Cairn-ish Content: Odd Math

I like Cairn. Into the Odd and it's children have been a breath of fresh air to me.

I also like combat. The default assumption in Cairn about combat is generally "avoid it". Even so, as it's bound to come up sometimes, I would like to build up good intuitions for how the math shakes out. (For reasons I'll elaborate on later. But forget that for now, on to the goods.)

Basics

Let's start with a simple table of average dice output for each damage die:

	d4	d6	d8	d10	d12
Average	2.5	3.5	4.5	5.5	6.5

This is a well understood result: the average roll is about half the dice total (slightly better because numbers) and when you increase the dice total by 2, you increase the average roll by 1. Nice!

Armor

The oft-cited conclusion based on the above result, however, is that in terms of averages, each point of Armor a defender adds is like reducing the attacker's dice by 1 type. Which is true.... sort of. Look at this table of average rolls by die plus armor defense:

Die	A0	A1	A2	A3
d4	2.5	1.5	0.75	0.25
d6	3.5	2.5	1.66	1
d8	4.5	3.5	2.63	1.88
d10	5.5	4.5	3.6	2.8
d12	6.5	5.5	4.58	3.75

In terms of raw averages, this means that each additional Armor point is actually slightly less than reducing the attack die by one. Maybe the difference is slight enough that you will say "close enough!", and that's fair, but there's another element to this that deserves to be considered.

A few heuristics

In normal, Armor-less combat every attack always results in HP loss. And that means that it's only a matter of time before a defender's HP will be drained and they'll be forced into the territory of making Critical Damage saves. There's a couple of simple heuristics we can use to think about that:

• Maximum number of rounds to CD save: in Armor-less circumstances, that's just the HP of the defender plus 1 (because you're rolling a minimum of 1 every round)
• Minimum number of rounds to CD save: take the maximum possible roll on the dice, and divide the defenders total HP+1 by that number. This is the fastest* path to victory. (More on this later.)
• Average number of rounds to CD save: take the average possible roll, and divide the defenders total HP+1 by that number. Under normal circumstances, how long will it usually take to eat through that HP?

We could probably draw a nice little graph for any given match up and see how two entities compare to each other on these metrics. Obviously if your opponent's average rounds to HP reduction against you is lower than yours to it then you're in trouble. (If their average rounds is lower than your minimum rounds than you're really really in trouble.) What's more (and a bit cumbersome to compute at this stage), the higher the minimum number of rounds actually is, the less likely that possibility becomes (rolling two 6's in a row is only a 1/36 chance, but three is 1/216)

So what does this have to do with armor? Well, loads. Because just a single point of armor sends your maximum rounds to HP reduction to infinity, grows your average rounds, and also grows your minimum rounds. Lets look at a simple example. Consider two entities with the following stats:

HP 9, d6 (sword)

By our heuristics that means: 

Maximum rounds: 10

Average rounds: ~3.14

Minimum rounds: ~1.83

Great, so since we can't actually subdivide rounds, on average it's going to take about 4 rounds for one party to tap flesh on the other, and the absolute minimum is 2.

So what happens when one party picks up a single point of Armor? First let's assume "armor is like reducing the attacker's die by one" and see how that compares:

Maximum rounds: 10

Average rounds: 4

Minimum rounds: 2.5

That's pretty bad! It's like moving from 50/50 odds to a certain loss! But that's assuming a point of Armor is equivalent to reducing a die by 1; what's the truth? Let's look:

Maximum rounds: Infinity

Average rounds: 4

Minimum rounds: 2

The average rounds is the same, sure, but check out the minimum rounds! It's functionally the same as the no armor encounter! Obviously the details matter: the defender still has the upper hand here, and we're not actually looking at any odds related to CD saves, but the important thing is that Armor is playing a less significant role than you've been told. In fact, there's another factor that's important here that deserves considering: morale.

The will to fight

Players don't make Morale saves. Enemies do. That means that, while a player dropping to 0 HP likely means nothing mechanically speaking (except for a pretty cool scar!), a monster dropping to 0 might actually flee in terror. This is huge, in fact. Because it means that:

• The player should calculate their heuristics based on HP, not HP+1
• A lone monster has to make a WIL save and a STR save if they drop below 0 in a single hit (and if you know anything about probability, that sucks.)

So really, players have a meaningful upper hand here--in more ways than one!... but how much?

Put a pin in that. We're going to come back to it.

Multiple attackers

Ok, so 1v1 is a pretty lame representation of combat; I'm guessing most parties have at least 3 players, not to mention any hirelings. So ideally we're ganging up, and maybe only at the worst does it sort of shake out to combatants pairing off in little 1v1s.

In Cairn (and it's kin) we handle multiple attackers by rolling every relevant die, and taking the highest roll. The best way to explain what that does to your odds is to link you to this lovely video. 

tl;dw when you roll multiple dice of the same type, and take the highest result, the average works out to  n/(n+1)+.5, where "n" is the number of sides on your dice. In grug-speak that means the average of a single dice is half of the dice plus .5, the average of two dice is two-thirds plus .5, the average of three dice is three-fourths plus .5, and so on. That's quite an elegant result! 

What it amounts to is that the more dice you roll, the more your average rolls start to skew closer and closer to your maximum rolls (with a diminishing return, of course).

So stacking a second attacker on an enemy is great! And stacking another attacker is good! And stacking another attacker is, well, okay I guess.

(At one point I built a bunch of tables in Google Sheets mapping out the precise averages based on different combinations of attackers, their attack dice, and the defenders Armor. But whoops! I lost that sheet, and it was a ton of work, and I don't have it in me to redo it. I'm sorry.)

Alas, what I can tell you about it is that the amount additional attackers increases your roll in absolute terms, naturally, is better the bigger dice you're rolling: two attackers with d4's is basically a +1. But 2 attackers with d12's is a +2. And that drops off quickly.

What's a bit more significant is that the odds of rolling minimum values plummets with multiple attackers, and if you've been paying attention that's a big deal where Armor is involved!

A simple case study

So with these ideas in mind, let's take just a moment to look at a match up and think about it's implications. Lets suppose two stat blocks:

Weakling

HP 3, STR 8, d6

Thug

HP 8, Armor 1, STR 10, WIL 10, d8

Bearing in mind those heuristics, how does this look? Well, it ain't pretty for that Weakling. With a d6 against that Thug's armor, they're looking at their best bet being 2 rounds to punch a hole through that HP. (We could be precise about how likely this outcome is: they would need to roll a 5 or greater on each roll, and to do that twice in a row. That's a 1/9 chance... ewww.) More likely, it's going to take them about four rounds. Ouch.

What about the Thug? Well, there are decent odds they can down this Weakling in a single hit!! (Rolling a 4 or more, so 50%! And if you're already on the bad side of that CD save, so losing any more strength is scary!). A second hit is a near certain death sentence.

Neat. What happens when you add more attackers? Well, squinting our eyes a bit, we just imagine the weakling getting max damage as a virtual certainty, and the outcome, pretty obviously, is that the players will probably "win" and either kill the Thug or scare them off, but one of the Weaklings is going down. Maybe permanently, but as long as their starting STR is high enough, we're looking at virtually certain attribute loss, and then you're right as rain for whatever comes next!

Plunging headlong into insanity

That's all well and good, but I want more. Enough with this maddening guess work! I want precision!

Long story short, I wrote a program to actually simulate a very bland 1v1 combat and report the odds. Crazy! I know! How long did you spend on this?! I'd rather not say.

Here's the code. (I'm reasonably confident in it, but in releasing it to the world I hope better minds--or at least more maniacal ones--can find any mistakes I may have made.)

And here's a spreadsheet with some results on display!

You can peek over at the code to get an idea for what kinds of stats each of these entities has to ground the results, but here are some of my take-aways from spending altogether way too much time on this:

• In an actually, truly, even matchup, a player has a little better than a 50% chance of winning. Player advantage is real, but it's pretty small in a raw 1v1.
• Winning that DEX save in the first round (or conniving to make sure it's not necessary) is a significant boost if the fight is somewhat even, but it's woefully inadequate in the face of...
• Even small increases in STR and HP can make a foe dramatically more difficult to defeat. Single armor points can make a foe dramatically more difficult to defeat.
• The first Armor point is eerily close to just reducing the attackers die by one. But my prior analysis with regards to the effect of additional armor points holds true. Ha! HAAA!
• Detachments are no joke. Do not joke with detachments. I could have spent another section just talking about detachments but it would just be a very long-winded: don't mess with detachments.

To summarize: yes, you have got to stack the odds. Impair your foe, enhance your attacks, negate the DEX save, try to bypass combat and hit directly to attributes. And given the tools players have to their advantage (morale saves, multiple attackers), it's only fair that the monsters should get things like insta-death Critical Damage effects to make sure you respect them.

All in all, these are delightful results in validating that Odd-like combat is all it claims to be, in it's core; and what's more it's impressive to me how much gradient is able to exist within such aggressively capped numbers!

Acknowledgements

Aside from the obvious ones--Chris McDowell, Yochai Gal, and all others who inspired them in making such an interesting and elegant game--I want to thank my brother.

This started out as us just getting on a video call and rolling some fake combat encounters to get a feel for things when we pulled various levers. I'm glad he was willing to sit and fake-play with me for a couple of hours!

I hope even he is surprised with how far down the rabbit hole I eventually went...

* But why?!

I suspect that the fully indoctrinated among us will be baffled as to why I should care so much to do all this work.

Part of it is purely academic. The math in these games is simple enough that simulating it felt within my grasp, so curiosity kept me going. (And it was fun code to write!)

But the other reason is something of a personal trauma: part of what drove me to Cairn in the first place was to run away from the worst behavior of 5e play culture. So many times was my careful prep in the interest of "balance" ruined by poor intuition for just how powerful a 5e character can get!

I love to homebrew, and I love to come up with cool magic items and other goodies, and I needed to reassure myself that I wasn't going to accidentally do that to myself again the first time I granted a player some kind of Growth or power up. I needed to know that a little extra armor or a slightly stronger sword weren't going to send my players absolutely sprinting into the ceiling that Cairn's attribute-caps present.

I hope that I'm not opening a pandoras box to invite the min-maxers in. What a horrifying thought.