Wednesday, November 13, 2013

Basic weapon DPS theory


This is a "Playing Better" segment focusing on weapon choices.

Basic Math

The basic theme running through all games is that you need a balanced set of stats to maximize values that are determined multiplicatively. As an extreme example, assume damage multiplied by attack speed = damage per second, so A x B = C, and assume buying A is as costly as buying B. You get an optimal result (maximum C) when A = B, for example, if you could buy 6 units of "weapon power" you would get the best result when A =3 and B =3 so A x B = 9. An imbalanced scenario, for example A = 2 and B =4 would yield C =8 and even worse when A = 1 and B = 5 where C = 5.

Example: Mirana

Of course, reality isn't quite so simple, but the principle still holds in general. Let's pick Mirana, for example, at level 20 she has 106 damage and 1.07 attacks per second for a DPS of 113. Ignore her 16% attack speed buff from leap. Her basic set of early game items would maybe consist of Phase Boots, Drums of Endurance and Ring of Aquila, bringing her to a DPS of 186. By level 20, assume she wants to maximize her damage and she could buy two items. The four main items of note are Daedalus or MKB (attack damage) and Mjollnir or Butterfly (attack speed). Common wisdom says you need to pick one from each group. I've crunched some numbers and put them in a table below.

Daedalus MKB Mjollnir Butterfly
Daedalus 612
MKB 607 550
Mjollnir 653 601 585
Butterfly 620 554 565 525

Mirana DPS output of different weapon combinations.

I think the first thing to note is that the variance in DPS output between the choices isn't particularly great. This is because even without advanced items, her core and base stats give her a respectable amount of damage and attack speed already. This means that you should be picking items based on the utility they offer, rather than the raw potential DPS they can output.

Mjollnir and Daedalus seem to be the best combination - not only is it the highest DPS but also offers good AOE from chain lightning and even more DPS from static shield. Building its other counterpart - Butterfly and MKB - offers 20% less damage done and no AOE, but gives you mini-stun, true-strike and evasion. The necessity for utility like this has to be judged on a match to match basis. If you're facing an opponent with a Butterfly then the MKB is immediately better than the Daedalus.

Even some of the weirder combinations that sound terrible aren't really that bad when you think about it, because their damage penalty isn't that great.

Double Mjollnir seems to be in the category of "joke" builds with its 44% chance to proc chain lightning per attack, but you might use it against a team with a lot of summons or multiple Linkens holders (the chain lightning attack breaks the Spellblock, even on secondary bounces). It's not just the increased proc chance that's at work here, it's also the increased attack speed, so there's a synergistic effect - a single Mjollnir has a 61% chance of proccing a chain lightning in a 2 second attack window, while a double Mjollnir has a 91% of proccing a chain lightning in a 2 second attack window. Also note that in the last patch Mjollnir chain lightning got buffed to a 900 search radius, so you're going to be hitting everyone in a teamfight when it procs, dealing significant damage.

Double Daedalus is actually quite commonly seen on Kunkka, because of its interaction with Tidebringer (he only cares about raw damage, not attack speed, and the proc on MKB does not help him) and the damage output is pretty competitive with the other options, which I found surprising.

There is pretty much no reason to ever go double Butterfly or double MKB unless you're trolling the enemy. Maybe double MKB with its 58% chance to stun on hit against a Pugna who has an Aghanims. Double Butterfly is nice with the 51% dodge but don't be surprised if your enemies buy an MKB immediately to put an end to your trolling...

Other notable mentions

Special mention for Desolator and Cuirass if you felt like it would help your team. Taking an opponent's armor from 12 to 0 increases DPS by 71%. Mirana's own DPS with this combination however only sits at around 560, which is on the low side but still competitive with the rest of the item choices in the table. The big advantage of this combination is the fact that it affects towers and buildings making sieges a lot faster and safer, and of course the fact that all other heroes on your team will benefit from this, potentially making this far more powerful than the other options as long as you have at least one other physical DPS oriented ally.

In reality, maximizing DPS isn't the primary concern for heroes. With the amount of disable and escape effects in DOTA2 and the already high lethality of most heroes, most of the time you're better off purchasing utility items that allow you to shrug off disables, lock down opponents or get the jump on them. This means items like BKB, Force Staff / Blink Dagger, Scythe of Vyse or Orchid, etc. You will find that Ursa for example more commonly builds a Scythe of Vyse rather than a Daedalus. But it's good to know the theory anyway.

Appendix (mechanics)

Mjollnir overrides all other unique attack modifiers (UAMs) only when it procs, so it stacks diminishingly with itself.

Daedalus crits also overrides other weaker or equivalent crits when it procs, so it stacks diminishingly with itself.

MKB bash is not an UAM to begin with, so it has independent proc chance (fully stacks) and can double proc (applies two separate instances of +100 damage in a single attack). This 100 damage is magical in nature and thus does not interact with Daedalus. Magic immune targets ignore the stun and bonus damage.

Butterfly evasion stacks diminishingly with itself as of the latest patch.