Even on Path, and this is one of the most math-heavy games in the mainstream style.
Concepts in this post should all be covered in a basic topology class. Some you might even find in set theory, wouldn't know, never took that.
The question was raised, what is the definition of a "build"? The relevant part of my answer is below.
- Path build n-space is discrete. It's also closed and contains a (very
large, but) finite number of elements.
- Passive tree major or minor changes have some effect sometimes. Big
nodes are game-changing, small nodes have cumulative effects, but the
only logical way to resolve this is for no nodes to be valueless.
- Skills are different. Support combinations are different, to some
extent. As above, there can be only one logical resolution to this. Both
these spaces don't necessarily behave all that well if you pick the
wrong definitions, so don't pick the wrong definitions.
- For the most part, rare item stats have less impact, and similarly
with statball uniques like Kaom's Primacy. On the other hand, there are
build definers like Facebreaker. Such items, then, each correspond to a
single dimension in Path build n-space, or we can say that Path build
n-space is the product space of several spaces, some of which are
Boolean in nature and correspond to items. But even statballs have their
impacts: physical, spell, both? How much total MF? And so observe: it
makes sense to have each individual stat mapping to some interval in R,
and the sum total of rares/statballs occupying some point in R^m.
That's...clearly not all different builds the way the term "build" is
generally used, though a strict definition should define them that way,
so let me introduce the term "setup". Each element of Path build n-space
is a different setup.
Luckily for us, setups tend to cluster in different regions of each of
the parent spaces that contribute to the product space. Just as an
example, on the skills space, taking a subset of all setups which
contain the active skill element LA, a lot of xMP, chain, WED, etc are
found, while setups containing MPD and splash are conspicuously absent.
(If you're wondering, I defined a compact R^6 where "support gem does
not work with skill" means that element just maps back to 0 on that
dimension. There's a lot of (LA,0,0,0,0,0) in it. Not necessarily the
most efficient method, but I'll take that trade-off for the
well-behaved-ness of R^6 any day!)
>> I believe it is possible to define a metric on Path build n-space. Just saying. This is important.
But going back to the idea of clusters, that's basically a finer - or
coarser, sorry I'm on a timer here not going to figure that one out -
version of each of the mentioned spaces. Each element in Path build
cluster n-space is a different build. Alternatively, for each cluster
choose a point near its centre* to be the element in the corresponding
space, and thus define builds.
*this is why the metric was important
Problem solved, right? The only fuzzy bit is the clusters one, and I'm
not happy with that either, but how else are you going to draw an
arbitrary line? The second method might be better, as in "I'm playing a
setup that's close to theWombo's build on Path build n-space".