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There was a post in this forum by someone who wanted to measure the effectiveness or efficency of a unit. He was shot down by the cruel guns of math - you can just use a formula. You gotta try a probabilistic approach. So, I have taken up the challenge given that I like math problems.
Well, I've been thinking of an approach to this problem and it relies on a few premises, which I shall lay out as follows:
GOAL: We want to measure the RELATIVE EFFICIENCY OF 40k UNITS. Efficiency may be defined as the Unit Efficacy divided by the Point Cost.
Unit Efficacy (Effectiveness)
Since it is practically impossible to measure the effectiveness of a unit in SHOOTING, due to the ranges and etc, I will onyl focus on the close combat effectiveness of a unit.
By unit efficacy, I refer to the damage output a unit is able to make in Close Combat.
Why dont we just calculate the expected damage output and compare from there?
Now, let me get something clear.
IT IS VERY HARD TO MEASURE THE ABSOLUTE EFFICACY OF A UNIT IN CLOSE COMBAT.
By this, I mean that too many variables such as squad leaders, special rules etc... makes it nearly impossible to determine the damage output of a squad reliably. Plus, this doesnt take into account modifiers etc.
The biggest problem however, is that of the opposing unit. Who are we fighting against. Are we fighting against Eldar? Orks? Marines? Nids? If so, how big is the unit? This can change the combat outcome quite significantly, for example, Eldar strike first causing 5 casualties to you 10 man marine squad. You fight back but with less attacks as half you squad is dead. But you calculated your squad's damage output against a unit assuming you stuck first or at the same time. So does that mean your squad damage is half of what it was? Of course not! We still cause the same amount of damage Orks or Chaos Marines. Just that initiative order has screwed things up.
Hence, this model is FLAWED
So how do we circumvent this problem?
WE STANDARDISE THE COMBAT
WE MEASURE THE RELATIVE EFFICACY OF UNITS.
WE USE PROBABILITY, NOT AN EXPECTED CONSTANT VALUE!
Hence, to calculate the efficacy of a unit, say a squad of 30 Ork Boyz, no leaders, in close combat, we make them fight theoretically against a STANDARD REFERENCE UNIT of say a 10 man Space Marine Squad with no leaders.
Consider the combat.
So, we calculate who strikes first. WE ASSUME NO CHARGING BONUSES but weapon bonuses still count. This is to standardise the combat.
The marines hit first, causing X hits.
Of these X hits, Y wounds are caused.
Orks make a save, of which Z still are unsaved. Z models removed as a casualty.
So, 30 - Z orks survive from an original unit of 30.
Then the orks counter attack, causing A hits.
Of these A hits , B wounds are caused
Determine the winner (who scores the more unsaved wounds, I believe) and record result
So what now?
Ok, so you know how to find the winner of this theoretical value.
We want progress this further. I want to find how many times in 100,000 combats (theoretically) the Orks will win.
Well, we can run a massive simulation through Excel to determine the winner or do a bit of math to find the win rate of the attacking unit of Orks.
So we get the win rate, say 33.333% or a third, meaning one third of the time we win and two thirds of the time we lose OR draw.
A side note
We can express the number of wounds inflicted as a binomial expression and do some uni maths there to find the probability distribution curve - i.e. percentiles of winning. But that really is complicating things.
We do the same calculation for another another unit, using the same standard reference unit. Say a unit of 5 Assault Terminators with Lightning Claws vs. the reference unit.
We find the win percentage.
We calculate this win rate for every single unit in the Warhammer universe.
***@!#@ what? Are you #$@# kidding?
Yeah I am. Sort of.
To make a reliable relative comparison of effectiveness, we need to really consider EVERY single unit and the permutations available. So if we just compare effectiveness between two or three units you can only have a reliable measure of how good a unit is in relation to those few units, not the game. Its analogous to the big fish in a small pond idea - we cant make judgement based on a puny sample set.
So is it possible to calculate the relative efficiency of a unit in relation to the entire game?
Nope, unless you want to be stuck coding for 10 years trying to get all the the exact permutations of wargear and the like. There are literally thousands of different possible configurations of wargear and unit size and game rules etc...
What do we do?
We can make a relative comparison across a codex, say Space Marines. We pick out the units that we are most likely to use or the ones we like and figure out the most popular combos, discarding the most useless combinations.
This way we can measure relative efficiencies quite usefully.
Compiling the stats:
Ok, we spend 8 hours placing values into excel, working out hardcore math.
We get a table of relative win percentages vs our standard reference unit.
Nobz with Power Klaw -> 90%
Boyz Mob (30) with Choppas -> 88%
Boys Mob (29) with Choppas -> 87%
Now, we divide each percentage by the points cost as per the simple forumla:
Efficacy Co-efficient = (Percentage Win Chance x 100) / Points Cost
Apply a sort to this data, to get an idea of what unit is the most efficient.
We are finished. Pick the best units in your army and happy fighting!
Wait! Is that it?
Yep. Although there is a MAJOR problem we havent dealt with.
The issue of SHOOTING.
We have assumed so far that combat effectiveness, a la both shooting AND CC are measured in this process.
We have not taken into account the shooting capbability of the units. BEAR THAT IN MIND> this is just for close combat. Shooting is a whole can of worms I DONT WANT TO TOUCH!! ARRGGHH! YUK!
I will address that later in another thread.
If you use this method to pick units, bear in mind you are short changing yourself. You are not considering shooting. However, this relative efficency measure is just for scholarly interest.
All this for nothing?
Not exactly. Close combat armies such as Orks, Nids and Chaos have a whole permuation of weapons to choose from are their shooting sux (except for Marines). This measure is thus more accurate for them, as their actual combat effectivness or damage output taking into account shooting is not much different than the CC effectivness.
To cap it off:
We want to find the relative combat effectiveness of a 40k unit in Close Combat and their efficiency.
1. Consider the combat.
2. Determine the winner
3. Record Results
4. Repeat steps 2 to 4, say 1,000,000 in Excel.
5. Repeat steps 1 to 4 with another unit
6. Caclulate each unit's win rate.
7. Apply this formula Efficacy Co-efficient = (Percentage Win Chance x 100) / Points Cost
8. Rank them
9. Pick your units and win games
An after word
This Ranking is absolutely RIPE for further development. I see it maturing into an actual Combat Rating sometime into the future.
If you have the time and are able to, you can experiment with the math by applying weighting coefficients where needed and the like.
Some advanced tweaks to this include:
1) Adding your own shooting formula to the mix.
2) Do the same steps three times, each time replacing the Standard Reference Unit with another unit with the same profile except for the initiative. We do the math each time with different initiative - one higher - one the same - one lower. We average the final efficiency scores to intrinsically within our rating the unit's effectivness versus a WHOLE RANGE of units.
I hope this helps you in your gaming. And can someone PLEASE do the math and post the stats?
HOPE THIS IS DESERVING OF REP
Doing it using probability is essentially doing the experiment an infinite number of times, so why not use that? You can still do it against standard units and work out efficiency per points cost
I think probability is required in every single metric of the WH40k games.
What you are saying is essentially what I have done with my method.
Well probability gives us an expected outcome. If you ran the test n number of times, then the outcome would tend to the expected outcome as n tended to infinity. So instead of doing the test a million times, you could just use the probability of what will happen, which is "equivalent" to doing it an infinite number of times.
I like your thinking though. Do you do much programming at all? I quite like programs which run simulations many times to see if we can create conjecture, which later we attempt to prove analytically. If this sort of thing appeals to you then you should check out this book, it's pretty damn awesome
C++ for Mathematicians
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Is there any relation between mathhammer and mathcore?
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I don't get most of the mathhammer. I've always tested much more of an english than a math guy. English doesn't mean spelling. Cheers ...
Yeah, im a pretty decent programmer.
I dont think I have explained and refined my concept that well in this post. I just sat down and wrote this straight out of my head without that much development. If that makes this seem a little bit confusing and contradictory, then sorry. I will return to this later. Some of my main points I should have made I didnt put in, and some points that werent really that important were expanded.
So yeah, a really hidge podge article.
Anyway, the main thesis remains the same.
It is better to use a percentage chance of victory than a damage output measure for the basis of determining victory. The main point I failed to mention is that probability distributions should play a bigger part in the analysis of unit effectiveness. i.e. How reliably can a unit win?
I totally failed to integrate expectancy ranges of units' victory rates etc. into my method.
I tried to replace this reliability and probability idea with all the experimenting concept, which I think does the job reasonably nicely.
I WILL rewrite this article in the next week to make it much more theoretically sound and a much simpler method. (I have my final exams on now - i should be studying!)
BTW. Stomehambey and Wulfbanes, if we could work together on this, that would be cool. Just PM me.
I think the traphole remains in that it is clear to the writer how things should go on, but is not so clear to the ones reading. :o I'd be glad to help on that though.
Finished my own exams last week, meaning I'm back to having time to do all kind of stuff. Good luck with your own exams.