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Comprehensive bolt-thrower mathhammer: Bolt or Volley 
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Slave on the Altar

Joined: Sun Nov 11, 2012 3:41 am
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Since your bolt throwers work the same as ours, I thought I would post a link to a mathhammer I put up on the forum of your favorite neighbors, the Asur.

http://www.ulthuan.net/forum/viewtopic. ... 60#p784160

It gives a detailed comparison of bolt vs volley for various armor, toughness and wounds values of targets.

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Erik


Sun Nov 11, 2012 3:44 am
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Thank you eriktheguy, to come and pay us a visit. :D
Welcome to D.net and don't you forget to bow to the True King. ;)

There's a difference between RBT and RBT: some work vertically while the others work horizontally. :roll:
Otherwise, you might be interested in some of our latest discussions in RBTs worth a Shot?
with some hyperlinks included.



Myself, this is what I advocate:

Simplified rule:
Just multishoot everything except T6 monsters with armour.

If you want the more detailed rule, here is it:
The Rule of Multi Four, Single Five:
Add the Toughness and the Wounds of the target (the higher, the harder).
Subtract its Armour Save (the higher, the easier).
The higher the total, the harder the target!
If the result is 4 or less, multi shoot. If it is 5 or more, single shoot.

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Sun Nov 11, 2012 5:10 am
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Slave on the Altar

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The linked thread has been updated to show some interesting figures. How likely are you to penetrate 1, 2 or 3 ranks against various targets? Also, bolts work better than volleys on brettonians.

This point on Brettonians violates your rule of 4s and 5s. In fact, your rule tends to send bolts against higher wound enemies which is a mistake. Bolts are great against any enemy with a low W score in ranks, because the chance to penetrate ranks against 1w models is much higher. For this reason, bolts are better against Brets, standard T3 2+ armor knights, empire knights, and chaos knights (assuming you shoot at ranks or down the flank). These situations violate your rule. Your rule correctly predicts that bolts are better against WoC Juggernaught knights, demigryphs and arachnaroks. It fails for several monsters against which bolts are better (hydras, HPAs, anything with high toughness and no armor).

Regarding the orientation of your RBTs vs ours, I think I prefer horizontal. Easier to load and less to block your view as you pan across the horizon. I popped over to the thread you linked. I think the RBTs make our lists in different ways than they do yours. Your core has much better shooting, and already gets some armor piercing potential. High Elves are forced to make up for these weaknesses with RBTs in a shooting based list. In any case, our magic contains some great shooting support.

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Erik


Sun Nov 11, 2012 7:23 pm
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Modelling wise, I agree with eriktheguy. The vertical RBTs are unbalance and look like they would shoot either directly into the ground or straight up in the air.

All mine are the older style. Not that I ever actually field them ;)

eriktheguy wrote:
Bolts are great against any enemy with a low W score in ranks, because the chance to penetrate ranks against 1w models is much higher.


Are you factoring in chance to hit? At long range you're only hitting once with every two shots on average. With a single bolt, that gives you a 50/50 chance of even getting to the to-wound roll, but with 6 bolts you are likely to get three hits.

Mathematically speaking, at long range (4+ to hit) against T3 2+ a volley will score 1 wound, compared to a single bolt having slightly less than 50% chance of scoring a single wound.

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Sun Nov 11, 2012 9:02 pm
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Slave on the Altar

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Brad wrote:
Modelling wise, I agree with eriktheguy. The vertical RBTs are unbalance and look like they would shoot either directly into the ground or straight up in the air.

All mine are the older style. Not that I ever actually field them ;)

Are you factoring in chance to hit? At long range you're only hitting once with every two shots on average. With a single bolt, that gives you a 50/50 chance of even getting to the to-wound roll, but with 6 bolts you are likely to get three hits.

Mathematically speaking, at long range (4+ to hit) against T3 2+ a volley will score 1 wound, compared to a single bolt having slightly less than 50% chance of scoring a single wound.

Most bows are fired only from the verticle position because it is important to aiming; I've read online that bows are only fired horizontally in movies and games. Crossbows are almost always fired horizontally. Key reasons include easier loading (the bolt sits on top of the bow) and easier to hold (you can hold it from below more easily without the bow or string getting in the way). You can also fire a horizontal crossbow more easily from a prone position or from behind cover.

You're absolutely correct that if you want to calculate the chance of inflicting at least 1 wound, taking the to-hit chance into consideration is extremely important. All of my presented figures assume 100% hit rate. The average number of wounds does not differ between the two firing options from the to-hit chance. You can multiply any 'average number of wounds' or 'chance to get at least X wounds' value on the tables I provided by your hit chance, to get a new 'average wounds' or 'chance for X wounds' value that is accurate.

If you're shooting a target with only 1 or 2 wounds left, then it would be worth crunching some more numbers to see whether bolt or volley is the best option. I expect that for light infantry the volley is always the best option, period. The volley would also be a better choice for poorly armored monsters. There might be some uncertainty with armored monsters and various forms of heavy cavalry.

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Sun Nov 11, 2012 10:24 pm
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eriktheguy, let me say, first of all, that I think it's absolutely fantastic that you try to bring your topic on here as well. Considering the work and quality of the work done, it's great you're willing to share it with a fluff-rival :)

You made a table with "chance to kill at least X" for the single bolt, but you didn't do so for the volley shots. And that, I think, is the most interesting part since that allows one to calculate the best odds of killing 1, 2 or more. A single bolt may score higher on average, but with an "all or nothing" single shot you automatically have a bigger chance to get nothing in return. There's 2 remarks on that:

1) An important aspect to include is "chance to fail". While it may be nice to kill 3 models in a single shot, you have to take into account the chance to get nothing. Part of this is visible in your table with "chance to kill at least one". If you take the inverse of that, 45% in case of the bret Knights, you see the chance to get nothing from a single shot.

2) It would be interesting to compare that to volley shots. Now, to calculate the chance to fail on a volley shot is fairly easy. Just take the chance to fail, power 6.
But we can take it a step further... Using a binomial distribution (I'm sure you know it) you can calculate the chance to score 1, 2, 3 or more wounds. And by consequence, the chance to score "at least X" wounds. This is important in strategy. It sometimes happens I just want to thin out a unit and go for whatever gives me the best average kills. But sometimes I really need to get that 1 or 2 kills to take the edge of a unit.

I built this handy tool to help out with such math:
http://warhammer.orderoftheathanor.eu/C ... lator.html

Now, let's say we fire 6 shots at bret knights on close range. Then we have a 77.86% chance to kill at least 1. That's better than the single bolt. We have more chance to kill at least 2, and more or less the same chance to kill at least 3.

So how comes the bolt still scores higher on average? Because, once it hits, it has a chance to kill more than 3 with better odds than a volley. But ... The odds of getting that are less than 10% which isn't something I'd dare to rely on. Now the single bolt has a 45% chance to fail completely, whereas the volley has 22.14% chance to fail completely. That's why, even against Bret Knights, I might favor the volley.

Anyhow.. That's why it's interesting to compare "chance to score X kills or more" between both options :)

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Sun Nov 11, 2012 10:58 pm
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Slave on the Altar

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Since there is demand for it, and using some good ideas posted by Daeron, I think I'll get to work on some numbers for 'chance to kill at least X' for the volley option.

Against knights with 2+ or better armor. Assuming the usual 1/2 hit chance, 2/3 wound chance and 1/2 armor chance, the volley shots wound 1/6 of the time, and therefore fail 5/6 of the time. The chance for all 6 shots to fail is (5/6)^6 = 33%. For a bolt you hit 1/2 the time and wound 5/6 of the time, for a fail chance of 58%. Volley wins for guaranteeing at least single wound.

Against 1+ armor the volley gets through armor 1/3 of the time, giving a 1/9 to wound chance, an 8/9 failure rate, and an (8/9)^6 = 49% chance for 0 wounds. Volley still beats bolt.

If we add T4 to the 1+ armor we get a 1/12 chance to wound, a 59% chance for 0 wounds, effectively a tie with bolts.

T4 with a 2+ armor save is a 1/8 chance to wound, a 44% chance for 0 wounds, volley beats bolt.

Probably worth rearranging the spreadsheet to remove some less interesting dimensions/information and add this.

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Erik


Mon Nov 12, 2012 12:22 am
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D.R.A.I.C.H.ed.
eriktheguy, you can claim that you've reached the hall of fame of the Druchii!
We might keep you alive here a little longer, he hee...

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Mon Nov 12, 2012 4:46 am
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Slave on the Altar

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Calisson wrote:
We might keep you alive here a little longer, he hee...
I've not read much Druchii fluff, but I've read enough to know that this isn't good...

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Mon Nov 12, 2012 5:06 am
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eriktheguy wrote:
Most bows are fired only from the verticle position because it is important to aiming; I've read online that bows are only fired horizontally in movies and games. Crossbows are almost always fired horizontally. Key reasons include easier loading (the bolt sits on top of the bow) and easier to hold (you can hold it from below more easily without the bow or string getting in the way). You can also fire a horizontal crossbow more easily from a prone position or from behind cover.


When I say the Druchii RBT is unbalanced, I mean that the upper arms are about twice as long as the lower arms. This not only looks incredibly ugly but would unbalance the tension in the strings, making all shots fly upwards at a fairly extreme angle. It would be all but impossible to achieve any hits with this weapon except through sheer luck.

IMPORTANT NOTE: I am not a physics or ballistics expert, so I may be 100% wrong and would be happy to be proved such. Nothing, however, will make me believe that it isn't a hideously awful figure.

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DE: 1/0/0
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Mon Nov 12, 2012 9:24 pm
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Slave on the Altar

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Updated with some corrections, volley IS better against knights than a bolt. Oops. Added figures for bolt firing through ward and regen saves.

Will soon update with chance to do at least X wounds with volley.

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Mon Nov 12, 2012 11:21 pm
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If you want a hand with the maths, I'd gladly help. Though.. my experience is that many mathhammerers like to compare notes after the results :)

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"I move unseen. I hide in light and shadow. I move faster than a bird. No plate of armour ever stopped me. I strike recruits and veterans with equal ease. And all shiver at my coldest of whispers."
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Mon Nov 12, 2012 11:39 pm
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Slave on the Altar

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Daeron wrote:
If you want a hand with the maths, I'd gladly help. Though.. my experience is that many mathhammerers like to compare notes after the results :)

Macro enabled xlsm file: http://www.mediafire.com/view/?mb8slx0y5vb3sla
You might need office 2007 or later to use it. Has all the math, and the functions I used to produce them. Use alt+f11 to look at the functions I defined (that's how I view them in office 2007 anyways). You might have to enable macros and I don't know if your computer will think it's a virus.

I inserted functions for calculating to-wound chance and chance to get a wound through armor, Wound and Armor respectively. For armor you put in the value of the save (ie 1 for a 1+).

Bolt1 gives chance to kill 1 model for given toughness, wounds, wardsave
Bolt2 for chance to kill 2 models
Bolt3 for 3 models
BoltAvg for average number of WOUNDS caused

Volley function gives avg number of wounds from a volley for given toughness, armor.

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Tue Nov 13, 2012 12:38 am
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eriktheguy wrote:
Updated with some corrections, volley IS better against knights than a bolt. Oops.
Nice try. Who are you working for? :lol:

More seriously, I made some researches years ago, I don't even remember what I used.
Like you, I came up with nice tables... which nobody but a computer can remember during a game.
This is why I tried to simplify as much as I could, even if the result is not mathematically accurate. I'm satisfied with close enough results.
My rule "multi 4, single 5" is supposed to be rather easy to memorize. In the rare cases when it predicts the wrong setting (single/volley), you can check that both settings are very close anyway.

If you came up with such a simplified rule, that would be highly apreciated by many gamers who don't want to memorize endless tables of figures, or to memorize a list of th units which deserve the single bolt.

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Tue Nov 13, 2012 4:48 am
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Brad wrote:
When I say the Druchii RBT is unbalanced, I mean that the upper arms are about twice as long as the lower arms. This not only looks incredibly ugly but would unbalance the tension in the strings, making all shots fly upwards at a fairly extreme angle. It would be all but impossible to achieve any hits with this weapon except through sheer luck.

IMPORTANT NOTE: I am not a physics or ballistics expert, so I may be 100% wrong and would be happy to be proved such. Nothing, however, will make me believe that it isn't a hideously awful figure.


I'm not an expert in this area either but i happen to have spent far too much time looking at different weapons articles online. Although it may look slightly weird with uneven arms on a bow, it does work, and quite well at that, as can be seen in the traditional bows of Japan, the Yumi, which have quite similar proportions in length to that of the bolt thrower.


Tue Nov 13, 2012 9:46 am
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