How my Kings of War Formula worked

The topic of a points formula for Kings of War has reared its head again. It comes up from time to time and the truth is that there is no fixed formula where you type in the statlines and special rules and it gives you the points cost. We actually use a lot of comparisons between existing units, and a ratio for the points cost for a troop vs a regiment vs a horde (e.g. an infantry troop is worth 70% of the points cost of its regiment). I discuss it in the *ahem shameless plug* Clash of Kings book.

One of my first tasks when joining the Rules Committee was to try and come up with such a formula. I work in data modelling and have plenty of experience with complex spreadsheets, so I gave it my best shot. 

I failed.

I tried for a month in my spare time (obviously not every minute of my spare time, but it was a significant timesink and about as much work as I’d spend on a typical data model at work!). In the end I had something that… kinda worked… I guess… if you squint hard enough… The only special rules it could handle were Crushing Strength and Piercing, and it wasn’t particularly great when a unit had both good ranged and melee attacks. When it came to other special rules… that’s the point where I gave up.

There are plenty of ideas about how the formula could work. Sadly I’ve long since lost the copy I worked on, but I do remember how mine worked. I’m going to break a few of the high level calculations down and illustrate the problems I ran in to. 

Note: If anyone doesn’t realise, I use the asterisk character (*) to show multiply by and a forward slash (/) to show divide by.

Unit Score

The calculation essentially came up with an arbitrary “score” for the unit. If I remember rightly, the basic Kingdoms of Men Shieldwall regiment had a score of 11.34632 (ok, the decimals are made up but it was eleven point something), and a unit that was twice as good as a Shieldwall would be worth 22.86924.

If we assume that we want the Shieldwall to be 100 points in-game, then we divide 100 by 11.34632 to give us a points multiplier of 8.81. Then we can take any score pumped out by the formula, multiply it by 8.81, and we get the points value for a score of that unit score.

(Points Value) = (Unit Score) * 8.81

Calculating the Unit Score

The unit score consisted of three calculated elements; the offensive value, the defensive value and the speed multiplier;

(Unit Score) = (Speed Multiplier) * ( (Offensive Score) + (Defensive Score) )

Let’s take the right hand part first; (Offensive Score) + (Defensive Score). I came up with separate scores for how much damage the unit can deal and how much damage the unit can take. These are first added together. They were essentially a total of the average damage that the unit could deal and the average damage that a unit would take, based on how prevalent crushing strength, piercing and various Defence values are in the game.

The other part of this equation is the speed multiplier. The faster a unit is in Kings of War, the better it is all round. We assume that the basic speed in the game is 5, so anything that is speed 5 is considered the norm. If something has a speed of 4 then it is worth less than an otherwise identical speed 5 unit, so its total score needs to be brought down a touch. In this case the multiplier for speed 4 might be 0.8. If we wrote a unit that was identical to the shieldwall, but had speed 4 instead, then we would multiply the shieldwall’s score of 11.3 by 0.8 to give us a unit score of 9.0. This would then correspond to a points value of 80 points. 

On the flipside, if something was faster than speed 5 then it needed a positive speed multiplier, such as 1.2 for a speed 6. The whole speed multiplier table might look like this:

Speed	4	5	6	7	8	9	10
Multiplier	0.8	1.0	1.2	1.4	1.6	1.8	2.0

So a unit that was speed 9 would multiply its total offensive and defensive score by 1.8, to give its total unit score.

Already we’re seeing an issue.

How do you determine what those multipliers should be?

The ones above are just an educated guess. It means that any Speed 7 units are balanced with each other, but how do you know that 1.4 is a good multiplier for speed 7 rather than 1.3 or 1.35 or even 2.0? They might be balanced with other speed 7 units, but would they be paying too much or too little compared to units of other speeds?

To take it to extremes, if we set the multiplier for Speed 6 to be 1.5, that wouldn’t be balanced in my opinion. I would never pay 50% more points for a speed 6 unit that was otherwise identical to a speed 5 unit. 

The issue with this table is that it relies on human intuition, which is what the formula is intending to get away from. We could look back to existing unit types, find similar or identical units with slightly different speeds and try to figure out what the ratio between them is, but that again assumes that the original points value, the one derived from human intuition, is correct. There’s no way to be certain of that and it kind of undermines the entire point of the formula if one of its main calculations is based on human intuition.

Special Rules Multiplier

The missing part of the Unit Score calculation was something that I didn’t even attempt – a multiplier for the unit’s special rules. Some special rules, such as Crushing Strength have a measurable effect on a units performance and can be included in the offensive or defensive score calculation. Others, such as Nimble, don’t have a direct impact on the offensive/defensive scores, but actually impact on everything. Fly for example, gets exponentially better the faster, stronger and tougher a unit is. Fly on a troop of Orc Ax would be fairly lacklustre compared to the same thing on a dragon. I wouldn’t pay 30 points for Fly on a unit of Orc Ax since the speed holds them back so much, but 30 points for fly on say  a heavy cavalry unit? Hell yes!

The unit score calculation should therefore be something like:

(Unit Score) = (Speed Multiplier) * ( (Offensive Score) + (Defensive Score) ) * (Special Rules Multiplier)

Each special rule would have to have its own formula for determining its multiplier. A very simple one might be Inspiring which had a fixed multiplier of say 1.2. The better a unit is, the better Inspiring is since not only does it inspire itself, but it’s able to stick around longer if it’s an otherwise better unit.

A special rule might not be a fixed multiplier either. Nimble and Fly could be exponentially better and have to be done more like the speed multiplier above, where the higher the Speed and Offensive/Defensive scores, the larger the multiplier is.

Nimble provides another issue – it not only needs to multiply the overall score of the unit, but the score for its shooting output since nimble units ignore the moving & shooting modifier. 

We’ve not touched on how a special rule loses its worth the more common it is. Ogres have Brutal on nearly every single unit, however that +1 from brutal is only ever applied once per combat. It doesn’t matter if one unit or three units are in a combat with Brutal; the effect is the same – it’s still a single +1 modifier. I’d happily pay the 15 points for the Brutal artefact in an army that doesn’t have the rule, but I would not pay 15 points for every single Ogre unit to have it. The same applies for pathfinder; there’s only so much terrain on the board for you to benefit from. At some point you’ve filled every available forest with your units, so it’s wasted on all your other units.

On TOP of all this, we come back to the same issue that plagues the speed multiplier. Whatever that calculation is for the special rule, it ultimately comes from human intuition. If that baseline human intuition is wrong then the calculation is also wrong. Units with the same special rule might be balanced with each other, but they wouldn’t be balanced with units without the special rule. If we had the multiplier for Brutal be 1.5, then any units with Brutal would be balanced against other units with Brutal, but I sure as hell wouldn’t pay 50% more points for Brutal. Equally if we had the multiplier for Yellow Bellied be 0.5 then they would cost half of their equivalent units without yellow bellied! That would be ridiculous!

Extremes, obviously, but illustrating the point!

Formulas don’t work

The whole point of using a points formula is to remove human intuition and mistake. You could sit there for weeks and months deriving rules and multipliers for every single special rule in the game. Ultimately, at some point in the equation, there’s going to be a baseline number that’s derived from human intuition. That basic cost for a special rule, that basic assumption for how many De6 units are in army lists so how valuable your Crushing Strength(4) is, they all come from human intuition and are susceptible to error.

If someone wants to go ahead and write an amazing formula that spits out perfect points values according to all the units we’ve entered in the game, then please do so. I don’t personally think it’s a productive use of time, unless you’re doing it purely for academic reasons, but if it works then we will use it.

One of the guys illustrating that we don’t use a formula, therefore the game is “grossly” unbalanced, used the example of a Knight Horde that he calculated to be 10 points less than it should be. Regardless of the calculation he used (which was wrong), even if something it’s 10 points undercosted, that’s not enough to swing the game. An imbalance of 3.9% is nothing. Nearly every single unit is going to be ever so slightly under or overcosted, if just because we round points values off to the nearest 5. Overall it’s going to work out to a roughly even cost, and even if you were to take only units that are undercosted, as long as that slight imbalance is not enough to override player skill, we’re not bothered.

Perfect balance is impossible. We’ve got it pretty damned close without a formula, and those problematic units that we under or overcosted significantly, we’ll address in annual tournament pack updates.