On Math and Modeling — Ursa’s Earthshock vs Fury Swipes

Some of you might remember this post on Reddit full of Dota mechanics math from late last year.  Shortly before the whole DBR storm came up, discussion about this post popped up independently from a couple different sources, so I thought I’d chime in.  There’s this big disagreement over whether theorycrafting has its place as a useful way to improve your play.  My position is yes, it can be a great way for us to test whether our judgments about the game are correct.  But it should always be used to supplement human judgment and never as a replacement for human judgment.  This is particularly true for Dota because it is curiously one of the few RPG-esque games that emphasize qualitative decisions over quantitative decisions.  So let’s move on to the subject at hand:

s = time spent hitting target, d = basedamage, t = total attacks = s×(1+as)/170

total damage = 15t² + t×(15+d)

This means Ursa’s damage scales quadratically with both the time he can spend hitting one target and his attack speed, meaning that if he worries about getting outcarried he should probably completely ignore pure damage items and mostly build attack speed for DPS. This also means that evasion is particularly effective against Ursa.

For level one Fury Swipes you only need to replace 15 with 7.5, so Fury Swipes doesn’t actually scale that well ~ which makes me wonder, why do people build Ursa (lvl 10) 1-4-4-1, and not 4-4-1-1, since his slow is almost as strong as Brewmaster’s (but much more spammable) and he can burst down pretty much anyone with that build and a blink dagger, with no fear of them getting away due to 4 seconds of 55% slow. Or even when jungling, 4-2-3-1 should still be better than 1-4-4-1. (I’d also get all other skills before lvl 2 ult)

For example the burst damage (Earthshock + Overpower attacks) of lvl 10 Ursa with 4-2-3-1, vlads and blink is 240+12.5×4² + 4×(12.5+76+1083×0.05) = 1010, and he’ll obviously get more auto attacks in, since the slow lasts for 4 seconds, while the 4 Overpower attacks are done after 1 second, so overall it should be enough damage to kill everyone at that point within a few seconds.

I haven’t actually tried those builds, but in theory they look much better than the standard “max Overpower and Fury Swipes above all else”. I actually think Ursa might be a decent hero if you max Earthshock early.

(emphasis mine)

Ignore the first part if you want.  The summary is that attack speed is more important to Ursa’s overall DPS than straight damage.  I don’t think this is of huge importance to us since most of Ursa’s midgame purchases tend to be oriented around reducing his susceptibility to kiting, particularly blink dagger and BKB though there are other, somewhat less popular options.

What’s really interesting here is the actually rather obvious but often overlooked fact that Fury Swipes suffers from diminishing returns.  At first glance it looks obvious.  The first rank gives 15 additional damage per swipe and each additional rank only gives 5 additional damage per swipe, but the problem with viewing it this way is that we don’t care that the second, third, and fourth points of Fury Swipes are weaker than the first; what matters is how they stack up against points in the other abilities.  A more mathematical way of looking at it is that the first point establishes the y-intercept, and every additional point establishes the slope.

For another example, take Gyrocopter’s Rocket Barrage.  The first point of Rocket Barrage gives you 11 damage per rocket, and each additional point only gives you 4 damage per rocket.  Looking at it this way, skilling Rocket Barrage past the first point looks like a waste.  But if you work out the math, each additional point of Rocket Barrage is worth 120 extra damage, which is actually pretty strong scaling.  Of course, the value of that 120 damage depends on you being able to position yourself so it’s directed exclusively towards an enemy hero, but it’s not a trivial damage gain despite 4 being so much smaller than 11.

Regardless, the guy’s math appears correct and Fury Swipes does indeed have weak scaling.  But it’s a peculiar skill because the amount of damage it adds depends on how long you can continue attacking the same target and scales in a funky way.  If we want to straight up compare it to Earth Shock we have to create a scenario and see how the damage plays out in two different builds.

For Ursa there are three relevant scenarios: jungling, Roshan, and killing enemy players.  Fury Swipes will hands-down beat Earthshock in the first two, but we’re going to put them aside for now.  As for the last, we’re going to use a scenario where Ursa gets an Earth Shock + X number of attacks on a target.  At the end of those attacks, we tally up the damage of the two builds.  The lower the damage, the more likely the target managed to escape.  Ursa’s big counter is being kited, so what we’re focusing on is putting out as much burst damage before the target escapes to safety.

We’re going to use very simplified builds that completely ignore Overpower and Enrage.  We assume whatever rank of Overpower we have is the one we need to manage however many attacks we’re landing.  The only difference between the two builds is one has 1 point in Earthshock and 2 in Fury Swipes and the other has 2 in Earthshock and 1 in Fury Swipes.  For the sake of simplicity we’re going to assume that the target has a sufficient armor level that it’s physical and magical damage reduction are identical.

Each point of Fury Swipes adds an additional 5 damage per attack.  What this looks like over 4 attacks is this:

1st Attack — 5 Damage

2nd Attack –10 Damage

3rd Attack — 15 Damage

4th Attack — 20 Damage

——————————

Total — 50 Damage

Incidentally, each additional point of Earthshock also adds 50 damage, so we know if an encounter lasts 4 attacks the two builds are equivalent in damage output, and if the fight lasts longer than 4 attacks, Fury Swipes pulls ahead.

One thing to keep in mind is that there’s a maximum number of attacks our simulated fight can reach because eventually the target will run out of HP.  If we’re looking at a 1Earthshock/2 Fury Swipes build at level 10, and assume that Ursa’s base damage at this point is 60 and the target has 1,000 HP, it will take Ursa 9 attacks to chew through that HP ((10 * 9^2 + 9*(10+60))*.75 if you want to check my math).  Realistically, this is probably an underestimate of his potency, since it assumes no Vlad’s, no Phase Boots, no other items of any kind, and probably undershoots natural damage growth (it also ignores Earthshock damage, and shorts him 2 skill points; 1/4/2/1 = 8).

So point-for-point, Earthshock is stronger in fights of 1 to 4 attacks, Fury Swipes in fights of 5 to 9 attacks.  But we’re still neglecting the fact that Earthshock provides a slow and the potency of the slow grows 10% each rank.  How do our builds compare if the additional point of Earth Shock allows for just one additional autoattack during the fight?

Assuming again a base damage of 60, the 5th attack will do 285 extra damage.  That pretty definitively outpaces the growth of Fury Swipes, and it ignores any additional item benefits that will only tip the scale in favor of Earthshock.  Earthshock’s slow could also allow allies to catch up to the target to add their own damage or contribute to the CC chain.

You might say that Earthshock is a bad skill because it’s too hard to hit with.  This is an outdated notion that ignores the buff Earthshock received in 6.75 in reducing its cast time from .5 to .3.  To continue the comparison with Brewmaster’s Thunder Clap from the original quote, Earthshock actually has a .1 shorter cast time and only a slightly smaller radius.  If Thuder Clap is accepted as an integral part of Brewmaster play I see no reason the same should not apply to Ursa.

Now returning to the question of jungle and Roshan viability.  I think there’s a compelling case to be made for a laning Ursa to max Earthshock before Fury Swipes, but that ignores that Fury Swipes is a big component to Ursa’s damage against neutrals.  Can a build with only 1 point in Fury Swipes jungle quickly enough to be viable in most games?  I don’t know.  You’d have to time the competing builds to get a good idea of just how much efficiency you’re losing, and then make a judgment call on whether the loss is worth the benefit.  If a 1/3/1/1 start isn’t viable, maybe a 0/3/2/1 would be an acceptable compromise.  But that’s just another set of tests and another judgment call.

As for Roshan, is the fastest Roshan possible that big of a deal for Ursa?  He’s going to be off the map for quite a while either way, and either his opponents will think to guard the Rosh pit or they won’t.  I guess there’s the possibility that a build that skimps on Fury Swipes might make the fight take just long enough that the opponents will get suspicious and stop it, but is that possibility significant enough on its own to justify a different skill build?  Again, it’s a judgment call.  Granted, it’s one we could approach statistically if we had enough samples of early game Ursa matches.

One of the most subtly cool things about Dota is that it appears to be designed from the ground up in a way that emphasizes qualitative decisions over quantitative decisions.  For instance, the Assault Cuirass vs Heart comparison that gets brought up in the Reddit post.  If the items were straight stat sticks, it’d be easy.  You want survivability?  Find where you are on the chart and pick the appropriate item.  But in actuality we have to weigh in the worth of the Heart’s HP regen passive, specifically is our character evasive enough to be able to activate it?  Are the Assault Cuirass’ attack speed and armor reduction auras something our team needs to amplify our damage?  There’s rarely a straight comparison to be made.  Nearly everything is situational.

But breaking things down to math is still important when we need to re-evaluate our intuitions.  Shortly after Keeper of the Light came out, I was involved in a brief argument over the value of Chakra.  They said it was an outdated skill now that item-based mana solutions are more readily available than they were in the path.  I pointed out that in terms of sheer additional mana/sec, Chakra blows everything out of the water.  Now maybe that amount of mana is overkill for your team, or maybe Keeper just isn’t a good fit for your lineup.  It still helps to have an accurate mental image as to what 300 mana (minus 85 if you’re worried about net) every 16 seconds actually is.

Anyway, with skill builds being mentioned here and in the upcoming API patch, next post will likely look at how we can categorize skill builds in a more useful way.

 

Correction: As many people have pointed out, Fury Swipes applies to the first hit.  I wasn’t sure whether it would or not and couldn’t find confirmation, so I tested it before the post but my test was in error.  The correct values should be in place now, assuming I caught everything.