Sunday, 30 August 2015

Mass Targets

I've just come back from a week's holiday in the Lake District with no Internet Connection, so to satisfy my cravings I'm going to address Leland's questions in the comments on archery versus mass targets. Here's the first:
I concur with Delta's analysis about treating range as reducing target size, but I think this falls apart somewhat when firing at long-range dispersed targets. I think of the dispersed target as a big area that the archer is just trying to drop an arrow into somewhere. The target is quite large, so it's not hard to hit the area. But what's important here is, how much of the target area actually has a target in it? And also, what's the effect of angle of incidence -- what you really want is the projection of the actual target area into a plane orthogonal to the arrow's direction of flight (also allowing for potential overlap/shadowing of individual targets). If (to choose a simple example) the target area is 50% real target and 50% empty space, then it's simple in your system: just apply an additional -3. So I think you're baking in some assumptions about density of the massed troops. Personally I think 50% density is a bit high, but that's probably because we have different mental images of the targets.
Firstly I'll recap with the help of some illustrations. If you shoot at a target which is twice as far away, then that is the same apparent size as a target which is half the size (which is a quarter of the area):
The further warrior is twice as far away as the near warrior and the warriors are about the same area as the targets. The further target (twice as distant) is the same apparent size as the nearer central green section, which is only 4 studs instead of 8. Hence hitting more distant targets is simply like shooting a smaller target.

The distribution of shots is a bivariate normal distribution (i.e. both x and y are a bell curve), so 1000 arrows shot into a target would look like this:
As you can see, once you get into the four central squares, the distribution is even. If you hit one of these four central squares, whereabouts you hit is completely random. If we highlight the best 50% of shots you can see that the distribution of these is completely random:
For shooting at massed target shots there is the chance of hitting the area with the target in, and then the chance of hitting something within that area. If you have less than a 50% chance of hitting the target area, then all you need to know for hitting an actual target is what proportion of the area is filled with target and add in the appropriate penalty. My eldest son described it as being like "shooting at a target with holes in it". If half the area is empty, then that's a -3 penalty.

I don't have any armies to hand, so I was going to take photos of miniatures (Lego figures are not proportionate) but it's tricky to see as my figures are all on painted bases, so I've taken this approximation of people as x 9.5'' x 1'6'' x 5'9'' and made stacks of Lego bricks with the same proportions. I've given a gap of 2 studs - 1'6'' - which seems pretty close when you're holding a sword. According to Wikipedia this is the spacing of a tight phalanx.

Viewed head on, only the front rank can be seen:

However, from a distance, arrows will come in at an angle, so present a target something like this:
For a maximum range shot, you're shooting at approx 42 degrees, and the arrows are coming down at approx 48 degrees, and from this angle the troops look like this:

At this angle a 5'9'' man only presents the same size target as a 4'5'' man, but as the angle has increased the amount of overlap has reduced to almost zero. The area to hit has gone up from 95 sq ft to 131 sq ft. Half range would be shooting at 12.5 degrees, coming down at 14 degrees, and at this angle the area would be 114 sq ft. Hence shooting at range has actually increased the target area.

In contrast let's consider the ultra tight synaspismos formation the phalanx adopted when under "extra pressure, intense missile volleys or frontal cavalry charges". This is no spacing between figures, and only one stud between rows. At 48 degrees there is now only 112 sq ft presented to the arrows of the enemy, this might result in a small -1 to hit penalty, but overlapping shields would result in a more significant advantage.

Now a phalanx is tighter than you'd expect - if the spacing between troops was doubled as per a loose phalanx, then the troops would occupy four times the area, and when the arrow comes down onto the phalanx three quarters of the area would be empty. This would apparently give a -6 penalty to the archer, but on the other hand the troops would cover a much larger area.

For comparison, Here's loose, tight, and ultra-tight formations together in a single photo:
My assumption is that the area the archer is shooting into is either in loose order, so the whole of the target area is occupied by troops, but only one quarter of the area has a person in it (from the arrow's perspective), or it is in tight order and the troops cover only a the central quarter of the target area.

That is we either have loose order:
Or we have tight order:

And either way, assuming the archer has less than 50% chance to hit, the troops cover a quarter of his target circle, and hence -6 is a good penalty.

Mass Targets or Individual Targets?
Leland also asked:
I guess one question is, where do you transition from shooting at an individual target, vs. at an area that is n% full of targets?
Due to the logic behind how I derived mass target shooting, I'm limiting archers to a minimum of 11  needed to hit mass targets (i.e. minimum of 50% chance to hit). Hence at close range it can become better to switch to individual shooting.

The chance to hit is the best of individual shooting (with range penalty) and mass shooting (-6 fixed penalty, min 11).

For example, at 120' range a +11 archer only needs 7 to mass hit troops with a parry of 12. This is below the lower limit for mass target shooting (11), so they need an 11. For individual shooting it would be a -12 distance penalty (-3 per range category), so a 13 to hit. Hence they stick with the 11 target.

If the troops were 80' away the archer only needs a 10, so they can now switch to individual shooting. 

Hitting Important Targets
A third question was:
Another thought that occurs -- you could have archers shoot at one individual target in a mass, and if they miss by, um, not very much (depending on the density and other modifiers) you could rule that they hit SOMEBODY, just not the guy they were aiming for. 
Now imagine that one of the targets is a special target, an important NPC perhaps. Determine the number of troops around him and roll to see whether he is targeted. E.g. if he is one of 20, a 1 on a d20 would pick him. I'd say one in a hundred maximum. If he is selected then we can see that the rest of the troops are now irrelevant and you simply make your attack roll against him. If it misses him it misses everyone. It he is not selected then you make the attack roll against the normal troops.

This same method can be used for mounted troops - fifty fifty chance of targeting the mount (depending upon the size of the mount).

Field Experiments
Now back to considering Crossbows. My youngest is keen to to do some field experiments with his toy Crossbow...
We're only shooting an apple juice carton, and the biggest danger is loosing the bolt in the hedge!


  1. I read this a while ago but haven't had time to put a coherent comment together. Thanks for the additional detail on your thinking; I think a big difference in our notions was that in my mind I was thinking of (1) longer-range fire such that the incident angle of the incoming missile was closer to 45 degrees; and (2) targets being more dispersed than you have (perhaps even more than the "loose phalanx").

    Something else to consider: I believe it's true [citation needed] that generally speaking error in missile fire is not distributed with the same variance in range and angle. That is, the impact pattern doesn't approach a circle so much as an ellipse, with the major axis along the line of fire (errors in range/elevation tending to be larger than errors in azimuth/alignment). But that's probably more detail than is really worth modeling, particularly in the absence of concrete data, and even if true may be a small effect for experienced archers.

    W.r.t. the single important figure, I was thinking that the important figure was distinguishable in some way ("shoot the guy in the black armor with the death's head helmet!") such that you're shooting at one guy with normal ranged attack penalties (ignoring the effect of a mass target), but if that shot misses there's still a chance that the shot happens to hit some hapless underling nearby. I agree if there's no particular reason to target that one guy then a random check is sufficient.

    In any event, thanks a lot for amplifying your thinking on this issue.

  2. Thanks Leland for your comments.

    I think we agree on the angle of incidence (in my post I have it as 48 degrees at max range, which is even steeper than 45 degrees, due to wind resistance) but I agree that if you had more dispersed targets than this you’d have less chance of hitting, which would be an extra penalty. For example, double the separation = one quarter the density = -6 penalty. This implies that open formation is best against missile fire, but then it opens you up (literally) to cavalry charge. Sounds like there’s some complex trade-offs.

    On variation in y versus x due I decided it wasn’t worth investigating, though as I have the code I could always look another time if I’m still interested.

    On targeting individuals, I was going to explain that when the chance is <50% then even if you’re targeting that individual then the statistics say it’s actually entirely random who you hit. So the system I outlined is correct – except - you’ve prompted me to spot a deficiency in the method I outlined – if you target someone then because of the arbitrary number aspect of my solution you end up with a different chance of hitting that individual when he’s in the crowd and when he isn’t, and it could in fact be easier to hit him in the crowd.

    So you’ve prompted to me to try harder :-)

    What we have to work with is as follows - you know the chance of hitting the individual, and the chance of hitting the crowd. The chance of hitting the individual is much worse than hitting the crowd (due to the range penalties). You want the chance of hitting the individual to be unchanged by the presence of the crowd, and you don’t want to change the chance of hitting the crowd just because you’re aiming at an individual in the crowd. So the revised system is this - roll to hit the individual, if you miss then reapply exactly the same roll as an attempt to hit the crowd.
    That's simple, and seems to give exactly the results that you'd want.