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Mathhammer you have not seen before!

Joined
Feb 6, 2012
Messages
44
#1
I noticed that all mathhammer posts I have read so far on this and other forums always compute the Expected Value of result of a close combat or ranged attack etc.

I have yet to see a thread which takes into account the VARIANCE

For those not entirely sure what the word means, here is an example.

Suppose we have a unit which has a ranged attack which can wipe out 216 wounds worth of enemy models anywhere on the table top with no saves of any kind allowed. For this attack to work the player must roll 2 sixes on 2 dice (ie. 1/36) chance.

Suppose also that we have another unit which can wipe out 6 wounds of enemy models anywhere on the table top with no saves of any kind allowed. This attack auto hits always.

Using conventional mathhammer we would get that both units do 6 wounds on average. While it is not clear which unit is better (I would tend to say that the second one is), it is clear that these are very different units. We would say that the second units damage output has zero variance (its always 6), while the other unit has a very high variance. (almost always 0, and very rarely 216).

How does this apply to real-life (ie. actual warhammer uints)? Consider our beloved 20 grave guard 5 wide with hand weapon and shield vs 5 frenzied chaos knights charging our grave guard. The mathhammer would tell us that the knights cause 5.6 wounds and we cause 1.2 wounds. We lose by about 4.6. However, we don't need math hammer to figure out that most (if not all) kills we get against the chaos knights will be because of killing blow. Therefore, this combat has an EV of 4.6 but also a high Variance. (if we don't do any killing blows we will get crushed next turn, if we cause say 2 killing blows we will proabably kill the knights next turn).

What does all this mean? In general, it is better to have low variance then high variance. In some situations though high variance tends to favour VC (or any other unbreakable army/unit) over live armies. The reason is simple. Say that on average our unit X loses to our enemies unit Y by 1. If this particular combat would have no variance, we would always lose by 1. However, if the variance was large (ie. the results oscillated a lot), then (for example), the result would be between us winning by 6 and our opponent winning by 8. If we lose by 8 we have now lost 7 more models then expected (from combat res anyway). This is bad but not a disaster for us usually. On the other hand, if our opponent losses by 6 and is not stubborn/steadfast he/she might have a problem. This means that grave guard (for example) are slightly better then mathhammer would have us believe because of their higher variance. It is possible to take variance into account exactly when computing the effectivness of a unit via traditional mathhammer but would be very tedious.

Our wraith's chill grasp attack is very interesting because it is a very low variance attack compared to its regular attacks. Given a particular situation during a battle, we may opt to use one attack or the other not only based on the Expected value but also the need for higher/lower variance, or just whatever we feel like ;)

Our (almost useless I think) Master Strike is even worse then previously thought becaue of it high variance.

I could expand more on the mathhammer but I am getting a little sleepy and don't even know why I wrote all this...
 

Tawg

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Jan 16, 2012
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#2
Yeah, Math-hammer often chooses to ignore things that could go a far bit in explaining things better.

I often find the best way to figure something out is just to see it in action, since then you get a feel for it, a sort of knowledge that is not easily obtained by simply crunching numbers.
 
Joined
Feb 4, 2012
Messages
14
#3
@Wojmir
Very good points. I have actually been looking for ways to compute some of those numbers. Have you tried doing it in excel or mathlab?
 
Joined
Feb 28, 2010
Messages
165
#4
my maths isn't very good. i see warhammer as an advanced scissor paper stone game. The only time i use maths is when i add magic items and choose magic. Ie. Sword of swift slaying is good on a vampire but for the same points i could put it on my lord but get more out of it.

I definatly do agree though. If i ever did go down the dark immersion destroying role of mathammer i would definatly look at consistancey. Much easier to plan tactics
 

N1AK

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Apr 21, 2008
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#5
I think most comparisons ignore variation because the complexity added provides less benefit than the complexity added detracts. You're absolutely right that in general players prefer to minimise variation (less randomness to foil plans) but that in other, rare, situations threat range (generally extended by high variation) is beneficial.
 
Joined
Jan 14, 2012
Messages
212
#6
rusty said:
@Wojmir
Very good points. I have actually been looking for ways to compute some of those numbers. Have you tried doing it in excel or mathlab?
I thought mathlabs were for making drugs not crunching numbers. Har Har methlabs :thumbsup:

Poor jokes aside,
I see mathhammer as quite a valuable resource when comparing units in your own army book, but not as valuable when trying to find an average result from any given combat. (too many variables from possible magic etc)

Mathhammer comes into play a lot more in 40k than fantasy (from my experience atleast). I enjoy mathhammer as a tool to put myself in good meta-game positions with versatile and or available units/accessable magic support.
Army fluff is the counter to mathhammer. Not every fluffy army is necessarily crunched down to mathhammer standards of army list building.

My word of warning is to play with models you like rather than only using ones that have the best "mathhammer scores" :thumbsup:
 

The Dead of Night

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#7
TBH I don't necessarily use math to pick which units I take, I'm more likely to use it to see the likely outcome of combats. If I'm going to charge a particular unit of mine into the front of a particular unit of my enemy; what will happen?

You can get the answers from experience of course. It can take a while though. You have to set up each combat multiple times so that you can be sure it wasn't a fluke for example. Depending how often you play and against how many armies, that can be difficult.
 
Joined
Apr 14, 2011
Messages
113
#8
Variance is why Dark Elves and High Elves kick butt, along with anything else that can re-roll. Hatred keeps the variance in your favor, as do things like BSB rerolls and Cold Blooded for Lizardmen. Even taking that into account I rather agree with drmooreflava: including a full breakdown of the variance is likely to result in a lit of people loosing the real meaning within the math, getting confused/bored, and wandering off.

Pyre
 

Dklyn

Crypt Horror
Joined
Dec 21, 2009
Messages
550
#9
It's always important to recognize how variation in one part of the combat (such as random number of attacks) can drastically alter the remainder of the combat. Variation increases as the total number of rolls you need to make decrease (and of course with less rerolls) but because of the generally high number of attacks thrown around in 8th, it's usually not too big of an issue. In essence, you're looking at variance as the opposite of reliability which is usually the case. You can actually ignore a fair bit of variation since it, in theory, has equal chance of being good or bad (or at least bad or neutral).

In your example, I think the issue isn't so much not accounting for variation, but misrepresenting. The double 6's causing 216 wounds is useless expressed as a wounds/turn (as it's not representative of how it works). Rather than that, I would say this attacks works 1 out of every 6 games (and certainly cleans the table when it does!)

You can actually calculate the effect of KB, reroll and poison fairly easily (take 1/6th of the successful hits, and count as wounds that failed AS, take the 5/6th of the hits and treat as normal, multiply by chance to wound, chance to fail AS, then add to the 1/6th from earlier) and its fairly predictable (in my experience), i.e. I would say its fairly reliable (i.e. low variance). While I agree there are lots of aspects of the game that have "high variability", I don't think KB (or poison) fall into that category. I would say it's easier for it to benefit from good luck, but not by enough for it to be hard to rely on. Using your example, GG would inflict (assuming no burrows and champ included) would be 1.72 wounds (I think knights have 2+ as?), not really all that much higher on the whole. Even if you got lucky and landed 3 KBs (instead of the predicted 1), the odds are nearly the same as landing 3 wounds that resulted in failed AS.

How our army reacts to "variance" is well, kinda variable. In essence, I think what you mean by this is how well our army reacts to a bad CC phase and I think it actually depends more on unit size than on anything. If a varghulf loses combat by 5 (or more) he will die (0 variation, or 100% reliable). If an enemy single model loses by 5 or more, he still has a chance to stay (at least 1/36, ~1/18 if near BSB), and even if he flees only a 50% chance of losing the model (of course, the results of fleeing would be an instance of incalculable variability as it can be really bad, or neutral, or maybe even good depending on the table). In any case, the lone model has a chance to survive compared to ours which doesn't.
While lone models are definitely disfavored, the R&F are more favored the more models you have in a unit (since that means a bad CC will result in a lower % of the unit wiped out... this is why lone models are disfavored, they only have a few wounds).


Having said all this, there is a fair bit of unpredictability about the game, which is why it's more fun to play than doing stats hw. In fact, I tend treasure the moments that defy the odds. I think it's valuable to recognize what items/rules have low incidence of working, but high payouts and even include a couple in the list (just in case they save your bacon ;) ) but not overload on them, lest you get trampled in the likely scenario of nothing working (see my friend's skaven lists :D ).
 

N1AK

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Apr 21, 2008
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#12
Dklyn said:
You can actually calculate the effect of KB, reroll and poison fairly easily
Actually Poison and Killing Blow are two the rules where (when used to best effect the variance is key).

Whenever you're rolling a lot of dice, most results do nothing and a rare one (6 for poison or KB) has a big impact variance has a big role to play.

2 Ghouls charging a Giant should get a wound (mainly due to good odds of getting a poison hit) however the odds of not wounding at all are still high.
 
Joined
Jan 2, 2010
Messages
43
#13
The trick with using parametric statistics to explain/predict results in a variable based game is stats can be adjusted to show what ever you like in reason of the specific test used. low statdard deviation (1) will produce highly variable results and be less reliable while high standard deviaton (3) will most likly tell you what you need to hear. maybe not what you want to hear.

In short unless your a stat PHD or particular brilliant Masters of science on the subject there are far too many variables to account for. K.I.S.S; is the best way to look at mathammer (Keep it simple stupid).
 
Joined
Feb 8, 2012
Messages
19
#14
I think Mathhammering has it's place but I math and dice IMO are two different things.

Take Craps for example: If you don't throw the dice a certain distance then your roll is no good. Why is this? The further a dice rolls , the more bounce it has , the less likely the person who rolled the dice can have an outcome on what the dice end up.

In 40k/WHF I see the following:

1. A player will pick up his dice, shake them in his hand or a cup and then carpet bomb the table.

2. Another player will pick up his dice off the table, won't shake them in his hands and will drop them a few inches from the table.

What's the difference between the two? Player 1's dice will roll around and whatever the dice end up on, was completly random. While Player 2's dice won't bounce as much thus the outcome isn't as random.

In addition, I wonder if the surface of a table matters?

I don't know if the above really has a difference or not but I do believe that how a player rolls the dice or drops them does have an effect, albeit minimal that it skews the statitics of mathhammering.

Lastly, you can never discount LUCK.

It's good to know what the odds are or have a good idea of what they are before you charge a unit or allow yourself to be charged but the dice gods don't always smile upon you.
 
Joined
Feb 11, 2012
Messages
3
#15
What simulation gives you is the probability of winning combat. So you may have a skeleton horde that, on an average turn loses by 2. However if it wins every 3rd turn by 2 (maybe fear does its job for once) and that gives it a 50/50 chance of breaking the enemy then you've a very different outcome being forecast than "lose by 2" - you may even have near a 100% chance of winning the combat.

To do that properly we'd need to simulate the combat in its whole, rather than just look at the average performance of a unit for a round of combat. However WFB is pretty complex so the simulation isn't that easy to do once you have to start worrying about crumble, leadership tests, parry (which is subltly different to regen or wards), base sizes, charge bonuses etc. I've done odd bits and pieces but its tricky to get it accurate.
 

rothgar13

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Jan 26, 2012
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#17
Variance adds a whole lot of mess to the typical math-hammer calculation. At that point, Excel and Matlab are your best bets for getting good data. Personally, I find that it's overkill - rough ballpark numbers tell me most of what I need to know. Variance would be nice, but anyone with a decent grasp of statistics and probability can pick up on the fact that re-rolls shrink your variance and are thus underestimated by the average outcome, for example.

As an aside, the fact that math-hammer posts and threads always attract posts from people who don't appreciate or understand it is a never-ending source of amusement for me. Don't you ever change, you glorious math-hammer bashers. You give me a chuckle every time. :D
 
Joined
Feb 6, 2012
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#18
DoubleSkulls said:
What simulation gives you is the probability of winning combat. So you may have a skeleton horde that, on an average turn loses by 2. However if it wins every 3rd turn by 2 (maybe fear does its job for once) and that gives it a 50/50 chance of breaking the enemy then you've a very different outcome being forecast than "lose by 2" - you may even have near a 100% chance of winning the combat.
This pretty much summarizes what I wanted to say.

By the way I was drunk :suck: when I worte my post so sorry to all if it does not make sense :)
 
Joined
Feb 11, 2012
Messages
3
#19
Tim said:
See it as poker, you can have the worst hand but still bluff a win out of it.
Its a bit like Texas Holdem, you know enough to have fairly good odds, but they are odds not facts. So not really bluffing, just knowing the relative strength of your hand and playing the odds appropriately (so betting the game on a combat you expect to lose 9/10 isn't as good as betting it on a combat you are 50/50!)
 

Count Darvaleth

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#20
Interesting. I haven't ever really considered mathhammer when writing my own lists, but have sometimes used it to work out specific actions when discussing in-game tactics.

Looking at variance might be erring on over-the-top, but at least it's not as bad as that guy who tried to design the mathematically perfect Space Marine list a few years ago, using all kinds of formulae. He ended up stopping because there are simply too many variables, but the sort of half-born list was completely wacky; filled with Land Speeders using flamethrowers, IIRC, and even had a Thunderfire Cannon. :slapface:

Variance is possibly too deep, but overall reliability is a good thing to look at.
 

Bishop

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#21
It's good to know what the odds are or have a good idea of what they are before you charge a unit or allow yourself to be charged but the dice gods don't always smile upon you.
This is important. This should be taken into account during game play. It will allow you to quickly (if you are good at math, or have practice) figure out IF you should charge something; or double charge something. Or if you get better (on average) results vs. Unit A or B, if there's more then one target.

Yes, it's the average / expected result... and yes there will be time when it will be grossly wrong. However, over the course of time it should average out.

I don't do too much mathhammering during list construction. That should be more about tactical flexibility and overall strategy.
 
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