Math Hammer

[daedalus81]

I was recently asked about how I come up with some of my numbers and to elaborate on the math so I figured I'd make a post about it.  I'll save calculating specific outcomes like the odds for rollings two sixes in four dice for a later post.

 

So the obvious parts.  Each face of a dice has an equal probability to be rolled (assuming good dice) - 1/6 or 16.666% repeating, but we'll call it 16.7%.   If I need to roll a 3 or better then my chance is four faces (3,4,5, and 6) times 16.7% or 66.7%.  You can express this mathematically with simply 4/6 (four faces).

 

We can easily apply this stage to our hits, wounds, and saves.

 

 Let's say we have a model that does 5 attacks, hits on 3s, wounds on 4s, has -1 rend, and does 2 damage.  We are fighting an opponent with a 4+ save.

We can calculate this whole outcome like so:

Attacks = 5
ToHit = 3
ToWound = 4
Rend = 1 (I express this as a positive number for simplcity)
Save = 4
Damage = 2

(Attacks * ((7 - ToHit)/6)) * ((7 - ToWound)/6) * (1 - ((7 - Save - Rend)/6)) * 2

With the numbers in it:
(5 * ((7 - 3)/6)) * ((7 - 4)/6) * (1 - ((7 - 4 - 1)/6)) * 2

 

This part takes the number of attacks and determines the total hits.  We determine the faces that hit by subtraction the desired roll from 7 -- remember a to hit of 3+ means we succeed on 4 faces, so 7 minus 3 equals the 4 faces we desire.

(Attacks * ((7 - ToHit)/6))

This takes the total hits from the previous section and multiplies them against the odds of wounding.

((7 - ToWound)/6)

This takes the total wounds and multiplies them against the odds of passing through the armor.  As you can see we're still subtracting from 7.  So, if the opponent has a 4+ save then there are 3 faces it can succeed on.  We reduce that by 1 for the rend so now only 2 can succeed.  The odds of success is 2/6, which means the opposite of that is the odds we want. So we subtract 2/6 from 6/6 or 1, which leaves us having 4/6 effectively.

(1 - ((7 - Save - Rend)/6))

And the final part multiplies the result by 2 for the damage.

 

Mortal wounds / no armor / etc

If you want to handle cases when you don't roll a dice you change the value so that the end result is 1.  So an automatic wound would have a ToWound value of 1, because 7 - 1 = 6 and 6/6 = 1 - 100% of the wounds will succeed.

If you have mortal wounds then your rend would be a 6, but if you're computing in a formula than any result below 0 should just return 0.

If a target has no armor we can use a save value of 7 (7-7 = 0).

 

The big picture

To measure overall effectiveness I calculate a model's set of attacks 5 times.  Once against each common armor value (-,6+,5+,4+,3+).  Depending on how you want to look at things you can chop off - and 3+ for a more standard view.  But as long as you compare like to like you'll get a decent picture.  I attached a quick excel sheet with some formulas in it.

Eventually i'll have a web app that will calculate more complex scenarios with synergies et al.

https://docs.google.com/spreadsheets/d/1ov-q_VSmIpx33pvEh3zeE0RlQrbg9yGAkH2axgi5x_8/edit?usp=sharing

[BaldoBeardo]

Any chance of a binomial distribution graph for damage output...? [emoji14]

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[Lucio]

9 minutes ago, BaldoBeardo said:

Any chance of a binomial distribution graph for damage output...?

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You can only do that if you're charting the probable number of casulties a unit would inflict on a known target, as the maths above only calculate p

[BaldoBeardo]

You can do it for wounds to produce discrete data points. Gets a bit complicated as you do nCr to determine unsaved damaging hits, then divide into possible damage profiles (e.g. 1, D3, 2 etc.) then reorganise into wound order.

I was working on something similar but my scripting is VB but I only use Google sheets these days so it was a ball ache.

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[MrCharisma]

Very insightful Daedalus. How do you apply this in your gaming... are you using it to choose warscrolls?

[daedalus81]

6 hours ago, BaldoBeardo said:

Any chance of a binomial distribution graph for damage output...?

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Yes, but that is going to take some hard coding.  Some units will be really difficult (monsters with lots of random attacks and damage), but I may force average rolls on those random effects.  

1 hour ago, MrCharisma said:

Very insightful Daedalus. How do you apply this in your gaming... are you using it to choose warscrolls?

Not to choose scrolls, but to build a fundamental understanding of the mechanics of units.  

For example chaos knights come with two weapon options.  The lance obviously has a charge bonus, but is much weaker otherwise.  With this I can say that lances can produce 12 wounds in 4 rounds of combat with 1 charge (games last 5 turns and you'll need to get to combat).  Regular weapons can average about 13 wounds.  If only 3 combat rounds are fought it is ~10 vs 10.

With this I can analyze my games and measure how often I get the charge and how often I am in combat, which will help me determine (after many games) if lances might work better with my play style.  Or i can simply decide that I desire the potential burst damage regardless of any downsides.

[MrCharisma]

Perfect, thanks for that.

I run a bunch of statistical analysis on a fantasy football game but have yet to apply this methodology to gaming. 

I might run your code against my Empire / Free Guild to see what comes out...

[BaldoBeardo]

Probably the K.U.T.A. I need to dive into Google script...

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[The Jabber Tzeentch]

Good write up. I've used this method often to get a better understanding of a units average damage output compared to its maximum damage output which I think is an important consideration. Some units can seem very powerful at first until you do the math and realise the quality of their attacks is low.

[Arkiham]

Cheers for this 

[daedalus81]

Oh!  One more thing - rerolls!

Let's say you have a 3+ to hit and re-roll 1s.  We know that a 3+ equates to 66.7% (4/6).  That means 33.3% of that "missed", however, we can only reroll ones, which is 16.7% of the 33.3% (roughly).  Our to hit is still 66.7% so we multiply that against the faces that can be rerolled (16.7%), which equals 11%.

In other words we hit on 3s, but also 66.7% of our 1s become hits as well.

How do we represent this on the excel sheet?  Just the same way that 3+ to hit means 4/6 we can convert the fraction into a usable number.  If we add our chances to hit we get 77.7% (67.7 + 11).  

4/6 is 66.7% and 5/6 is 83.3% so our number is somewhere between a 4 and a 5 (3+ and 2+).  We can use simple algebra to convert it:

x / 6 = .777
x = 6 * .777
x = 4.662

But we can't plug 4.662 in, because the sheet uses die rolls to represent the odds, so we have to subtract our result from 7 leaving us with 2.338. This shows that a 3+ to hit is somewhere between 2+ and 3+.

We can test it with a single attack:

These are the values for the common rerolls:

Reroll 1s
2+	1.17
3+	2.34
4+	3.5
5+	4.67
6+	5.83

Reroll all
2+	1.17
3+	1.67
4+	2.5
5+	3.67
6+	5.16

What you see above are effectively the equivalent die rolls for those effects.  

[daedalus81]

I lied - the psychology of dice rolls

A situation came up in another thread where a -1 rend produced same damage or better than a -2 rend, but people still want the -2 rend weapon.  Let's look at an example:

.

 

The -2 rend weapon does less except where it ties on 3+ armor and does just a little better on 2+ armor.  If you're facing so much 2+ armor that you need a 12% increase in damage then by all means do it.  But other than that the -1 rend is "the better choice".

How can this be?

Let's roll 1000 attacks for each weapon:

-2 rend hits on 4s so we're left with 500 dice.
It wounds on 3s so that 500 gets reduced to 333.
Then our opponent saves on 5s (3+ -2) leaving us with 221.

1000 -> 500 -> 333 -> 221
 

-1 rend hits on 3s -- 1000 to 666.
It wounds on 3s again so 666 to 443.
And our opponent reduces (3+ -1) 443 to 221

1000 -> 666 -> 443 -> 221

Red is when you roll the dice.  Green is when your opponent rolls dice.  The same outcome, but it differs in who is in control of the dice.  People seem to prefer their opponent roll less dice instead of they themselves rolling more.  Why?  Don't you want to be more in control of the outcome?

Now most weapons are not like this and the results will be a lot closer and the choices far more difficult.  Just don't let rend (and spectacular dice rolls) blind you.

[Arkiham]

So, to add a ward save the enemy has you'd need to add an extra dice in, let's say 5+ so that's 16.7*5=83.5%. In your original equation it would be 

(5 * ((7 - 3)/6)) * ((7 - 4)/6) * (1 - ((7 - 4 - 1-5)/6)) * 2 

Correct?

[daedalus81]

No - "ward" saves come after multiplication of wounds.  So you would tack onto the end whatever passes through the ward as the multiplier.  So a 5+ would be the equation below.

(5 * ((7 - 3)/6)) * ((7 - 4)/6) * (1 - ((7 - 4 - 1)/6)) * 2 * (4/6)

[Taketheskull]

I do this sort of analysis frequently to work out the damage output of units and find it really useful, however, I would caution about getting lost in the maths.

It is easy to calculate all sorts of sick combos that will wipe your opponent off the table before they've said boo, but executing them on the battlefield is a completely different game (and the best part of i imho).

The ability to be able to deliver damage at the right time in the right place with the right buffs is going to be more decisive then mathhammering the highest damage output combo. To do this you need to take into account all sorts of factors deployment, terrain, buff delivery, movement, weapon range etc

And then you still have to get those six-sided cubes of crushed dreams and broken hope to behave!!!

[daedalus81]

Absolutely.  There are also tons of abilities that you can't put math to (well you can, but not easily).  So you'll just have to decide on your own if the Vexillor is worth the extra points.

[Taketheskull]

20 hours ago, daedalus81 said:

Absolutely.  There are also tons of abilities that you can't put math to (well you can, but not easily).  So you'll just have to decide on your own if the Vexillor is worth the extra points.

Hehe, well he is armed with a hammer...