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### World of Mathcraft: Physical Damage vs Physical Defense

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renticletape

Senior Member

Its has already been tested and well proven in general by practice that building resistance and HP is more efficient than building damage and attack speed throughout most of the game on Bruisers vs Melee DPS in 1v1 instances. Its simple, the longer you live longer, the more damage you deal throughout the duration of your lifespan. However there are some question that I would like to have answered.

For example:
Just what is the efficiency ratio? Is it linear? Does it cap off at some point? If DPS scales better later on in the game, at what point does it turn around? Whats an optimal split between usage of gold for offense and defense items?

To answer these questions, it would be impossible because there are way too many variables and conditions and things to take into account to determine this. The game is far too complex with the amount of champions, their abilities, the amount of items, 5v5 match ups etc. Using specific champions and specific items also doesn't work well either because champions themselves have too many variables and things to take into consideration like passives, cool downs on their abilities, duration of their steroids, mana and other resources, etc etc etc and builds vary from one champion to another. However, all we are really interested in here is the efficiency of building defensive with HP and resist vs building offensive with damage, crit and attack speed.

In order to give a general answer to these questions, we must first create a simplified mathematical model that we can use to gauge the efficiency of damage vs defense. Given that base damages and base survivability is free, this model would have to take that into account. The other things this model would need to take into account is the gold efficiency of the major stats which effect defense and offense. The other thing needed to be taken into consideration is the math used by the damage to calculate effective DPS and effective HP.

First lets look at Effective HP. Effective HP is how much defense you have total. Taking into consideration Attack Speed, Attack Damage and Crit Chance, we can expand DPS into the formula:

DPS = AttackDamage*AttackSpeed*(1+CritChance)
assuming each hit has a chance to crit for double damage.

The formula used in LoL to determine effect dps after mitigation is:

EffectiveDPS = DPS*
100/(100 + Armor)

Also since DPS is per second, the duration of which the champion is DPSing also must be taken into account. Thus:

EffectiveDamage = EffectiveDPS*Duration

where duration is how long the champion can DPS for in seconds.

Given a certain amount of HP, Armor acts as a multiplier of that HP. Doing some quick algebra on EffectiveDPS, from this we can derive the formula:

EffectiveHP = HP + Armor*HP/100.

Using these 2 metrics, we can gauge the total power level of a champion using this formula:

ChampionPowerLevel = EffectiveHP + EffectiveDamage.

With this metric, if we were to put 2 identical champions up against each other to auto-attack each other to death, the one with the higher powerlevel would be victorious and thus be stronger than the other. This is simply because EffectiveDamage negates EffectiveHP as a Zero Sum rule thus the champion with the highest total sum will be stronger - effectively.

So using ChampionPowerLevel as a comparison metric for DPS and Defense stats of a champion, we should now look into the gold efficiency that goes into raw Attack Damage, Attack Speed, Crits, Armor and HP. Using actual item data we can get the gold efficiency of an item using this formula:

GoldEfficiency = StatBonus/TotalGold

For this we will use B.F. Sword, Recurve Bow, and Agility Cloak as samples for gold efficiency on DPS items and Chain Mail and Giant's Belt as samples for defense items. The gold efficiency data samples yield the following stats:

CritChanceEfficiency: 0.000217
AttackSpeed
Efficiency: 0.000381
AttackDamage
Efficiency: 0.0273

ArmorEfficiency: 0.0643
HP
Efficiency: 0.387

If we were to take into consideration the amount of gold a champion has spent on damage items assuming and even 1/3:1/3:1/3 split between Crit, AS, and AD, we can derive this formula:

EffectiveDamage = (BaseDamage + AttackDamageEfficiency*TotalGold/3)*(BaseAttackSpeed + AttackSpeedEfficiency*TotalGold/3)*(1+CritChanceEfficiency*TotalGold/3)*Duration

That is quite a long equation but it computes a simplified approximation of how much damage you effectively get given your base damages and the amount of gold you spent on stats for damage. We will need to use this when we gauge powerlevels given certain amounts of gold.

Similarly, now given a certain amount of gold we can spend on defense items and assuming an even 50:50 split between Armor and HP we can compute the EffectiveHP using this formula.

TotalHP =
BaseHP + HPEfficiency*TotalGold/2

EffectiveHP = TotalHP + (TotalHP)*(BaseArmor + ArmorEfficiency*TotalGold/2)/100

Now using these formulas which take factor in Gold Efficiency, we can model how efficient it is to maximize a champion's powerlevel while minimizing the total gold spent. I figured the best way to represent this is with a graph that compares powerlevel of 2 different builds (1 focused on defense, the other focused on offense) and the amount of gold put into the build.

But first lets start with some basic stats we can plug into this model. For this I am going to invent a champion called Bob. Bob has No abilities, No Ult and No passive. He is a melee DPS and only has melee Auto-Attacks. His base stats at lvl 18 are as following:

BaseAttackSpeed - 0.9
BaseArmor - 50
BaseHP - 2000
BaseAttackDamage - 100

We will also assume that he will last for about 10 seconds for total Damage calculations.

Duration - 10

By holding all these variables constant, we can graph his powerlevel to gold graph given 3 different build options.
Build1 (red solid line): 0% gold spent on offensive stats, %100 gold spent on defensive stats
Build2 (blue dotted line): 100% gold spent on offensive stats, 0% spent on defensive stats.
Build3 (green dashed line): 50:50 gold split between offensive and defensive stats.
169523

So what does this graph say for Bob? From this case, we can see that Tank Bob was able to achieve a power level of over 9000 with about 4200 gold. While on the other hand, DPS Bob would have had to farmed up over 9000 gold just to get to the same powerlevel as Tank Bob just so he can compete and possibly win a 1v1. Tanky DPS Bob who split his gold evenly between defense and damage stats only needed to farm up a little over 6700 Gold to reach a power level of over 9000.

So what can we conclude from this? Well in terms of this simplified model, we can determine that Defense stats are more than twice as efficient gold wise than DPS stats assuming gold is distributed equally between HP and Armor and AD, Crit and AS.

Basically:
HP + Armor is 2.143x more gold efficient than AD + AS + Crit on Bob.

Of course this doesn't apply to everything. Bob is a fictional champion. This is a simplified model which doesn't take into consideration every possible factor in this game. We could also tweak with many of the parameters to this model, play around with base stats, change the duration, and even change the distributions of gold to see what efficiencies they reveal. Playing around with these parameters have shown that in most situations that defense stats is by far more gold efficient in most situations than damage stats. This model can answer the questions I originally posed in a general way.

However, don't mis-interpret this model. This model makes many assumptions, the main one is that the powerlevel is only used as comparison metric for gauging a single champion's usage of gold to fight against himself. I am not sure if this model can be used to compare efficiency stats across different champions with different builds. This model also doesn't take into consideration other DPS or Tank Stats such as Flat % damage reduction (Mastery/Alistar ult/Kat or Kass passives) or % Armor reduction (Last Whisper etc). However, thats the limitation of this model. Though these effects would skew the powerlevels a bit and efficiencies a bit but not enough to fundamentally change the model. You could spend more gold on Armor Pen or Just spend more gold on Flat armor conversely and it would negate the changes.

Anyways. I thought this would be an interesting incite as to why defense builds are so strong and why they work. One of the other interesting factors is that if you build Offensive items, you can farm gold faster meaning this may close some of the gold efficiency gaps between the two.

Edit:
As someone suggested in this thread, they were interested in seeing the model applied as a product of Damage and HP instead of a Sum. My sum model is based on a zero sum rule which is applied after a certain amount of time so a comparison would basically mean which champion would do the most damage and have the most left over HP meaning a champion who is purely tank (zero damage) would have some powerlevel as tanking as would a pure DPS (zero HP) would also have a powerlevel. A product version, as I have considered would make sense in gauging who would win if the fight prolonged over any duration of time but would mean a pure tank with no damage would have 0 power level as would a pure DPS who would still do damage but have no HP and would die after a single hit. This made little sense in comparing gold efficiency ratios as you would end up with 0/0 for comparing 2 different extremes but would make more sense in comparing which would always win a 1v1 over any prolonged amount of time.

ChampionPowerLevel = EffectiveHP*EffectiveDamage.

Anyways, I made a graph using a product instead of a sum and the same data I used for Bob and it gave interesting results:
169522

This shows that Tanky DPS build is greater than both pure DPS and Pure Tank build in situations where fights can be prolonged over any duration of time.

This could represent a better model but can easily give misleading results as in extreme cases, a champion with very high dps but near zero survivability can seem more stronger than a champion with very low dps but extremely amounts of survivability. In practice, this has been shown to not be true at all.

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xYoloSwagKid69x

Senior Member

what the nerd

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All Who Oppose

Senior Member

Okay, I REALLY wanted to read this....but ah.....
It's like, 10PM and I'm kinda tired....

Needs moar tl;dr
+1 for effort

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PCQWLUNF

Senior Member

Quote:
renticletape:
EffectiveDPS = DPS*Armor/(Armor + 100)

Shouldn't this be (1 - Armor / (Armor + 100) ) ?

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Lord Tolkien

Senior Member

...but what about MR?

Despite that, good work. I have to say, I'm impressed.

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renticletape

Senior Member

Quote:
Arcaria:
Shouldn't this be (1 - Armor / (Armor + 100) ) ?

That computes how much reduction is applied. Take 1-X to get the actual damage dealt.

For example with 200 armor, you get 66.66% reduction which is actually 33.3% Damage being dealt.

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ShadyGrave

Senior Member

sorry could only read some of it before my eyes got tired. but it was fascinating to read anyways. props to putting this together.

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MV101

Recruiter

AD * ASpd = total damage done, not damage per second. Damage per second would be AD/ASpd since 1 attack speed is 1 attack per second.

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LOLno

Senior Member

Don't forget unique passives, as well as lifesteal.

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renticletape

Senior Member

Quote:
MV101:
AD * ASpd = total damage done, not damage per second. Damage per second would be AD/ASpd since 1 attack speed is 1 attack per second.

And you do multiple attacks over time. Over that time is how much damage done.