Friday, November 20, 2015

Quantity vs Quality: A Mathematical Approach

Over the past several articles I have been looking at when to give units toughness bonuses and when to buff units with psychic abilities.  These have been dancing around the edges of a much bigger question, "When it comes to units is quality better than quantity?"  I have been working on this problem directly, and I'm finally able to share these results.

When people compare two units they often look at each units chances to do damage or take damage.  These percents can be used to calculate an 'expected value' or an 'expected average', and this is what most people compare.  For instance, a space marine has a 4/9 chance of wounding a guardsman with shooting, so 10 space marines will cause on average 10*(4/9)=40/9=4.44 wounds.  This unit of space marines would be comparatively better than any unit that caused less than 4.44 wounds.  One of the problems of using this method is that it doesn't tell you a whole lot about when two units have the same expected value.

An example unit:
 BS 4 units hit twice as often as BS2 units, so in theory having twice as many BS2 shots should 'even out' with a BS4 unit.  Here are the percents for one shot at BS4 vs two shots at BS2



If we apply a bit of basic probability we can compare the two units more directly.  The BS 4 squad will have more hits 29.63% of the time, the two units will get the same number of hits 44.44% of the time, and the BS 2 unit will have more hits 25.93% of the time.  Now we see that the BS4 unit has a distinct advantage.  Why is this?  The expected value for the two units was the same, but remember 'expected value' is better described as 'expected average'.

The reason for this difference is that expected value is dealing with continuous probability, with an infinite number of dice being rolled; since we are playing in the actual world with a limited number of dice our results are going to be slightly different.

As the sizes of the two units increase our percents change.  For instance, if we have two models that are BS4 and four models that are BS2 then there is a 35.16% chance that the BS4 models will do better, a 32.92% chance that they will tie, and 31.92% that the BS2 models will do better.  We see that the percents for both sides wining increase and the chances of the two units getting the same number of hits decreases.  As the number of models in each unit approach infinity (and twice as infinitely many BS2 models) the chances of a tie will approach zero.  When this happens the chances of each side winning will approach 50%

Does this apply to more than just ballistic skill?
Yes.  This applies to all aspects of 40k, and dice games in general.  A chance that is a 'sure thing' is usually better (especially on a small scale) than multiple chances at a 'long shot' when all other things are equal.

What does this mean with regard to your army?
All things being equal a unit with lower stats is generally not going to perform as well as a unit with higher stats even if their expected averages are the same.  The perfect example is a squad of 10 Astra Militarum veterans compared to a squad of 20 Renegade guardsmen.  These two units cost the same points, and even though these two units will have the same expected average of hits from shooting the veterans are going to perform better more often than the renegades.  As these two squads get bigger they will become more or less even, but the renegades are always going to be catching up.  If you are comparing point values, a squad of veterans is (slightly) under costed compared to renegades and the renegades are (slightly) over costed compared to the veterans.

**Note: when I say slightly over/under costed I am talking in terms of fractions of points**

So all cheap units are trash even in large numbers?
No.  I could directly compare the veterans to the renegades because they have the same weapons and their ratio of ballistic skill matched the ratio of their costs.  Most units aren't going to match so perfectly, and slight differences in point values will make more of a difference than the percents shown here.  For instance renegades compared to veterans may be (slightly) over costed, but renegades compared marines are incredibly under costed.

Then why write the article?
As a whole we don't need to directly compare units to gauge quantity vs quality.  The previous examples help visualize the math.

How can my horde army remain effective?
Force Multipliers.  As I discussed recently units with worse stats are better targets for force multiplying buffs such as psychic powers or a character's special rules.  Let's use the comparison of 2 BS 2 models vs 1 BS 4 model.  If both units get twin linked for their weapons this dramatically changes the comparative performance of the two units.  The expected hits for the BS 4 unit increases to 8/9=0.8889 and the BS2 unit increases to 10/9 or 1.1111.  Looking at the actual probabilities the BS4 unit will have a 17.56% chance of getting more hits, there is a 46.09% chance that the two units will get the same number of hits, and there is a 36.35% chance that the BS2 unit will get more hits.

Putting all of this together:
Small, elite armies are going to be slightly more effective in low point games.  As more and more models/units are added these differences between a horde unit and an elite unit are going to disappear.

Even in a large scale battle this principle is still important.  A small firefight can take place between two isolated units.  As both sides take casualties this principle is going to become more and more relevant.

Hopefully this article has been helpful, and you have a deeper understanding of probability.  This weekend I am going to try and finish up all of the conversions for my drop pod.  My friend Phil (who has been featured in several battle reports) just purchased a bunch of new models, so we are going to drink beer and put things together.


  1. I think the last point is the most important. I've noticed that in small games it really pays to have strong units because it's hard for your opponent to kill them. It might make it slightly harder to wipe out your opponent if he's got hordes but he definitely won't kill you. Whereas the bigger the game gets, the more useless your elite squads are. It doesn't matter if your 500 point squad is a deathball in a 3000 point game because your opponent is just going to cpncentrate on wiping it out. But if you bring like a 300 point squad in a 1200 point game, your opponent probably doesn't have the firepower to take them down easily.

    1. I agree, the last point is the most important: he is going to drink beer and do the hobby with his friend. :)

    2. The funniest part about this is that my group of friends all started playing 40k back in middle school when Phil broke his leg and was stuck at home all summer. We are re-living one of the best times of our lives, but this time with beer.

    3. Haha, not what I meant but equally important. What's the point of playing these wargames if you aren't going to have a little fun doing it? :)

      I think I'm going to get a chance to play the Betrayal at Calth boxgame with a bud this weekend which looks like a fun, relaxing game. Haven't played it yet but reading through the rules they seem to make for quick, easy games.

    4. I am interested to hear about how it goes, especially if the games are quick. I have been considering forming a board game club at my school, and this could be a great introduction to 40k.

    5. I'll let you know if I play. The rules are a lot more simple than 40k. And the first mission is designed for newbies because it's just 10 marines vs 10 marines in a race to get to the other side of the board. So that should be easy enough to teach people.

    6. Hey Andrew, played through the Betrayal at Calth campaign with my friend yesterday and I have to say it was pretty fun. For starters, it's $260+ of models for only $150. So that's always a plus.

      It took us most of the day to play through the 6 campaign missions including going out to eat, but I'd say we averaged about an hour (or less) per mission. Some of them are pretty quick. The first mission is a great intro because it's just one tactical squad vs one as a way to break the gameplay in slowly. The rest are more complicated and are genuinely unique. They were all fun with the exception of mission 5 which was pretty much a dud in our opinions. If the traitor player is smart, I don't see any way he can lose. Other than that, the missions were pretty fair - we split 3 to 3 and each one could have gone either way apart from mission 5.

      There are a few vague rules (in usual GW style) but the gameplay is pretty streamlined and much, much simpler than a typical 40k game. It's quite a bit different from 40k from the perspective of a 'veteran gamer' but if you were trying to get some kids into this kind of board game in general, it would make a great intro. The basic mechanics are all there in smaller killteam-type gameplay.

  2. It's interesting to see the breakdowns. However, there are so many factors I generally just keep concepts like this in the back of my mind. This last weekend I was playing a 2v2 with random teams (my thousand sons + friends salamanders vs Tyranid carnifex spam + grey knights) 1k points.

    Now I have found my invisible prescience maxed out helcult to do quite well against hordes, SM, and termies and for my "Psychic star" to be quite durable at low points games. Unfortunately the GK player randomly decided he needed anti horde and brought 1 wyvern. His team then rolled first turn. Ruh Roh.

    Turn 1 before I could get powers up lost 24 cultists, turn 2 another 21. Well there goes my tarpits. I feel this game definitely has a good rock paper scissors going right now. People are complaining about Tau being so nuts but I saw a necron player with canoptic wraith spam rush him down easily. Funny times.

    1. That is the whole idea. These concepts are great as guidelines, but obviously in the thick of a battle adaptation is important.

  3. - Really intersting article. Buffing large units can be a good idea, as you said, but we have to consider the cost of the buff (FI: if I add a psiker to twin linking my cultist, or reroll to wound, how many cultist could I had had instead? Is this worthing the cost? I should check...)

    - Rose Stenborg, I like thousand sons but I have no idea about how it can be possible to field them. What list do you use, usually?

    - We have HH too, but they are tooooo many models to put togheter and paint for a "fast" game...

    1. Since the stats for a cultist are the same as an upgraded renegade the break points are also going to be the same. A level 1 sorcerer should be with a squad that costs a minimum of 158 points, a level 2 with a squad costing 137 points, and a level 3 with a squad costing 126 points.

  4. I was looking at the Mont'ka leaks about the Primaris Psyker formation. I have been butthurt ever since they took away IG Allies of Convenience, so much so that I simply run some AM Wyrdvanes CtA allies (lets just call them unruly slaves) for the warp charges. I think I will be using this formation a lot combined with CSM. It could act as a force multiplier as well as giving your TRUE CSM psykers SOOOOO many dice to play with?

    1. Yeah, those new formations look pretty fun. I wish renegades had psykers that weren't terrible.