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You could also look at wounds to points ratio based on a standard armour save. For instance, a grey hunter has a 3+ armour save, 1 wound, and costs 15 points. On average, it takes 3 wounding hits to kill him provided he gets the armour save. So, we get a wound to point ratio of .2, which isn't bad at all!

Now let's look at a Paladin. 2 wounds, 2+ armour save, and 55 points base. It'll take 6 wounding hits to cause a wound, and 12 to put him down. This brings his wound to point ratio to .218, which is better than the grey hunter, as you'll get more mileage out of the paladin before he dies. Of course, a melta gun will still ruin a Paladins day just like it will the grey hunter's.

Now how about an ork. 1 wound, 6+ armour, and 8 points. It'll take It'll take .833 wounding hits to take him down, making his wound to point ratio .104. Not great, but when you can take them en masse, they cause carnage in sheer weight of numbers. You can get get 7 orks for the price of one paladin, that's 7 guys the paladin has to chew through, but the paladin will likely walk it off in the end because he is FAR more resilient than the orks!

You can look at all sorts of numbers, but this just give you a baseline probability. Tzeentch may or may not like you rolling 6s all the time!

 Barksdale wrote:
It is common knowledge that the AC is better versus av 10 and and av11, when compared with the las. Right? Let us take a look.

This is a bad example, for three reasons:

1) The 'average outcome' approach already gave you the answer. Your table might not include it, but it's very simple to realize the difference between AP 2 and AP 4 and adjust the average outcome numbers appropriately. So that first analysis already contains all the information you need to come to the conclusion that the LC is better at single-shot kills than the LC, doing all the extra math after that point just confirms what you already know.

2) Your AV 10 example is kind of misleading in that you're using a 3 HP target when most AV 10 targets are only 2 HP. If you drop the target to 2 HP the "surprising" difference in favor of the LC is greatly reduced as the AC is more likely to inflict death by HP loss. IOW, the real result in the AV 10 case is a lot closer to the 'average outcome' answer.

3) You're skewing the answer by only looking at single-turn probabilities. What if we consider two ACs firing for two turns vs. two LCs firing for two turns? The ACs are more likely to kill the AV 10 target (even with 3 HP) than the LCs. So if we're buying two heavy weapons the math favors the AC, since with only a roughly 25% chance of either pair of guns killing the target in the first turn we're likely to be looking at two or more turns of shooting. And, again, this is what the 'average outcome' approach already told us.

So it's not the situation I asked for, where the basic approach gives you one answer but a more complicated analysis says the opposite. Instead it's exactly the problem I mentioned: looking at average outcomes gives you the answer, doing a more detailed analysis just means you spend a lot more time to get to the exact same answer.


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Ruphi wrote:
A deeper analysis- Ork boys will be shot at more than grots because they pose more of a threat. On average boys get shot at 6 times more than grots making grots more survivable because there are less chances of them taking a wound than the boys.

No. This has nothing to do with the differences between different statistics you can look at, it's just your arbitrary assumption that boyz are six times more likely to get shot at.

We are assuming that any amount of dice we roll during a game..or our lifetime..is a great enough pool for all probablities to be represented. We are assuimng that if we roll a 6 sideed die 6 times..we'll get a single 6 and a single 1. That isn't necesarily the case.

That all being said, not looking at a weapon over the duration of a game seems kind of silly. I always assume 6 turns of shooting. And once i'm actually playing I flub all my dice rolls and my mathhammering gets a kick to the junk.

ender502

 Ixidor13 wrote:
Honestly I think simhammer.com's Resiliency-per-Point is the best way to look at ability to take damage against a given weapon.

Base resilience = 1/(((To Hit) * (To Wound) * (Fail Save) * (Fail FNP)) / (Wounds))

Divide this value by the cost of the model to get its resilience-per-point (RPP)

Base resilience per point = ((Base resilience)/(Point Cost)) * 100

This will give you an empirical method of comparison between units vs. a given weapon (normalized for cost). For example: A Plague Marine has an RPP of 224.9 against IG lasguns and an Ork Boy has an RPP of 100 vs. an IG lasgun.

But at high strength/low AP, an Ork actually beats out the Plague Marine. Look at a lascannon: Plague Marine: 11.2, Ork: 30.

Wow, thanks.
That seemed to me to be the best way to rate models.


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 Barksdale wrote:
In fact, take a look at the probabilities. When taking only one heavy weapon, versus av 11, the las is over 100% better, for only a 16% increase in point cost when taken in a standard infantry squad (70/60). For two heavy weapons the las is 47% better. Three heavy weapons the las is 21% better. Four heavy weapons the las is 6.5% better. Five heavy weapons the las is just about identical, but can also deal with a ton of things that the ac cannot. For over 5 heavy weapons versus av11, weight of fire starts to have its impact and the ac becomes more effective.

Another thing to consider is that the first AC shot might blow up the vehicle. If it does, the second AC shot will never fire. This is even more of a PITA when your working with an assault cannon with 4 shots.

A few years ago I built a simulator for this. Its still 5th edition, but you can use it for the general concepts.


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 Ixidor13 wrote:
For RPP: Note that all of this is normalized by point, so while a Plague Marine is unquestionably better than an Ork boy stat-wise and equipment-wise, throwing your Ork allies against lascannons is better that using your Plague Marines against them. Not because Orks can take the hits better per-say, but because you will lose less effective strength per round of shooting against you when you throw the Orks against the lascannons.

And this works because points are a finite resource in each game.

So finding the most resilient/choppy/dakka models per point is really helpful.
There are differences between them. This is because the point values were given by game designers and not computer algorithms. They were arbitrary values assigned to each unit.

The point of the RPP/CPP/DPP exercise is to see where the mistakes were made by game designers and use those units.

 Peregrine wrote:

1) The 'average outcome' approach already gave you the answer. Your table might not include it, but it's very simple to realize the difference between AP 2 and AP 4 and adjust the average outcome numbers appropriately. So that first analysis already contains all the information you need to come to the conclusion that the LC is better at single-shot kills than the LC, doing all the extra math after that point just confirms what you already know.

I think that you are missing the point. Using a binomial model is not telling you that the las is better at single shot kills (and you are right that is obvious). It is telling you that, for a given av and below a threshold number of heavy weapons, compared with the ac the las is better (has a higher probability of) at destroying vehicles period. Whether that is accomplished by acing it with a single shot or knocking off hull points or double immo results, it doesn't matter. Except maybe 2HP av10 as you point out here:

 Peregrine wrote:

2) Your AV 10 example is kind of misleading in that you're using a 3 HP target when most AV 10 targets are only 2 HP. If you drop the target to 2 HP the "surprising" difference in favor of the LC is greatly reduced as the AC is more likely to inflict death by HP loss. IOW, the real result in the AV 10 case is a lot closer to the 'average outcome' answer.

I note that the difference in probability is about 3% (between the ac and the las), resulting in about 15% greater effectiveness for 1 ac (compared with 1 las) versus a 2HP av10 vehicle. Not too shabby.

 Peregrine wrote:

3) You're skewing the answer by only looking at single-turn probabilities. What if we consider two ACs firing for two turns vs. two LCs firing for two turns? The ACs are more likely to kill the AV 10 target (even with 3 HP) than the LCs. So if we're buying two heavy weapons the math favors the AC, since with only a roughly 25% chance of either pair of guns killing the target in the first turn we're likely to be looking at two or more turns of shooting. And, again, this is what the 'average outcome' approach already told us.

Now you would be skewing the answer by assuming that those heavy weapons did not receive any return fire during your enemy's turn, and are still around to fire a second round. With this binomial model you can say that in any given round of shooting (turn 1, turn 6, or anything in between) these are the probabilities of accomplishing X or Y with Z number of weapons. It could be extended to see what weapons over several turns assuming everything is still firing. Anyway, it sounds like a fun exercise.

 ender502 wrote:

That all being said, not looking at a weapon over the duration of a game seems kind of silly. I always assume 6 turns of shooting. And once i'm actually playing I flub all my dice rolls and my mathhammering gets a kick to the junk.

Ditto.

 Peregrine wrote:
So it's not the situation I asked for, where the basic approach gives you one answer but a more complicated analysis says the opposite. Instead it's exactly the problem I mentioned: looking at average outcomes gives you the answer, doing a more detailed analysis just means you spend a lot more time to get to the exact same answer.

Except that using expected value operators does not give you the same answer. But I guess you need to be interested in what the numbers are telling you. Don't get me wrong. I still think expected values are useful. Playing on expected values will yield decent results. Worrying about little things like maximizing probabilities in a given list is a fun exercise (for me). But beyond us mathematicians I don't think most people need to bother really.

 labmouse42 wrote:

Another thing to consider is that the first AC shot might blow up the vehicle. If it does, the second AC shot will never fire. This is even more of a PITA when your working with an assault cannon with 4 shots.

I don't really want to go into the math too much, but this is actually irrelevant to the model.

If you get a destroyed result from the first shot, in game you just resolve as such, destroyed. Mathematically, you still need to take into account what happens with the other shots, even if it has no impact on an actual event in game.

 Peregrine wrote:

 Barksdale wrote:


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Ruphi wrote:
A deeper analysis- Ork boys will be shot at more than grots because they pose more of a threat. On average boys get shot at 6 times more than grots making grots more survivable because there are less chances of them taking a wound than the boys.

No. This has nothing to do with the differences between different statistics you can look at, it's just your arbitrary assumption that boyz are six times more likely to get shot at.

It is measurable, not as definite as I put it where you have a solid 1/7 shots that will be shot at grots, as each player is different, but you can sample data from target priority and technically add it too your model.

It would just be modeled Of all shots fired, how many shots are being fired at unit X over unit Y. Of the shots on unit X would be the simulation everyone is all ready discussing.
It boils down to tactics and putting math behind it. If I have a unit they cannot answer easily, are they going to 1: throw much more fire into it where the rest of my army will kill it, or will they ignore it (Land Raiders).

The measurable calculation shifts from how many guys will die when I fire to how many casualties will I take over the course of an average game.

A non 40k example would be - choosing what type of cereal you want to eat. Then instead of looking at that you realize that you have the choice of bacon and eggs. Either way one thing is going to get consumed, but if you chose bacon and eggs over cereal you are going to consume less cereal.

 Barksdale wrote:
[

 labmouse42 wrote:

Another thing to consider is that the first AC shot might blow up the vehicle. If it does, the second AC shot will never fire. This is even more of a PITA when your working with an assault cannon with 4 shots.

I don't really want to go into the math too much, but this is actually irrelevant to the model.

If you get a destroyed result from the first shot, in game you just resolve as such, destroyed. Mathematically, you still need to take into account what happens with the other shots, even if it has no impact on an actual event in game.

Why?

Once your shot has destroyed the target, all other shots are null and void. Perhaps were talking about different things here, but if your looking to see the probability of a multi-shot weapon destroying a target, you can not give a percentage without including that variable.

In the case of a multi-shot weapon all the shots are fired at the same time. Each roll is just as likely to destroy the vehicle, which could result in multiple explodes results. All these have to be counted to get an accurate average.

In the case of single shot weapons for the simulation if the item is destroyed on the first shot great, if it takes 5 shots not so great, but they will all average out, which is what people are trying to determine. The model breaks down to more or less,
I shot did it die? Y/N
Y, good add 1 to the tally, shoot again
N, oh well, shoot again
Ok, divide kills by total shots, the more shots I have the more accurate my simulation is.

 Ixidor13 wrote:
Honestly I think simhammer.com's Resiliency-per-Point is the best way to look at ability to take damage against a given weapon.

Base resilience = 1/(((To Hit) * (To Wound) * (Fail Save) * (Fail FNP)) / (Wounds))

Divide this value by the cost of the model to get its resilience-per-point (RPP)

Base resilience per point = ((Base resilience)/(Point Cost)) * 100

This will give you an empirical method of comparison between units vs. a given weapon (normalized for cost). For example: A Plague Marine has an RPP of 224.9 against IG lasguns and an Ork Boy has an RPP of 100 vs. an IG lasgun.

But at high strength/low AP, an Ork actually beats out the Plague Marine. Look at a lascannon: Plague Marine: 11.2, Ork: 30.

I'll give a plug to this. By the way, resilience is simply the average number of shots of some reference weapon needed to kill the target. So, for a non-FNP MEQ against a BS4 Bolter, the value is 9, wheras an Ork without cover would have a value of 3.

You could pair this with a similar calculation of expected GEQ, MEQ, TEQ and vehicles killed per turn of shooting and CC to get a reasonably inclusive feel for overall effectivenss. It would be interesting to see how well these metrics compare to actual use on the tabletop.


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Another point: Barksdale's analysis is misleading in an unmentioned way. It notes that the Autocannon has a higher expectation for both glances and pens, but ignores the fact that the Lascannon is AP2, and therefore gets +1 on the damage table. That is what is driving that difference.

So, the result isn't so much about the value of the distribution against the mean for understanding a random variable, it's about being careful to account for all the relevant factors.

 MrEconomics wrote:
You could pair this with a similar calculation of expected GEQ, MEQ, TEQ and vehicles killed per turn of shooting and CC to get a reasonably inclusive feel for overall effectivenss. It would be interesting to see how well these metrics compare to actual use on the tabletop.

I'm working on this now during my downtime at work.

Earlier this week I wrote a program that parsed out all the unit values from army builder and imported them into a database. (What, you didn't think I would manually enter 500+ units, did you?)
Now, I just need to write a program to calculate Choppa/Dakka/Resiliance values from those numbers. Given my workload I expect to have time in a week or two.

Yes, I do expect there to be some differences. As I mentioned earlier in this thread, those are arbitrary values assigned by a gamer. They will be close, but like someone hanging a painting without a measuring tape -- its hard to get it excally in the center of that wall.

This thread highlights two important issues associated with applied mathematical analysis: picking the right tool for the job.

Just reporting averages can misleading. Usually in warhammer, what we actually care about is the chance to do at least x wounds. For example, if we're shooting at a squad of 5 marines, what we want to know is the probability of getting 5 or more wounds (ie wiping out the unit). Knowing we'll kill 4 dudes on average is much less informative than knowing we'll wipe out the unit 40% of the time. Glances are another good example of this: 2 glances might be pointless, what we really want is 3 or more, and 7 glances would be no more useful than 3. And yet the fact that you can get 7 glances (potentially) could skew the average.

The main problem with mathhammering is that basic averages are pretty easy for anyone to calculate and so they are used a lot. But sometimes they aren't really informing you in the way you'd like.

For people who can do basic matrix algebra and know how to use excel, a 'markov chain' (see wiki) can do almost any calculation you like (since dice rolls aren't correlated with each other). In some situations you'd need to 'nest' it.

Overall, this can be great for comparisons. If you're equipping a unit of long fangs, depending on your meta, point for point you might sometimes want missiles, and sometimes lascannons.

However, this still hides a lot of stuff. Strategy, movement, cover, etc etc etc.

 labmouse42 wrote:

 Barksdale wrote:
[

 labmouse42 wrote:

Another thing to consider is that the first AC shot might blow up the vehicle. If it does, the second AC shot will never fire. This is even more of a PITA when your working with an assault cannon with 4 shots.

I don't really want to go into the math too much, but this is actually irrelevant to the model.

If you get a destroyed result from the first shot, in game you just resolve as such, destroyed. Mathematically, you still need to take into account what happens with the other shots, even if it has no impact on an actual event in game.

Why?

Once your shot has destroyed the target, all other shots are null and void. Perhaps were talking about different things here, but if your looking to see the probability of a multi-shot weapon destroying a target, you can not give a percentage without including that variable.

I apologize if we are talking about different things. But you definately need to include all those possibilities in the model, whether it s the first shot, or the fourth shot. If you are interested, pick up a first year statistics textbook, and look up the bionomial distribution play around a bit, and try it out yourself.

 MrEconomics wrote:

Another point: Barksdale's analysis is misleading in an unmentioned way. It notes that the Autocannon has a higher expectation for both glances and pens, but ignores the fact that the Lascannon is AP2, and therefore gets +1 on the damage table. That is what is driving that difference.

So, the result isn't so much about the value of the distribution against the mean for understanding a random variable, it's about being careful to account for all the relevant factors.

This is a good point. The reason I posted up expected number of glances and pens is that is how most people are using expected values. Anyway, the real question you should be asking is not what the expected number of destroyed results for a given number of weapons or shots is, but how many weapons do you need to take to give you at least an X percent probability of destroying the vehicle.

There is a balance I think needs to be mentioned.

Sometimes I just want to take a whopping great power axe on a marine SGT. The emotional element in the game gets a little lost when choices are purely determined by statistical analysis.

Having said that I think no appreciation of unit effectiveness, the opposite of heavy maths hammer, is folly.

Final point is your own reputation among the gaming community. If your playing slow because your doing calcs, or you always show up spamming hard hitting units, I'm probably going to politely decline a friendly game.