Should monsters have a fixed max hp or variable max hp?

[Leafsnail]

I think firstly we need to define what we're referring to when we say "increased variance". I believe the only sensible thing that could be increasing in variance is the number of hits it takes to kill a monster. This is going to be, roughly speaking, monster HP/damage (obviously this isn't a mathematically rigorous expression at all, but I think the basic role of max HP is correct unless rounding does something weird). Going from fixed HP -> variable HP simply changes this formula so that instead of being multiplied by a constant it's being multiplied by a distribution with a source of variance. Provided this source of variance is a) independent and b) has the same average as the constant that came before it I believe this will always result in an increase of variance.

This is not the same as crate's example - in that he's comparing multiplying a distribution by a constant (which would multiply the variance by 2^2 = 4) and adding another similar but independent distribution to itself (which would multiply the variance by 2, so it is indeed true that 2d6 has a lower variance than 2*d6, but note that they both have a higher variance than d6). Something more comparable would be, I dunno, 2*d6 vs d3*d6.

e: I've realized that variance doesn't actually mean what people intuitively think it means, though, something more like the standard error or standard error squared might be more appropriate. The same basic argument holds though.

[crate]

I'm using the mathematical definition of variance (i.e. square of standard deviation) here, and the variance of 3.5-1d6 is not the same as 1d6-1d6 (you need to compare things with the same mean, see below). In fact the latter has twice the variance. (3.5-1d6 has variance roughly 3, 1d6-1d6 has variance roughly 6).

For the additive case what you care about is really something like (stddev/mean), since that's giving you the width of the distribution in a meaningful fashion (you need some kind of scale; while the variance of 1d6 is in absolute terms less than the variance of 2d6 that's not actually useful here, and if you look at the probabilities of the possible results it's pretty obvious that the 2d6 case has less "variability"; the results tend to be closer to the average. The average of the 1d6 case is 3.5; you get a result within 15% of that 1/3 of the time, or about 33%. The average of the 2d6 case is 7; you get a result within 15% of that 15/36 of the time, or about 42%. It's true that something like 100+d6 has less variance than 0+d3, but you really don't even care about the difference between 106 and 100 in most cases while you absolutely care about the difference between 1 and 3 a lot of the time.)

I took a shortcut and just changed the things I'm comparing to have the same mean to begin with, to simplify things, and then continued to use that in the subtraction case.

I would believe that variance in the mathematical sense here would translate into "variability" in the crawl sense (e.g. how many turns it takes to kill a monster) pretty much just fine (again, assuming you're looking at things with the same mean), but if someone would like to argue otherwise you are free to do so.

It is perhaps true that 1d6-1d6 is not a good example to look at (you can't do the same scaling by using the mean, since the mean is 0), and I was not clear enough in my previous post that I was only comparing things with the same mean (i.e. that I was comparing 3.5-1d6 to 1d6-1d6).

The important difference with addition compared to subtraction is of course how they change the mean of the distribution you're looking at. Addition increases the mean, so even though it also increases variance the important thing (stdev/mean) decreases. Subtraction reduces the mean, so this isn't the case; the increase in variance does actually make the distribution wider in a meaningful sense.

[ragnarokchu]

Personally they should shorten the variable HP between monsters, something like 9 to +20 is an extreme variable, something like 13 to 20 would make more sense.

High amounts of variable HP should exist as a challenge/tactical thing to consider and same thing with fixed max hp with certain monsters. (A super hard monster always have exactly 100 hp, this other monster has exactly 10 hp and is extremely hard to hit/high melee defenses but you can always one shot them with a wand or other stuff like that. Vice versa something like super chaotic monsters that can have 1 hp or up to 200 that spawn at you in large waves so your lucking it out in fighting them in a prolonged period.)