The Difficulties of Making it Worth it's Points

Greetings Gentlereaders,

One of the biggest distinctions in talking about units in 40k, especially in light of our recent podcast, is how to make units "worth their points."  From a design perspective, this is a much more difficult task than most of us would probably think.  Listen to how our discussion grew beyond its stated bounds and began re-designing the whole game from the ground up.  Thinking more about the basic mechanics of the game brings me to think the statistic system of the game make calculations of value damnably complex.  Hopefully you'll be willing to go through why 40k is so frustrating from a mathematical perspective and how that makes designing 40k such a difficult task.  If you're not up to delving into it with me, let's just say that the core rulebook's rules causes drastic shifts in the value of weapons and armor from target to target.  That means that making a rule for determining what a stat point is worth is possible, but quite complex.

--Beginning of Math--

Okay, so the first thing we need to look at is our assumptions of what makes a unit worth its points.  A unit needs to have a certain amount of strength, toughness, leadership and an appropriate armor save of its points.  When we look at units we can compare the dire avenger to the tactical marine and see a lot of differences in statistics and rules, and that's without looking at the supplementary units that each has access to.  That's a mind-boggling question that I might go into later, but for now let's look just at the units in a vacuum.  Would the dire avenger be 'worth its points' if it went up one point per model and took on the statline of the tactical marine?  The problem with figuring that out is that we are all assuming there is some formula to determine the how much each stat point and weapon characteristic is worth in terms of points.  The problem is that that formula, as far as I can tell doesn't exist, so let's solve that.

The most basic aspect of the game to look at mathematically is damage taken and dealt.  Each of these combines WS/BS, model's/weapon's S and model's/weapon's AP values and compares them to the target's T, armor/invulnerable/cover save and possibly FnP to give us a probability that any given shot will result in a lost wound.  Again, we're leaving out things like multiwound models and instant death for simplicity's sake, so this isn't as complex as it could be.  That being said we've still got an equation with six variables to work with. Let's start building this equation with shooting.

We know that the rule for rolling to hit is that if your d6+BS is at least seven you hit.  That works really well for most models, who have BS 1-5 because you can just you have a  (BS)/6 probability to hit.  Sadly, we can't just move on to evaluating strength if we want a complete formula.  For simplicity sake we could lump all BS higher than five into five, but then we're giving some models 'free stats' because, rare as it is, there are times a BS 8 model will hit that a BS 5 model won't.  Alright, let's try to keep this moving.

Looking at strength, we know that a simple fraction won't work because we need to know the target's toughness to know anything about whether our strength value is useful (e.g. S3 shooting at T4 or T7 makes a huge difference).  Okay we know we need a number over six because we're using a d6 to roll: (X)/6.  Now we need out variables inserted: (S-T)/6, well that won't work because that means centurions can't wound each other.  So we need some constant in the mix to make it work out: (S-T+C)/6.  Being as we know centurions wound each other on 4+, let's add in a three and see if that holds in our one case: (S-T+3)/6.  Let's start with bolters; (4-5+3)/6 = 2/6 = 0.333, good it holds and does so with strength five through seven weapons, too.  The problem comes in with strength eight and above weapons ([<7>0.83], which means that if you had a weapon that was strength eight would always wound and higher strength weapons would generate more than one wound per shot.  Looking the other way, we get that our equation holds for strength three weapons, but for strength two we get (2-5+3)/6 = 0/6 and we know that doesn't work.  We should get 1/6 by game rules.  Strength one weapons should have a 0/6 chance, but our equation gives us a negative chance of wounding and that doesn't work either.

I fall to pieces...

Looking into the 'fancy mathematics' of the problem we do have a solution.  When if you remember questions like 'solve for Y if Y = 8x+7' you know what a function is.  You plug something in, mess with it and get something else out.  We're not just putting in one variable (something that can change, e.g. S, T, Sv) and we don't have a rule that works for each roll we're trying to calculate for any values we want to put in.  Now we need to get into another type of function called a peicewise function, which essentially says that we set up different rules for different values of our variables.  Looking at  our simplest test, rolling to hit  we can and already did construct a simple rule for if our ballistic skill was 1-5 before.  Now we need to write one for the rest of the scale (except zero): if BS > 5 (5 + (BS - 5)/6)/6 so if we combine that with our original rule for ballistic skill we can get one step that applies to all of values of BS.

Probability of hitting
dependent on firer's BS

if BS = [0, 1, 2, 3, 4, 5], then (BS)/6
if BS = [6, 7, 8, 9, 10], then (5 + (BS - 5)/6)/6

We get the chart to the left to display the probability of hitting based on ballistic skill.  As you can see, we have a steady return from increasing ballistic skill until we hit BS5, after which the slope drops significantly.  This shows us that, even without considering the prevalence of things like overwatch, not all points of ballistic skill are created equally.

Then we do something similar with strength and toughness, but our function needs to become more complex because strength doesn't mean anything without an opposing toughness value.  Because we have both strength and toughness that factor into our probability of wounding, we have two independent variables and one dependent.  Normally this would be illustrated in three dimensions (S, T, Prob) and look like a sheet billowing in the breeze, but for simplicity's sake let's just use our previous example of shooting at toughness five models to see how different strengths measure up.  For clarity our function would be

Probability of wounding a T5 model
dependent on weapon's strength

if (S-T+3)/6 is less than 0, then prob = 0
if (S-T+3)/6  equals 0, then prob = 1/6
if (S-T+3)/6 is greater than or equal to 5/6, then prob = 5/6
otherwise prob =  (S-T+3)/6

Just like with ballistic skill we can see that the value of each point of strength isn't created equally.  In our example, each point of strength above seven is valueless as it doesn't increase our chance of wounding.  If we were looking at a multiwound model, then strength ten would have some value, if and only if the model had multiple wounds remaining, but strength eight and nine would still be worthless.

Unlike the ballistic skill question, we can't answer how much a point of weapon's strength (or model's toughness/AV) without knowing the prevalence of its opposite.  In a world of T5, S1 is useless, and S2 and S3 are exactly the same, but that isn't how things are.  There's a wide spectrum of units available each with their own values without thinking about how many of each unit would be taken by a player.  For instance, does the existence of howling banshees effect the value of a heavy bolter as much as the existence of dire avengers?  I would argue not, simply because dire avengers are more likely to be taken and  are more necessary to winning most games than banshees are.  Without knowing how prevalent units will be in actual games we can't put a prevalence value on stats to weight them and determine the value of a point of the opposing stat.

--End of Math--

So we need to know that actually happens in a game of 40k, how prevalent charges (and thus overwatch dropping the value of additional points of BS) are, how often specific units are taken and how prevalent specific stat values are so we can price opposing stats appropriately.  But we can't forget that how we price things will change how much people take them and we need to start the loop all over again.  There are ways to finish this and find an optimal value for each stat, but that's going to take more math than I can stand right now.