Radical Alternatives: How SHOULD Size and Speed/Range Affect Chance to Hit?

[Anthony]

Originally Posted by vicky_molokh

Hmm. Talking to a shooting enthusiast resulted in some support of the quadratic hypothesis: he said that typically, someone with a 50% hit rate at 100m would have a 12½% rate at 200m against the same target under similar conditions. But in an RPG context, that seems to lead to hit chances quickly approaching 100% and 0% outside a relatively narrow range band, and I'm not sure how to feel about that.

For people approaching their accuracy limits against unmoving targets, hit chance really does vary like that. The thing is, the reasons for missing at short ranges are not accuracy related; they're related to time to aim vs target motion, rushing the shot, and bad visibility.

[Eukie]

The ballistics-simulator-with-tabletop-roleplaying-applications Phoenix Command has a number of factors that determine the final to-hit chance of an attack, primarily:

• The actual angular dispersion of the shot, which is a function of skill, range, stance, aim time, darkness penalties, wind, etc.
• The mechanical accuracy of the firearm, which introduces a range-dependent minimum angular dispersion.
• The uncertainty in lead estimation location when the target is moving, which introduces a minimum angular dispersion that's dependent bullet flight-time and target speed.
• The uncertainty introduced by erratic target movement, which increases the minimum angular dispersion for lead estimation based on the proportion of time the target spends not moving in a straight line.
• Once the smaller of the above has been calculated to determine the actual minimum angular dispersion, the target size is added to that factor to determine the actual hit chance.

In GURPS-y terms, this is broadly min(Skill + modifiers - SM, MOA, X+Y)+SM, where X is the lead estimation error and Y is the erratic target error. Conveniently, the physical effects this models are such that X is calculated using the logarithm of target speed and projectile time of flight: in other words, the standard Speed modifiers can be used in a calculation with this level of detail.

Originally Posted by Anthony

For people approaching their accuracy limits against unmoving targets, hit chance really does vary like that. The thing is, the reasons for missing at short ranges are not accuracy related; they're related to time to aim vs target motion, rushing the shot, and bad visibility.

I found the suggested hit probabilities at various ranges for US Army soldiers in a Field Manual and used it to determine a (parametrized) shot dispersion at various ranges. The dispersion angle actually narrows with range, suggesting that they take more careful aim at longer ranges.

[Anthony]

Originally Posted by Eukie

I found the suggested hit probabilities at various ranges for US Army soldiers in a Field Manual and used it to determine a (parametrized) shot dispersion at various ranges. The dispersion angle actually narrows with range, suggesting that they take more careful aim at longer ranges.

A variant I was considering for ranged combat is adjusting turn length at long ranges and not adjusting skill.

[vicky_molokh]

Originally Posted by Anthony

A variant I was considering for ranged combat is adjusting turn length at long ranges and not adjusting skill.

I found a way to model that using a series applications of a logarithmic scale for almost everything, but that project's first steps went so far as to no longer be GURPS at all. But at least the issue of turn scale and RoF bonus interactions seems solved in my project.

[Humabout]

Vicky, you make the assumption that shots fired approximate a uniform distribution, which is not the case. They will be clustered more densely around the actual target and become less dense as you look further out. So the size of the target won't have a quadraric effect on hit chance.

Let's assume a normal distribution of distance from bullseye with a standard deviation of 1 unit. If the target is 1 unit radius, the shooter has about a 68% chance of hitting. If we double the target's radius, he now has a 95% chance of hit. Adding another 1 unit to radius increases his odds of hit to 99.8%. He'll never hit 100%.

Now that addresses target size. Distance effects I'm not sure of. It would definitely increase the standard deviation of the final spread, but at what rate, I don't know. The shooter's own ability to be on target plus the effects of atmospheric conditions would be factors, though.

[Anthony]

Originally Posted by Humabout

Vicky, you make the assumption that shots fired approximate a uniform distribution, which is not the case. They will be clustered more densely around the actual target and become less dense as you look further out. So the size of the target won't have a quadraric effect on hit chance.

It's generally going to be a chi-square distribution, but that comes very close to quadratic once hit probability is low.

[Agemegos]

Originally Posted by Anthony

It's generally going to be a chi-square distribution, but that comes very close to quadratic once hit probability is low.

It's a bivariate normal, so the chi-squared distribution simplifies to an exponential distribution. In the general case the pattern is ellipsoidal, but assuming that the vertical and horizontal dispersions are uncorrelated and have the same variance you get the result that the chance id a given hit falling more than a distance from the centre of the pattern falls off exponentially with the distance.

[Eukie]

Originally Posted by Agemegos

It's a bivariate normal, so the chi-squared distribution simplifies to an exponential distribution. In the general case the pattern is ellipsoidal, but assuming that the vertical and horizontal dispersions are uncorrelated and have the same variance you get the result that the chance id a given hit falling more than a distance from the centre of the pattern falls off exponentially with the distance.

To be explicit:

P(Hit) = 1 - exp((2σ/x)^-2)

Where x is a measure of the target's projected area (typically this model assumes that the target is circular) and σ is the standard deviation of the projectile at the target ('projectile' should be interpreted liberally, and includes light beams and even the view through a camera).

Often we'll want to decompose σ into two parts: the angular dispersion α and the range r, or in other words:

P(Hit) = 1 - exp((2αr/x)^-2)

Multiplication is clunky, so we may want to define:

A = ln(αr/x) = ln(α) +ln(r) + ln(1/x)
P(Hit) = 1 - exp((2*exp(A))^-2)

P(Hit) is now a function of only one variable (A), which is a simple sum. This is broadly what GURPS already models: a ranged attack's skill, Acc, and other modifiers is ln(α), the Size Speed/Range table gives ln(r), and SM gives ln(1/x). Conveniently, 3d6-roll-under is a rough but workable approximation of 1 - exp((2*exp(A))^-2).

Where things get tricky is when we want to be more accurate about σ.

[vicky_molokh]

Originally Posted by Eukie

In GURPS-y terms, this is broadly min(Skill + modifiers - SM, MOA, X+Y)+SM, where X is the lead estimation error and Y is the erratic target error. Conveniently, the physical effects this models are such that X is calculated using the logarithm of target speed and projectile time of flight: in other words, the standard Speed modifiers can be used in a calculation with this level of detail.

That looks like a very promising direction, but I feel confused by the function as currently posted: SM being included as a negative in one of the candidates, but as a positive outside, and the fact that X+Y seems completely non-interacting with skill level (shouldn't skilled shooters be better at estimating lead)?

Also, the derivation of Y, erratic target error, is of high interest for any combat system.

[Eukie]

Originally Posted by vicky_molokh

That looks like a very promising direction, but I feel confused by the function as currently posted: SM being included as a negative in one of the candidates, but as a positive outside, and the fact that X+Y seems completely non-interacting with skill level (shouldn't skilled shooters be better at estimating lead)?

Also, the derivation of Y, erratic target error, is of high interest for any combat system.

Consider it "skill + modifiers (not including SM)" if you will. As for lead estimation, it's not unreasonable that skilled shooters might be better at it, but I've found very little on the topic of how it improves with skill: just that for the average tank gunner, the standard error is 30% of the range.

Originally Posted by Ulzgoroth

That's a bizarrely sweeping move to make to address the problem of smooth-bore shot instability. Considering that smooth-bore muskets and rifles are contemporary at certain tech levels, and rifles do not have at all the same problem (though they do have their own problems), it really doesn't make any sense.

If you want a special rule for muskets (and maybe unrifled cannon) give them a special rule.

There's the Tactical Shooting MOA rules, but if you want to be really specific about it, you could also make it an inherent trait of the firearm itself: no need for special rules for special weapons, just a fact that some weapons are very accurate and precise are some are not, and the final skill can't exceed Inherent Accuracy Limitation + SM.

This is also, conveniently, a way to make it so that weapons that can be aimed accurately aren't necessarily fast to aim.