Thursday, 27 April 2017

The Math of Warmachine Pt 1: Rolling to Hit.

1D6, 2D6 or 3D6 when rolling to hit? 

Super complicated. 

Warmachine has a beautiful system in the form of the 2D6 mechanic, allowing for a lot more dynamic in gameplqay when it comes to rolling, as well as some interesting patterns when determining what constitutes a "hard" roll to make. The maths presented here might help you to better understand when you should increase your odds, when you should boost, and when the dice should roll favourably without help.

So, what kinds of math do you need to understand? For the most part, it's just simple frequencies and percentage math. It's worth noting that there is a huge shift in dynamic between 1D6 and 2D6, but also that 2D6 and 3D6 follow a similar pattern to each other. Since we use 1D6 very little I'll just include this graph with a short explanation. This is usually called a "Uniform Distribution". 

With that out of the way, we get into the meat of this article: 2D6 and 3D6 dice maths. Unlike 1D6 math, 2D6 and 3D6 rolls can have a multitude of different rolls which produce the same result. For example, 2D6 can roll a 1 and a 5, or a 2 and a 4, or a 3 and another 3 in order to roll a six, while 1 dice can only roll a six one way. What this means is that the values are distributed less evenly and will tend towards the middle more readily; it looks more "smooth" and is usually described as a unimodal distribution. This trend is very visible in the graph to your right.  You can see that the same trend is noticeable for 3D6, and it is less 'sharp' than the 2D6 graph is. The more dice you add, the smoother this curve will become. You'll also notice that the ends of the graph slope off harder the more dice you roll. This means your chance of rolling extremely high numbers are lessened, but your chances of rolling very low dice are also lessened.

Improving The Roll

So, now we come to the crux of the issue what is the correct way to improve this dice roll? A problem with asking this question is that it isn't very sensible: If you need to improve a dice roll substantially, boosting and increasing your RAT/MAT are both equally valuable, so you should use whichever answer is most effective or most efficient, or what you have available to you at the time. This doesn't mean we can't look at how these modifications to your dice roll influence your chances, however; and with this in mind, there are two ways to increase your odds. While they are functionally the same, I've differentiated them as "improving consistency" and "improving raw odds".

 Improving Consistency.

You might've noticed in the graphs above something that I described as "smoothing", or that the more dice you add, the more your roll will tend towards the average, middle for the number of dice you're rolling. (Add 3.5 to your average for every dice being rolled to determine the average number). Therefore, you produce more consistent, higher rolls simply by increasing the number of dice you roll. You can see how 2D6 and 3D6 differ in the two graphs below. The orange line represents your chance of rolling an 8 or better - refer to the final column's orange mark if you're unsure which value you're looking for to see what you score. You can see that you only hit an 8 on 2D6 about 40% of the time, while you could hit an 8 almost 85% of the time on 3D6. The contrast is stark. 

You'll notice that these graphs are produced so that the percentages increase as you go up in value, rather than being a unimodal distribution the way they're shown in the first set of graphs. That's because, for our purposes, we need to roll higher, meaning that even though 6 and 15 are technically an equal roll, we will tend to prefer the 15. 

Improving Odds

The Alternative is decreasing the number you need to roll, either by increasing your MAT/RAT or your opponent's DEF, or both. We usually refer to the act of narrowing the gap as a "swing". So, let's look at reducing an 8 to hit to a six to hit, on 2 dice and determine what kind of difference it makes. 

 As you can see, your chance of hitting increases to about 70%. Not quite as drastic as the difference between 2D6 and 3D6, but +2 to hit still makes a notable difference. 

What's the difference? 

So, we've seen how adding dice impacts the roll, as well as how changing your roll's benchmark impacts it, but how do we tell what the tangible difference is? There's a problem with asking this question, in that a +2 to hit impacts a roll in one way that is clearly inferior to a an additional dice, but the same would not be true of a +6 to hit - The problem with comparing the two is that the added dice adds a spectrum of results, such that sometimes it will perform worse than the +2 to hit (approximately 1/6 times, when your boost dice rolls a 1), sometimes it will perform the same (when it rolls a 2), and most of the time it will outperform a +2, where it could roll a 3, 4, 5 or 6. However, if that dice was competing with +6 to hit, suddenly the dynamic shifts in favour of improving your odds. At that point, the only time where 1D6 is even on par with +6 is when it rolls a 6, and is inferior in every other instance. 

What Should I aim for? 

Generally speaking, you should aim for less than a 7 to hit on 2D6, and if you can, less than a 10 to hit on 3D6. If you're rolling an 8 on 2D6, boosting to a 6 is always advisable. Otherwise, you might be looking for a spike - As an example gun bunny spam will routinely try for 12s or may even go for 14s, in the hope that just a few of them break through. Ideally you will increase your odds, but if you're left with no other option, it's better than nothing. Always quickly evaluate the dice roll before making it, and don't get tilted if you roll poorly. Just because you should hit a boosted 8 80% of the time, doesn't mean you will hit 4 out of 5 8's. 

Exceptions to note:

Sometimes it's worth boosting even if you wouldn't normally need to. An example could be a stormclad who finished off a light 'jack in two hits, with only an infantry model in melee range. You wouldn't normally boost to hit a 7, but if it nets you an extra kill through e-leaps, you might as well. It may also not be worth boosting a 7 to hit if your normal dice attack wouldn't be strong enough to crack armour - it'd be better to take a chance on the roll, miss and buy another, than to boost to hit and fail to crack armour. 

Outgoing:

Some of this math may have been poorly explained or I may have misrepresented some of the math. For the most part, I did my best to present it clearly and concisely, but given that it is not a field of expertise for me I feel I may have made some mistakes. If you'd like clarification, or would like me to make an addendum where I've made a mistake, feel free to comment below and let me know how bad at this I am! If people are interested, I may also go into the topic of standard deviation when doing dice rolls, but since I tried to keep this (very) beginner, I decided not to broach the subject.