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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
etc.. etc...
Determine the winner (who scores the more unsaved wounds, I believe) and record result
Repeat again
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.
Ok, now?
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.
And then?
We calculate this win rate for every single unit in the Warhammer universe.
***@!#@ what? Are you #$@# kidding?
Nope.
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.
Such as:
Nobz with Power Klaw -> 90%
Boyz Mob (30) with Choppas -> 88%
Boys Mob (29) with Choppas -> 87%
so on...
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.
A summary:
To cap it off:
We want to find the relative combat effectiveness of a 40k unit in Close Combat and their efficiency.
METHOD:
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?
phr0z3n
HOPE THIS IS DESERVING OF REP
Last edited by phr0z3n; October 30th, 2008 at 07:34.
Does anyone feel like commenting?
Hmm, nothing really wrong with it. It is however truly undoable. As you said yourself, every parameter should be calculated for, if you want to do it like this.
There is of course a mathematical easy way. *yay*
In most games, unit efficiency can be seen as 'how much it kills' before it goes down. Right? So, unit efficiency could be measured as: 'points obliterated'/'points costed'. Seeing every turn, the damage output of each model is theoretically the same (that half the unit is shot up doesn't change anything to how much damage a single model does).
Parameters that do occur are then, for example:
1. Type of enemy units.
Which of course can be simplified as well!
Type of enemy units. For this, you'll need an army list of the army you're referencing against. For example, a shooty Marine army. You make the following calulation:
1. Add every stat from each model (note this is all infantry, etc, no tanks so far), sorted by type of stat. So WS goes with WS, S with S. Include bonuses from gear.
2. Divide total stats by number of models. You'll get fractured numbers most likely, this is okay.
3. Count the number of no-armour save weapons, divide by the number of models. This gives a percentage of instant wounds.
4. Calculate both Shooting and melee from 1 model you created, against infinite model of this 'standard unit'. You'll get a fractal number of the amount of kills you'll make in the shooting and the assault fase.
5. Calculate 1 enemy 'standard model' against infinite models you're testing, to see how much would die from shooting and assaulting.
6. Do the same steps for an enemy standard vehicle against your unit.
7. Multiply standard vehicle results (in killed and get killed) with the number of enemy vehicles. Same for standard unit results. Mediate with percentagal weight of the number of vehicle and infantry models.
8. Divide the enemy army lists' cost by the number of models total.
9. Multply model costs with damage you cause, divided by your own model cost by damage enemy causes to find out your efficiency against the enemy army.
Cheers,
Wulf
How is my method undoable?
This method is by far the most accurate method that can ever possibly constructed of evaluated close combat effectiveness. I am not kidding. I can prove it mathematically to you if you ask.
Plus, all it takes is a couple of minutes in Excel and voila!
My method has implicit within it an empirical model of unit effectiveness, where as yours is a completely inaccurate guess at how unit stats will translate into combat effectiveness. You build into your method and assumption that stats = direct effectiveness. In that case, a Necron would be more effective and efficient that a Space Marine. It is not.
So, although your method is short, it is highly inaccurate.
Hell, since i can prove my method is the best, might as well stick in in the finished tactica
Last edited by phr0z3n; October 30th, 2008 at 07:34.
As you said, you simulate battle against 1 standard enemy unit. Your enemy usually puts exactly the things in front of your units that you don't want there, units that are better against them, meaning they'll loose efficiency. Like a predator tank in front of a bunch of shootaz, instead of those tasty marines in the sights.
The way things I did are not unlike yourself, The difference I'm making is where you calculate against a specific enemy, I calculate against the enemy's total army. It gives a reasonable amount of damage you expect to deal and to take per model.
My method is unwholy less inaccurate then yours, which I can prove as well. I calculate chances, you measure outcome. As your method is based on the factor of chance, it'll be biased, no matter how many times you run it. Of course my method has it's own problems, but as far as I can tell, I've done exactly the same 'who wins' and 'how much' calculations, and I did them for the entire enemy army.
I've been playing Mathhammer for a while, calculating my units for tournaments, and I must say that if my 'standard army' had a lower rating than the enemy's, I've been in a disadvantage, because stuff just wouldn't die, etc, leaving me to play objectives without killing too much of the enemy. However, I knew this in advance, giving me a playing advantage.
I never feel lucky with the dice, so I play objective when the odds are against, and I can play a lot more freeform when the numbers are in my advantage. Now Warhammer, truly ain't Mathhammer, as in a game of chance, 50% will always be luck, leaving almost all our calculations worthless.
They do however prove their use that you 'get x kills and make unit y flee', or 'I can take that tank with v shots'.
All in all, Warhammer is not a game laid out for practising math on, no matter how much we want to.
I did however like the attempt to catch the entire system. Rep for that.
Cheers,
Wulf
Wulf, check the thread in the finished tactica forum for my comments.
A great model for CC in general. There is none better then I have seen so far. As a studying mathematics education student, Rep for that.
It is respectable that you realise that this method is not foolproof, and it doesn't include shooting.
However, I would also like to point out, that if you truly wanted to make a formula for determining a units 'effectiveness' with the definition of effectiveness being as above, but considering every game aspect/mechanic, then you would also have to consider all of the possible contexts of the unit and how they would affect the outcome of combat/shooting etc... a task which is as impossible and silly, as trying to calculate infinity itself.
Again, you made it clear that you weren't trying to make a formula for a units overall effectiveness, but simply its effectiveness in CC. To that effect, you seem to have done an excellent job. I just wanted to make it clear to anyone else less mathematically inclined that this is NOT a suitable method for comparing units accurately within the context of a full 40k game. It is however a suitable method for calculating the effectiveness (chance of winning) in a single combat round between two single units of models (making sure to add in the modifiers and other factors affecting the calculation).
For example if I wanted to calculate the efficacy of a squad of orks against a squad of guardsmen in a particular round of combat, I could use this method. However, if I was looking to give my orks a general efficacy rating for every combat against every opponent then this doesn't work, and any 'reasonable' calculation of such a thing is theoretically impossible.
Basically, the efficacy value you calculate for a unit, against another unit in a particular round of combat, only applies to those two units, in that round of combat... The efficacy rating will change when the factors involved change. Eg. if the unit of guard above wins combat with 5 losses say (by some lucky miracle)... and then next turn continues on to assault another unit of orks exactly the same as the first but in cover. You then have to recalculate, because 2 factors have changed, the orks gain the initiative for being in cover, and the guard squad now has half as many members... Obviously a half strong guard squad who has lost the initiative for going first, will not have as high an efficacy 'value' as the guard squad who was at full strength with the initiative for going first.
Bad example I know =P, but I'm sure you all get the idea.
Last edited by Smurphy; November 2nd, 2008 at 16:01.
40k Armies:
Orks: W7/D2/L3, Necrons: 6/0/1, Eldar: 4/1/2, Marines: 0/0/0
Fantasy Armies:
Ogres: 1/0/1, Bretonnians 0/0/0
Exactly, Smurphy. It is impossible to completely measure the effectiveness of a unit in the context of the entire game, a problem I have found in other models of unit effectiveness. This is about as far as you can get in terms of measurement of effectiveness without compromising accuracy.
BTW. It was not my intention to devise a model for overall effectiveness in the context of 40k.
Last edited by phr0z3n; November 3rd, 2008 at 03:20.
1984
"Or you could just play a number of times with a unit against a few different armies and see how they do, much easier, less time consuming and it factors in the one thing math hammer can never do, the human factor. Seems to have worked quite well so far (just a thought) "
First thing, playing games of Warhammer takes WAY longer than spending a couple of minutes on the math. You still get a idea of how well a unit works on paper just as well as you do on the table top. Plus, Mathammer is all about deciding which unit to try out on the battlefield. It narrows down the choices so you dont have to experiment and playtest all the time. And yes, all of us knowledged Mathhammerers know that mathhammer isn't about winning Warhammer purely by army lists and maths. No one can obviously expect to win simply by fielding units and having them charge randomly across the battlefield. Therefore, we have never attempted to factor in the "human factor" in mathhammer - hence you are wrong if you believe Mathhammerers are a bunch of complete idiots, Rikimaru.
Mathhammer about maximising your potential on the table top, reducing your dependence on luck and enabling the 'human factor' to help you win. Honestly, I'm just as good at strategy as analysing units' effectivenesses through math. I emphasise good tactics just as a good, efficient army list - i.e. army composition and your play style are equally important. If you are suggesting that I have completely missed the point, Rikimaru, then you are utterly wrong.
phr0z3n