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.

Kotov Syndrome and your Death Clock

Kotov Predicted Haley 3 Nerfs. 

A recent thread on the Cygnar Facebook Group sparked me to get back into blogging. In the thread in question, A person decried Warmachine's current state as being relatively low risk, likening it to chess. Now, I'm not going to talk about what he said here, but I will use it as a launching pad to talk about the concept of Kotov Syndrome, and how you overthink your turns. In this post, I will explain Kotov Syndrome, how players fall victim to it, and some strategies to deal with it. In doing this, I hope to create a resource which players can draw upon in order to improve their time spent playing.

Grand-Master Kotov.

Kotov Syndrome 

In 1971, Chess Grand-master Alexander Kotov penned a book titled Think Like A Grandmaster, in which he described a situation in which a player in a bind considers his options intensely, when he's unable to find a way to solve his situation, he panics and makes an exceptionally poor move that could cost him the game.

Death Clock 

This concept of Kotov Syndrome translates to Warmachine perfectly; I've seen it be the undoing of many a tight game. You try to plot out that last ditch assassination run, you're down on time and can't play the long game. You end up spending just a little too long on the planning stage and clock out before you can clinch the game. Often, the things that cause this event are small things that are easily curtailed through practices early stages.

Playing Practised

One of the things new and veteran players alike fail to respect when playing Warmachine is the importance of a planned and practised deployment and first moves strategy. Some things will happen the same way every game. For example, an Amon or a Vyros1 list will almost always deploy the same way, as neither list truly cares about which pieces it loses, nor about the terrain thanks to mobility. Furthermore, certain things always occur in the same way. For example, Amon's Choir will always walk forward, put a prayer on 'jacks, and then reposition forward. With these things in mind, these simpler lists can strive to complete their deployment within a minute. Well practised players can complete their first turn in just two minutes, often meaning they've completed their first turn with a 5 minute lead on their unpractised competition.

Playing Clean

But, practise isn't just pure board positioning. Sometimes it's having the values you need to know in your head consistently. It seems like such a small time waiting for your opponent to look up his defence, but you could be surprised to learn how much time an inexperienced player might wast trying to determine what he needs to hit a target, or to damage it. You can see this in good players, as they can go to roll an attack out without even having to ask what they need to hit. After a while, you just know what you need to hit that charger. This is something you will learn by simply playing pick ups against a variety of opponents and models, and it is important that when you play these pick up games, you play them timed.

Playing Timed 

I would like to believe the biggest mistake a competitive player could make is not playing casual pick up games on death-clock. I don't play casuals on death-clock (except if I play Haley3, where I force myself to play clock in order to get the discipline to play her quickly), but unlike someone who's reading this in order to learn how to play your time efficiently, I don't play slowly. In fact, an opponent has more than once told me I play and talk to quickly on timed turns. So, if you struggle under time pressure, I highly recommend playing your games timed. Not just a few, or against comfort opponents, all of them.Not only will this improve you through practise and experience, but it will also teach you the value of decisiveness. 

A side note, If your opponent isn't up to play timed games, that's OK. You can just say to your opponent that you would like to time yourself, and that he doesn't have to be bound by the clock. Most players will be compliant about swapping the clock and give you no issue.

Playing Decisive 

Probably the biggest thing players who struggle with death-clock need to learn is to play decisive. Decide on a plan of action and play it out quickly. If you're going for an assassination run, do everything necessary to facilitate the run, and then try to follow through. If it fails, then plan out further actions. Another big thing to learn is not to tilt if your plan gets screwed up, or if you make a mistake. You compound your mistakes by ruminating on them and it is often best to laugh it off and look at a new plan of action. The less time you spend crying over split dice the more time you can cram full of activation goodness.

Conclusion

Playing under clock is probably the most beautiful thing about truly competitive turn based games. It forces players to play quickly and can really improve overall player involvement as it takes less time between each turn. The other value to mention when playing clocked or quickly, is that you can play more games in the time - I  can often get in four games in a day if I'm playing from midday to nine in the evening, and it's led to me being far faster at the game than my play quality would allow. 

There are limitations to consider, of course, for example, being to decisive can lead to you being an idiot. I am an idiot. I've several times had clear assassination runs, but tried to play too fast and missed out on easy wins against good opponents. So, take everything here with a grain; it's important to play quickly, but it's also important to play well. Just remember that you don't have to trade one to have the other.