Threat Radius II

The previous post attempted to define a value-multiplier based on a unit's range based on the threat radius said range accorded the unit. This is all fine and dandy, except that some ranges are more important than others. In particular, the fact that standard infantry move 6".

The 6" move enforces a degree of granularity on what might otherwise be a continuum of range, because it governs exactly how many shots a shooter with a certain range will be able to plug into an approaching Ork before said Ork is forcibly putting his choppa into yonder hurty bits. Thus, while we can quantify the proportional increase of threat radius with range on the one hand, we can compare it to the practical application, the number of firing opportunities, on the other.

To consider firing opportunities, let's start with a hypothetical Guardsman in the exact center of a common 6' x 4' board, and give him a heavy weapon of range x. Not being able to move and fire will obviously simplify the situation here. For a target, we provide an Ork in one of the corners of the board, 43" away from the fellow with the gun. Roll for first turn, then rock on.

Going first will clearly only matter in the case of a 48" range. Assume that the Ork can judge distance well enough to stay out of range if he wants to, this means that he will move to 1/2 an inch beyond any maximum range of <48" the turn before he starts his run.

Then they're off and racing! The Ork enters max range at a full 6" move, then runs a further d6", until he is in assault range. Against a foot-bound foe who is determined to be in assault as quickly as possible, the guardsmen will get:

Range		Firing Opp
48		4
42		3.722222222
36		3.092592593
30		2.416666667
24		2
18		1
12		1
6		0

Note that the firing opportunities depicted as decimals represent the average of probabilities incurred by running. For example, when using a 36" range weapon, the Ork has a 90.75% chance of making it to choppa range before the guardsman gets a 4th shot, but a 9.25% chance of rolling poorly enough for run rolls that he is able to take it.

What does this tell us? Well, the utility of low ranged move-or-fire weapons is clearly extremely limited. Equally, the number of firing opportunities in this hypothetical example increases linearly, rather than by the square - since we are dealing with an Ork moving in a straight line towards a static target this is hardly surprising. Since an actual game gives a whole extra dimension to move in, I am inclined to somehow take an average of the number of firing opportunities presented in this linear system and the ratio of increased area provided by extending range in two dimensions. If I normalise firing opportunities to the number of opportunities at 24" range, just like I normalised the area ratio to 24" range, then this should be doable.

Range	Area Ratio	Firing Opp	Norm FO	Average
48	4	4	2	3
42	3.0625	3.72222	1.86111	2.46180
36	2.25	3.09259	1.54629	1.89814
30	1.5625	2.41666	1.20833	1.38541
24	1	2	1	1
18	0.5625	1	0.5	0.53125
12	0.25	1	0.5	0.375
6	0.0625	0	0	0.03125

And that is what we get!



By plotting the area ratio, normalised firing opportunities and the average of both, we end up with a delicate curve in betwixt the two. So in this hypothetical rating system, shots with a range of 12" are worth 0.375 as much as shots with a range of 24", and shots with a 48" range are worth 3 times as much as 24" ranged shots.

So does it work? Will it give a sensible result for comparison of weapons with different ranges? I don't know! I guess I'll have to give it a try...