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Amphitheatre Unchained loot tables

October 21, 2012 Leave a comment

I updated my post about the unchained version of the Amphitheatre of Karutonia with the loot tables for all three bosses: click here.

I guess there’s a tiny possibility that some items remain undiscovered, but I doubt it.

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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.

Hello world!

December 20, 2011 1 comment

Welcome to my blog. My intention is to post useful information about Age of Conan, an MMORPG inspired by the work of Robert E. Howard.

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