Monday, February 15, 2010

Character Math: Damage per round

Today I'm starting yet another article series, this time devoted to the calculation of a player character's statistics. The idea is to provide ways to measure and compare different aspects of a PC's performance, much like the post on attack and defense normalization, a while ago. I have a few cool formulas lying around to quantify stuff that hasn't really been explored in depth... but I should start from the beginning. So this post will focus on damage dealing, which is pretty much all that is looked at, nowadays. You will find some basic definitions and formulas, as well as guidelines for calculating damage in some common special scenarios.

Most message boards have adopted the term "Damage Per Round" (DPR) as a measure of the amount of damage that can be dealt during a game turn. Although I can't tell for sure where this expression originated, it does look awfully similar to the "Damage Per Second" (DPS) commonly used in MMORPGS.

  • Damage per round (DPR): Average damage inflicted by a character to a single enemy in a round. It takes into account hit rates and extra damage from critical hits. DPR typically measures the abilities of a single character, so bonuses from other party members are ignored.

We usually clasify DPR in two types, depending on whether it involves a character using spendable resources or not: At-will DPR, and Nova DPR.

  • At-Will DPR: Damage per round that can be achieved by using exclusively at-will powers. Bonuses from daily or encounter effects do not apply, and spending of action points or other consumable resurces is not considered. Exceptionally, an encounter power granting encounter-lengh effects can be considered a valid modifier because of being permanently active.

    At-will DPR is the most commonly used metric in optimization boards. It isn't the most realistic estimation of actual PC damage, since the contribution of encounter and Daily powers is usually very significant. On the other hand, it's relatively straightforward to calculate, and a reliable indicator of damage in the worst case scenario for a damage dealing PC.
  • Nova DPR: Damage that can be dealt in a round with all the tools that a single character has at his disposal, including action points and daily powers. Used to measure a character's maximum offensive potential.
In addition, any DPR value can be against a single target (the default assumption) or against multiples (Area DPR).
  • Area DPR: When attacking multiple enemies, damage that can be dealt to each of these enemies in a round. Area DPR measurements should specify what kind of area or how many enemies can be covered: "Area DPR (area burst 1)" or "Area DPR (2 targets)".

Calculating DPR

The DPR of a character making a single attack each round is equal to the average damage of that attack. For scenarios with more than one attack per round, see below.

Average Damage of an attack.

The following calculations assume that a character is attacking a standard skirmisher monster of his own level. If you want to find out DPR values against different types of enemies, you will need to adjust Hit Rates accordingly.

Average Damage= (Hit rate * Average Hit Damage) + (1 - Hit Rate) * Average Miss Damage) + Crit Rate * Extra Crit Damage

Or, abbreviated:

AD= HR*Dh + (1-HR)*Dm + CR * Dc

  • Hit rate (HR), chance to hit a Skirmisher Monster of the same level. Can be calculated from here. If an attack has an intrinsic bonus to the attack roll, add it to the HR (+0,05 per extra point).

  • Crit Rate (CR), chance to score a critical hit.

    Roll to crit - Crit rate
    Crit on a 20 - 0.05
    Crit on 19-20 - 0.1
    Crit on 18-20 - 0.15

  • Average Hit Damage (Dh). To calculate, add the average value of all rolled dice, and any damage modifiers that apply.

    Die - Average value
    d4 - 2.5
    d6 - 3.5
    d8 - 4.5
    d10 -5.5
    d12 -6.5

    The Brutal property adds 0.5 to the average value of a die per point of brutal.

  • Average Miss Damage (Dm). Usually 0, but can be half the hit damage (Dh/2) for certain Daily attacks, or a fixed falue in certain cases.

  • Extra Crit Damage (Dc), or how much more damage is dealt by a critical hit, on average, in excess of the average damage of a normal hit.

    Dc = Dh - Average Crit Damage.

    (Average Crit Damage is the sum of all maximized damage dice, plus damage modifiers, plus extra crit dice).

Multiple attacks

When your character can make several attacks during a single round, calculate separately the average damage of each one and add them together.

Conditional attacks

When you have an attack that only triggers after hitting with a previous power, multiply that attack's Average Damage by the Hit Rate of the previous attack.

Attacks that trigger on conditions not completely under your control, such as a Fighter's Combat Challenge should not be considered on DPR calculations.

Automatic damage

If an attack deals automatic damage in addition to, or instead of a conventional attack roll, that damage is directly added to the total Average Damage.

Multiple rolls

If an attack allows you to roll twice and use the highest value, the effective Hit Rate and Crit Rate are increased. You can calculate these new values, HR' and CR' from the original HR and CR as follows:

HR' = HR * (1-HR)
CR' = CR * (1-CR)

Once-per-turn damage bonuses

Most striker classes have features that deal extra damage once per round. When calculating DPR, you can add this extra damage to Average Hit Damage only if you are making a single attack per round. When making multiple attacks, calculate this extra damage separately, as follows.

Striker damage contribution = Chance to hit once * average extra damage

The chance to hit once (H1) depends on the number of attacks and the Hit Rate HR:

2 attacks: H1= HR * (1-HR)
3 attacks: H1= 1 - (1-HR)*(1-HR)*(1-HR)
N attacks: H1 = 1 - ((1-HR) ^ N)


  1. Assassin's are an interesting case here. Their class feature deals damage on a miss or on a hit and the value isn't just a flat 1/2 either. I once read somewhere that you want to invoke your shrouds every other round, but my own math seems to suggest that the optimum damage output is to hold off until you have maximum shrouds and then invoke. Interestingly, though it complicates the formulas, the assassin's shroud doesn't really have complicated math. After all, you can weight the chance of hitting on an invoked attack and multiply the full damage by a percentage and the -1 shroud damage by the other percentage, and add that to flat average damage value-divided by the number of rounds that you needed to build up for that.

  2. Going strictly by the math shown here, assassins achieve maximum damage when they blow up the full 4 shrouds, because of the greater miss damage. However, there is a variable that can't really be factored in these formulas: the chance that someone else in the party kills your target. If that happens, you end up losing many turns' worth of extra damage.

    Blowing up shrouds every 2 turns is a decent compromise of risk vs reward, as you get some miss damage, but do not lose more than one shroud if an ally crits the target or whatever. You could also enshroud targets that your party isn't attacking right now, to make one big strike later. Yet another thing that should be taken in consideration is that delayed damage is worth less for your party - as you want to kill monsters as soon as possible.

    Assassins are also frequent victims of another mathematical nightmare for damage calculators, which is damage wasted in overkills. But that depends hugely on context, and it's probably not worth it to take into account.