Strength in Numbers: A Warmachine Math Article

Hey there Warmachine community, my name is Robert and my username on the forums is Reik. I’ve been following Warmachine since the release of MkII and while I haven’t been able to play as much as I’d like, I have been able to put a fair amount of time in to the Probabilities behind the game. I went to school for a degree in Mathematics with a focus on Statistics and am currently working as an Actuary. If you aren’t familiar with what an Actuary does it’s basically a Statistician that focuses his work in the field of insurance.

I’m writing this article because I’d like to share my knowledge of probability and statistics with the Warmachine community to help us all make more informed decisions. In episode 122 of Muse on Minis the crew talked about gut feelings that revolve around certain ARM and DEF “breakpoints” in the game. I wanted to tackle these Gut-Feelings in this article using what I’ve learned. I’ll start by going over some basic probability related to the game, then tackling the feelings.

Basic Probability

The first thing I’d like to go over with you guys is some of the terminology I’ll be using in this article. When describing the distribution of dice, one of the easiest ways is to talk about the Probability Mass Function (PMF) of the distribution. The PMF can be expressed as: p(x) = Pr[X=x] which means the probability that our distribution X is equal to some numerical value x. For instance Pr[2d6 = 7]=6/36 with 2d6 being our distribution X and 7 being our value x. By looking at this probability for all possible values of x we can see our entire distribution. The PMF of 2d6 can be seen in the graph below.

 2d6PMF

The second method to describe our distribution is through the Cumulative Distribution Function (CDF). This function can be expressed as: F[x] = Pr[X≤x]  that is, the probability that our distribution X takes on a value less than or equal to the value x. The CDF of 2d6 is represented in the chart below.

 2d6CDF

It should be noted that the CDF evaluated at x is equal to the sum of all values of the PMF at or below x. That is, the CDF at 3 would be expressed as Pr[X ≤3]=p(2)+p(3) 

Finally we’ll be talking about expected values of distributions notated as E[X]. This is technical way to describe the average outcome of an event. For example, E[2d6]=7.

The Defense-Based Gut Feeling

Due to the Law of Total Probability which states that the sum of all possible events must have a probability of 1 and the fact that our distribution X must be either less than or equal to x or greater than x, the sum of their probability must be 1. We can use this to end up with the following equation: Pr[X≤x]+ Pr[X>x]=1. Substituting in our CDF equation and moving a term around we are left with: 1-F[x] = Pr[X>x].

Let’s say we have a MAT 6 trooper trying to hit a DEF 12 model. The probability of him hitting is equal to the probability of rolling a 6 or higher. Because there are no possible results between 5 and 6 on 2d6, rolling a 6 or higher is equivalent to rolling above a 5. Therefore: Pr(Hitting Given I need x)=Pr[2d6>x-1]=1-F[x-1]. Using the chart above we can see F[6-1] = F[5] = 10/36. Therefore, our probability of hitting DEF 12 with a MAT 6 trooper is 1 – 10/36 = 36/36-10/36 =26/36. This comes out to approximately 72%. A chart below shows the probability of hitting for all results of a 2d6. This chart ignores the rule that double 1s always miss and double 6s always hits. We will continue to ignore this rule throughout the article as it only affects extreme cases

2d6Hit

If we look at the chart of our CDF above, we can see that the slope of our CDF curve is greatest at the result of 7. This is because the result of 7 is the most likely to happen at 6/36 and adding this result to our previous CDF at 6 gives the largest increase in value. In both directions our slope decreases symmetrically as we get further away from 7. This means that the closer we are to a roll of 7, the more change we see in our CDF and our resulting probabilities to hit. For example, if we compare the probabilities to hit on a 6 and 8 of 72.2% and 41.7% respectively, we see there is a drop in hit percentage of 30.6% We see this same 30.6% when we compare the probabilities to hit on a 7 and a 9. The difference between a 4 and 6 on the other hand is only 19.4%, and a 8 to a 10 is 25.0%

Because of this, the greatest change in probability resulting from a +2 bonus to hit or DEF will be 30.6%, occurring from going either from a 6 to an 8 or a 7 to a 9. Since the average MAT seems to be around 6 and the average RAT around 5 without aiming, this suggests that going from DEF 12 to 14 will results in a 30.6% decrease in your chance to be hit by both shooting and melee. This could be one of the reasons why it feels like it’s such a big jump going from DEF 12 to DEF 14.

The Armor-Based Gut Feeling

The Armor-Based Gut Feeling needs to be broken down in to three categories: Single-Wound Troops, Multi-Wound Troops, and Jacks/Beasts. In this article I will not be going in to the Multi-Wound Troop category as it’s a fair amount more complicated topic. This complication is due to the fact that damage done over their hit boxes disappears, making the focus not on how much damage they take per swing but on how many swings it takes to kill them. For instance, the change from 2d6-2 and 2d6-3 isn’t that big against 8 wound troops since it’s pretty likely you’ll kill them in 2 swings either way, but 2d6-3 compared to 2d6-4 could results in a third attack, making the model much more survivable.

Single-Wound Troops

The damage roll for single-wound troops functions identically to the to-hit roll except that the attacker must exceed the ARM instead of meeting it. Because of this, we can use the same concept applied in the Defense-Based Gut Feeling.

The only thing that makes this harder to evaluate is that the POW from attacks varies much more than the MAT or RAT of models. Against POW 13 or 14 attacks, the rolls to kill needed against ARM 19 are 6 and 7 respectively. Because of this, bumping our ARM value up 2 points yields us that 30.6% probability change that we saw earlier. However, against POW 10-12 attacks, an ARM increase in the range of 15 to 17 or 17 to 19 would be much more significant than 19 to 21, as they would be mostly immune to those damage rolls already. Because of this, and that fact that most anti-infantry shooting seems to fall in the POW 10-12 range, I would say for single-wound troops based on what you expect to get shot at by the most the biggest improvements to ARM values would be either 15 to 17 against POW 10, or 17 to 19 against POW 12.

Against non-charge melee attacks we can apply the same shooting concept, but against charging infantry we’re going to see a difference caused by the boosted damage roll. The table below shows the probabilities of killing based on the dice roll needed for 3d6.

3d6Hit

The largest change a +2 bonus to damage could cause here is 25.0% which occurs between 10 and 12. The change between 9 and 11 and 11 and 13 are both 24.1% though, which is pretty close. I honestly don’t know a good average POW for charging infantry, but I’m gonna make up a number and say POW 12 is a good melee average. Using POW 12 as our average, the 25.0% probability change occurs when you go from ARM 21 to 23, however the changes caused by going from ARM 20 to 22 and ARM 22 to 24 are almost as large. ARM 19 to ARM 21 would be the biggest gain when being charged by a POW 10 infantry model, but other than that I can’t really give any good evidence that it’s the most effective ARM increase for single-wound infantry against a charge or any other boosted damage roll.

Jacks/Beasts

Some of the concepts I’ll be using in this section actually originate from the fact that I work in the field of insurance, so I’m going to start this off with an analogy. Let’s say you’re a car insurance company that wants to protect itself from small losses. Because of this, you put a deductible of $500 on your insurance policy. Whenever a loss occurs, you subtract $500 from the loss amount and pay that amount. If the loss is less than $500 however, the policyholder doesn’t end up paying you because the loss goes negative.

Now imagine you’re Captain Sebastian Nemo building an Ironclad. You want to protect this Ironclad from small attacks, so you put 18 ARM worth of armor plating on the jack. Whenever the jack suffers a damage roll, you subtract 18 from the total damage to determine how much the Ironclad takes. If after subtracting the 18 the damage roll goes negative, the attacker doesn’t end up healing the Ironclad because the damage roll goes negative.

While this analogy might seem a little out there, deductibles on insurance policies and ARM values on models function almost identically. That is, the amount of loss an insurance company incurs follows the equation: Loss=Max(Total Loss Amount-Deductible,0) and the damage a jack or beast takes from an attack follows the equation: Damage=Max(Total Damage Roll-Armor,0)  . This allows us to apply some concepts that have been thoroughly explored for the purpose of insurance mathematics to Warmachine.  I do have to make one assumption however because deductible and ARM values differ in one way. You will never have a negative deductible, which would be a deductible that increases payments for every loss, but if your ARM value is lower than their POW or P+S of the attack then the attacker will add the difference to their rolls. Because of this, I will only be looking at cases where the ARM is greater than or equal to the POW of the attack.

The first concept we’ll explore is expected payment per loss or E[X-d]+. What this represents is the expected value of X subtracting off d, but only the positive part. That is, if X is less than d, you replace X-d with 0. This concept is equal to the amount of damage you can expect a jack to take from an attack where X is 2d6+POW and d is the ARM of the jack.

The second concept we’ll explore is called the Loss Elimination Ratio (LER). This term is used to measure the impact deductibles or other features have on insurance policies. For example, let’s say in a given year for a policy you paid $1,000,000. Going back and looking at all of the losses you incur you realize that if you had a $500 deductible on all of those losses you only would have paid $800,000 over the year, then that deductible would have a LER of 20%. That is, it eliminated 20% of the original $1,00,000 losses you incurred.

When comparing LERs, it’s important not to look at the strict difference between the LERS like we did with the probabilities in the defense section. Looking at our example above we see a $500 deductible had a 20% LER. Let’s say we look back again and see that if we had a deductible of $750 then we only would have paid $600,000 in claims, giving a $750 deductible a 40% LER. Using basic subtraction one might say that going from no deducible to a $500 deductible has the same impact as going from a $500 to a $750 deductible (20% – 0% = 20% = 40% – 20%), however that is not the case.

When comparing two deductibles, you must use one of the deductibles as the base in which to calculate the LER off of. Using $500 as the base we see that increasing the deductible removes $200,000 of our base $800,000 losses, giving it a 25% LER. Therefore, the increase from $500 to $750 has more relative effect than going from no deductible to $500. Here’s a table and graph for the LER of 2d6 assuming that the ARM of the model is at least the POW of the attack.

LER

LERChart

We see that while the LER increases at a slower and slower rate, the relative remaining loss reduced is increases at a faster-than-linear rate. Because of this, every point of ARM has a higher impact than the previous. Therefore, the statement the crew made on going from ARM 20 to 21 felt like a bigger jump than ARM 19 to 20 is definitely true. As far as ARM 19 to 21 being a more significant jump than other ARM jumps though, there’s no real support with regards to jacks and beasts.

 

 

 

Author: Reik

Robert received his Undergraduate degree in Mathematics with a focus on Statistics and a Minor in Actuarial Science. During the day he works as an actuary for a major insurance carrier, and in his free time he eschews painting his miniatures to create spreadsheets based on the six-sided die or spend time with his wife and cats. He plans on playing his paintless CoC army as soon as enough of the faction is released.

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