Damage Balance Analysis

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Damage Balance Analysis

#1 Post by DarnYak »

Preface:
- There was a lot of data entry involved so there's a reasonable chance some of the numbers are wrong, or even that formulas are wrong, so don't be afraid to call me out if something's off
- These are all raw numbers assuming a normal perfect hit - nothing except standard hero reduction, AoE's that diminish based on range assume its dead center, if its over time it assumes the full thing hits, etc.
- There's also some left out assumptions where damage increases over time, for example Incinerate calculates only its first cast even though its damage goes up and mana cost goes down on consecutive hits
- I tried to stick with mostly instant or "guaranteed" damage, but there's a few exceptions I threw in just for comparison (example: frost spire)
- Ultimates are left out entirely
- My goal is not to homogenize all damage skills, that would be bland and completely disregard factors such as range (and lack of it and associated risks), chance of missing, the hero's other skills, side effects from hitting, etc.
- If you have an observation that hasn't been said already, please don't be afraid to voice it. I might not have seen it or it might otherwise provide interesting insight. Similarly, if there's some additional data you're interested in seeing I'll consider providing it based on the time/difficulty to put it together.

This batch of charts represents rank 6 skills on a level 30 hero.

One number I use is something I'm calling a "Balance Ratio" for lack of a better term. It's quite simply just { Average Hero Health / Hero's Health }. This number is relevant because for equilibrium a skill's damage should be multiplied by this value. For instance, Average health is 1000, Caster has 800 (BR=1.25), Tank has 1250 (BR=0.8 ). Using a base damage of 100, and all other stats equal, the two heroes will kill each other at the exact same time if their personal version of the spell's damage is multiplied by the BR, so caster's would do 125 damage and Tank's would do 80 damage, killing each other off in 10 hits.

Alright, the charts:
Numbers01.png
(Same as the other thread)
Single Target Damage = The total damage from a perfect hit from one cast of the spell
- Nice to see that single target effects are generally above AoE types.
Maximum damage per second {Damage / Cooldown} = Assumes recasting on cooldown
- Top of this list is pretty dominated by 8s cooldown spells
- Pheonix Strike and Incinerate are pretty close for the top AoE spells, Frost Spire is actually fairly low for DPS despite being the top damage per cast aoe's
Damage per mana {Damage / Mana Cost} = Damage done per point of mana spent
- Best and worst spell here on the same hero
- Again, generally single target skills on top, partly due to better damage
Time to regenerate mana { Mana cost / Mana regeneration }
- Mana regen is actually pretty similar across all heroes (1.6 to 2.1 mana/s), so this is pretty much just a reflection of mana costs
Numbers02.png
Scaling per cast { Rank 6 Damage / Rank 1 Damage } = Reflects how much investing skill points benefits burst
Scaling per cast { Rank 6 DPS / Rank 1 DPS } = Reflects how much investing skill points benefits potential dps
- DPS chart is almost identical to the damage chart, unsurprisingly the only differences are the few spells with cooldowns that change
Glass cannon { Damage / Hero's Health } = How much of their own life a hero would take by nuking themself
- Doesn't really say much, just a vaguely interesting number
Factored cast { Damage / Balance Ratio } = Scales the actual damage to try to derive what the equilibrium base damage would be
- No cooldowns factored in here, so don't read too much into it
- Soulstrike pulls even further ahead!
- In general, its all the melee single target fatty attacks on top
- Average total health is ~1300, so ~325 damage would be 25% of health per hit
Factored DPS { DPS / Balance Ratio } = Scales the DPS to try to derive what the equilibrium base DPS would be
- Probably the most overall significant number for pure balancing
- Fatty nukes clearly reign here
- Higher numbers here should represent the winner of a sustained fight if this is all that's involved
Numbers03.png
Ignore the Crimson Squall damage numbers, it adds in the 3 variants of his nuke as if they were 3 separate spells

Starting Stats {Starting Str + Agi + Int}
Growth {Str + Agi + Int Gain per level}
Burst Potential {Sum of Damage belonging to this hero} = This number's relatively flakey but it tries to represent getting all of a hero's damage spells off on a target back to back
- The top 5 can be pretty safely disregarded since they involve channeled or delayed spells, hence not that bursty
- IS could almost kill himself with one round of casts!
Factored Burst Potential {Burst Potential / Balance Ratio} = Tries to put tankiness relative to burst potential into some perspective
- Grav can really smash
- 'of Average Health' column is broken, its not important anyway
- 'Of self' column is broken, and I'm not sure what it means anyway since it factors in the hero's health twice (one in the ratio, then one for self targeting)
-- Actually, it comes out identical to the unfactored damage % of average health, so it doesn't mean anything!
DPS Potential {Sum of DPS belonging to this hero} = Similarly flakey to above, assumes being able to chain cast all the dps spells disregarding channeling and whatnot
Factored DPS Potential {DPS Potential / Balance Ratio}
- Probably the second most important number in all these charts (only second because its a bit unreliable)
- Heroes with higher numbers ought to win in even contests


This is really only the first step in a proper analysis, since an arguably bigger factor is comparing heroes with their different item builds. For today, however, this is all I'm going to be doing.

The excel sheet is attached, here's a brief explanation of it:
Sheet1: Has the basic stats to play with, messing with these numbers shouldn't break anything and will propagate throughout the spreadsheet
Sheet2: Raw hero data (leave alone)
Sheet3: Raw ability data (leave alone)
Sheet4: Something about healing that's outdated (this was originally the healing data excel file)
Sheet5: A lot of intermediate data (don't touch)
Sheet6: Some charts I was toying with until I realized they really didn't show anything
Sheet7: The results, you'll need to manually resort columns if you change anything

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Re: Damage Balance Analysis

#2 Post by DarnYak »

Here's some updated charts to include all heroes and every non-ult damage skill I could reasonably calculate a number for. Sorry about the shrunken font, no other easy way to keep them on one page.
Numbers04.png
Numbers05.png
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Re: Damage Balance Analysis

#3 Post by DarnYak »

So here's data with items mixed in. I'm going to call these "well rounded" item sets because...well, they seem to be. Item list was provided by watermelon (all items are +5).

Str heroes: Angelstone, Devil's Smirk, Hydra Mail, Dragonscale Mask (Total: 12705 gold)
Str heroes with int nukes: Angelstone, Glacial Wall, Sorcerer's Robes, Dragonscale Mask (Total: 11760 gold)
Agi heroes: Raider's Orb, Leather Armor, Rogue Gloves, Executioner's Hood (Total: 11130 gold)
Int heroes: Angelstone, Glacial Wall, Sorcerer's Robes, Dragonscale Mask (Total: 11760 gold)
Numbers06.png
Numbers07.png
Average hero health is now ~2300. The agi heroes are pretty far behind since their builds give them no health (harpy has a paltry 800 health). Fatties are just short of 3000 and casters hover around 2200.

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Re: Damage Balance Analysis

#4 Post by DarnYak »

Here's the last charts I'm currently planning on creating. This is the maximum damage output item builds, ie focus entirely on you're main stat forsaking any items purely for survivability.

Str heroes: Devil's Smirk, Ring of Might x 2 (only 3 items, Total: 10370 Gold)
Str heroes with int nukes: Mindgem, Glacial Wall, Sorcerer's Robes, Dragonscale Mask (Total: 11300 gold)
Agi heroes: Dryad's Orb, Rogue Gloves, Helm of Champions, Dexterous Ring (Total: 11940 gold)
Int heroes: Dryad's Orb, Glacial Wall, Sorcerer's Robes, Gloves of Evocation (Total: 12300 gold)
Numbers08.png
Numbers09.png
Average health is now ~1850, with fatties up around 3000. BR here is pretty out of whack, fatties are down at about 60% where casters are around 140%.

Also, here's the updated excel sheet with these last numbers.

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Re: Damage Balance Analysis

#5 Post by DarnYak »

So it turns out I wasn't quite done yet. After posting a crapton of numbers, I'm not sure they had any sufficient context to explain anything.

Also random note: I'm using average health, but I probably should be using median (in the last data set its ~1800 average, ~1400 median - with no items its ~1250 average and median).

The BR and its application to Damage or DPS is, in my opinion, the most important number in trying to balance. What they mean exactly is this: the Factored Burst Damage (that's just BR * Damage) is the amount of damage that would have to be done in return to that hero to match the hero's burst damage as a percentage. Using similar numbers to before, Average health is 1000, Tank has 1500 health (BR: 66.7%), Tank's nuke deals 200 damage (bringing the target down to 80%). For it to balance out, a hero with average health would need to deal 300 damage (bringing tank down to 300), which is equal to the damage divided by the BR (200 / 0.667 = 300). Taking this further to double check it, a non-average hero health would have to be scaled further, such as a caster with 800 health (BR: 125%) would need to deal 375 damage. 375 / 1.25 = 300, so Tank dealing 200 damage and Caster dealing 375 damage are in equilibrium.

Factored DPS is similar, it represents the damage per second the hero would have to be taken from an average health hero for their health percentages to remain equal and die at the same time. While the DPS number is less accurate then burst damage for real gameplay, its more meaningful for any engagement that is at all sustained.

So with that, here is two hideous charts comparing heroes' factored dps.
Hero Comparison - No Items.png
Hero Comparison - Max Damage Items.png
This goes by heroes' total dps, partly because listing skills wouldn't fit and partly because its the most telling imho. Each row represents a hero's matchup against the hero in the corresponding column. Green is balanced, red means the hero strongly wins, blue means the hero severely loses. The exact formula for the numbers is { (Hero's Factored DPS - Target's Factored DPS) / Hero's Factored DPS } - its roughly a severity of overkill when positive and when negative its how much higher it would need to be to even out.

Note that I chopped out some heroes that had either useless data (Inf, Red Swashy), or no damage data.

Edit: Gotten this question twice so some clarification, the very left part of the chart is just the raw data (its just dps/factored dps), which is then plotted out against itself in the colored chart. So basically just ignore the left 3 columns unless you want the raw data.

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Re: Damage Balance Analysis

#6 Post by DarnYak »

Ok, one more. The same as the previous charts, except sorted by strength heroes instead of alphabetical. I think it says something.
Hero Comparison - No Items STR Sort.png
Hero Comparison - Max Damage Items STR Sort.png
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Re: Damage Balance Analysis

#7 Post by Executor »

I believe an important variable should be considered for hero balancing: income. This is similar to the jungler or pusher classification. Even though allies may share gold, I think a team’s overall income can have a significant outcome in the game. Also, an increase in income might indicate an increase in experience per second. Heroes such as Grim Hag (using Blade Rain with or without Raptor Feast) and Behemoth (using Maw of Death and/or Miasma) have a considerable advantage in this regard. Overall, more income and experience leads to more dps and survivability.

There are other complicated things to consider such as the growth rates of hero stats and their contribution to the dps of abilities…

Question 1: Should total armor and health regeneration be considered for the Balance Ratio?

Strength heroes tend to use strength items as it contributes to both dps and survivability. Similarly, Agility heroes tend to use agility items, which somewhat contributes to survivability through armor. In my experience, strength heroes feel overpowered for various reasons.

Question 2: Should auto-attack dps be considered in the damage balance analysis?

This applies especially to Mystic Swashbuckler, High Oracle, and Tactician.

I’ll continue to examine and think about all of this as time permits.

Edit: I guess income may balance out in the team selections, but I wonder if there is enough variety of heroes to compete in income.

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Re: Damage Balance Analysis

#8 Post by DarnYak »

Income is obviously very important but since its rather difficult to calculate it has to be left out of these models. If we had actual game data we could get statistics that way but that's not realistic in wc3.
Question 1: Should total armor and health regeneration be considered for the Balance Ratio?
Armor doesn't effect skill damage, so in terms of these charts no. Health regen should be tied very closely to overall health percentage so its safe to leave out as well.
Question 2: Should auto-attack dps be considered in the damage balance analysis?
Yes, but its beyond the scope of these spreadsheets. The mix of casts interrupting the swing timer, channeled spells, and positioning requirements for auto attacks make any prediction about their dps wildly inaccurate. Yes, that does exist to some extent already, but I feel adding in auto attack damage is probably going slightly too far into inaccuracy that the render the data useless. Where they are now is they give a rough approximation of damage but its a discrete enough number that it can be useful in the context of the overall hero for balancing purposes.

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Re: Damage Balance Analysis

#9 Post by Executor »

Another perspective of Balance Ratio would be BR = (Health * DPS)/(Avg. Health * Avg. DPS), which means that a hero with average health and average dps would have a BR of 1. This simply adjusts the original BR by a scale factor and changes the context to something that is not purely DPS nor Health but both. In other words, there is no need to change your formula for BR since all you really need is Health * DPS. You can ignore this information if you want lol.

For now, intuition and experience are my best methods for balancing heroes, but this data is also needed to make stronger arguments. You have already provided useful information about strength heroes. It is possible to balance purely through calculation, and usually people rely on computers to simulate results…so this would be a good step in that direction. I also enjoy a game where the rules are more clearly stated. It would be a pain to figure out this stuff on my own, so thanks.

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Re: Damage Balance Analysis

#10 Post by DarnYak »

Executor wrote:Another perspective of Balance Ratio would be BR = (Health * DPS)/(Avg. Health * Avg. DPS), which means that a hero with average health and average dps would have a BR of 1. This simply adjusts the original BR by a scale factor and changes the context to something that is not purely DPS nor Health but both. In other words, there is no need to change your formula for BR since all you really need is Health * DPS. You can ignore this information if you want lol.
Yea, I had considered doing this, except average DPS would be applying to itself and strictly speaking average dps isn't necessarily a good balance point. Desired DPS could be used and derived from average health, and similarly an ideal TTL or a preferred health % per nuke. All effectively different ways to get to the same result. I stuck with the method I choose because this way I can look at a nuke that I feel works well, and then try to balance the other nukes relative to it.

And I'm assuming people are going to wonder who the midpoint is. Which isn't quite accurate, since I'm not picking the most balanced hero, but zeroing in on a rough range where I think things work well. Blackguard and Reek are the most centered of the heroes.
For now, intuition and experience are my best methods for balancing heroes, but this data is also needed to make stronger arguments. You have already provided useful information about strength heroes. It is possible to balance purely through calculation, and usually people rely on computers to simulate results…so this would be a good step in that direction. I also enjoy a game where the rules are more clearly stated. It would be a pain to figure out this stuff on my own, so thanks.
Yea, as I keep saying there's a lot of factors not numerically reducible, but lets face it damage is the most important stat in the game and comparing them to each other isn't far fetched. As I'm in the middle of doing the changes now, most of the numbers I end up agreeing with based off experience and especially in the context of each other.

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Re: Damage Balance Analysis

#11 Post by DarnYak »

Alright, so I wanted to do a quick update post for the new revised balance numbers just to show how things have changed (for better or for worse, I still don't know yet!). So here's the previous colored charts again, no items & max damage items.
Revised Numbers01.png
Revised Numbers02.png
The distinct str favoring pattern is still there, but significantly reduced. Quite a bit more green all around. By the way, without the item health changes the second chart looked almost identical to the previous version.

I also want to share another chart I made and found a bit more useful. As you may have guessed from the other thread, I threw in an arbitrary "fudge factor" number to allow human bias into the numbers. The cynical view would be that the factor exists to make the chart more green and justify balance. This isn't strictly wrong, as in revising the numbers it would make me ask "does this spells mechanics justify an adjustment to the calculated value of the spell" and largely exists to bridge the gap between the different spell mechanics while making it visually clear that its ok that the raw damage isn't the same.

Fudged formulas:
{Fudged Skill DPS = Factored DPS * Square Root(Fudge Factor)}
{Fudged Hero DPS = Sum(Fudged Skill DPS) * Fudge Factor }

Why square root the skill's fudge factor? Quite often, the fudge factor there is simply the success rate of a spell hitting. However, a 50% success rate doesn't justify a doubling of damage, that would get out of hand quickly (taking it further, 10% success would balance out at 10 times normal damage, aka a guaranteed kill). I don't square root the hero fudge factor because those are less about odds of success and just a rough approximation of effectiveness.

So here are the fudged charts, once again same data as before, first without items then with maxed dps items:
Revised Numbers03.png
Revised Numbers04.png
Lots of freaking green. The heroes that stand out tend to have fairly clear reasons for standing out (support heroes largely). It's hard to say how accurate this chart is at representing balance at all, but I found it useful as a factor in balancing numbers. And just for clarification, this chart was actually significantly green even prior to the first round of balance changes.

And one final note, I had actually tried to make a formula to roughly balance skills with longer cds vs spammy skills beyond their basic dps (to account for the benefits of burst), but those numbers didn't come out to anything useful.

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