Cairn-ish Content: Odd Math, Part 2

I'm back!

My last entry on this subject was such a tease, and going the route of writing code to simulate combat odds was quite a rabbit-hole. (It's just the sort of thing that I can lose myself for hours on).

In any case, the big revelation came from some helpful advice on Reddit (shoutout to u/south2012!) to use a Monte-Carlo method with simulated combats instead of a a pure-deterministic approach. That unlocked the ability to run simulations with completely arbitrary stats, and never run into the really thorny programming cases.

Today, there are two questions I want to dive deeper on:

1. If I can mechanically improve a character in just 1 way, what should I prefer?
2. How much does impairing enemies and enhancing attacks shift the odds?

Mechanical Improvements

Now, Cairn isn't really about mechanical advancement. Even so, Cairn has numbers, and they matter. Players will want to equip themselves to increase their odds of survival!

The question though is what gets you the most bang for your buck?

For this question I started by taking a matchup of two similar stat blocks. Then, I ran the simulation again for the same matchup with various augmentations for the player: +1 STR, +1 DEX, +1 HP,  an increased weapon die, and +1 Armor. (Based on early results, I also checked every possible weapon die increase, every Armor increase, and a few points more for each stat). I did this for a few different matchups, comparing same-stat matchups, and a few against a somewhat stronger opponent.

Now, I recognize that a player isn't really in control of increasing their stats, so if we're asking the question "what should I do to buff my character?" then we can't readily answer "you should increase your HP!". Even so, I think the experiment is warranted, because it's useful to know what increasing a stat by 1 actually get you.

But also, if that bothers you, the good news is the stat increase is the least helpful thing*. These were kind of funky experiments, so I'm just going to summarize the data.

• Armor is generally the most advantageous, usually by a lot. Like, a 10 to 15 percent increase in the chance of winning a fight. What's even more interesting is that the rate of improvement actually increases if you already have Armor! E.g. going from 0 Armor to 1 Armor is a smaller bump then going from 1 Armor to 2 Armor! (Unless you're already pretty beefy)
• Buffing your weapon is next, though if your foe is particularly Armored then the weapon is actually more impactful than the armor. I'm not quite sure why this is the case; I don't have an intuition for it.
• HP is next. If your stats aren't too different from your opponent, increasing HP by 1 increases your odds of winning by close to 5%. And it takes roughly +3 HP to match a single +1 to Armor. (This makes some sense if you consider that if fights usually last around 3 rounds, then one point of Armor might be buying you 3 extra HP, effectively)
• STR is next, and its actually pretty pitiful: it takes +2 or +3 points of STR to match +1 HP, and +4 or +5 to match +2 HP. This also makes some sense, since typically if you're taking damage to STR it's too late anyways.
• DEX is practically useless. I only included it for completeness anyways, since technically a DEX save at the beginning of the fight influences the outcome. But consider that you could increase your DEX, or you could just manipulate circumstances to skip the DEX save and go first (the equivalent of just boosting your DEX right up to 20! I want to evaluate what that accomplishes in the future...)

This is great news! Because Armor and Weapons are things you could theoretically buy!

Ok, so let's address that asterisk. Consider a roughly average, brand new character: they've got about 3 or 4 HP, about 10 STR, no Armor, and they wield a basic d6 weapon. It's meaningful to recognize where the headroom actually is. You can only boost that Armor 3 more points. You can only increase that weapon by 3 more die increments. But you can increase that HP and STR by quite a bit. Now, I don't think it necessarily follows logically that this must correlate with what gets you the most value, but it sure is interesting. At the very least, as I've laid out above, more HP eventually equates to Armor, so the fact that you can theoretically increase your HP by a lot more than you can increase your Armor means that it can eventually outcompete the Armor.

Recalling previous data: maxing out your Armor can let you fight up one "tier", so it has to be remembered that, ultimately, maxing your Armor isn't really enough to take on the world (and what happens when the bad guys have Armor, huh?).

We could also ask how much weapon damage increase it takes to fight up one "tier", but I think the next section will give us a pretty big clue on that.

Impairing and Enhancing

This particular exercise was a delight.

Last time, I established that it only takes a few point of STR and HP advantage (plus a die increase) to totally outclass an opponent. (That's a good thing so far, because it means that even with a pretty low cap on stats--18--you can still achieve plenty of gradation in matchups.)

The next question is how Impairments and Enhancements can shift the odds. Let's start with the stat blocks I used:

Weak
3 HP, 3 STR, d4

Average

6 HP, 6 STR, d6

Sturdy

10 HP, 10 STR, d8

Tough

14 HP, 14 STR, d10

Boss

17 HP, 17 STR, d12

Each of these constitutes roughly a distinct tier, in which facing an opponent from just one tier above you is near-certain death.

(I also used variants of each of these with 1, 2, and 3 points of Armor)

Now, here are some charts about how various matchups here can go (the number is % likelihood of winning the fight):

As you can see, in general, in an even matchup the player has better odds. Usually around 58%. Great; that's review so far. Let's move on to looking at what Impairing an enemy does.

Impaired foes

An Impaired attacker rolls a d4 instead of their usual die. For these simulations, I assumed that the enemy is Impaired every round.

Already this is extremely interesting. If we ignore armor for a moment, we should see that Impairing doesn't do much if you are yourself pretty weak. It's not even enough to punch up a tier! But, as the player themselves gets strong/better-equipped, the effect of impairing an opponent becomes more pronounced.

And look at what happens when you have armor. 

The results there are dramatic, but they shouldn't actually be surprising; a d4 is small, and Armor points go up to 3. That means that if you've got full armor and they're rolling a d4, their average damage is just 0.25. Is it any surprise then that the weakest player character with an improvised weapon can take on even the strongest opponent under those terms? (...as long as they don't have much armor).

Let's look at enhanced attacks.

Enhanced attacks

The definition of "Enhanced" varies a little between Odd-likes, but this is "Cairn-ish Content", so I'm relying on the way it works in Cairn: using a d12 instead of your usual die. For these simulations, I assumed the players are Enhanced every round.

This looks a little more like a flat improvement across the board, though it's slightly better for a weaker attacker. Again, this is intuitive: if you're already wielding a d10, going up to a d12 isn't much of an increase. What's more surprising is how small the improvement is, even for that usually weak attacker. Essentially, you become competitive one tier up and that's it.

Enhanced + Impaired

For the sake of completeness, I wanted to also check what happens if you can do both:

Quite a lot it turns out! Obviously, if you've got plenty of armor you can take on pretty much anyone. If you don't you're still pretty well off, unless your stats are totally in the gutter to begin with.

Further inquiry

This has been a gold mine for the mathematical corners of my brain. In the future I want to dig more precisely into:

• What happens when you can guarantee that you get to go first?
• How big of a bad guy can you take down when you gang up?
• How do Detachment rules affect matchups (particularly when the Detachment actually has relatively low stats)