Friday, June 30, 2017

Mathhammer: Percentages Lie

The most infuriating thing I hear when people talk about list building and wargear options is something like, “option a is 33% more effective than option b and costs only 5 more points!”  I call this the fallacy of using percentages in a nominal system.

This isn’t just a 40k problem, I see it in every sort of RPG, MMO, card game, etc where players have to pick from differing options using limited resources.  But since this is a 40k blog, my examples are going to be from 40k for the sake of being understood universally.

So what is the problem?  The problem is, you don’t build lists based on percentages, you build them based on points.  In the crudest terms, how many MEQ a gun kills per turn is worthless information.  If gun A kills 1 MEQ per turn and gun b kills 1.33, if you don’t then divide by the points cost, you aren’t actually making any sort of rational choice.  Since we are limited in points, any choice comes at the expense of another choice, and making efficient choices is how you win in the list building phase.  In a balanced game, there would never be a single weapon that was good against every type of army for an affordable amount of points ::cough::grav cannon::cough::.  However, even figuring out killing power divided by points isn’t the whole story.

You could build a list using killing power divided by points to only pick the most efficient choices, but you’d probably lose.  There are some armies that simply don’t have great options across the whole killing spectrum.  They might shred vehicles.  They might kill MEQ.  But they have to pay through the nose to kill hordes.  If you’re trying to field a balanced army you may have to bring inefficient choices in order to actually have a well rounded force.  For example, let’s say an army has a hyper efficient lascannon platform.  You could bring an army bristling with these lascannon platforms.  You might has 25% more lascannons than any other army could bring to the table for the same points.  

Is that even good?  I’d argue that those extra lascannons are not helping.  What you SHOULD do is economise by using your cheap lascannons to fill out your anti-vehicle slots, and then use the points savings to bring killy but points inefficient units that shred infantry.  This is an oversimplified example, ofcourse but it demonstrates that idea that simplying making a list of nothing but points efficient units won’t check all the boxes.  Unless, ofcourse, your codex is broke and simply better at every role than everyone else.

Another thing to consider is the value of nominalizing these numbers.  “33% better at killing MEQ!” sounds great.  How many extra MEQ dead per game is that?  Oh well Gun A kills .65 MEQ per turn and Gun B kills .87 MEQ per turn.  Cool, gun B MIGHT kill one extra marine per game, if you get perfect conditions every turn.  Is that really worth paying ANY extra points for? If we value a Marine at 13 points, you might argue that so long as the gun costs less than 13 points it is worth paying for.  We know, ofcourse, this is a nonsense way to look at list building.

Yes, I just “proved” that big percentages of small numbers is still small numbers.  Please deliver my nobel prize for mathematics any time now.  But that is important to consider!  When people use percentages, they are trying to confuse you!  One of the writers at 3plusplus made an awesome series of charts demonstrating the killing power of all the weapons for each faction.  This was great.  It showed how many wounds per turn a weapon would do against a variety of targets ranging from GEQ to Land Raiders.  Fantastically useful information… which people proceeded to draw insanely bad conclusions from because they put too much value in comparing percentages of small numbers and not enough on looking at the nominal amounts.  

In conclusion, beware percentages.  Big sounding percentages differences are often used to exaggerate what are actually small nominal differences on the table top especially in regards to higher variance, single shot weapons.  Efficiency in list building is extremely important but don’t let the desire to be hyper efficient cause you to make mistakes.


  1. Agreed.

    I also think that another thing that is ignored in a pure looking at mathematics point of view is questions like range of results. By this I simply mean that a weapon that does D3 damage is on average exactly the same as a weapon that does 2 damage. Except, it is not. If you are shooting a target with 2 wounds, the D3 damage weapon actually averages out to less than 2 wounds because on occasion you roll 1 damage, and 3 damage is effectively capped at 2 damage. Also, with the availability of command point re-rolls against a target with more than 2 wounds you could argue that a D3 damage weapon is more likely to more wounds because you could choose to re-roll the damage result.

    Similarly with the war the morale system works, there is an in game difference between a weapon that kills 1 model every turn versus one that does 3 kills every third turn. The first will force morale tests every turn which is devastating against low leadership units as they will relatively frequently fail the test and lose additional models, while the other is better against high leadership units that can effectively ignore suffering one loss a turn.

    Additionally another thing to consider is the likelihood that you will get to actually USE a weapon. By this I mean that if a unit can be given special close combat weapons that make them twice as good at a mere 20% increase in cost, how much more effort is your opponent going to invest in killing these models? Similarly if you have some really good shooting weapons, what lengths will your opponent go to to shoot them off the table / lock them down in close combat?

    In general I think there has been also a lot more effort put into making different weapons of different value based on the target, making this kind of analysis really hard, and very conditional. I have seen some suggestion that for Imperial Guard for example the only good weapon option is the plasma gun. Unless of course, your opponent is running a termagant horde with tervignons and venomthropes...

  2. Speaking of Termagant hordes with Tervigons, those are great!

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