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Wednesday, January 12, 2011

Back to Basics: Mathammer Take 2

After the last article about mathammer was published and well received by Dariokan we got another e-mail in about the stuff! So I thought why not, part 2. This article by Mathieu Girard-Reydet is a bit more advanced and has a nifty spreadsheet attached which I cannot upload to google documents I'm afraid... so if anyone knows how to attach excel sheets or upload them so everyone can access them, please let me know so this can be done! Without further ado, here is the article: Mathammer Take 2!

Hello Kirby and contributors!

I enjoyed reading the "Back to Basics: Mathammer" and wanted to add a few comments and thoughts. I am bad at statistics (so bad, in fact, that I couldn't pass any mathematical or statistical-driven modules while studying economics at university). Hopefully, W40K only require a basic understanding of probabilities so this is within my reach.

1) How to add up probabilities up to 100% ?

Let's assume, for simplicity's sake, an Imperial Guard HW team of 3 missile launchers, and try to calculate the probability to score at least 1 hit. Each guardsman has a BS3, meaning 50% hit rate each. If adding up the hit rates, i.e. 3x50% as one would assume, we would end up with a 150% hit probability. As every gamer will tell from experience, it is possible to roll 3 die and get 3 1s, so this is not correct.

The correct rationale is an iterative one. If you hit with the first dice, fine, the target is achieved. Otherwise, let's roll the second dice and check the result. Then only, if not success is scored, let's roll the third dice.

From a mathematical standpoint, it translates into:
  • Chance to hit with dice 1: 50% hit rate
  • If dice 1 fails, chance to hit with dice 2: chance to fail with dice 1 x 50% hit rate = 50% x 50% = 25%
  • Chance to hit with die 1 and 2: 50% + 25% = 75% 
  • If dice 2 also fails, chance to hit with die 3: chance to fail with die 1 and 2 x 50% hit rate = (100%-75%) x 50% = 12.5%
--> Consolidated chance to score at least 1 hit with 3 die: 50% + 25% + 12.5% = 87.5%

Obviously, the more die thrown, the better success chance. However, you notice that the marginal success factor (i.e. additional consolidated chance to score a single success with each additional dice thrown) is decreasing. This means that, once you've reached a certain critical mass of die thrown, the impact of each subsequent dice gets smaller and smaller. This is the law of diminishing returns, well known by athletes around the globe: one needs to practice harder and longer for any additional improvement in one's performance.

2) What's the point of mathhammer is real life gaming?

Mathhammer is a great tool to fuel forum arguments, but a basic understanding of probabilities can help you become a better W40K player. The gaming resolves around throwing (lots of) die, meaning that, at it's core, it is a game of probabilities. Therefore, knowing what outcome one can expect from a statistical point of view helps addressing target priority. This means, for instance, deciding what gun to point at what target depending on your objective (suppression or destruction). This is in most circumstances rather easy, and 3++ offers much well-thought guidance with this regards.

More importantly, this also means understanding whether one is placing enough firepower to realistically achieve their objective, and planning accordingly. If you know that you have a 50% chance of preventing that Land Raider full of TH/SS Terminators from smashing into your line next turn, you'd better have an answer in them in close combat. Therefore, you know you're in trouble if you don't have a tarpit unit.

Of course, the Land Raider example is a caricature, but replace it with an IG Chimera full of Melta Vets, or a Rhino ready to zoom in and contest that vital objective on Turn 5, and you get more mileage from mathhammer.

Basically, the statistical analysis called "mathammer" is a decision-helping tool. It helps you make informed decisions that are sound according to the backbone of the game mechanisms. Making those informed decisions will, in turn, make you a better gamer. Of course, deviation exists, so be aware that, sooner or later, you'll experience or witness outcomes outside of the statistical norm.

3) Mathhammer and army list building

Mathhammer helps on the gaming table, it also helps building army lists. The first and obvious effect of statistical analysis is deciding what tool is best for a specific task, or set of tasks. For instance, when building an Imperial Guard army, what is the best suppression fire tool against AV10-12 hulls? Again, the answer is easy in this case, but how about suppression fire in a Tyranid army?

Once the tools for a job are set, one must decide how many of them are required to realistically implement the desired strategy. Drawing on the previous example, how many Autocannon HW teams are needed to realistically suppress 4 AV11 hulls per turn?

I mentioned the law of diminishing returns before; it must be kept in mind when building army lists. Applied to W40K, it means that any additional unit or weapon of the same time you bring to the table will contribute towards the achievement of your strategy, but less than the previous one. It also has the hidden cost of not taking another unit which could fill a different role or strategy. The FoC mitigates this by providing tools for different strategies in separate FoC slots but this is not always the case. Necrons players know this dilemma very well: for Wraith unit taken, a Destroyer unit is forfeited. TH/SS Terminators, Stenguard and Dreadnought all compete for the Elite slots of a SM army.

Here, the root of the problem lies with the notion of "realistically achieving one's objective". A 100% achievement is obviously out of scope, but what accumulated probability represents a good enough chance of achieving a task or role? Everybody perceives a "good enough chance" differently, there are therefore as many different answers as W40K players around the world. Let's call the Law of Pareto (named after a Portuguese 18th Century economist), also called the "80/20 law", to the rescue. One can consider a 80% probability of outcome fair enough, as gaining the 20% remaining chances of output would require at least 4 times as much additional effort as what you did to achieve the 80% output. Statistical analysis of approximations abides by this, as an 80% coefficient of correlation (R=0.8) is the minimal milestone of any meaningful regression.

However, calculations reveal achieving an 80% probability of output is very resources (points)-consuming, to the points where it may because detrimental. I will therefore suggest that any probability around or above 67% (basically, 2 chances out of 3) is a "good enough" chance of success, although your mileage may vary.

This means, for instance, that it takes 4 autocanon to realistically suppress an AV 11 hull (69% chance of suppression) while an IG HW unit armed with autocanons only has 58% is suppressing said hull (roughly equal to the chance of a Lascanon-armed HW unit). Being to effectively suppress 4 AV 11 a turn means
  • either investing a lot into Autocannons as each HW unit has roughly 60% of success at its job
  • or supplementing with with more effective weapons such as meltagun, tough the shorter range requires a different strategy...
To summarize, statistics are important to understand
  • what you can expect from the list you're designing, and tailor your strategy accordingly
  • how to optimize target priority, depending on the tactical situation during the game
You'll find attached a file that I've designed to assess probabilities to either suppress or destroy vehicles based on the weapon chosen. It's far from being complete so far as it was started with the Elysean army in mind, but I'll add more weaponry as time goes by.

Mathieu Girard-Reydet
Again a big thanks to Mathieu (and Dariokan) for these articles which give a good insight into the math behind the game without making people go too fuzzy brained. Hopefully can get that spreadsheet up soon!

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