dropping from the Crate of Random Oddities, which you could get by handing in the Sacrificial Marks from the Khemi Halloween quest (Halloween 2014). Buff: −35 chance of being staggered. Unlike all the other mounts in this category, this one requires just 7 seconds to fully regenerate its energy (others require 9 seconds).
  • Tarpani Stallion: quest reward from a quest that becomes available in your guild city at renown level 20. In the process of doing the quest, you’ll have to pay 200 gold. Buff: −35 chance of being staggered, +590 protection.
  • Purebred Tarpani Stallion*: from the item shop (1890 Funcom points).
  • Stygian Halfbreed*: quest reward from a quest that becomes available in your guild city at renown level 15. In the process of doing the quest, you’ll have to pay 200 gold.
  • Swift Shemite Horse*: sold by the horse vendor in your guild city (from renown level 10); costs 150 gold.
  • Purebred Shemite Horse*: from the item shop (1575 Funcom points).

We also have two mounts that were introduced with the achievement system:

  • Hyrkanian Riding Camel: you get this mount by reaching 1000 achievement points. Mounted buff: −20 chance of being staggered. Unlike the other mounts in this category, its total sprint time is just 17 sec (same as for level 40 mounts). Apart from that, it has the same speeds (and turning time) as the swifter subgroup of this category (i.e. like the Fleet Black Riding Camel and the Swift Hyperborean Siege Mammoth).

  • Swift Cimmerian War Mammoth: you get this mount by reaching 7500 achievement points. Mounted buff: +1900 HP, −40 chance of being staggered. In terms of speed, turning time, and stamina, it behaves exactly like the Swift Hyperborean War Mammoth mentioned above.

One thing that I haven’t investigated in my tests is which mounts can do decent siege damage; I imagine that mammoths, rhinos and armored horses can do siege damage, but I’m not sure about e.g. the Tarpani Stallion and the like.

Level 40 mounts

This category seems to consist of nothing but plain ordinary horses. They are named simply Y Horse, where Y is one of the eight horse colors (see above). You can buy them from the horse vendors in hub cities (75 silver each); from the item shop (495 Funcom points each, except the White Horse, which costs 1125 Funcom points!); and from the veteran vendors (3 veteran tokens each). The mounted buff is: −20 chance of being staggered.

Siege mounts

These are noticeably slower than other mounts, and are particularly slow at turning. Their mounted buff is: +1900 health, −40 chance of being staggered.

  • Blue Snow Mammoth, Purple Snow Mammoth*, Red Snow Mammoth*: sold by horse vendors in the hub cities (Conarch Village, Old Tarantia, Khemi). The blue one has also been included in some subscription offers.
  • War Mammoth: was included with some pre-order and subscription offers.
  • Killer Rhino: was included with some pre-order and subscription offers.
  • Purebred Killer Rhino: available from the Hand of Glory faction vendor in Ardashir (requires rank 4; costs 10 gold); also from the item shop (990 Funcom points).
  • Purebred War Mammoth*: from the item shop (990 Funcom points).
  • Tuskripper War Mammoth*: from the item shop (990 Funcom points).

Basic mounts

This category includes the (non-swift) camels and the recently introduced Reaver’s Steed. Their main feature is supposed to be that they are available from very low levels (Reaver’s Steed at level 1, camels at level 20). Reaver’s Steed is a little faster than the camels, which is why there are 2 rows for this category in the table above. The buff you get while riding them doesn’t give you any stat bonuses.

  • Reaver’s Steed: from the item shop, where it appears as Reavers [sic] Stallion (costs 1080 Funcom points).
  • Brown Riding Camel, Dark Riding Camel*, Pale Riding Camel*: from the item shop (1575 Funcom points each, except the Pale one, which costs 1890 Funcom points). The Brown one is also included in some subscription offers.

Money drops from bosses, continued

November 17, 2012 1 comment

This is a sort of continuation of my previous post about the amount of money that drops from bosses in epic instances of old-world playfields.

As I wrote there, it would be interesting to see how the amount of money depends on the level of the mob, so I did some more testing with mobs of other levels. One problem here is how to find several bosses of the same level, so that you won’t have to spend too much time waiting for respawns and the like. There are of course plenty of level 80 bosses in Kheshatta, and plenty of level 50 bosses in the Field of the Dead, but other levels are not so convenient. Well, you could be killing level 75 Hong Gildong bosses in Kheshatta pretty conveniently as well, but I decided not to bother as the difference in loot from level 80 is probably small enough that I thought it would take an annoyingly large sample to be able to say anything reliable about it. (Though as we’ll see later, it isn’t necessarily that bad.)

So I mostly focused on lower levels. The Wild Lands of Zelata are pretty convenient; in the Nemedian camps in the east of the playfield, you can find five level 38 bosses and one patrolling level 39 boss; and in the camps further to the south there are three level 30 bosses. I was also killing some level 20 and 21 bosses in the area just south of Tesso. The problem with those is that not all of them drop money — generally the idea is that animal mobs don’t drop money, although some of them in fact do. For example, the level 21 Ancient Wolverine drops money, but nearby you have a level 21 Furious Wolverine which doesn’t drop money. Anyway, these bosses respawn reasonably quickly, so you can keep cycling through these for as long as you like without too much waiting for respawns.

I also found several level 66 bosses in Thunder River — this is a suitable level because it’s approximately half-way between 50 and 80 (for which we already have plenty of data). For example, there’s Tarakwi, and also several Pict Chiefs (which are just mini bosses in the normal instance of the playfield, but proper bosses in the epic instance; there is no concept of mini bosses among epic mobs). The annoying thing about these is that each of them is accompanied by at least two trashmobs, and fighting three level 66 group mobs (one of which is a boss) isn’t completely trivial, even with a level 80 character — you have to CC them and so on, same as you would with a level 80 boss. What is more, many of these trashmobs are ranged, so they aren’t really suitable for kiting.

If you want to miss out on all the interesting mathematical stuff, you can jump directly to the results 😛

Taking b = 3a into account

As we already saw in the previous post, the amount of money dropped by epic bosses seems to be distributed uniformly in the range [a, b] where a and b of course depend on the level of the boss. We also saw that it furthermore appears that b = 3 · a in all these distributions; this was seen there in the data for level 50, 80 and 82 bosses, and the data I’ve collected since then for levels 21, 30, 38, 66 follows the same pattern.

So by now we can feel pretty comfortable in assuming that b = 3a. This assumption opens up some new possibilities in estimating the parameters of our distribution. The mean μ of our distribution, μ = (a + b) / 2, can now be expressed as μ = 2a = 2b/3; and from this we can also express a and b in terms of μ, namely as a = μ/2 and b = 3μ/2.

Now, remember that we denoted the sample minimum by W and the sample maximum by Y. Obviously the sample minimum cannot be less than the population minimum; in other words, aW. Together with a = μ/2 we get μ ≤ 2W. Similarly, the sample maximum cannot be greater than the population maximum; thus, bY. Together with b = 3μ/2 we get μ ≥ 2Y/3. So we obtained a lower and an upper bound on μ:

2Y/3 ≤ μ ≤ 2W.

We could similarly obtain bounds for a and b by taking into account that μ = 2a = 2b/3; we would get

Y/3 ≤ aW and Yb ≤ 3W.

In the previous post, we saw various estimators: Xa and M as unbiased estimators of μ, and A and B as unbiased estimators of a and b. Those were derived without the assumption that b = 3a, so they can sometimes result in estimates that are outside of the new bounds we’ve derived here:

  • A should be in the range [Y/3, W]; from the definition of A we can see that it can’t be greater than W, but it can turn out to be less than Y/3. This happens when Y > 3W · n / (n + 2).
  • B should be in the range [Y, 3W]; from the definition of B we can see that it can’t be less than Y, but it can turn out to be greater than 3W. This happens when Y > W (3 − 2/n). In fact it can be shown that when this happens, the previous problem (that A is too small) will also happen at the same time (but the converse is not always the case).
  • Xa can be either too small (less than 2Y/3) or too big (greater than 2W). Given a sufficiently large sample (large n) and sufficiently bad luck, you can get an Xa that’s very close to W or very close to Y and thus quite far from the range [2Y/3, 2W].

These things aren’t just theoretical, they actually happened in some of my measurements for loot from level 50 bosses.

For M, on the other hand, it can be shown that (if b = 3a) it will always be in the range from 2Y/3 to 2W. For example, if M were greater than 2W, we would have (W + Y) / 2 > 2W, therefore W + Y > 4W, therefore Y > 3W; combining this with the fact that Wa, we get Y > 3a = b, which is impossible, as Y is inevitably ≤ b. We can similarly show that if M were less than 2Y/3, this would also lead to a contradiction.

So if you know that b = 3a, and you want to estimate a or b, it would be better to estimate them with M/2 and 3M/2, respectively, rather than with A and B.

Bayesian estimation

I found this very interesting article about estimating the parameters of the uniform distribution. These things are no doubt all very well known to real statisticians, but for a dilettante like myself this was a new and fascinating thing. The article I linked to works with the assumption that a = 0; we don’t have that, but we have b = 3a instead, and we can use a method analogous to theirs to derive a Bayesian estimator for our situation.

Suppose that we are estimating a (we could do just the same for μ or b, of course — it’s really all the same since we saw earlier that they are linked by the fact that μ = 2a and b = 3a). We already saw that it must lie in the range [Y/3, W]. Now, the idea about Bayesian estimation is that we don’t want to commit ourselves right away to any particular value of a. Rather, we’d prefer to think of all of them (from the range [Y/3, W]) as possible, but some of them might seem more probable to us than others would. You might say that we’ll act as if a itself was some sort of random variable. Let’s start by thinking that all a‘s are equally probable:

p(a) = 1 / (WY/3) if WaY/3, and 0 elsewhere.

We’ll denote our sample X1, …, Xn more briefly by X. We know that each Xi is distributed uniformly in the range [a, 3a]; thus, given a concrete value of a, we have

p(Xi | a) = 1 / (ba) = 1/(2a)

And since the various Xi are independent of each other, we have

p(X | a) = p(X1, …, Xn | a)
= p(X1 | a) … p(Xn | a)
= 1 / (2a)n.

From this conditional probability we get the joint probability

p(X, a) = p(X | a) p(a) = 1 / (2a)n / (WY/3).

And from this we get the unconditional probability of X:

p(X) = ∫Y/3W p(X, a) da
= … = (W1 − n − (Y/3)1 − n) / [(WY/3) 2n (1 − n)].

Now we use Bayes’s formula to get the conditional probability of a given X:

p(a | X) = p(X, a) / p(X)
= (1 − n) / an / (W1 − n − (Y/3)1 − n).

Or, in other words, if we originally considered all values of a to be equally likely (this was reflected in our original distribution p(a), which is also called the “prior” distribution, because we had it before we saw the data*), we now, having seen the data X, consider some values of a more likely and some less likely than before; this is reflected in the new distribution p(a | X), which is also called the “posterior” distribution, because we obtained it after seeing the data.

[* Well, not entirely — we had to see the data to compute the values of W and Y that we used in our definition of p(a).]

If we still insist on estimating a with just one concrete number rather than with a distribution, we can take the expected value of our posterior distribution. In other words, this is a bit like taking a weighted average of all possible values of a, in which different values might have a greater or smaller influence, depending on how likely we consider them to be. So we get

E[a | X] = ∫Y/3W a p(a | X) da
= … = W · (n − 1) / (n − 2) · R / (R + Zn − 2),

where we introduced Z = 3W/Y and R = 1 + Z + Z2 + … + Zn − 3.

Calculating this with the computer is manageable enough, but it’s sufficiently messy that I have no idea how one would go about analyzing it theoretically. I made some experiments with it and got the impression that it looks unbiased enough, and its standard deviation seems to be typically around 10% smaller than that of M, so from that point of view I guess you could say it’s a tiny improvement over M.


Now let’s see some results. The following table shows, for each level:

  • how many bosses of that level I killed (i.e. n, the sample size);
  • the minimum and maximum drop (sample minimum W and sample maximum Y);
  • the sample midpoint, M = (W + Y) / 2, which, as we already saw in the previous post, is a very nice estimator of the population average μ;
  • the Bayesian estimate of μ, which we derived in the previous section;
  • and finally the lower and upper bound on μ which we saw at the beginning of this post: 2Y/3 ≤ μ ≤ 2W.
Level No. of
Estimates of avg. drop μ Bounds on μ
Midpoint M Bayesian Lower
































































The numbers in the table are shown in silver and are rounded to two decimal digits (i.e. to the nearest copper), but my underlying data was actually measured in tin (i.e. 4 decimal digits). The only exception to that were the measurements for level 80 and 82, where I was mostly ignoring the tin and writing down just the amount of silver and copper that dropped; thus the numbers in our table are probably slightly underestimating the actual amount of money that drops.

A further note about the results for level 82. Initially my sample minimum there was W = 12.63, which is less than one-third of the sample maximum, Y = 38.7696; that is, it gives us Y/W = approx. 3.07, whereas at all other levels we had Y/W slightly below 3. I think it’s more likely that I have an error in my records than that the distribution of money drops at level 82 is different than at all other levels.

This discrepancy can’t be explained just by the fact that I rounded down some of my measurements to the nearest copper (i.e. by ignoring the tin). My W = 12.63 was obtained from a kill in which I was in a group of 3 players and received 4.21 silver (and 3 × 4.21 = 12.63). Due to the rounding down to the nearest copper, it’s possible that I actually received as much as 4.2199 silver, in which case the actual drop would have been 3 × 4.2199 = 12.6597. But that would still give us Y/W = approx. 3.06.

In fact 12.63 is well below even the sample minimum at level 80 (which is 13.17). So I’m inclined to think that I either mistyped some digit, or perhaps we had 4 people in the group instead of 3 (which would give us a drop of 4 × 4.21 = 16.84, which would be a perfectly unremarkable amount).

So if we remove the 12.63 measurement, we’re left with n = 88 measurements at level 82, of which the smallest is now W = 13.06, and this is what the last row of the table above is based on.

As we can see from the bounds on μ, our sample sizes are big enough to give us a pretty accurate estimate of μ (the average money drop at that level), and in fact the bounds give us disjoint ranges even at adjacent levels (for example, at level 38 we got 2.12 ≤ μ ≤ 2.23, whereas at level 39 we got 2.27 ≤ μ ≤ 2.40).

We can also see that the distribution of money drops at level 82 seems to be pretty much exactly the same as at level 80 — the sample minimum, maximum, midpoint, as well as the estimates and bounds on μ, are very close. On earlier level, similar sample sizes were sufficient to show a clear difference between the average drop at one level and at the next level (e.g. 20 and 21, or 38 and 39); so if there was a difference in the average drops at level 80 and at level 82, we should see it clearly as well. (Probably the average drop at level 81 is also the same as at 80 and 82, but I only killed a handful of level 81 bosses, so I can’t get good estimates for that level.)

Let’s draw a chart showing how the average drop grows with the level:

It sure grows quickly (up to level 80, that is); so we might think it grows exponentially with the level. To test that, we can use logarithmic scale on the y-axis; if the growth was really exponential, the graph should now be a straight line:

We can see that the growth is a bit too slow to be exponential. Let’s put the x-axis on a logarithmic scale as well:

Now we see a really nice straight line. And if you have log(y) = c × log(x), this means that y = xc — in other words, the growth is polynomial rather than exponential. Doing a linear regression on the (log level, log μ) pairs suggests that the exponent c is approximately 3.13.

I guess that money drops from trashmobs, and from mobs in normal instances of the playfields (as opposed to epic ones), follow similar characteristics as well.

If you want to kill bosses for the sake of farming money, I think your best bet is to farm the level 50 bosses in the southern part of epic Field of the Dead. There’s lots of them in a small area, they respawn quickly, and they are quick and easy to kill. At lower levels, the average drop is so much lower that you’d have to kill a lot more bosses in the same amount of time to get the same income, and you simply won’t find that many bosses sufficiently close together. At higher levels, you get more money per kill, but each kill takes you so much longer that it ends up giving you a smaller income again (or you need a group to speed things up, which decreases your income again).

Categories: Age of Conan, Statistics

Shadow Imp Lord

October 21, 2012 7 comments

The Athyr-Bast encounter in Wing 3 of the Black Ring Citadel contains a few elements that must have either been placed there to deliberately misdirect the attention of players as they were first figuring out the tactics for the fight, or that were perhaps originally meant to play a more significant role in the fight but the developers changed their mind about that later. As is well known, the main challenge in this fight comes from the Shadow Imp Lord, who spawns in the centre of the room when you begin the fight. He oneshots people at random unless the character that is on top of his aggro list stands close enough to him; so you always need to have a tank on top of his aggro list and standing next to him. But you can’t use just one tank for this purpose, because the Shadow Imp Lord puts an unholy invulnerability debuff on any players nearby, and this debuff gradually stacks higher and higher, so that the tank standing next to him is getting hit harder and harder; thus you need several tanks, and when the current tank’s debuff stacks high enough, you send the next tank in, he takes aggro and the previous tank then moves away from the Shadow Imp Lord until his debuff expires, and so on. In principle, two tanks are enough for this, if they are reasonably well geared and if they know what they are doing; but to be safer, many pug raids use 3 or even 4 tanks for this.

As soon as you start the fight (and the Shadow Imp Lord appears), a steady stream of small imps begins spawning from all sides of the room and walking towards the Shadow Imp Lord, where they disappear once they reach him. The little imps can be DPSed and, with a bit of effort, killed before they reach the Shadow Imp Lord; but there’s no obvious reason to do so, and nowadays people very reasonably ignore the little imps altogether. In the early days after Wing 3 was open (in update 1.04), people had the idea that the little imps buff up the Shadow Imp Lord when they get consumed by him, and that this is why you should kill them (or at least most of them) before they reach him. But the little imps were too numerous and had too much HP to be killed, at least without having practically your entire raid dealing with them and almost nobody on the boss herself.

This is where the other two mysterious elements of the fight come in: the torch and the braziers. There’s a torch near the central square (where the Shadow Imp Lord appears once the fight begins), and several braziers along the walls of the room. One player can pick up the torch and then click the braziers to light them, and while a brazier is alight, no imps spawn from that side of the room (or at least much fewer than normally — it’s been a very long time since I’ve been in a raid that tried this, so I don’t remember the details). After a while, the fire in the brazier goes out, and the imps spawn normally again. So people naturally got the idea that you should assign a player to pick up the torch and then move around the room all the time, lighting the braziers so as to keep the overall number of the little imps as low as possible.

Carrying out this idea was made harder by the fact that picking up the torch generates a lot of aggro on Athyr-Bast; in other words, she’ll tend to attack the torch-holder no matter what the other players are doing to her. Nowadays some pug raids use this mechanic to reduce the risk that the main tank on her will lose aggro (e.g. because they put all guardians and DTs on the Shadow Imp Lord, leaving only conqs to tank Athyr-Bast, and many conqs have (or used to have, until the recent revamp) relatively weak aggro). But admittedly this mechanic looks like exactly the sort of thing that Funcom would implement to make the “proper” tactic, with lighting the braziers, harder to implement: if the torch-bearer is walking around the room all the time, and you practically can’t pull the boss off him, this means that the main part of the raid has to move around the room with him as well if they want to keep DPSing the boss. Additionally, it’s a challenge to prevent the boss from killing the torch-holder, because if she kills him, there’ll be no way to keep the braziers alight and prevent the small imps from spawning.

So this would be quite an interesting tactic — considerably more complex (and harder to execute) than what we have now, when we ignore the braziers (and often also the torch) altogether and just tank and spank the boss in a fixed place all the time. All that’s missing here is a reason to be interested in killing the little imps before they reach the Shadow Imp Lord. So perhaps the developers initially did intend the little imps to buff up the Shadow Imp Lord’s damage to the point where it would be impossible to tank him, even with 3 or 4 tanks; or perhaps they intended it to make it impossible to use more than 1 tank on him at all, because, from what I remember of the early post-1.04 days, swapping aggro on the Shadow Imp Lord was unusually difficult, much harder than on other mobs (nowadays swapping aggro on him is no harder than on other mobs), and perhaps the devlopers’ initial idea was that it would be impossible altogether.

But in practice, neither of these two things is the case. It is (and always was) possible to swap aggro on the Shadow Imp Lord and thus use more than one tank on him; and he never got noticeably buffed no matter how many little imps he consumed. So either they didn’t implement the missing mechanic here (to have the little imps buff up the Shadow Imp Lord), or they balanced things incorrectly, or they deliberately introduced the braziers and little imps (and TBH the torch as well) for no other reason than to confuse players.

But anyway, is it really the case that the little imps don’t buff up the Shadow Imp Lord? If they don’t, why would there be, early in the post-1.04 period, such a widespread idea that they do? (Of course it’s possible that they don’t buff him now but that they did buff him back then. I wasn’t analyzing combat logs back then so I can’t check.) And in fact, even in more recent times, I occasionally had the feeling that the Shadow Imp Lord hits a bit harder as the fight progresses. So I started to wonder if perhaps the little imps do buff him up a bit, just not enough to really make a difference (or to require a change in the tactics).

Then I remembered that I have a good way of analyzing this. A few months ago we killed Athyr-Bast with a 6-player group for fun: two tanks on the Shadow Imp Lord, one on Athyr-Bast, two healers and one DPSer. (In fact you could do it with 5 people — remove the extra DPSer and the only difference would be that the fight would last longer. Perhaps you could even do it with 4 people by removing one of the healers, but that would require a bit more care and some luck.) This fight took about 20 minutes; so if the little imps were buffing up the Shadow Imp Lord at all, surely this would be noticeable over such a long period. So I now processed the log file from that fight and came up with the following chart:

This chart shows us one point for each hit that the Shadow Imp Lord did; the x-coordinate is time when the hit occured (in seconds since the start of the fight), and the y-coordinate is how many points the hit was for. You can see the typical zigzagging shape — as a tank goes in, he gets hit harder and harder because of his stacking unholy invulnerability debuff, until the other tank takes aggro. But you can also see that there is no long-term growing trend in the strength of the hits; after nearly 20 minutes, the Shadow Imp Lord wasn’t hitting even a tiny bit harder than at the start of the fight. So I think we can safely say that the little imps don’t buff the Shadow Imp Lord at all.

P.S. This post gets funnier if you mentally replace each occurrence of the word “imp” with “pimp” 😛

Level 70-80 world-drop sets

September 29, 2012 20 comments

A few months ago I made a few posts about the old world-drop armor sets and included the screenshots of all twelve level 40–69 sets. I was wondering whether I should try farming the level 70–80 sets as well; initially I thought it would take too much effort, but eventually I decided to do it after all. It provided a pleasant enough diversion from other boringly repetitive activities such as farming hardmodes.

(If you don’t want to read my ramblings, you can jump straight to the screenshots.)

Farming epic Kheshatta

I did all the farming in the epic instance of Kheshatta (although a few bosses in the 70–80 level range could also be found elsewhere, particularly in Atzel’s Approach). Although these are group bosses, they can also be killed by a single player, at least by soldier classes, as long as you use enough CCs (don’t forget the stun from doubletapping forward) and kite them when needed to recover health. You can even solo kill the level 82 Bat Demon Lord boss that way, even though it hits pretty damn hard; in my experience, the best thing to do was to kite it in a fairly big circle in the area left of where it floats before you pull it; the sloping and uneven terrain there seems to hinder the bat’s pathing a bit and thus slows it down, making it easier to kite and recover health. But solo killing the bat is really more trouble than it’s worth.

The main problem with soloing these high-level bosses is that it takes a relatively long time to kill them; with my DT it usually took me a little over 6 minutes to kill a level 80 boss in the Ghanatan area in the southwest of the playfield. (Killing the level 75 Hong Gildong bosses took me about the same amount of time — although they have a bit less HP, they do magical damage and I don’t have much protection gear, so I had to kite them more.) If I grouped up with a healer, even if it was a crappy healbot that didn’t do much damage by himself, the same level 80 boss could be killed in little more than 2 minutes, simply because I didn’t have to kite and doubletap.

Another problem with farming these bosses by yourself is that many of them are surrounded by trash mobs, e.g. the ones in tents in the bat camp area. With a group of 2 people, this is no problem — one of them pulls mobs from the tent and runs away; the other player aggroes the boss from this group, moves him out of the way and starts fighting him; the first player keeps running until the other mobs reset, whereupon he can join the first player in fighting the boss. If you are alone, separating the boss from the trash mobs is a lot more difficult because by the time the trash reset, chances are very good that the boss resets as well. You have to try hitting him or irritating him so as to keep him aggroed a bit longer than the trash mobs. So generally, when I was farming alone, I limited myself to bosses that can be pulled without aggroing any trash: the two level 80 ones in the first group of camps in the Ghanatan area, and two level 75 Hong Gildongs near the city of Kheshatta (one near Onyx Chambers and one northeast of the city).

I feared it would be very difficult to find someone to team up with for farming epic Kheshatta bosses, since hardly anybody goes there nowadays. However, it turned out to be easier than I expected; I could often get a healer by advertising in global. Of course, if you wanted to find a full group for this, it would probably be next to impossible. Fortunately the way these bosses are balanced now makes it easy enough to fight them with 2 people and there’s really no need for a complete group. It even turned out that you don’t really *need* a healer for this; on a few occasions I teamed up with an assassin and it went OK as well — between the two of us we could keep the bosses CCed so much that my potions and self-heals were able to outheal the damage I was taking.

(It was interesting to see how thoroughly unfamiliar many people are with epic Kheshatta farming nowadays. For example, the concept of pulling the trash so that I could then separate the boss from the rest was new to many people, and some of them had remarkably many problems implementing it. Yes, when I say “pull the mobs and run away until they reset”, I mean that you should sprint the hell away from there, not take two timid steps backwards and wait until the mobs maul you! And when I say “pull any mob from inside the tent”, I don’t mean that you should walk into the tent with your 4k HP and your green light armor and expect to be able to run out of there alive while five epic mobs are whacking at you… It’s as if some people didn’t know about ranged attacks, and remarkably many healers seem to lack a bow or crossbow. Grrrr.)

Epic Kheshatta in the olden days

Farming epic Kheshatta felt very nostalgic and brought back many memories from the early months of the game. I suppose that nowadays much of the game’s population consists of people who weren’t playing in 2008 or early 2009 and thus don’t even see what the fuss is about — why the heck would anyone get the idea of going to epic Kheshatta in the first place? And yet in those early months of the game, the level 80 blue armor sets you could farm there were for all practical purposes the best armor you could get. They were practically just as good, in terms of stats, as the dungeon sets from places like Atzel’s Fortress and Onyx Chambers; and the only armor better than that was purple gear from raids, which few people had access to in the early months of the game.

Many people would join with a character below level 80 and get some XP from the boss kills, which was particularly valuable in those days when the amount of solo content was insufficient to comfortably level you through the 70–80 range (unless you wanted to grind villas a lot). I spent so much time in epic Kheshatta farming groups with my barbarian that she got the entire level 80 Ravager’s set before even reaching level 80 🙂

Another good reason for going to epic Kheshatta was money. In the early days of the game, bosses in epic playfields dropped a lot more money than they do now. If a group of 6 players killed a level 80 boss in the epic instance, it could easily drop 15–30 silver for each of them. You could get a similar amount by solo killing a level 50-ish boss, for example in epic Fields of the Dead. And I heard of people who would go at level 40 to farm level 20-ish bosses in epic Wild Lands of Zelata to get enough money for their horse. I wasn’t really aware of the money-making potential of epic bosses until relatively late; when I did, I spent a few hours farming epic Fields of the Dead and got the impression that the money from it was quite a bit better than from gathering materials for sale. But I was hampered by the fact that my character was a juggernaut guardian, her DPS was therefore complete crap and killing those bosses was fairly slow.

In any case, some time in the summer of 2008 Funcom then nerfed the money drops from bosses in epic instances, to approximately 1/4 or 1/5 of their previous values. This made farming epic bosses (killing level 50-ish bosses alone, or level 80-ish bosses in a 6-player group) quite unattractive from a money-making point of view. However, nowadays, since you can farm epic level 80 bosses just fine as a group of 2 players instead of 6, the money aspect is non-negligible again. In terms of gold per hour, I suspect it’s a viable alternative to doing MoA farming runs in Kara Korum. I still wouldn’t really want to try farming substantial amounts of gold this way, though.

In my experience, epic Kheshatta farming declined significantly from early 2009 onwards, probably as people got more accustomed to doing dungeons and raids, but what really put the last nail in its coffin were the itemization changes in 1.05. Previously, epic Kheshatta gear was, in terms of stats, almost as good as the best blue dungeon gear; after 1.05, blue dungeon gear was still decent but epic Kheshatta gear turned into utter crap. So epic Kheshatta farming was quickly turning into a very time-consuming way of farming very unattractive gear, and it’s no surprise that people gave it up.

I couldn’t help noticing how much easier epic Kheshatta is nowadays than it was in those early days of the game. Back in mid-2008, it was a perfectly respectable challenge for a full group of 6 people. I guess this is partly because we have better gear and AA now, partly because the bosses probably got nerfed during the various rebalancing changes, especially following 1.05; but partly it might be because we use crowd control abilities so much more. At least I do — back in those early days, I hardly ever used CC at all. Maybe PvPers and people who played squishy classes were more used to doing CC, but I really only started paying attention to CC after the expansion was released — there the solo mobs were hitting hard enough that CCing them really made a significant difference.

Another thing that used to make epic Kheshatta farming harder than it’s now is that mobs sometimes failed to reset properly. For example, one player would pull mobs from a tent and kite them until they reset; meanwhile someone else would extract the boss from this group and start fighting him; and the trash mobs that were supposed to reset and go back to their tent would sometimes go to the players that were fighting the boss instead. Sometimes this would happen even if they took care not to hit those other mobs with splash damage as they were running past the players, etc. These problems seem to have been fixed, and I haven’t noticed any difficulties with the mobs resetting nowadays. Indeed sometimes they are more eager to reset than I would have liked — for example, bosses would sometimes reset while I was kiting them (in what I thought was a reasonably small circle) to regain health.


The sets are shown in alphabetical order, and the class with which each set is traditionally associated is shown in parentheses after the set name. I included a few comments about other similar items below some of the screenshots.

Baleful (Dark Templar)

In typical Funcom style, the relationship between the weight of the armor and its appearance is tenuous at best. This supposedly heavy armor set looks fairly similar to (and just as leathery as) the light armor Zephyrous set (level 40–69 ToS set).

Beastfury (Bear Shaman)

This is very similar to the Wildsoul set (bear shaman level 40–69 set); one notable difference is that Beastfury Helm has horns, which the Wildsoul Helm lacks.

Crow Feather (Ranger)

Crow Feather Jerkin and Tasset use the same model as Canach Scales and Breeches (barb light armor from the Iron Tower). Crow Feather Helm uses the same model as Coronal Helm (priest light helmet from Sanctum).

Empyreal (Tempest of Set)

This set is worn by many Stygian soldier NPCs. (The higher ranks seem to prefer Skyshear, though — the ToS level 80 dungeon set.) The level 40–69 PoM set, Beatific, is in a similar style.

I can’t help thinking that this curious combination of the flat-topped hat with the slutty chest is unlikely to get you taken seriously anywhere outside of a Railroad Conductors Gone Wild audition.

Midnight (Assassin)

Many of these items are similar to ones that drop in mid-level dungeons or as quest rewards in Tarantia Common District; see one of my old posts for more about those items. But they aren’t exactly the same; the Midnight set is more dark blue instead of really black.

Nadiral (Demonologist)

Ravager’s (Barbarian)

Ravager’s Harness (the chest part of this set) uses the same model as the Royal Hunting Jerkin, a level 76 green chest piece (barbarian light armor). I’m not sure where exactly I got the latter one; I suspect it’s some sort of quest reward. Ravager’s Helm uses the same model as Corybantic Helm (from the level 40–69 barb set), Spellguard Helm (level 80 blue BoP PoM/ToS helm, but I have no idea where one gets it), and Bossonian Bascinet (ranger helmet from Awar in Atzel’s Fortress).

In the ‘front’ picture, the character’s hair appears to be clipping very badly through the helmet.

Resolute (Guardian)

A disappointingly nondescript set, in which you’ll look little better than in green crafted full-plate armor :S Resolute Chestguard uses the same model as Deadmercy Chestguard (rare blue drop from the Trapped Ghost, Slaughterhouse Cellar) and Bloodsire Harness (heavy chest from Sanctum). Resolute Helm uses the same model as Bloodsire Helm (heavy armor, Sanctum) and Rookwarden Helm (from Jovus in the Crows’ Nest). Resolute Boots and Tasset use the same model as Watchman’s Boots and Tasset (guardian level 40–59 armor).

Resplendent (Priest of Mitra)

The chest uses the same model as the Sanctified Jerkin (a level 16 light armor chest, IIRC one of the quest rewards in Tortage night quests) and the Dustcovered Jerkin (green chest from the Trapped Ghost, Slaughterhouse Cellar), and IIRC there’s also an early 70-ish quest reward based on the same model. But the main feature of this set is surely the sexy blue miniskirt. There’s at least one other item based on the same model, Elegiast’s Britches, which is a low-level quest reward, but I don’t remember exactly from where. (Damn, how I wish that was still available!)

Vanquisher’s (Conqueror)

The helmet has an interesting crest which looks better from the side:

This set is very similar to Zingaran Doomed Path set (ranger culture armor), though not exactly the same. Vanquisher’s Tasset uses the same model as Mordec Greaves (which drop from Rorik in Atzel’s Fortress).

Voidseeker (Necromancer)

The Voidseeker Robe seems to have worse clipping problems with boots and thighs than other robes of a similar style.

Wildfire (Herald of Xotli)

I had to remove the boots for the back picture because they clipped too badly with the robe. Apart from that, the boots themselves are pretty nice and I’m not aware of any other boots that would look exactly the same. So they deserve a separate screenshot:

What else is in the loot tables?

In addition to the armor from these twelve sets, the loot tables of bosses in epic playfields include various bind-on-equip items (not just armor but also weapons, shields, etc.). In one of the recent posts, we saw the list of twenty such items that drop from level 50 epic bosses; here’s a similar list for level 80 and 82 bosses:

Name Level Description Seen it drop at level
80 82

80 guardian polearm x x
Bloodrill Pants

80 demo/necro pants x
Bloodspittle Girdle

80 BS belt x

80 melee 1HB x
Celeritous Slippers

80 cloth feet with str x
Dawnmist Mantle

80 melee cloak x
Deathscream Arrows

80 arrows x x
Deathvigil Tunic

80 sin chest x
Diamondmind Opus

80 magic tali x

80 2HE, str, con x
Earthlink Boots

80 BS feet x

80 wis tali x x
Hollowheart Tunic

80 BS/ranger chest x
Howler Hide Cloak

80 melee cloak x x
Impskin Tunic

80 demo/necro chest x
Malignant Crusher

80 2HB, str, con x
Mark of Hate

80 DT tali x x

80 magic staff x

79 magic staff x
Nightwhisper Crossbow

80 xbow x
Ornvyth’s Bane

80 1HE, str, con x
Razorspike Chestguard

80 plate chest x
Scarabshell Tasset

80 BS legs x
Shadowslick Treads

80 sin feet x

80 sin dagger x
Skysplendor Shafts

80 bolts x

80 demo/necro staff x

80 soldier shield x x
Steelbrawn Sabatons

80 plate feet x
Swiftsinew Shoes

80 barb feet x x
Thornwarden Hauberk

78 heavy chest x
Thundernorth Girdle

80 heavy belt x

IME the epic open-world bosses generally drop items that are at most 2 levels above or below their own level; but unless there is some mistake in my notes, I’ve seen the level 79 Mindspire drop from level 82 bosses. I haven’t seen any level 77 items from level 80 bosses, though.

Of course it’s quite possible that some items are still missing from this list, if they never happened to drop for me during the time I was farming epic Kheshatta bosses. And it could be the case that some of the items that I’ve only seen dropping from level 82 bosses can also drop from level 80 bosses (or vice versa).

Some of these items also exist in a few lower-level versions, which drop from bosses of a suitably lower level (e.g. Howler Hide Cloak, Mark of Hate, Soulfeast).

A bit of statistics

In my post about the level 40–69 sets, we saw how to compute the probability that you’ll need to kill a certain number of bosses to get all the r items you want, assuming that the bosses have a loot table consisting of n items (and each item is equally likely to drop). So if you want to read about how to compute this, just click the link to that old post and read it there.

In the charts included in that post we used n = 96 as the size of the loot table, i.e. we pretended that it contains just the twelve armor sets and none of the other bind-on-equip items. But now we’ve seen that the number of these additional BoE items seems to be usually around 20, so we can repeat our analysis with n = 116 instead of 96. As for r, the number of items that we want to get, there are various interesting values to choose: r = 1 (if you’re missing just one part to complete your set — a familiar experience for many farmers like myself!), r = 4 (if you want to farm the BoP parts of one set, and will buy any missing BoE parts on the trader), r = 8 (if you want to farm one whole set), r = 20 (if you want to find all the items that aren’t part of any of the sets), r = 48 (if you want to farm the BoP parts of all 12 sets), and r = 96 (if you want to farm all parts of all 12 sets).


(all the results are for a loot table of n = 116 items)
r (no. of items we want) Avg. (± std. dev.) number of kills needed No. of kills needed to have a certain probability of getting all r items you want
10% 50% 90%
1 116.0 ± 115.5 13 81 266
4 241.7 ± 137.5 97 213 422
8 315.2 ± 142.2 162 289 502
20 417.3 ± 145.1 259 392 607
48 517.2 ± 146.0 358 492 708
96 597.0 ± 146.3 437 572 788
116 618.9 ± 146.3 459 594 810

By the way, in the special case where r = n, the Wikipedia refers to our problem as the “coupon collector’s problem”. One of the interesting results in its Wikipedia article is that the average number of kills needed in this case is n · Hn, where Hn is the n-th harmonic number. In our case, for n = 116, it turns out that H166 is approx. 5.335, and multiplying this by 116 really gives us approximately 618.9, which matches our result from the above table.

An encouraging conclusion

As you can see, the amount of farming required is considerable even at small values of r. In a way, this is discouraging — even if we limit our interest to just 8 out of 116 items, we still have to farm 50% as much as if we wanted to get all 116 items!

But seen in another way, it’s actually encouraging. Nobody would regard the goal of farming one set as unreasonable — after all, that’s exactly why we were doing all that farming in epic Kheshatta back in 2008 and early 2009. I farmed the complete Resolute set for my guardian back then, as well as the complete Ravager’s set for my barbarian. Now, a few weeks ago, when someone saw me looking in global for a healer for epic Kheshatta farming, they asked me why I wanted to farm it, and seemed to be impressed by my goal of farming all 12 sets. Perhaps they thought me somewhat crazy for even attempting something like that.

A statistically very naive person might imagine that farming for 12 sets will take 12 times as long as farming for one set; a moment’s thought would convince him that this is not the case, as farming for one set will likely also give you a few pieces of various other sets, and so on; so farming for 12 sets will take less than 12 times as long as farming for just one set. But how much less exactly? Our intuition isn’t of terribly much help with things like this. But now we can see from the above table — just compare the rows for r = 8 and for r = 96 — that farming for all twelve sets takes less than twice as much effort as farming for just one set!

So there you have it. If you wouldn’t think a player crazy for farming one set, it would hardly be decent to think them crazy for farming all 12 sets either 😛

Money drops from mobs

September 12, 2012 5 comments

Many mobs drop some money in their loot bags. Usually these are pretty small amounts, but bosses in epic instances of playfields can drop a substantial amount. In fact, in the early months of the game, farming bosses in epic playfields was the major source of income for some players. If you killed a level 80 boss in epic Kheshatta with a group of 6 players, you could expect around 15 to 30 silver per player; and you would get a similar amount if you killed a level 50 boss in epic Field of the Dead alone. I even heard of level 40 players farming level 20-ish bosses in epic Wild Lands of Zelata for money with which they would buy their first horse.

A few months after release, Funcom nerfed the money drops from epic bosses by a factor of 5 or so — the same kills that would formerly net you 15–30 silver will now give you just 3–6 or so. Since then, farming epic bosses for money doesn’t make much sense. However, recently, as I was farming epic Kheshatta bosses for the sake of vanity gear (screenshots of which will be published in a forthcoming post as soon as I manage to transfer a few characters to testlive so I can take the screenshots there), I noticed that if you kill these bosses alone or teamed up with just one other player, the amount of money you get is still pretty decent. I probably made more than 50 gold while farming for my 12 epic Kheshatta armor sets.

So I became interested in the amount of money dropped by the bosses. Obviously this is a random number; but what can we say about its distribution? I started writing down the amount of money for each boss kill and ended up with the following statistics (all money amounts are in silver):

of kills
Std. dev. Minimum Maximum
75 8 22.38 7.77 11.09 30.53
80 96 26.45 7.73 13.17 38.72
81 7 28.25 9.37 14.10 37.71
82 80 23.42 7.38 12.63 38.70

Now, obviously we don’t have enough data for level 75 and 81 to say anything reliable there. And even for level 80 and 82, where we have a decent amount of data, we see that surprisingly the average drop from level 82 bosses is smaller than from level 80 bosses — this is surely a sign that we still have insufficient data to estimate the averages reliably enough. Still, we have enough data to plot a rough histogram. We divided the range from min to max into 10 smaller ranges (of equal width); the histogram shows, for each subrange, how many kills gave us an amount of money that falls into that particular subrange:

I was expecting to see something bell-shaped, like a normal distribution — but we see that this is not the case. The values around the average don’t really seem any more likely than those closer to the extremes. And the extremes themselves, the maximum and minimum, are suspiciously close to a ratio of 3:1 — a nice round number which we would hardly dare to expect from e.g. a normal distribution, but which wouldn’t be surprising if it’s a uniform distribution and the developers deliberately chose its range so that max = 3 × min.

Farming level 50 bosses

Now, it’s a pain in the ass to get more data for the above table, since killing level 80 epic bosses takes a lot of time; and I don’t intend to kill any more of them now that I’m done collecting my epic Kheshatta armor sets. So I started killing level 50 bosses in epic Field of the Dead instead, hoping that the distribution of money drops there is the same (just with smaller values, of course). Here are the results after 500 boss kills:

of kills
Std. dev. Minimum Maximum
50 500 5.14 1.48 2.56 7.67

These are just the sort of things that you would expect from a uniform distribution. A uniform distribution on the range [a, b] will have the average μ = (a + b)/2 and the standard deviation σ = (ba) / sqrt(12). So if we take a = min = 2.56, b = max = 7.68, we see that the average should be 5.12 and the standard deviation should be 1.478, which is very close to what we see in the table above. And we can also see that the maximum is roughly three times the minimum, b = approx. 3a, similar to what we already saw above in the case of level 80 bosses.

The histogram also shows how close we are to a uniform distribution:

Of course there’s some variation from column to column, but that’s only to be expected.

So, at this point we can speculate that the amount of money dropped by an epic boss of a given level is distributed uniformly in the range [a, 3a], where a depends on the level. It would be interesting to kill a few bosses of other levels to get a better idea of how a depends on the level (my guess is that it’s an exponential function of the level), but that would be pretty time-consuming as it’s hard to find a lot of bosses at the same level. Level 50 is a bit of an exception there, because you have the Fields of the Dead with 13 level 50 bosses in a small area; for most other levels you’re lucky if you can find one or two bosses at that level.

Incidentally, I managed to kill 120 level 50 bosses in epic Fields of the Dead in 90 minutes of farming (I think with a bit of care you could do it faster still). At an average of 5.14 silver per kill, this gives you an income of 4.11 gold per hour — not bad at all compared to things like doing quests in Kara Korum or farming resources for sale (at current prices on the Crom server at least). But it does assume that you’re the only player farming there; and I saw other people killing those bosses more often than I had expected.

By the way, while I was killing the level 50 bosses, I also wrote down which blue items dropped that weren’t part of any of the level 40–69 world-drop sets. I ended up noticing the following 20 items (they’re all bind-on-equip):

  • Battlebrawn Necklet
  • Battlespite
  • Backbreaker
  • Blacksever
  • Bladebrave Tunic
  • Bloodblight Bolts
  • Bloodrighteous Belt
  • Bloodpurge Boots
  • Compendium of Many Hurts
  • Corpseskewer
  • Howler Hide Cloak
  • Mark of Atrocity
  • Mark of Hate
  • Mindward Robe
  • Painthreaded Leggings
  • Sightshift Mantle
  • Sleekspeed Leggings
  • Soulfeast
  • Sparkfrost
  • Vigorspine Tunic

The last time I added a new (previously unseen) item was after approx. 380 boss kills. So it’s entirely possible that there might be a few more such items dropping there that I haven’t seen even after 500 boss kills.

Anyway, the number of such out-of-set items in the bosses’ loot table is potentially interesting if you want to estimate how long you’ll have to farm them for. In my post about the level 40–69 sets most of the calculations simply pretended that the loot table contains just the 12 × 8 = 96 items from the sets and nothing else; now we see that a more reasonable size of the loot table would be around 116, not just 96.

More about the uniform distribution

Suppose that we have a sample of n points, X1, …, Xn, taken independently from a uniform distribution in the range [a, b]. The mean of such a distribution is μ = (a + b)/ 2 and its variance is σ2 = d2 / 12, where d = ba. (You can easily derive these formulas for μ and σ2 by yourself from the definitions of mean and variance, or you can look them up in the Wikipedia.) In practice we don’t really know any of a, b, μ and σ — we just know the sampled values X1, …, Xn, and the interesting question is how to find out something about a, b, μ and σ from our sample.

We can start by computing things like:

  • Sample average: Xa = (1/n) Σ1 ≤ in Xi.
  • Sample variance: S2 = (1/(n − 1)) Σ1 ≤ in (XiXa)2.
  • Sample maximum: Y = max {X1, …, Xn}.
  • Sample minimum: W = min {X1, …, Xn}.

Since our X1, …, Xn are random variables, the above-listed sample statistics Xa, S2, Y and W are random variables as well. What can we say about their distributions?

As is well known, the sample average is distributed approximately according to the normal distribution with the expected value μ and variance σ2 / n. Since its expected value is μ, this means that the sample average is an unbiased estimator of μ. The standard deviation (square root of the variance) tells us that on average, the sample average will deviate from the correct value of μ by √(σ2 / n) = σ / √n.

How much is that in practice? In the case of our 500 level 50 boss kills, this standard deviation amounts to approx. 6.6 copper; for the 96 level 80 boss kills, the standard deviation was approx. 79 copper.

If we move on to the sample variance, S2, it is similarly well known that its expected value is σ2; in other words, it is an unbiased estimator of the population variance. (That’s why we had to divide by n − 1 when computing S2; if we had divided by n instead, which might seem to be intuitively more reasonable at first sight, we would have obtained a biased estimator that would slightly overestimate the population variance.)

Next let’s take a look at the sample maximum, Y. We can see that its cumulative probability function will be

P(Y < y) = P(X1 < y, …, Xn < y)
= P(X1 < y, …, Xn < y)
= P(X1 < y) · … · P(Xn < y)
= [(ya) / d]n.

We now have to take its derivative with respect to y to obtain the probability density function:

pY(y) = n (ya)n − 1 / dn.

Now we’re in a good position to calculate the expected value of Y:

E[Y] = ∫ab y pY(y) dy
= … = bd / (n + 1).

You can see right away that this is a little less than b, the true maximum of our distribution; but the discrepancy gets smaller if we take a bigger sample (i.e. a bigger n). And intuitively it certainly makes sense that the sample maximum, Y, underestimates the population maximum b, because sometimes Y is below b (if all your samples Xi are below b), but this doesn’t get counterbalanced by cases when Y would be above b (because this is impossible — none of the Xi can be greater than b, so Y can’t be either). So we see that using the sample maximum Y, to estimate the population maximum b will actually cause us to underestimate it. Soon we’ll see how to get a better estimator.

Finally we have the sample minimum, W. Here we can use a very analogous way of thinking as we did for Y above and we would end up with the following result:

E[W] = a + d / (n + 1).

In other words, on average, W overestimates the population minimum a by just as much as Y underestimates the population maximum b.

Estimating a and b from W and Y

Now, as is often the case in statistics, we can try to derive estimators of a and b by pretending that our Y and W happened to achieve exactly their expected value and then solving the resulting equations for a and b. In other words, we have the pair of equations

Y = bd / (n + 1)
Z = a + d / (n + 1).

Remembering that d = ba and solving these equations for a and b gives us the following estimators:

A = W − (YW) / (n − 1)
B = Y + (YW) / (n − 1).

Again, intuitively this makes a lot of sense; we know that Y underestimates b, so to get a better estimate of b we must add something to Y (and similarly we must subtract something from W). We know that Y was a biased estimator for b because on average it underestimated it; but what about B, is it still biased or do we finally have an unbiased estimator now? To find out, we have to compute its expected value:

E[B] = E[Y + (YW) / (n − 1)]
= E[Y n / (n + 1) − W / (n − 1)]
= E[Y] n / (n + 1) − E[W] / (n − 1)
= … = b.

So we see that B is an unbiased estimator of b, and we could similarly show that A is an unbiased estimator of a.

Estimating μ from W and Y

Now, remember that the mean of a uniform distribution is μ = (a + b) / 2. Since A is an unbiased estimator of a, and B is an unbiased estimator of b, it follows that (A + B) / 2 is an unbiased estimator of (a + b) / 2, i.e. of μ. Let’s call it M:

M = (A + B) / 2 = (W + Y) / 2.

This is the second unbiased estimator of μ we’ve seen today — the first one was the plain old sample average, Xa. Which of these two estimators is better? The fact that they are unbiased really just means that their mistakes in both directions tend to balance each other out on average (i.e. they err just as much by overestimating as they do by underestimating); this doesn’t yet tell us anything about how big these errors are, on average. For that we need to compute their variance. We already saw earlier that the variance of Xa is D[Xa] = σ2 / n = d2 / (12 n). But what about M?

The variance of M is a bit more tricky to compute. We saw that M can be computed as W + Y, but these two variables are obviously not independent of each other (e.g. because we know that WY), so we can’t say that D[M] = D[W] + D[Y]. Instead, we can start by computing the joint probability distribution of W and Y. The cumulative probability function is

P(Y < y, W < w)
= P(Y < y) − P(Y < y, Ww)
= P(Xi < y for all i) − P(wXi < y for all i)
= [(ya) / d]n − [(yw) / d]n.

To obtain the probability density function, we need to take the partial derivative with respect to y and w:

p(y, w) = ∂2 P(Y < y, W < w) / (∂yw)
= … = n (n − 1) (yw)n − 2 / dn.

(All of this of course only makes sense when w < y. If w is greater than y, the probability P(Y < y, Ww) is 0, as the sample minimum cannot be greater than the sample maximum. In that case P(Y < y, W < w) = P(Y < y) and since it no longer depends on w, its partial derivative with respect to w will be 0, so that p(y, w) will also be 0 there.)

Now we’re in a good position to start computing the variance of M:

D[M] = E[(ME[M])2]
= E[((W + Y) / 2 − (a + b) / 2)2]
= (1/4) E[(W + Yab)2]
= (1/4) ∫∫awyb (w + yab)2 p(y, w) dy dw
= n (n − 1) / (4 dn) ∫∫awyb (w + yab)2 (yw)n − 2 dy dw.

This looks a bit hairy, but it becomes a lot easier when we substitute the variables a bit. Let’s introduce u = yw and v = w + yab. Our integral turns into

D[M] = n (n − 1) / (4 dn) ∫0d duuddu dv (1/2) v2 un − 2
= … = d2 / [2 (n + 1) (n + 2)].

Remember that our other estimator of μ, namely Xa, had a variance of D[Xa] = d2 / (12 n). So we can see right away that M has a smaller variance by approximately a factor of n / 6. Or in other words, if we use M instead of Xa to estimate μ, our errors will be on average √(n / 6)-times smaller. Or in still other words, if we want to make sure that our average error will be sufficiently small, we need a much smaller sample size (i.e. n) if we use M than if we use Xa.

Estimating a and b from Xa and S2

We saw earlier that for a uniform distribution, the mean is μ = (a + b) / 2 and the variance is σ2 = (ba)2 / 12. If we solve this pair of equations for a and b, we see that a = μ − σ √3 and b = μ + σ √3. Since we know that e.g. Xa is an unbiased estimator for μ, and S2 is an unbiased estimator for σ2, we can plug them into these formulas for a and b to get another pair of estimators for a and b:

A’ = XaS √3
B’ = Xa + S √3.

Are the new estimators unbiased? Let’s try:

E[B’] = E[Xa + S √3]
= E[Xa] + √3 E[S]
= μ + √3 σ = b,

and similarly we could see that E[A’] = a. So these new estimators are also unbiased, just like our previous ones (A and B) were.

An interesting question would be which pair has the smaller variance; but when I tried to compute the variance of A and B, I couldn’t get any elegant results, it all quickly turned into a horrible mess.

However, there is one other argument against using A’ and B’: it’s perfectly possible that A’ might turn out to be greater than W, and B’ can be less than Y. And if that happens, you’ll feel a bit silly having estimated the population maximum with a number that is smaller than your sample maximum. This cannot happen with A and B.

Wrapping up

Now that we know a bit more about uniform distributions, we can go back to our experimental data:

Statistic Boss level
50 75 80 81 82
n (sample size) 500 8 96 7 80
W (sample min) 2.56 11.09 13.17 14.10 12.63
Y (sample max) 7.67 30.53 38.72 37.71 38.70
Xa (sample avg.; estimator of population avg.) 5.14 22.38 26.45 28.25 23.42
D[Xa]* 0.066 2.74 0.79 3.54 0.82
M (better estimator of population avg.) 5.11 20.81 25.95 25.91 25.67
D[M]* 0.0072 1.86 0.19 2.62 0.23
A (estimated population min) 2.55 8.31 12.90 10.17 12.30
B (estimated population max) 7.68 33.31 38.99 41.65 39.03

*Note: actually, the formulas for D[Xa] and D[M] require d, which is ba; we of course don’t know the true values of any of these parameters, so we used BA instead.

The standard deviations √D of Xa and M are useful as they give us an idea of how (un)reliable our estimates of μ are, and also how much more reliable M is than Xa. You can see that the expected error of our M-estimate of the average drop from level 50 bosses is barely 72 tin!

Level 40-69 world-drop sets

July 6, 2012 10 comments

You might remember my post from some time ago about the old world-drop armor sets. Well, I got the idea that it would be nice to have screenshots of all 12 level 40-69 sets, so I’ve lately been doing some farming in epic Fields of the Dead and eventually managed to get all items from all twelve sets. Unfortunately they take up too much inventory space, so I’ll probably delete them soonish.

Anyway, without further ado, here are the screenshots; above each screenshot, there’s the name of the set and of the class with which it has traditionally been associated.

Beatific (Priest of Mitra)

Brimstoned (Demonologist)

Corybantic (Barbarian)

Dark Ember (Herald of Xotli)

(Note: for the back picture, I had to remove the legs and shoulders as they were clipping very badly with the robe. As you can see, some clipping problems with the character’s chest remain.)

Eidolon’s (Necromancer)

Heretic’s (Dark Templar)

Pathfinder (Ranger)

Twilight (Assassin)

Vindicator (Conqueror)

Watchman’s (Guardian)

Wildsoul (Bear Shaman)

Zephyrous (Tempest of Set)

How much farming is needed?

Each of the 12 sets discussed in this post consists of 8 items, so that’s a total of 96 different items. So you’ll need to kill at least 96 bosses to get them all; but it’s likely that it will take more than that, because sometimes you’ll get the same item several times. So what can we say about the number of bosses you’ll have to kill to collect all 96 items?

This is a nice little exercise in combinatorics. Let’s say we have a loot table of n = 96 items that we’re interested in, and that every time we kill a boss, one of these items drops, and they are all equally likely. What’s the probability that we have to kill exactly k bosses to get all n items?

Killing k bosses results in a sequence of k items looted from them; since there are n possiblities for each of these items, there are a total of nk loot sequences from these k kills. They are all equally likely. Now we want to count those that contain each of the n items at least once, and where no shorter prefix of this sequence contains all n items (otherwise we could have stopped sooner than at k bosses).

So there must be some item, let’s call it x, that dropped for the first time only at the very end of the sequence; before that we have a slightly shorter sequence of k − 1 drops from among the remaining n − 1 items (i.e. all items except x; let’s call these items y1, …, yn − 1), with the additional constraint that each of these n − 1 items must occur at least once in that shorter sequence.

How many such shorter sequences are there? There are a total of (n − 1)k − 1 sequences of length k − 1 over a set of n − 1 items; from these we have to subtract those which lack y1, and also those which lack y2, and so on; and now we find that we subtracted those which lacked *both* y1 and y2 twice, instead of just once, so we have to add them back again; and so on. This is what is known as the inclusion-exclusion principle, and it leads us to the following formula:

Σ0 ≤ i < n (−1)i binom(n − 1, i) (n − 1 − i)k − 1.

This is the number of sequences of length k − 1 from a set of n − 1 items that contain each of these items at least once. Now, at the beginning we started with item x, which we could have chosen in any of n ways, so we have to multiply our sum by n; and then, to get a probability, we have to divide it by nk, which is the total number of sequences of length k from our set of n items.

Now, you might remember that half of the items we’re talking about here are actually bind-on-equip instead of bind-on-pickup, so strictly speaking you don’t have to farm them by yourself. You could just focus on farming the bind-on-pickup parts and once you have all those, you can then buy the missing bind-on-equip parts on the tradepost (assuming that some other player is selling them). How does that affect our calculations regarding the number of boss kills we need?

Let’s say that the loot table still has n items, like before, but we’re only interested in r specific items among those n items. A very similar line of reasoning like before now leads to the following formula:

Σ0 ≤ i < r (−1)i binom(r − 1, i) (n − 1 − i)k − 1.

And similarly as above, we have to multiply this sum by r and then divide it by nk to get the probability we’re looking for.

Note that our initial formula is just a special case of this second one, and you can get back to the initial formula by setting r = n.

I’m not sure if our formulas can be simplified still further (by replacing the sums with something more elegant), but in any case for the purposes of calculating these probabilities our current formula is enough. The following chart shows the resulting probability distribution at n = 96 and for r = 96 (if we want to farm all items) and r = 48 (if we want to farm just the bind-on-pickup items and will then buy whatever bind-on-equip items we’ll be missing by then):

Now that we can calculate the probability of each k, we can also calculate the expected value of k (i.e. the weighted average of all possible values of k, in which each k is weighted by its probability). In our case this turns out to be approx. 494 for r = 96, and 428 for r = 48. In other words, on average you’ll have to kill 494 bosses to get all 96 items, or 428 bosses to get just the 48 bind-on-pickup items; but you can of course be lucky and get them sooner, or you can be unlucky and require a lot more than 494 boss kills. (It turns out that if you kill 876 bosses, the probability that you’ll have obtained all n = 96 items by then exceeds 99%.) So from this point of view, the idea of focusing on farming just the bind-on-pickup parts isn’t actually a huge improvement.

[If you look at the above-mentioned formulas for P(k) more carefully, you can see that they are really a combination of r different geometric distributions. This also allows us to derive a nice direct formula for the expected value of k, which turns out to be:

E[k] = r n Σ0 ≤ i < r (−1)i binom(r − 1, i) / (i + 1)2.

Thus at r = 1, where we’re interested in just one item among n possible drops, the expected value of k turns out to be, unsurprisingly, simply n. At r = 2, it turns out that the expected value of k is 3n/2.]

In practice, of course, we’ll have to kill a bit more bosses than our calculations so far have shown, because their loot tables also contain some items that we aren’t interested in (e.g. weapons, or armor pieces that aren’t part of sets). We could in fact take this into account with our second formula by setting n to some value greater than 96 (to include the “uninteresting” items from the loot tables) and then keeping r at either 96 or 48, as before; but I haven’t tried to do so as I don’t have any clear idea of how many such uninteresting items there are.

I haven’t tried to keep track of how many bosses I actually had to kill to get my 96 items, but my impression is that it was probably less than 494, so I think I was relatively lucky there. And in fact the last item I got was a chestpiece, so the plan to focus on farming the bind-on-pickup parts (and buying any missing bind-on-equip parts at the end) wouldn’t have done me any good in this concrete case, because by the time I got my last bind-on-pickup part, I already had all the bind-on-equip parts as well.

[P.S. Another reason why I found this calculation interesting is because I was wondering if I should try collecting the level 70-80 sets as well. There, killing each boss takes quite a bit more time (unless you’re doing it with a group), so I wanted to get a sense of how many kills would be required to complete all the sets. I don’t think I want to try solo killing level 70-80 group bosses 494 times. :P]

So far all our calculations have been done for n = 96; but it is also interesting to ask what happens if you want to collect all items from a loot table of some other size. For example, here are distributions for n = 5 and 10. At n = 5, we’re dealing with sufficiently small values of k that it’s still easy to see the discrete nature of the distribution, so we’ve shown it as a bar chart.

Another very interesting question is: how does k (the required number of boss kills) depend on n (the size of the loot table)? The following chart shows some answers to this question. It shows the expected value of k (as a function of n), as well as the number of kills required if you want to have a certain chance (25%, 50%, 75%, 90%, 99%) of discovering all n items.

Or, if we zoom in a little on the smaller n‘s:

The Statistical Farmer

June 30, 2012 4 comments

I haven’t been doing very much resource gathering lately, but I used to do a fair bit of it in the past; to alleviate the boredom somewhat, I ended up collecting various statistics about it. They might be useful for other people as well, so I figured I’d post them here.

(A small disclaimer first: most of the statistics in this post are based on gathering done in Poitain in 2010 and early 2011. So I don’t claim that they still apply now and/or for other zones, though on the other hand I haven’t seen any good reason to believe that things such as drop rates might be any different nowadays or that they might be different in other zones.)

I was particularly interested in estimating the probability of getting rare resources. According to an old developer post in the testlive forum, this probability is 3% for T1 resources, 2% for T2 resources and 1% for T3 resources. I’m not completely convinced if that’s really the case; some of the results I got during gathering seem pretty incompatible with this simplified scheme, and I got the impression that each resource has a different probability of getting a rare.

Another very interesting development for gatherers like me came when they introduced the Fortuitous Vittles in the 1.06 patch. This is a consumable item that becomes available from the Brewmistress in your guild city tradepost when your guild renown level is at least 12. It costs 1 gold and gives you a 1 hour buff that increases your chance of getting rare resources. This of course leads to questions like, does it actually work? How much does it increase your chance of getting rares? And is it worth paying 1 gold for?

Well, I think the best way to start answering these questions is to gather lots of resources, count how many rares you got, and thus get a rough idea of what’s the probability of getting a rare resource from this node. So, without further ado, here’s a table with my results:

Resource Without vittles With vittles
Common Rare # common # rare % rare # common # rare % rare
Gold Platinum 5469 83 1.5% 11433 260 2.2%
Copper Tin 6058 189 3.0% 10486 475 4.3%
Electrum Illustrium 4100 87 2.1% 10045 321 3.1%
Iron Aurichalcum 1700 30 1.7% 1418 59 4.0%
Basalt Adamant 5676 63 1.1%
Duskmetal Blue Iron 3504 38 1.1% 914 21 2.2%
Oak Soulwood 1900 13 0.7%

Estimating the probability of rare drops

Now, before we continue, it might be worthwhile to remark that these probabilities can be tricky to estimate, especially because they are low. Suppose that you are interested in a particular resource and that each time you click on its node, the probability of getting a rare is p (as we saw in the table above, p will usually be somewhere in the 1% to 4% range). Now suppose you click nodes of this type n times. Then the probability of getting exactly k rares is binom(n, k) pk (1 − p)nk. (This is called the binomial distribution; see e.g. the Wikipedia or Mathworld for more.)

This is shown in the following chart for n = 100 and for p = 1%, 2% and 3%.

So, let’s say we clicked nodes of this type 100 times and got 2 rares in the process. But we don’t know the real value of p yet, of course — that’s why we started digging in the first place, to estimate p! So what can we say about p now?

Certainly it’s tempting to say that p = 2/100 = 2%. In fact, that’s what the table we saw earlier in this post is based on. And indeed no other value of p will give us a greater probability of k = 2 rares from n = 100 tries than p = 2% will. But as you see from the chart above, it’s also perfectly possible (just a bit less likely) to get 2 rares from 100 tries if p is 1% or 3% (or indeed any other value). So we can’t be very sure about p just yet; maybe the real p is 1% and we got lucky, or the real p is 3% and we were a bit unlucky.

Let’s keep digging until we’ve clicked the nodes n = 1000 times. What’s the probability of getting exactly k rares now?

You can see that the distributions overlap quite a bit less now, but they still do overlap to a certain extent. We’ll never be able to be completely sure about the correct value of p, but we can be a bit more sure now than we were at n = 100. Suppose that we got k = 23 rares from these n = 1000 clicks on nodes. Now we might feel comfortable in saying that p = 1% is most likely out of the question. If p were 1%, there would be a 95% chance that we’d get somewhere from 5 to 17 rares; so we would be very unlikely to get 23 rares if p were just 1%. But what about p = 3%? Of course it’s more likely that we get 23 rares at p = 2% than at p = 3% (and at p = 2.3% getting 23 rares would be more likely than at any other p, but we might speculate that the developers have chosen relatively ‘round’ numbers for values of p); but even at p = 3%, getting 23 rares isn’t that unlikely.

One way to approach this is by thinking in terms of confidence intervals. Given an n and a p, we can look for the narrowest range of k that accounts for some given amount of probability, e.g. 95%. As we saw in the previous paragraph, at n = 1000 and p = 1%, there’s a 95% chance that k will be in the range 5…17. At p = 2%, there’s a 95% chance that k will be in the range 12…29; and at p = 3%, there’s a 95% chance that k will be in the range 20…41. We see that these ranges overlap a little, so if our k falls into an area where the ranges overlap, we might be uncomfortable with committing to any particular p. As we keep digging and n increases, these ranges gradually stop overlapping; so we could decide to keep digging until these ranges stop overlapping.

Of course we could take a percentage lower than 95%, and then these ranges would be narrower in the first place and it wouldn’t take such a large n for them to stop overlapping (but then we’d also be less sure that we have the correct value of p). And on the other hand, we could allow other values of p besides 1%, 2% and 3%; we could consider p = 1.5% and so on; each of these would again have a range of its own, so it would take an even bigger n to prevent the ranges from overlapping.

(A concrete example: in the ‘without vittles’ section of the table at the start of this post, we saw that I got 5439 gold and 83 platinum; so if we take n = 5439 + 83 = 5528 and p = 1%, it turns out that there’s a 95% chance that we’ll get somewhere from 40 to 69 rares. So if the drop rate of platinum was really just 1% (like the developer post linked to at the start says), it would be very unusual to get 83 platinum from 5528 clicks on gold nodes.)

Or we could look at it from the opposite direction. Suppose that we are given an n and a k (a result of our back-breaking digging :P); now we find that for some values of p, this k would fall into the 95% range, whereas for p‘s that are too low or too high, our current k would be either above or below that range. In other words, we might say that at some values of p, our current k is ‘likely’ (in the sense of belonging to the 95% range for that p) whereas for other p‘s it’s ‘unlikely’. The values of p for which this k is likely again form a range, and we might simply report this range instead of (or in addition to) the value of p for which our current k is the most likely (that is of course p = k / n).

In our above example with n = 1000 and k = 23, we would find that this k lies within the 95% range for any p from approx. 1.56% to approx. 3.48%. This gives us a better idea of just how wide a range of possible values of p we must admit, if we want to be really quite sure that we’re discarding only such p as would make our current k really unlikely.

Another example: at n = 10000, k = 200, the range of p‘s that we get by this method is from 1.75% to 2.30%. Thus, after digging 10000 units of a resource, we can be pretty confident that we can estimate the probability of rares to approximately half a percent.

Do Fortuitous Vittles work?

Before we answer this question, let me rant a bit. Surprisingly many people seem to approach this whole business of gathering rare resources with an annoyingly poor grasp of probability and statistics. They buy the vittles and think of it as a guarantee that they’ll be shovered with rare resources during that hour. Then you get complaints along the lines of ‘I gathered for 1 hour without vittles and got 4 rare resources, then I gathered for 1 hour with vittles and got 2 rare resources, so clearly the vittles don’t work’.

What these people don’t understand is that because the probability of getting a rare is fairly small (even with vittles), there is considerable variance in the number of rares you’ll get from one hour to another. Thus it’s perfectly possible to be lucky for one hour while gathering without vittles, and to be unlucky for one hour while gathering with vittles, and to get more rares from the first hour than from the second.

We can illustrate this with the same sort of chart that we’ve seen above. Suppose that the vittles increase the probability of a rare resource from 1% to 2%. What’s the probability of getting k rares if you have clicked the nodes of this type n = 100 times?

I cut off the chart after k = 6 because all the remaining probabilities are so low (amounting to less than 1% all together).

As you can see from this chart, going from p = 1% to p = 2% has certainly increased our chances of getting rares, but outcomes with a small number of rares are still perfectly possible even at a higher p. Suppose you try clicking nodes 100 times at p = 1%, then you take vittles and click nodes another 100 times at p = 2%. What’s the probability that you will get no more rares after taking vittles than you did before taking vittles? It turns out that this probability is depressingly high — around 31%. So you might say that there’s a 31% chance that a statistically illiterate gatherer would complain that vittles don’t work for him.

But 100 clicks isn’t very much. If you’re gathering resources of just one type, you can easily click those nodes 200 times in 1 hour. If we repeat our calculation with n = 200, it turns out that the probability of getting no more rares with vittles than without is just 16%.

At n = 1000, this probability drops to approx. 2.5%. That’s not very much, but on the other hand 1000 units of resources is quite a bit. If you gather for 1 hour here and 1 hour there, and if you gather multiple resources at the same time, it might take you a week or more to gather 1000 units of a particular resource. And 2.5% isn’t that small either — it’s 1 in 40. In other words, if you’d have lots of gatherers gathering one week without vittles and then one week with vittles, 1 in 40 of these people would find that he got no more rare resources in the second week than he did in the first! Of course he’ll go and write a forum post to complain about it. But you presumably won’t hear from the other 39 for whom the vittles evidently worked fine.

In other words, yes, you can have bad luck. You can even have long streaks of bad luck. Every now and then someone will have a long streak of bad luck, and sometimes that someone will be you. This is not a sign that something’s wrong with the drop rates or with Funcom’s pseudorandom number generators. In fact, if such streaks of bad luck didn’t happen — that would be a sign that something’s wrong. (More concretely, it would be a sign that the probability of getting a rare on any particular click is not independent of what has been happening on other clicks. It would suggest that the resource nodes have evolved some self-awareness and memory, perhaps they have even been connected into a hive-mind, and it’s only a matter of time before they will take revenge on the players that are so inconsiderately whacking at them with their pick-axes. In Soviet Hyboria, basalt gathers you!)

Finally, let’s repeat the calculation for n = 10000. That’s a lot of gathering — several months’ worth, in fact, for most people. So if you gather 10000 units without vittles and then 10000 with vittles, what’s the probability that you’ll get no more rares from the second batch than you did from the first? This turns out to be approx. 1.5 · 10−7 %, or 1 in 669 million. So, if you were gathering diligently with vittles not for a week, but for several months, and got no better results than without vittles — then you’re welcome to complain. But no sooner.

* * *

So I guess the main point of all my statistical ranting in this post is that you need a large sample — a large value of n — before you can say anything decent about drop rates and such.

(Coming Soon™: a similarly grumpy statistical argument can be made about the issue of loot drops. I’m sure that your guild, like every other, has a sob story about how some particular piece of raid gear Just. Won’t. Drop. for them. Like, you’ve been raiding since 1960, barefoot and in the snow, it was uphill and against the wind both ways, and that particular piece of gear dropped just once (and the guy that won it has long since stopped playing, of course). And you and your guildies occasionally bitch about how Funcom’s loot tables are bugged and their random generators are unfair. Right? Right. Every guild has such an item. For me it was T2 guardian wrists. But you know what? This is perfectly normal. In fact it would be strange if you didn’t have any such mysteriously rare item.)

But really, do Fortuitous Vittles work?

Anyway, based on the results in the table at the start of this post, we can pretty confidently say that the Fortuitous Vittles do indeed increase the probability of getting a rare item. So in that sense they do work.

That leaves us with the question of whether they are worth the cost. This of course depends on such things as which resources you’re gathering, how efficiently, and what’s their price on the tradepost. (And of course you might not want to base your decisions purely on economic considerations. I know some people who, if they need material X, by golly they will dig material X, even if they could save time by digging the more valuable material Y, selling it, and spending the money to buy X on the tradepost. I never quite understood this point of view but I guess it means a lot to them to be using materials that they have gathered by themselves.)

I don’t think I can do much more than present a sample calculation, but in the end you’ll have to check the prices on your own server to decide which materials are worth gathering and whether you should be taking vittles or not.

Let’s assume that you start in an instance of Poitain with full nodes, that no other gatherers are interfering with you, and that you focus on gold, electrum and copper nodes. In my experience, this means you can click approx. 90 times on each type of node in 1 hour.

Thus, based on the drop rates from the table at the start of this post, we will gather, on average, the following:

  • without vittles: 87.3 copper, 2.7 tin, 88.1 electrum, 1.9 illustrium, 88.7 gold, 1.3 platinum;
  • with vittles: 86.1 copper, 3.9 tin, 87.2 electrum, 2.8 illustrium, 88.0 gold, 2.0 platinum.

How much is this stuff worth? Let’s take the current prices on the Crom server as an example: 100 copper = 16s; 100 electrum = 15s; 100 gold = 70s; 1 tin = 60s; 1 illustrium = 70s; 1 platinum = 82s.

Thus, what you gather without vittles in 1 hour is worth approx. 494s; what you gather with vittles is worth approx. 678s. If we subtract the 5% trader fee (which you’ll have to pay if you sell your resources on the tradepost), and if we furthermore subtract 1g in the case of vittles, we see that your hourly income was 469s without vittles, and 544s with vittles.

So from that point of view, the vittles are still worth using, but just barely; they were a lot more attractive in days when resources were selling at higher prices than now.

(Of course, another thing to consider when deciding whether to use the vittles is that, once you have taken them, you really should try to gather as efficiently as possible for the next 1 hour until the vittles run out, otherwise you’ll have wasted some of your money. Whether this is a good or a bad thing is, I guess, a matter of perspective. You might see it as a helpful encouragement to concentrate on gathering, or as an unwelcome constraint that prevents you from jumping into dungeon groups that might be forming around you in the next hour.)

Jealous prospectors

How likely is it that your gathering will be interrupted by mobs? I haven’t kept as much statistics on this as I did on the rare drop rates, partly because it was more of a hassle and partly because I wasn’t as interested in this. In particular, this means that the sample in the ‘with vittles’ part of the table below is too small to get a decent idea of the underlying probabilities. Nevertheless, here are the results:

Without vittles With vittles
Number of nodes clicked 26736 1329
Times interrupted 2040 9.8 % 94 8.5 %
Of which by minibosses 881 30.2 % 30 24.2 %

Thus, it doesn’t seem that the vittles affect being interrupted by the mobs in any way. When you click a node, the chance that you’ll get interrupted is about 10%; and if you do get interrupted, there’s a 30% chance that it will be by a mini boss.