CAP Math: Aspects and Equipment

#1
This is a thread to look at the math behind different aspects and equipment choices in the first beta release. There's going to be some math, but I'm going to hopefully keep it basic and understandable. To start with, though, I'm going to need to define some variables and terms:

DPS- Damage per swing. The expected average damage from one attack - you can calculate this by hand by adding up every possible outcome from a 1-20 roll, and then divide by 20. We'll mostly be talking about DPS gains, and most gains are going to be expressed as 0.05* some other number.

V- The average damage from a damage die roll, not counting anything else. The average for dice is the (the value of the die +1)/2, so a 1d8 has an average of 4.5, a 2d8 has an average of 9, etc. This is how much damage you can expect to deal on a penetrating hit.

M- The maximum damage from a damage die roll. The M for d8 is 8, the M for 2d8 is 16, etc. This is how much damage you'll deal on a penetrate.

B - The barrier value, Toughness or Resistance, that applies to this attack.

D- Dodge or Willpower, whichever is being used to avoid an attack, when talking about player defenses.

T - Your tier, needed to compare weapons.

Successful attack- Any attack that's a crit, pen, or hit. One in which you've actually succeeded on your roll against the enemy.

Hit - A non-critical, non-penetrating, still succesful attack: a roll that's triggering the A option of CAP.

"Displacing" - The net effect of losing an outcome possibility on a dieroll because it's being replaced by a better one. When you get +1P, you're "displacing" a hit in exchange for a penetrate. When you get +1C, you're "displacing" a penetrate most of the time - but if your C>=P to start, you're displacing a hit, instead.

Riders- Any effect that applies on a successful attack is a rider: most commonly these are conditions, but sometimes they're things like self-heals, too.
 
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#2
Now, let's get into the Offensive Aspects

+1A and +1P: Steady

First, it's worth noting that Tier 2 of steady adds up to +1A and +3P, but you can get +2A and +2P by repeccing between Precise and Powerful, so I'm not sure why Steady doesn't just remain the same throughout.

The math behind Steady is pretty simple: You're gaining one penetrate, which normally displaces a hit, but you're also gaining a hit. So your net gain is the effect of a penetrate - which is just V, the average damage on your dice, leaving your increase in DPS as 0.05*V.

So long as your P isn't already greater than your A, Steady doesn't care about your current CAP, or enemy defenses: it's just a flat expected % damage increase.

+1C: Deadly

For deadly, the math is a bit more complicated. The expected outcome of a bonus C is M, the maximum damage, but you also have to account for the displaced penetration - you're penetrating one time less for every 20 attacks.

As such, the expected DPS increase for Deadly is 0.05*(M-V).

Much like steady, Deadly doesn't care about your current stats as long as your C isn't already greater than P, it's just a flat expected % damage increase. But how does it compare to Steady?

Taking Steady and subtracting Deadly from it, we find:

0.05*V - 0.05(M-V)
0.05*(V - M + V)
0.05*(2V-M)

So the difference between the two is the difference between twice the average damage, and the maximum damage. Twice the average damage is always 1 higher than the maximum on any die, however. So Steady is better than Deadly by 0.5 damage per die of your attack.

Deadly gets a leg up if you already have so much C that you don't P to begin with, and start displacing hits - like if you're a tactician with the +6C buff, or an archer using -10P abilities, in which case the damage can be estimated to be more like:

0.05*(M-V+B) in cases where V>=B+4
or
0.05*(M-B) in cases where B=>V

The reason for this is better explained in the next section, where I talk about accuracy. In these cases, where you're displacing hits instead of penetrates, Steady often doesn't give any bonus penetrates at all, and can be rejected as a comparison.

+2A, Precise

One of the more appealing, general options is Precise, for +2A. The expected damage from Precise is complicated, as you're adding in hits that are mitigated by enemy barrier values. It's tempting to say this is V-B, but that isn't the truth: If you're rolling 1d6 damage against a barrier value of 4, your V-B is in the negatives, yet you deal damage on a roll of 5 or 6, leaving your expected damage value on a hit to be 0.5. Not great, but it's something! Each die roll to defense combination has to be calculated out individually, but we're mostly looking at that V-B term, and can make a few statements:

In cases where your MINIMUM damage is equal to or above the barrier value, V-B applies. So for a 4 toughness enemy, 4d6 ,or 2d12+2 both have V-B equal to the expected damage.

In cases where B is greater than or equal to M, our expected damage is a flat 0.

Everything else is a curve between the two where the actual expected damage moves away from V-B the smaller V-B is. My general rule of thumb is: as long as V>(B+4), V-B is a good stand-in for our expected value. The more dice we're rolling, the more true this is as well.

So how much damage does +2A add? It's simply the expected value of two hits, or 0.10*(V-B). The higher your average damage over the enemy barrier value is, the better accuracy bonuses are. It's also worth noting that accuracy bonuses help out by making your riders a lot more common - if you deal poison on a successful hit, your poison application chance jumps up by 10%, too. That's pretty good!

+2P: Powerful

The expected damage of a penetrating hit is pretty easy: 0.05*V, however, we're displacing a normal hits to do so. As such, our math instead looks like:

0.10*(V) - 0.10*(V-B)
0.10*[(V-(V-B)]

Again, we've got to do some estimates for the value of V-B, and can't just pull those variables out on their own unless V>>B, giving us...

0.10*B in cases where V>=B+4
or
0.10*V in cases where B=>V

We can basically sum this up as 0.10 x B or V, whichever of the two is lower.

The more your enemy barrier value is, the better penetrates are. This should be pretty expected! The breakpoint is roughly when B=0.5*V. Once you start losing half your damage to the enemy's barrier values, you'd rather get more P than A. There's one important note, though, which is that as soon as P=A, more P doesn't do anything for you. Seeing as P and A only start 4 apart, this is really easy to hit if you're not watching, and needs to be avoided.


+1 Damage Die: Vicious and Keen

Our last one is the most complicated, because it actually takes your CAP into account. +1 Damage Die has a greater effect the more often you hit and crit! Penetrates only matter if we're in that dreaded (V<B+4) zone, so we're going to ignore them for now, and assume every hit is dealing at least some damage to start with. Your expected damage bonus is:

0.05*C*(Maximum damage on one die)+0.05*(A-C)*(Average damage on one die):

For each die size this gives us:
d4: 0.2*C + 0.125*(A-C) = 0.075*C + 0.125*A
d6: 0.3*C + 0.175*(A-C) = 0.125*C + 0.175*A
d8: 0.4*C + 0.225*(A-C) = 0.175*C + 0.225*A
d10: 0.5*C + 0.275*(A-C) = 0.225*C + 0.275*A
d12: 0.6*C + 0.325*(A-C) = 0.275*C + 0.325*A

If we assume C=2, and A=14 (reasonable assumptions, I feel) we get:

d4: 1.65
d6: 2.70
d8: 3.50
d10: 4.30
d12: 5.10

If you regularly deal 0 damage with hits due to the enemy barrier value, this value drops: any damage that's also lost to 0 damage hits on variable damage will be gone, and then you need to take into account the value of penetrates. This is exercise is left to the patience of a far, far more patient reader.

Summary of Aspects, and DPS gains:

+1C: 0.05*(M-V)
+2A: 0.10*(V-B) , as long as V>=B+4
+2P: 0.10*B or 0.10*V, whichever is lower
+1A/+1P: 0.05*V
+1 Damage Die:
d4:
0.075*C + 0.125*A
d6: 0.125*C + 0.175*A
d8: 0.175*C + 0.225*A
d10: 0.225*C + 0.275*A
d12: 0.275*C + 0.325*A
 
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#3
Understanding Barrier Values and Dodge

Next it's time to look at the player's defenses. The expected value a player is going to suffer from an attack is:

Expected damage: (1-0.05*D)*(V-B)

...with, again, V-B having to be above certain values before things start getting wonky due to "expected" 0 damage attacks still dealing damage. The first term represents your odds of dodging the attack altogether. The second term represents the damage you'll be taking when you get hit. What's more interesting is to look at how things change as you pile more of a stat on. This answer may be obvious to some of you, but it's still good to do the math.

Looking at the change in damage from each +1 to D or +1 to B, we first write the equation out, and then derive (or just do the +1s by hand and subtract the original, either works).

Expected Damage: V-B-0.05DV+0.05DB

Damage change from +1D:
-0.05V+0.05B
% Damage Change from +1D: -100%/(20-D)

Damage change from +1B:
-1 +0.05D
% Damage Change from +1B: -100%/(V-B)

Some stuff we expect to see is here: The more damage you're taking, the better Dodge is. The more barrier value you have, the worse dodge is! The less damage you're taking, the better barrier values are! This should make sense to you if you look at the extremes: if you're taking 100 damage, +1 to toughness isn't going to help much, but +1 dodge will. if you're taking 1 damage, +1 to dodge will be minor, but +1 to toughness just makes you immune. So the goal, then, is to pick your focus and to get lots of it.

The breakpoint of incoming damage here is 20: if you're taking more than 20 damage, you want more dodge. If you're taking less than 20 damage, you want more barrier values. A lot of enemy damage values hover around this line, so that's pretty good. The toughest, nastiest enemies you'd rather have high dodge for (although that's also a riskier play!), and for big masses of small enemies, you want more toughness.

Now, the two aren't necessarily priced the same in all places. For armors, they're easy enough: +tier to Resistance, Dodge, or Toughness all priced the same. Off-hand shields are similar: +Dodge, Toughness, Resistance, or Willpower, with the medium shield as a +1/+1 to Dodge and Toughness as a weird middle point thrown in. Armors treat all all four of the defensive stats the same.

Traits, however, do not. If you go for high willpower, you get a bit of extra HP - willpower is the only option not available for armor. If you want to mix stats and pick up some Willlpower, you should take Willful or Defiant traits instead of getting the off-hand item that boosts it - it's just plain better.

The last thing to consider is Vigorous, which gives you +10 HP in exchange for 1 less toughness (and trading a willpower for a resistance, which is supposed to be a "fair" trade according to armor). This is a pretty easy calculation here - if you would get hit 10 or more times in a fight, you'd prefer the toughness. if you wouldn't, you'd prefer the HP.

Getting hit 10 times in a fight is pretty out there unless your party is doing a very dedicated tanking strategy, making Vigorous an extremely powerful choice for most characters. Heavy duty tanks will likely prefer Tough for the +2 Toughness, Evasive for the +2 Dodge, or Willful or Defiant for the +2 Willpower if they're ranged characters mostly worried about magical attacks. But anyone who expects to mix it up and not be the major target most of the time, or suffer from mixed attacks is absolutely taking Vigorous as their primary choice.


Summary
Expected Damage:
(1-0.05*D)*(V-B)

Damage change from +1D:
-0.05V+0.05B
% Damage Change from +1D: -100%/(20-D)

Damage change from +1B:
-1 +0.05D
% Damage Change from +1B: -100%/(V-B)
 

benso1

New member
#4
Nice math. I'm also looking at some of this stuff.

For +1 dmg dice, an easier way to compare it on penetrates and crits is that you deal about 50% more dmg on tier 1, and 33% more dmg at tier 2.

Also, from looking at the barrier values from the 9 enemies listed in the support files, 5 is about the average of enemy toughness with resistance having an average of 5.5
 
#5
Weapons and Off-Handers

Going to finally swing this back around with our look at offensive weapons and off-hand items. We're going to use sword and mace as a baseline - we've already compared straight accuracy vs penetration. However, it's worth considering that +4 pen is a LOT, and it's very, very easy to get into a situation where P=>A, at which point your penetration is wasted.

1h Sword vs 2h Sword

The first comparison I want to make is between the most basic 1H weapon, and the most basic 2h weapon.

Compared to the 1H sword, the 2H sword gets +2 penetration, and two bonus dice sizes, from d6 to d10. We're going to make the same assumptions as we did for Vicious and Keen above - you're always dealing at least 1 damage on a hit, so penetrates don't matter.

Our expected damage difference PER DIE per attack formula is:
0.05*C*(2hM - 1hM)+0.05*(A-C)*(2hV - 1hV)
0.05*C*(10-6) + 0.05*(A-C)*(5.5-3.5)
0.20*C + 0.10*A - 0.10*C
0.10*C + 0.10*A
0.10*(C+A)

Keeping the old assumption of C=2 and A=14, we get a net benefit of +1.6*(T+1) damage and +2P. So at Tier 1, we're looking at +3.2 DPS, at tier 2 we're looking at +4.8 DPS, etc. That's pretty damn hefty.

How does a 1h sword + offhand weapon compare? Not well - it gives less damage than the damage die difference does at Tier 1, which isn't surprising, as our average damage is going from 2d6 (7) to 2d10 (11).

The good news, however, is that the off-hand weapon gives a damage bonus to ALL actions. So if you're a spellcaster or ranged weapon user, it's a choice between +3 damage and +2P. +2P is worth 0.10*(V or B), whichever is lower, so that +3 damage looks really, really good in comparison. So off-hand weapons aren't viable for a two-axe warrior or a sword and dagger rogue, HOWEVER, they work great for any spellcaster or archer. A character who's using a lot of low-damage fast actions in particular is really going to like it, like a mana echoes ardent or a wildfire druid.

How about that polearm?

For a 1h weapon, the polearm gives you +1 die size and +1 reach in exchange for not getting +2 A compared to the sword. The +2A is 0.1*(V-B) damage.

If we compare the d6 to a d8 we now get:

0.05*C*(PoleM - SwordM)+0.05*(A-C)*(PoleH - SwordH)
0.05*C*(8-6) + 0.05*(A-C)*(4.5-3.5)
0.10*C + 0.05*A - 0.05*C
0.10*C + 0.10*A
0.05*(C+A)
And again, that's WITHOUT the (T+1) multiplier
So how does 0.05*(C+A)*(T+1) compare to 0.1*(V-B) ? The first ends up being around 0.8*(T+1). The second ends up being a lot less - 0.65 total at Tier 2, for instance. So the 1h polearm is very, very good.

Is the 2h polearm good? At this point, the damage is even, but you're giving up +2A in exchange for +1 reach. That's not something that can be mathed out, but that exchange doesn't look anywhere near as good.

How about that axe?

Now this is a tricky one. Compared to a sword, you're up +1 die size, +1 C, but down -4 A. Ouch!

We know each of these values already, and can set the C bonus in terms of T.

+1C: 0.05*(M-V) = 0.05*[(T+1)*12-(T+1)*6.5)] = 0.275*(T+1)
+1Die Size: 0.05*(C+A)*(T+1)
+4A: 0.20*(V-B) = 0.20*[5.5*(T+1)-B)] = 1.1*(T+1) - 0.20*B

This lets us try to find a breakeven point, using starting values of 2C and 14A:

Axe Bonus = Sword Bonus
0.275*(T+1) + 0.05*(2+12)*(T+1) = 1.1*(T+1) - 0.20*B
0.975*(T+1) = 1.1*(T+1) - 0.20*B
0.125*(T+1) = 0.2*B
B = 0.625*(T+1)

From calculations before, we know that high barrier values are bad for accuracy bonuses. So at Tier 1, if an enemy's barrier value is 1.25 or above, you'd rather use an axe. At tier 2, if their barrier value is 1.875 or above, you'd rather use an axe. If you have damage from another source, like a class trait, you actually want to *add* that to the barrier value to see the breakpoint: so if you have +5 damage from something, the tier 1 barrier value jumps to 5.25. As your A goes up from other sources, the axe becomes better - if your A isn't as high, the sword is better.

The short of it is: a 2h axe is very, very good, even with the massive -2A attached to it. But not everyone wants to use one. If you're ever using NON axe attacks, the axe is a heavy liability. If you have any sort of riders on your attacks, you'd likely prefer the 20% better chance to apply them! Likewise, if you have any bonus damage, the sword starts to shine again, too. This leaves the axe in a weird position, where it's going to be favored by healing druids, dual-attack rogues, warriors who prefer seasoned veteran over powerful, healer atlantas, kiai spiritualists, and support tacticians.

Is a shield worth it?

That's actually up for you to decide. No amount of math can justify offense vs defense, here. Is 1.6*(T+1) DPS worth giving up for +2 toughness? For someone who expects to attack most of the time with the weapon, I'd expect not. For a character who's going to be doing other things, absolutely.
 
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[NOM] Derek

Administrator
Staff member
#6
This is an incredible summary and analysis of the engine so far, and thank you so much for writing this up, @Karrius!

While there's a lot more that could be said, I just wanted to include a few considerations that can make this analysis even more complex.

One of those considerations is the "Cancel" Rider. Unlike other Actions that that place a Condition, Heal you, etc., Cancelling a Foe Action can be hugely valuable, depending on what your Hero ultimately prevents from happening. This comes down to the-moment to-moment, though, and relies heavily on player skill. However, making sure a "Cancel" Ride is reliable does put more value onto having a higher "A" than "P", for instance.

Another consideration is that having a P value that is greater than A isn't always wasteful. There are builds that focus on utilizing Sustains, Off-Guard, etc. to provide boosts to your A. An easy example of this is with a Seasoned Veteran Warrior build, who's accuracy will grow over time.
 
#7
Oh, yes, i hope i wasnt underselling my thoughts in it. As is i honestly expect accuracy bonus is the most general useful - its absolutely the ones support characters are taking, and for characters with big damage boosts or poison riders, it still stacks up well raw number wise imo. My personal preference would lean towards accuracy, just because i like consistency, if the pure damage wasnt so high.

For the A and P bit, i more think its a thing to be aware of, and a note having to be made for the calculations. Its less about the default stats and more what your A and P would be at any one particular time.
 
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[NOM] Derek

Administrator
Staff member
#8
Oh, yes, i hope i wasnt underselling my thoughts in it. As is i honestly expect accuracy bonus is the most general useful - its absolutely the ones support characters are taking, and for characters with big damage boosts or poison riders, it still stacks up well raw number wise imo. My personal preference would lean towards accuracy, just because i like consistency, if the pure damage wasnt so high.

For the A and P bit, i more think its a thing to be aware of, and a note having to be made for the calculations. Its less about the default stats and more what your A and P would be at any one particular time.
Oh, of course! Your note here (and whole analysis, really) is something that I think has a great deal of equally great information for new players!
 
#9
We've got new bonuses in the hero manual v0.3, so let's look at some!

To start with, we've got +1 damage die on all critical hits that deal damage. This is pretty easily summed up as being (Crit Chance * Die size), or 0.05*C*(die size), so:

+1 Damage Die to all critical hits:
d4:
0.2*C
d6: 0.3*C
d8: 0.4*C
d10: 0.5*C
d12: 0.6*C

From above, we also have:
+1Die Size: 0.05*(C+A)*(T+1)

And our older values of:
+1C: 0.05*(M-V)
+2A: 0.10*(V-B) , as long as V>=B+4
+2P: 0.10*B or 0.10*V, whichever is lower
+1A/+1P: 0.05*V
+1 Damage Die:
d4:
0.075*C + 0.125*A
d6: 0.125*C + 0.175*A
d8: 0.175*C + 0.225*A
d10: 0.225*C + 0.275*A
d12: 0.275*C + 0.325*A

So for our "average warrior" with C=2, and A=14 with a d10 weapon at Tier 3:

+1 Damage Die to Crits: 0.5*2 = 1
+1 Die Size = 0.05*(2+14)*(3+1) = 3.2
+1C = 0.05*(40-22)= 0.9
+1A/+1P: 0.05*22 = 1.1
 
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