Monday, 28 April 2014

Consistency lines

Preface 

Hey guys, hope you've been enjoying your reading so far. A while ago, I read Reecius' opening review for the Astra Militarum codex. He wrote something that's run with me for a while, and something I decided I'd finally post about here. The quote in question is this:

"The more variables you eliminate or reduce, the more consistent your results will be."
Now this statement in and of itself is obvious. An AP3 weapon should be better against a tactical marine than an AP5 weapon because the marine is denied his armour save. For example, of course you will fire your vengeance bolt at a marine rather than a poisoned 2+, because you remove a variable! Do you still do the same thing if the marine would get a 4+ cover? Well, yes you do because if he's getting a 4+ cover he might go to ground and that's always better than killing a marine. To be short, this post will feature a lot of mathematics (relatively simple - I'm going to guarantee some of the actual math is wrong, feel free to gripe.) and will focus heavily on not using final wound values, but instead, the gradient line of the wounds to determine how successful a wound is. 

Math 

Shooting math is very simple most of the time. Every value should be presented over 6, with very few (mostly to do with armour penetration) requiring more than a single dice. If you're familiar with basic math-hammer, you can skip this section - or not, because I know you'll be there to pick me up on my mistakes, I know you. (you're actually just skimming this section, praying that I've made a mistake!) Anyway, Lets have a look at some in game values and translate them into math. Let's do a boltgun Versus a tactical marine, nice and easy, relatable. 

A boltgun hits a marine on a 3,4,5 or 6 because of BS4, wounds on a 4,5 or 6 because of S4 versus T4 and causes an unsaved wound on a 1 or a 2. 4 of 6 instances result in a hit, 3 of 6 instances result in a wound and 2 of 6 instances translates to an unsaved wound. This can be easily represented as...

4/6 x 3/6 x 2/6. This translates into 24/216 which is 12/108, 6/54, 3/27 and then 1/9. This means, when shooting is exactly average, you can rely on your target tactical marine dropping to 9 boltgun rounds. Tell me if this is not clear enough. 

Let's say you actually shoot the 9 boltgun rounds needed to cause 3 wounds at him, and he saves all 3 - the lucky bastard! His chances of doing that are equal to his armour save (4/6) raised to the power of the number of wounds he had to save - in this case, 3, so (4/6)^3, or roughly 7.4% - a reasonable chance, actually. One in roughly 14 instances will result in him not taking a wound. 

What we're looking at here, is how to represent the fact that if one were to remove the armour save, how much removing that 1/~14 chance will increase your likelihood of causing a wound. For that, we need three things: Graphs, Gradients and Good ol' fashioned university holidays. 

Basic Shooting 

The Graph to the right represents the chances to wound for a range of weapons. The Dark Blue is a boltgun, Red is a lasgun, green is a hot shot lasgun, purple a heavy bolter and the cyan or light blue is a plasma gun. This graph only shows you what seems to be the attrition of wounds as you go through different rolls on the chart. You'll notice a spike for the hotshot lasgun, where it doesn't follow the trend of the other lines in its group - this just shows how much of an impact AP3 actually has on its function.

It is worth noting that technically, these graphs are not suitable for generating gradient lines because the X axis does not show a quantitative value, but a qualitative value. If it bugs you, simply treat "Shot" as X1,0, To hit as X2,0, To wound as X3,0 and after saves as X4,0. As for the meaning of the qualitative terms used, "Shots" is always 5 - I chose to use a baseline because the gradient for 2 plasma gun shots is going to be MUCH smaller than 20 boltguns, rather than just A LOT smaller. Also, 5 shots ensures the graphs are always a consistent scale.

Now we get onto the meat of this topic, Gradients and Consistency lines. Let's compare the gradients of different weapons. In the chart to the right, you can see that, compared to the other weapons, S7AP2 - a plasma gun - has a minute gradient value, being almost half that of a S4AP5 boltgun! Let's see why with some math. Of course, this shows that the ideal gradient is smaller - the smaller the gradient, the more horizontal the consistency line will be. 

As stated above, a marine dies to 11% of boltgun shots, or about 1/9 wounds from boltgun shooting. Some of the values are different for a plasma gun - while shots and "to hit" remain the same (the former for consistency's sake, the latter because your plasma gunner and your boltgunner will be in the same squad and thus, have the same BS. The differences come in wounding, and the fact that plasma guns ignore armour - this is huge because they completely ignore the armour save, effectively ignoring that 1/14 chance that 3 boltgun wounds won't kill a marine. 

Our values are for a boltgun are... 
4/6 x 3/6 x 2/6, or 24/216 (1/9)
Our values for a plasma gun are
4/6 x 5/6 x 6/6 or 120/216 (5/9)! 

This means roughly one plasma gun in two should wound and kill a marine - the odds are actually in the plasma gun's favour, at roughly 56%. This can be seen very clearly in the differences between the two graphs when they're isolated, as you can see to the right. For the record, the first graph shows a boltgun, with the dark red line being the gradient or consistency line of the shooting. 

The second graph shows a plasma gun's shooting. Notice how, in accordance with the remarkably small gradient line, the plasma gun's consistency or gradient line is closer to horizontal than the boltguns?  This is important. To make the matter of interpreting these graphs easier, the more horizontal the dark line is, the more likely the shot is to end in the result you want it to. 

In looking at graphing 40k in future, I implore my readers to use a consistency line, rather than a simple "hit, wound, save" line, like we saw in the first graph - it can be misleading. The Graph below shows all the shooting shown in the first graph, although this shooting is shown with consistency lines. Notice how, rather than blending in with the crowd until the very end, the Hot Shot Lasgun actually is distinctly above its peer weapons in terms of consistency? this is the result we should be looking for. 


Cookies for the person to comment and tell me what's wrong with this graph.

Twin-linked shooting 

So now then, let's go back to the inspiring quote for this post. Reecius wrote...

"The more variables you eliminate or reduce, the more consistent your results will be." 
Aside from the most obvious elimination - the armour save of the model, it is very hard to actually ignore a to hit or to wound roll, with almost no weapons having that capability. What we can do, is that second thing reduce our variables. The most readily available example of this is twin-linking a weapon. While not as good as increasing the number of shots in many cases - usually double shots would be better than a twin-linked weapon, twin-linking brings our variables into the fold. Suddenly, your chance of failing to hit with a single D6 at a 3+ drops from 33% to 11%. This can have a massive effect on the end result - for example, Twin-linked BS4 is actually more likely to hit than straight BS5 - the same being true of everything bar BS1, which is still about 5% worse than straight BS2. This is also why you can't expect to hit reliably with a non-skyfire weapon against a flyer. 

To our left is a graph detailing the same set of weapons, only this time the weapons have a higher to hit rating. (twin linked BS4, so .89 is the modifier rather than .67) What should become apparent quickly is that the difference between the final values is actually impacted so much so that the non S7AP2 shots don't cluster so much as they used to! I'm serious! have a look. Compare this graph to the other one. Go on, I'll give you a minute. 

Now, let's have a look at those gradients. It didn't make much of a difference for shooting that wasn't definitive (definitive meaning a shot that removes the armour save) Boltguns's gradient decreased .05, the lasgun's .03 hot shot lasguns dropped a massive .09, heavy bolters dropped below 1 at .98 (a loss of 9 points) and plasmaguns dropped from a minute .56 to an even smaller .32, or a loss of .24 points. Note that a plasma gun is now a third of a boltgun's gradient, not about half. 

This is excellent evidence of how as you eliminate variables, you become more able to produce consistent results

To see the differences in consistency lines, compare the graph to the right with the graph above, and I've provided a handy comparison graph for the boltgun and plasma gun as provided above, compared to now. 






Cover Saves 

Okay, so now we've seen how a positive variable adjustment impacts upon shooting. Let's look at how a negative one does. I'll have you note that these charts use a twin-linked weapon, so their values correspond to the immediate above, not the first set of graphs. 

The best way to throw a spanner in the works for a plasma gun's 5/9 kill ratio is to give it a cover save to chew through. Suddenly, your tactical marine is gifted with half his resilience in the form of his cover save. How does this affect the plasma gun's shooting, you ask? Let's have a look. Compare the graph to the right with the previous graph regarding the plasma gun. (included here for convenience.) Notice how many less wounds the plasma gun does? It drops from over 3.7 wounds to less than 2.5 with just a 5+ cover save! You can see this in the consistency line - the plasma gun's line now drops at a much sharper angle because of its newfound gradient.  Finally, one plasma gun now kills a marine 10/27 times, dropping from its old value at about 15/27!.



This change can be seen in the gradient value to the right - it has increased from .32 to .63 - almost double. Note that now, it is sitting at just under 60% of the boltgun's gradient value. 

The other interesting change here is the hotshot lasgun and the heavy bolter. Have a look at the graph above and you'll notice they cause the same number of wounds as each other, the heavy bolter being unchanged. This is because they're wound chance value has equalised to be the same. 

A heavy bolter's is
4/6 x 4/6 x 2/6
A HSLG is
4/6 x 2/6 x 4/6
They're literally the same, just with one roll requirement being swapped in each case. And this can be seen in their gradient values. 




An Ideal Case

Let's have a look at an absolutely ideal case of shooting consistency. A BS5 model shooting a plasma gun with twin-linked and re-rolls to wound, ignoring both cover and armour. 

Notice how close the green line is to horizontal? Its chances of hitting are 35/36 (or 5/6+(1/6 x 5/6), chance of wounding is 35/36 and it ignores saves completely (assuming no screamers are present) It loses about 5.4% of its original output, total. This results in a gradient value of .07. This is 1225/1296, as a comparison, a plasma gun with no boons outside of BS4 and S7AP2 should do 720/1296, assuming no cover. 

Absolute opposite

The absolute opposite of an ideal case, may at first seem like a screamer with a re-rollable 2+, but it actually isn't. It's a Necron warrior, or immortal - anything with RP. Why? They create a worse variable than an invulnerable save. It's the same as an invulnerable save because you can't ignore it, but it's worse than an invulnerable save because when one passes an invulnerable save, he can be called upon to take another until he is removed as a casualty. Not so with reanimation protocols - you have to hope against hope that the necrons all die, or else the squad can come back to haunt you. This is a fantastic example of a variable you can't actually reduce all that terribly easily (except by destroying the unit of course!) and you can't expect yourself to do so reliably.

Thanks for Reading!

Consistency lines have been fun to build and write about, and analysing these graphs has been a great passtime for me. I hope you find the information I divulge here useful, and creating your own gradient lines against non-generic marine targets. 

As always, you're welcome to email me at diaord@hotmail.com, or contact me as scipio africanus on dakkadakka if you're uncertain about anything you've just read. I'd love it if you left a comment in the comments tab, as comments really do tell me people are finding what I'm reading entertaining, and help me to reach a wider audience.