Theory crafting, formulas, and maths

trimsic_ESO

One of the questions that pop the most for a stamina player is the type of weapons we should use to have the largest DPS possible: double sword, double dagger, or double maces?

Two swords or two daggers?
Well, in order to answer this question, I propose to detail here the formula I use in order to determine how much damage I can deal in function of my crit rate (C), and my crit damage bonus (D).

If we don't have any crit damage bonus (D=0), a critical hit deals 50% more damage than a normal hit. If we have some crit damage bonus (D > 0), a critical hit deals 50% + Dx100% more damage than a normal hit:
Crit damage = normal damage x (1.5 + D).

Therefore, the basic formula to compute how much damage we deal in average is:
average damage = normal damage x [(1 - C) + C x (1.5 + D)]
average damage = normal damage x [1 + C x (0.5 + D)]

Now let's see what happen when we use two swords or two daggers.

With two swords, we can deal 5% more damage:
average damage = normal damage x [1 + C x (0.5 + D)] x 1.05

With two daggers, we can deal 10% more crit:
average damage = normal damage x [1 + (C + 0.1) x (0.5 + D)]

So we deal more damage in average with two swords if:
1.05 + C x (0.5 + D) x 1.05 > 1 + C x (0.5 +D) + 0.05 + 0.1 x D
C x (0.5 + D) x (1.05 - 1) > 0.1 x D
C > 0.1 x D / [(0.5 + D) x 0.05]

C > 2 x D / (0.5 + D)


So, two swords or two daggers?

1. If we don't have any crit damage bonus (D=0), using two swords always deal more damage in average than two daggers.
2. If we have a huge crit damage bonus (D=0.5: a crit hit deals twice as damage as a normal hit), using two daggers deal more damage in average than two swords
3. If we have a moderate crit damage bonus (D=0.125 or less: shadow stone, or CP), and if we have a crit rate of 40% or more, then using two swords is better than two daggers; if we have a crit rate of less than 40% (before the dagger bonus), using two daggers is better than two swords.
4. If we have an important crit damage bonus (D=0.25 or more), and if we have a crit rate of 67% or more, then using two swords is the best choice; otherwise, if we have a crit rate of less than 67% (before the dagger bonus), using two daggers is the best choice.

What does this mean?
If we have a low / moderate crit damage bonus, then:
- if we have a low crit rate, then we should use the daggers
- if we have a high crit rate build, then we should use the swords

If we have a very high crit damage bonus, then equipping two daggers is often the best choice, but not always - see below.

We can use the following formula to determine what is our best choice:
- two swords if our crit rate is greater than 2 x D / (0.5 + D)
- two daggers if our crit rate (before equipping the daggers) is lower than 2 x D / (0.5 + D)

Where D is our bonus to crit damage
- D is about 0.125 for the shadow stone
- D is either 0.125 or 0.25 for 100 points in precise strikes (I'm still determining whether precise strikes is additive or multiplicative)
- D is the sum of the two value above if we have both the shadow stone and 100 points in precise strike.


Two swords or two maces?
Under construction

Asayre

An interesting start @trimsic_ESO! I have a few comments
1. Two swords does not increase your normal damage by 5%. It increases the weapon damage component coming from your two swords by 5%.
2. I may be making a mistake here but the maths doesn't seem right to me at
average damage = normal damage x [(1 - C) + C x (1.5 + D)]
average damage = normal damage x [1 + C x (0.5 + D)]
3. What about a combination of sword and dagger?

Apart from that, good work! I look forward to your revised post and the section on two swords or two maces.

bigereard

Awesome work!
There's also NB's hemorrhage for the D equation
and since elfborn is multiplicative, i think precise strikes also multiplicative

If we really want to know which one is better, we must include the defender critical mitigation in D equation.
we have armor trait, sets, and champion passive that reduce critical damage, not mention the damage shield that completely imun.

I personally always favor sword over dagger
(and I personally almost always favor mace over sword in PvP)

Asayre wrote: »

1. Two swords does not increase your normal damage by 5%. It increases the weapon damage component coming from your two swords by 5%.
2. I may be making a mistake here but the maths doesn't seem right to me at
average damage = normal damage x [(1 - C) + C x (1.5 + D)]
average damage = normal damage x [1 + C x (0.5 + D)]

1. So is it stack additively or multiplicatively with other weapon damage buff? If it's only the weapon damage component, then we must add max stamina to the equation, isn't it?

2. I think the math is right

trimsic_ESO

No the two swords only buff the weapon damage part, not the stamina part, of total damage we can deal with our physical attacks. I'll fix the equation.

Basically I'll start with an evaluation of the actual damage bonus brought by two swords (the damage increase is not 5% as I initially thought).

damage with swords = damage caused by stamina + 1.05 x damage caused by weapon damage
damage without swords = damage caused by stamina + damage caused by weapon damage
percentage damage increase with swords = (damage with swords - damage without swords) / damage without swords

so:
percentage damage increase with swords = 0.05 x damage caused by weapon damage / (damage caused by stamina + damage caused by weapon damage)

If W is our weapon damage, and S is our max stamina, then:
percentage damage increase with swords = 0.05 x 10.5 x W / (S + 10.5 x W)
percentage damage increase with swords = 0.05 x 10.5 / (10.5 + S / W )

We can remark that we often have a max stamina roughly equal to 10 times our weapon damage. So an approximation of the formula above is:
percentage damage increase with swords is about 2.5%

With 33k max stamina and 3400 weapon damage:
percentage damage increase with swords = 2.6%

With 40k max stamina and 2400 weapon damage:
percentage damage increase with swords = 1.1%

When our max stamina / weapon damage ratio is low, having two swords is efficient.
When our max stamina / weapon damage ratio is high, having two swords is not efficient.