Math-hammer and You

What Is Math-Hammer?
I'm going to use Math-hammer as a term to describe the rudimentary application of statistics to 40k in an attempt to figure out the outcome of a discrete round of harming the other guy.

Math-hammer usually takes the form of something like this:

Number of shots * chance to hit * chance to wound * failed saves = Number of casualties.

Let's put this neat little model into action: I fire my 10 Dire Avengers into your Imperial Guardsmen that are in cover. The pertinent facts are:
1) Dire Avengers are BS4
2) Their guns are 2-shot, S4, AP5 jobs
3) Guardsmen have T3, and a 4+ cover save.

Let's turn that crank with the following data:
Number of shots = (10*2) 20
Chance to hit (at BS4) = 2/3
Chance to wound (S4 vs T3) = 2/3
Failed saves (4+ cover) = 1/2

Ergo: 20 * (2/3) * (2/3) * (1/2) =...
20*(2/3) = 13.333 hits * (2/3) = 8.8886 wounds * (1/2) = 4.44 Casualties.

Hooray! We just killed some Guardsmen!

Except Not
So, what's the problem with this model? There are a few: namely, the dice, the laws of statistics, and template weapons.

The Dice
Dice are not necessarily impartial random-number generators. Some people suggest that Chessex dice (the default, little rounded-corner boggarts) lean towards the 1 a little bit. There are also some nasty tricks you can do to manipulate dice. My personal bias involves just playing with the Chessex dice, and the hell with the bad rolls you get. Actively trying to rig your rolls is a punk thing to do, and should realistically get you kicked in the dice bag.

The bottom line is that the dice do not always behave in a perfectly predictable manner, but we can still make some basic assumptions about how they'll act any given time we roll them.

Laws of Statistics
Think about most the times you roll dice in 40k. Maybe it's an ordnance shot or blast weapon; then it's 2d6 and the scatter dice (...just TRY running some math-hammer for scatter. It's pretty rough). Maybe it's your Predator Destructor kicking out 8 shots in a turn. Maybe it's your tactical squad rapid-firing someone with about 16ish bolt-weapon shots. Maybe it's terminators on the charge with 15 attacks.

What do a lot of these have in common? It's not a lot of dice. Statistics works best with large samples; a good statistical sample has a minimum of 30ish trials. Some units can indeed pull that off in a single action (IE: Orks shooting/charging, Fiends of Slaanesh in assault, 30-man guard squad w/ First Rank Fire, Second Rank Fire), but most cannot. Some units cannot throw thirty dice in the course of the game, even if all seven turns happen.

What's that mean for you? Even through the course of a game, you cannot necessarily expect 'average' performance out of a given unit.

Template Weapons
Now, if we take a basic weapon that has fixed chances of hitting, we can run the rest of our formula from there, yes? Same goes for assault. The problem lies in predicting how many hits you can slap on a target with a flamer or blast weapon. Templates have to take into account how close models are, and the formation they are arranged in. If everyone is standing shoulder to shoulder with a 2" max coherency spread, it's a much different picture for blast weapons than if they're all in base-to-base contact with each other like a fresh Deep Strike formation, for example.

The best we can do for Blast Weapons is model their scatter, armed with the knowledge that 2d6 averages about a 7, and there's a 33% chance of getting a 'hit' result on the scatter dice. So, right off the bat we know a blast weapon will hit 33% of the time. 33% of the time, scatter will then average 7-BS, so we have some idea how far the center dot might go. About the only place we can use this knowledge is when we try to apply it to vehicles and how big their hull is; we have a better chance of landing a blast template on a Land Raider than a Killa Kan, for example.

Why use Math-hammer, then?
Why, indeed? Because it gives us a starting point for making decisions. Common sense tells us that one Guardsman is unlikely to beat up a thunder hammer storm shield terminator, but it IS still possible. On the other hand, some things get a little closer. Math hammer gives us tools to make a guess at the outcome of shooting or assault, and that's at least something to base your decision on.

The numbers are not always accurate. However, it's a darn sight better than just using wishful thinking and sacrifices to the 'dice gods' as a means of making decisions with your army's firepower.

Using Math-Hammer intelligently
Here are some suggestions for using Math-hammer to your advantage:

1) Know the limits of the stats
-You need 30 rolls to get an accurate 'sample size'; if it's below that than you're not really guaranteed an 'average' result

2) Know the limits of the dice
-Dice are not perfect random-number generators, either through manufacturing processes or human interference a la practiced rolling.

3) It's Still Math
-Math, when done right and mindful of its problems, is STILL a reasonably reliable when you follow the steps.
-At least, math is a reliable tool given an understanding of the tool. It's like a computer: it can only do what you tell it to do, how you tell it to do.

4) Know what the 'Average' means
-The 'average' result is the midway point, which means that IF you roll enough dice, THIS is what happens about 50% of the time.
-What you're really dealing with is a Normal Distribution (aka bell-shaped curve)*. You're hoping that your dice will land somewhere in the middle, but it is still entirely possible to see a given set of rolls fall way the hell away from average (hey, it's possible to see 5 1's, [0.01% chance, theoretically] and you have the same chance of seeing five 6's in a row as well)

5) Know that Things Can Still Go Wrong
-The dice are still fully capable of throwing complete outliers at you. We call this 'the wheels coming off', when you fire five bolters into a single guardsman in the open, and he still lives, for example. (Worse still, he kills 2 guys on the charge, and lives.)
-In other words, you care about math, but math doesn't give a flip about you, and can screw either of you over, or both of you, or neither of you. Crap happens. Deal with it.

6) Realize the limits of Math-Hammer
There are things Math-hammer simply will not do, and they are:
-Tell you WHERE to move
-Tell you what your opponent will do
-Gauge range
-Tell you how many of your infantry will actually be in range of your target
-Tell you a blessed thing about how many people you actually will hit with a template-based weapon

Math-Hammer CAN give you an idea on:
-How much fire you need to kill a target
-How much fire you need to force a morale check
-Whether or not you might make assault (remember, you just need base-to-base with ONE guy)
-The chances of immobilizing yourself in terrain
-The chances of you winning/losing an assault, and potential morale results

Conclusion
Math-hammer is a tool. No more, no less. It's not perfect, but it's a workable enough basis for making decisions about dice in the game. As such, I suggest a working knowledge of it, since this IS a dice game, and Math-hammer DOES tell us a little how the dice work. It's a more reliable basis than all the little pre-concieved notions that your brain tends to have.

Any kind of theory makes assumptions. The more you know about the assumptions, the more use you can get out of the theory. At the end of the day, theory itself is a tool. Theory simplifies the world, loses some resolution, and usually gives you a (reasonably reliable) method of making decisions and/or learning about an aspect of life, the universe, and everything.

Math-hammer just gives you a rough idea of what can happen when you throw the dice, and that's better than nothing, in my book. There are days where the average will happen every single roll, there are days where the dice will roll sub-par on a game level for you, but in the end, across enough games, the dice will roll average.**


*Disclamer: I'm being a little lazy here and linking to Wikipedia. Note that since Wikipedia is open to anyone who wants to edit it, anyone can (and in some cases, has) tweaked the data in an incorrect manner. If you really, truly want solid info, wikipedia is a good start, but then you'd better check out the sources they've listed in the article, and move from there.

**Unless you're a bloody dice cheat. Or the other guy is.

Your Own Worst Enemy: Cognitive Bias and 40k

Sorry about the pace of posts this month; I blame it on work + end-of-semester papers. Brainpower's at a premium, and I'm giving a nod to National Novel Writing Month* as well. Today, though, I'll be highlighting some of the ways the mind works, and how that impacts 40k.

Lazy Mind
Basically, the mind likes to take some shortcuts. A stereotype is a ready example, and we can all call to mind an easy one like 'jocks/blonds are dumb.' Is it true 100% of the time? No. Sometimes, though, a stereotype is true enough often enough for it to save us mental energy, and that's the core problem: the mind and thought tries to be as efficient as possible, and this is a fine way to cut corners.

Disclaimer
First, it's important for me to tell everyone that this is a very, very basic overview. Reading this is not an equivalent of a cognitive psychology course. It's food for thought, not qualification for diagnosis. Much of this is skimming theory, and I offer it for purely as food for thought.

Memory Biases
Your memory isn't always perfect. How? Here's some examples.

Choice-Supportive Bias: If you chose it, it had to be better than other options

Consistency Bias: You think that you always acted the way you act now

Ego-Centric Bias: Your memory is a little self-serving, you think you looked/did better than you actually did

Suggestability: Suggestions become memory of something that you actually experienced

Decision-Making Biases
Bandwagon Effect: If Everyone's Doing It, then Everyone must be onto something...

Base Rate Fallacy: IGNORING STATISTICAL DATA in favor of anecdotes

Bias Blind Spot: Tendency NOT to compensate for your own biases

Confirmation Bias: Selectively attend to data that agrees with you; essentially it's that you've made up your mind and are getting information to back you up and ignoring info that doesn't.

Deformation Professionalle: Tendency to look at things through your own profession and forget about a broader POV (...irony? Maybe, maybe not. I choose to ignore it, though.)

Expectation Bias: similar to the confirmation bias; you look for data that backs your point and ignore/attack info that disagrees with you

Irrational Escalation: A rational decision in the past doesn't mean you'll follow it up with an irrational decision in the future

Neglect of Probability: ...yes, do ignore the odds. Makes great sense.

Outcome Bias: Well, if you got results, then that's what matters, right? Surely the methods don't count...(Oh, wait. I think I touched on that when talking about Tourney Win = Street Cred)

Probability Biases
Availability Heuristic: Your memory of the data/odds is more relevant than the actual odds

Belief Bias: Well, conclusion > logic of an argument, in your mind. Basically, doing your homework is not as important to you as liking the answer.

Gambler's Fallacy: You think that the previous dice rolls have a damned thing to do with future dice rolls. (Look, rolling a bunch of 6's in a row has NO IMPACT on your ability to roll another heap of 6's. You are never 'DUE' more results)

Hindsight Bias: Hindsight's 20/20, but you did not necessarily know it all along. Sometimes it just HAPPENS, ok? Ok.

Observer-Expectancy Effect: Not only do you have an expectation, but you actively manipulate an experiment to produce the results you want.

Ostrich Effect: You just might ignore an obvious negative

Positive Outcome Bias: wishful thinking about how things will turn out for you. You think your odds are better, just because it's YOU you're talking about, and not that guy...

Regression to the mean, and ignoring it: Thing is, on a long enough time line things will go average. But, some people keep expecting extreme performance

Selection Bias: The conclusion gets distorted because you chose data in an odd way. Kind of like the tourney thing. Just take ALL the data and analyze it; do your homework

Closing Rambling
This is meant to be food for thought, largely. It's important to realize that there are lots of ways that you can botch an analysis of a situation. A lot of these ways may very well be unconscious, and hard to catch. In a sense, that's why I do some blogging; it helps me write things down and then come back to them. Frankly, it helps when you go back and try to pour over things once you've got some distance from them; you gain emotional distance through time. It also helps to have a clear record of the events.

Or, the short version: people have blind spots in the way they think. Some folks I talk to will pick something because "It Always Happens To Me." No, it doesn't. It happens to everyone. You're just being selective about interpreting the information. Honestly, strings of bad luck DO happen; one of my buddies was against MM/HF speeders 'because his meltas always miss.' No, you just tend to remember the extremes more often than you remember the good stuff. The thing is? That's a perfectly human reaction, and perfectly legitimate if not exactly logical.

Bottom line? Logic and impartiality are difficult things to pull off. You're human, you make mistakes. Admit it, try to compensate, and move on. Hopefully this list provides some insight into mistakes.

*NaNoWriMo's goal is to rough-draft a 50,000 word story, just to get people writing. It's followed by the Editing Month, go figure. I have about 5,000 words down. Not quite banking on making the other 45,000 by the end of the month.