# We're trying to analyze which monsters drop what in an MMO

After killing a monster there's a percent chance certain items drop that are unique to the monster, but sometimes the monsters are in groups and we only get the drops after killing all of them.

Monster A, B, C sometimes appear like [A,B] A with B but never [A] alone, but sometimes as [C,A].

What is it called solving this type of problem, for example isolating what monster A drops with a confidence interval? I think it's a linear regression or a multivariate linear regression but I'm not sure what mathematical tools I would use to solve this. Just stats or probability alone? I've toyed with R but my stats is not that good anymore. I'd like to know if this is within the realm of a problem a beginner can be able to solve with maybe a few weeks of study.

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Suppose that you have $N$ objects and $M$ monsters. Then your goal is to ascertain which monster drops which object. You can model your uncertainty by assuming that:

$p_{nm}$ is the probability that the $n^\text{th}$ monster drops the $m^\text{th}$ object.

Suppose that you observe that when you encounter monsters $n$ and $n'$ they drop object $m$. Depending on how the game works you can write down the probability that you observe object $m$ given that monsters $n$ and $n'$ were present. For example, if we assume that the object can be dropped by either monster then it must be the case that (a) monster $n$ dropped the object and monster $n'$ did not or (b) monster $n$ did not drop the object and monster $n'$ did. We can capture this probability as follows:

$\text{P(observed object is } \ m|\text{monsters are: }$n$,$n'$) = p_{nm} (1-p_{n'm}) + (1-p_{nm}) p_{n'm}$

You could then construct a likelihood function by multiplying the probabilities of observing all the objects that monsters dropped in terms of $p_{nm}$. Maximizing the likelihood function will then give you estimates for $p_{nm}$.

Obviously, the number of observations in terms of objects dropped and the combination of monsters that you have to observe in order to get reasonable estimates of $p_{nm}$ increases with $N$ and $M$. Depending on the design of the game the above can be an intractable problem. For example, you have 100 objects, 100 monsters and at any given time you face at least 5 different monsters. Then depending on how much you play the game your data may be very sparse as there are $\binom{100}{5}$ possible combinations of monsters you can face each of which can drop any of the 100 objects!

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Thanks for the answer. The monsters have a percent chance of dropping so I may fight 10 monsters and drop nothing at all. I need a way for saying X monster being in a group correlates with Y item being dropped, but with some kind of way to decide whether I have enough data to have confidence in that assessment. The data is being collected automatically as we play the game. Usually it's 2-6 monsters that drop about 3 or 4 items each, some rare, some not. –  Jazz Man Nov 29 '11 at 19:09
Logic stays the same. If you observe that two monsters $n$ and $n'$ do not drop the $m^\text{th}$ object then the probability of that happening is: $(1-p_{nm}) (1-p_{n'm})$ Include this probability as well in the likelihood function and maximize the likelihood function to estimate $p_{nm}, p_{n'm}$. You can also get confidence intervals for your estimates via maximum likelihood theory. –  varty Nov 29 '11 at 19:42