Weapon Discussion Thread

[Surrealistik]

@oharthstein: The product of multiple gaussians (i.e. the gaussian curve of each body part's damage multipliers whose probabilities are further modified by their location, with the torso probabilities having the highest probability multiplier and head the lowest) is gaussian. Further I very much doubt that this convolution of gaussians in terms of damage outcomes would more closely resemble a perfectly flat line than a gaussian curve, even if the curve is somewhat flattened.

[ohartenstein23]

... that's not how the math works.  A linear combination of Gaussian curves, each with a different mean value, does not result in another Gaussian.  Convolution would have nothing to do with it either.

[Surrealistik]

As in they coalesce around certain probabilities which are more likely than others, and this is reflected by the subsequent shape (which may not represent a bell curve, yes, this is apparent).

When you convolute/combine the shapes of those gaussian distributions, you would not get something that more closely resembles a flat line than a gaussian distribution.

I understand that people often like to be pedantic when talking about such things, but I feel the gist of what I was saying was pretty clear.

[ohartenstein23]

Here's a simple version of this model: let's assume you're using a 50 power base weapon, and that 50 is the mean damage it'll do on a shot to the body.  Let's add two more curves, one for a headshot with base damage 75, and one for a limb shot, with a base damage of 25.  Let's also say that when you hit, there is a 40% chance to hit a limb, 40% chance to hit the torso, and 20% chance to hit the head.  The resulting probability distribution function is the linear sum of three Gaussians - in the figure I've attached, it's obvious that this result is not a Gaussian function, and is much more flat.

[Surrealistik]

See above.

I clearly acknowledge that the combination is not going to represent a bell curve; however it resembles one (an elongated one, but nonetheless) more than it does a flat line to be sure, which makes the gaussian solution better.

Also what program did you use to generate that graph? Standard excel or something else?

[ohartenstein23]

I used Octave to create the figure - it's an open source version of MATLAB.

[ohartenstein23]

The problem is that it doesn't take much fiddling with the model to make it more resemble flat than bell-shaped, especially when you take into account more possible ways to hit.  Even as is, this function is described neither very well by Gaussian or flat distributions (metric: multiplying the probability distribution by one of the two possibilities we're arguing about and integrating over all possible damage values).  It really depends on what we want to model which one is better; the Gaussian picks up the centered-ness of our curve, but the flat is much better at describing the variance.  And making it more like real-life (i.e. adding more possible hit locations) is just going to flatten this more.

[Surrealistik]

You can tinker the inputs to skew the shape in any direction (pro-gaussian/pro-flat) you see fit, yes.

However a reasonable set of inputs with respect to probability of hitting + gaussian distribution of multipliers will ultimately lean gaussian more than flat, because reality itself tends more towards gaussian than flat outcomes when bullet casualties for example, are examined.

[Yankes]

You can tinker the inputs to skew the shape in any direction (pro-gaussian/pro-flat) you see fit, yes.

However a reasonable set of inputs with respect to probability of hitting + gaussian distribution of multipliers will ultimately lean gaussian more than flat, because reality itself tends more towards gaussian than flat outcomes when bullet casualties for example, are examined.

multiple flat distribution get avenge to gaussian distribution: https://en.wikipedia.org/wiki/Central_limit_theorem

[Surrealistik]

multiple flat distribution get avenge to gaussian distribution: https://en.wikipedia.org/wiki/Central_limit_theorem

For sure; that's the entire principle behind say rolling two dice; the two flat outcomes result in a gaussian one.

If you're trying to argue though that gun shot wound has flat variability IRL which culminates in the appearance of gaussian casualties over many iterations, I disagree. While there's no doubt that gun shot severity can vary greatly, on the whole, if you get hit with a bullet, you're much more likely than not to suffer substantial if not serious injury (while your odds of survival are significantly greater than death excepting certain firearm classes like shotguns, so J curving is improbable). A bunch of gaussian outcomes can also arrive at a gaussian outcome just as many flat outcomes average out to one.

[ohartenstein23]

Let's look at some data then:

The first paper I've attached looks at wounds to the limbs - if we abstract the length of hospital stay into the amount of damage done (just like the in-game model for wound recovery time), we see their results have a mean stay of 11.9 days, a range of 2 - 48, and a median of 8.  This is much more skewed towards towards shorter hospital stays, more like an exponentially decaying probability distribution, or a Poisson distribution.

On the other hand, a bullet penetrating the skull is fatal within 48 hours in ~75% of cases. https://library.med.utah.edu/WebPath/TUTORIAL/GUNS/GUNINJ.html

Finally, we can look at the distribution of where non-fatal injuries occur on the body: https://www.dtic.mil/dtic/tr/fulltext/u2/a570804.pdf.  This might not be quite as accurate for people trained in firearm use and aiming for center of mass, but the data shows a heavy skew towards what we would categorize in-game as low-damage shots.

If we take body shots as an intermediate between these cases, then we can re-evaluate the categories I used for my previous figure:  we leave the body shot damage as the Gaussian distribution, the headshot distribution becomes sharper and with a higher mean, and the limb shots should start at near 0 damage and decay with a long tail towards higher damage.  The properly weighted and summed distribution would have three peaks - one near zero, one skewed towards very high damage, and one in the center.  While not anywhere near flat, the Gaussian alone does not capture the very high and very low peaks at all.

[Arthanor]

Nice work with the figures, ohartenstein23!

Indeed, the sum of multiple linear distributions gives a more gaussian distribution (ex.: sum of multiple dice). It is very interesting to note (and see in those figures) that the sum of multiple gaussians lean towards the linear.

In fact, in that second figure, if we consider shots under 30 damage to be all equally useless (because of some armor on the target) and shots above 80 to be equally deadly (50 hp + 30 armor), the effective curve becomes even more flat as one can just use the "average probability of hitting under 30" for all points under 30, the "average probability of hitting above 80" for all points above 80.

You'd have to add pretty much identical distributions or have such a dominating one as to make the others pretty much irrelevant to still get a gaussian-like one.

[Surrealistik]

You need to include recovery times  (which tends to be significantly longer) as well as hospitalization times.

For example, total recovery (such that the person is capable of work, nevermind combat) averaged 65.7 days, while hospitalization averaged only 13.2 days: https://www.ncbi.nlm.nih.gov/pubmed/17626461

[ohartenstein23]

You need to give a credible reason why a single Gaussian encapsulates all the complexity of an individual bullet wound - adding in longer recovery times for light wounds to the real data just means the low-damage peak is slightly higher, not really changing the shape of the distribution.

You have convinced us that you believe the Gaussian model is best for how you play the game, but have not presented any data why it should be the default for others.

[Surrealistik]

My point is that considering only the hospitalization time can be distortive (for example, the hospitalization time for a superficial gunshot injury to a serious one might be comparable or only slightly different with both requiring surgery, but the recovery time for the latter might be far more substantial). If one were to judge wound severity (which is what really counts here, locational considerations are secondary/irrelevant) exclusively by hospitalization times, you may end up with inaccurate outcomes when compared to a better measure that captures the harm inflicted more faithfully.

I'm attempting to find good data concerning max, min, median and average full recovery times for generalized gunshot wounds as well as distributions/charts, etc.