MacroHard
09-19-2011, 05:06 PM
Contents
Preface - What and why
Damage Reduction and Damage Factor - How armor reduces incoming damage
Offensive Damage Amplification - The "sweet spot" for armor reduction
Effective Health and the Linearity of Armor - How the benefits of armor are linear (not diminishing)
"Diminishing" Returns of EHP - The illusion of diminishing returns
Incremental Benefits of Health and Armor - The benefits of 1 extra HP or armor
Lines of Equivalency - How much health a single point of armor is equal to
Optimization and Adjustments for Fixed Magic Damage - How to optimize your gold for maximum EHP for fixed incoming magic damage
Optimization and Adjustments for Scaling Magic Damage - How to optimize your gold for maximum EHP for incoming damage as a percentage of total damage
Health/Armor Balance Quick Reference - Easy to read rule of thumb that quickly summarizes all of the above
Conclusion - Meaningless generalities
Item EHP Tables - 60 individual tables itemizing the EHP and EHP/gold for every item in HoN
FAQ - Your questions answered
Preface
Armor is one of, if not the most, under appreciated and underrated attributes in HoN.
After searching through the threads, stickied posts, and google searches, I still could not find a guide truly dedicated to unraveling the mysteries of EHP specific to HoN. The closest guide that met my standards, by Malle, was mostly focused on comparing armor to shield blocking. While brilliant, it still didn't quite get the message out on the true value of armor. Malle's guide can be read here: http://forums.heroesofnewerth.com/showthread.php?p=444209#post=444209.
I am aware that these formulas are commonly available and have been hashed out many, many times before. The first few sections of this guide are just for ease of reference and overall consolodation. Please continue to the end of this guide for newer revelations.
Damage Reduction and Damage Factor
for positive armor:
DR = 0.06*A/(1 + .06*A)
DF = 1/(1+0.06*A)
for negative armor:
DR = 0.94^(-A) - 1
DF = 2 - 0.94^(-A)
These are the basic formulas that are the building block of effective health. All of these formulas are available on various threads and guides on the mechanics forum and honwiki. They are repeated here for your convenience.
Definitions
Health HP
Health is the in-game attribute that determines your capacity for life. When health drops to zero your hero dies. Maximum health increases with strength (19 health for every 1 strength), can be purchased directly, or enhanced by skills/spells.
Armor A
Armor is the in-game attribute that is used to calculate Damage Reduction. Higher armor corresponds to more damage resistance. Armor increases with agility (.14 armor for every 1 agility), can be purchased directly, or enhanced by skills/spells.
Damage Reduction DR
DR is how much, by percentage, incoming damage is reduced. For example, a DR value of 25% reduces one-fourth of damage. DR increases with armor. Both physical and magical armor work the same way, but against different damage types.
Damage Factor DF
DF is how much damage an attack actually does as a proportion of the listed damage. For example, for DF = 0.60, a 200 damage attack will only remove 120 health from the target. DF = 1 - DR.
Damage Reduction Formulas, Table, and Graph
Damage Reduction DR
DR = 0.06*A/(1 + .06*A), A > 0
DR = 0.94^(-A) - 1, A < 0
note: For this equation, since A < 0, -A will be positive
Table:
http://i53.tinypic.com/259it92.jpg
Graph:
http://i56.tinypic.com/256wpyq.jpg
Damage Factor Formulas, Table, and Graph
Damage Factor DF
DF = 1/(1+0.06*A), A > 0
DF = 2 - 0.94^(-A), A < 0
note: For this equation, since A < 0, -A will be positive
Table:
http://i56.tinypic.com/2gt8zrp.jpg
Graph:
http://i51.tinypic.com/2ep4yo8.jpg
Offensive Damage Amplification
When talking about DPS, all you care about is how fast you can eliminate the opponent. If the enemy has 20 armor, how much faster can I kill him with Shieldbreaker Level 2? If the enemy has -5 armor because of Pesti ultimate, is it really worth buying Sol's Bulwark? These questions can be answered by looking at when armor makes the enemy most vulnerable to physical damage.
This section identifies the "sweet spot" where armor reducing items and spells have the biggest effect. This sweet spot is typically between 3 and 5 armor. You can also see that armor reducing items and spells still maintain usefulness even into the negative armor range.
Definition and Example
Damage Amplification
Damage Amplification will be defined, for the sake of this guide, the rate of change of the Damage Factor with respect to Armor, divided by the original Damage Factor. It is essentially the increase of damage on a percentage basis.
For example, the enemy has an armor value of 15 and you intend to decrease it by 5.
DF(15) = 1/(1+.06*15) = 0.526 = 52.6%
DF(10) = 1/(1+.06*10) = 0.625 = 62.5%
The increase in Damage Factor is 9.9%.
However, the Damage Amplification is (0.625-0.526)/(0.526) = 18.8%.
By lowering the enemies' armor from 15 to 10 you will kill them 18.8% faster.
Formula
For negative armor,
DF = 2-0.94^(-A)
-dDF/dA = 2489*2^(A-1)*5^(2*A)/(20113*47^A)
-dDF/dA / DF = 2489*2^(A-1)*5^(2*A)/(20113*47^A)/(2-0.94^(-A))
For positive armor,
DF = 1/(1+.06*A)
-dDF/dA = 150/(9*A^2+300*A+2500)
-dDF/dA / DF = 150/(9*A^2+300*A+2500)*(1+.06*A)
Table, General Case
http://i52.tinypic.com/2cr8uc5.jpg
Graph, General Case
http://i54.tinypic.com/2cgld04.jpg
Table, Item Specific
This shows the total Damage Amplification from Shieldbreaker and Sol's Bulwark.
http://i51.tinypic.com/2ciiatt.jpg
Graph, Item Specific
http://i51.tinypic.com/dxjrqp.jpg
Effective Health and the Linearity of Armor
Effective Health, or EHP, is another commonly referred to factor along with Damage Reduction. EHP is more meaningful to reality however, as explained in the definition below. Once again these formulas are repeated here for your convenience.
Definition
Effective Health EHP
EHP is the amount of unmitigated damage you can take before dying. For example, it takes 20 attacks of 200 listed damage to kill a target with 4000 EHP. Due to this direct relation to unmitigated incoming damage EHP is the single best measurement of survive-ability. Both Health and Armor contribute to the calculation of EHP.
Effective Health Formula
EHP = HP/(1 - DR) = HP/DF = HP*(1 + .06*A), A > 0
EHP = HP/(1 - DR) = HP/DF = HP/[2 - 0.94^(-A)], A < 0
note: For this equation, since A < 0, -A will be positive
Tables
http://i53.tinypic.com/fw6ps0.jpg
http://i51.tinypic.com/2vnm7gy.jpg
Graphs
http://i51.tinypic.com/2wf30gw.jpg
http://i54.tinypic.com/2s6rgua.jpg
Linearity
As shown in the graphs, the benefit of armor is purely linear for positive armor even though damage reduction is diminishing.
Every additional point of armor gives the same benefit as the point before.
In other words, going from 5 to 6 armor has the same benefit as going from 45 to 46 armor. Not only is there no functional armor cap, there is not even an upper bound where armor becomes less effective.
To prove this with the formulas, let's take the example above with 2000 health:
2000 HP and 5 armor gives 2600 EHP
2000 HP and 6 armor gives 2720 EHP... a gain of 120 EHP
2000 HP and 45 armor gives 7400 EHP
2000 HP and 46 armor gives 7520 EHP... also a gain of 120 EHP
"Diminishing" Returns of EHP
While total EHP increases linearly with armor, the percentage of EHP gain compared to current EHP naturally decreases. The same applies for every attribute in HoN. For example, increasing Health (or EHP, or mana, etc) from 1000 to 1100 is a gain of 100 or 10%. However, increasing Health from 2000 to 2100 is still a gain of 100 but now only 5%.
For example, if you already have 20 armor, 1 additional point of armor will increase your EHP by 2.7%, NOT by 6%.
Calculation
%EHP gain is calculated by taking the dirivative of the EHP formula and dividing the result by the EHP formula.
For negative armor,
EHP = 1/(2-0.94^(-A))
dEHP/dA = 2489*2^A*47^A*5^(2*A)/(20113*5^(4*A)*2^(2*A+1)-20113*47^A*5^(2*A)*2^(A+3)+160904*47^(2*A))
%EHP = dEHP/da / EHP = [2489*2^A*47^A*5^(2*A)/(20113*5^(4*A)*2^(2*A+1)-20113*47^A*5^(2*A)*2^(A+3)+160904*47^(2*A))]*(2-0.94^(-A))
For positive armor,
EHP = (1+.06*A)
dEHP/dA = 0.06 (somewhat easier than negative formula!)
%EHP = dEHP/dA / EHP = 0.06/(1+.06*A)
Table
http://i55.tinypic.com/11qs400.jpg
Graph
http://i51.tinypic.com/2w1v6nq.jpg
Incremental Benefits of Health and Armor
This is the first step towards understanding how to maximize EHP. In order to optimizing the balance between Health and Armor, we must first identify exactly the benefit, in terms of EHP, from one additional unit of Health or Armor.
HP's incremental effect on EHP: (increase in EHP) = (additional HP)*(1 + .06*A)
(original EHP) = (original HP)*(1 + .06*A)
(new EHP) = (original HP + additional HP)*(1 + .06*A)
(increase in EHP) = (new EHP) - (original EHP) = (additional HP)*(1 + .06*A)
Armor's incremental effect on EHP: (increase in EHP) = .06*HP*(additional A)
(original EHP) = HP*[1+.06*(original A)]
(new EHP) = HP*{1 + .06*[(original A)+(additional A)]}
(increase in EHP) = (new EHP) - (original EHP) = .06*HP*(additional A)
Lines of Equivalency
For every given Health and Armor combination, there is a precise ratio at which X Health is equivalent to Y Armor.
Each "line of equivalency" is:
(additional HP)/(additional A) = .06*HP/(1 + .06*A)
Explanation
Set incremental EHP gain from Health equal to incremental EHP gain from Armor:
(additional HP)*(1 + .06*A) = .06*HP*(additional A)
(additional HP)/(additional A) = .06*HP/(1 + .06*A)
For example, if you currently have 15 armor and 2000 health, increasing armor by 1 is equivalent to increasing health by 63.2. To put it in more game-real quantities, increasing armor by 5 is equivalent to increasing health by 316.
Table
http://i52.tinypic.com/2u5x7jt.jpg
Graph
http://i51.tinypic.com/24q2uqt.jpg
Optimization and Adjustments for Fixed Magic Damage
By assigning in game currency, gold, to each incremental point in Health or Armor, we can calculate a single line of Health/Armor optimization.
It is important to note that this gold assignment is subjective. My calculations and graphs are based on 1 HP = 4.4 gold and 1 Armor = 110 gold.
Using this assumption, the optimal balance of Health and Armor is:
A = 0.04*[HP-(Anticipated Magic Damage Received)] - 16.67
Assumptions
Fortified Bracelet = 510 gold, 114 HP -> 4.47 gold/HP
Beastheart = 1100 gold, 250 HP -> 4.40 gold/HP
Ringmail = 550 gold, 5 A -> 110 gold/A
Although there are many items that give armor and/or HP, I am assuming that the portion of the cost of these items directly responsible for health and/or armor fall within these same ratios.
For instance, Frostfield Plate costs 4700 gold and gives 15 armor. The cost per armor is not 313. Rather, I am assuming that the cost associated with 15 armor is only 1650 gold. The remaining 3050 gold for this item go towards the other benefits for that item (int, mana, active, etc).
Calculation
(additional HP)/(additional A) = .06*HP/(1 + .06*A)
(1/4.4)/(1/110) = .06*HP/(1 + .06*A)
25 = .06*HP/(1 + .06*A)
A = 0.04*HP - 16.67
This works fine for when 100% of the damage you take is physical. However, there is a lot of magic in HoN. Therefore you must account for the post-mitigated magic damage you anticipate to absorb in a fight. This damage deducts from your health such that the formula becomes:
A = 0.04*[HP-(Anticipated Magic Damage Received)] - 16.67
Graph
http://i56.tinypic.com/2u70do2.jpg
Example
You have 2000 HP. Based on the enemy team composition, you expect to receive 1000 damage after reductions in a particular teamfight.
Using the 1000 damage line, you should have about 23 physical armor for an optimal balance between health and armor.
Optimization and Adjustments for Scaling Magic Damage
Similar to above, this section uses the same gold assumptions of 1 HP = 4.4 gold and 1 Armor = 110 gold.
This time, rather than a fixed amount of magic damage, the balance is calculated using the percentage of incoming damage that is magical.
Calculations
Let X = percentage of incoming damage that is magical (.25 = 25% of unmitigated damage received is magic)
Let M = magic armor
Amount of Magic Damage = X/DF = X/(1+.06M)
Amount of Physical Damage = (1-X)/DF = (1-X)/(1+.06A)
Effective Combined Damage Factor = X/(1+.06M) + (1-X)/(1+.06A)
Effective Combined EHP = HP/DF = HP/((1-X)/(1+.06A)+X/(1+.06M))
Value of HP = ((H+1)/((1-X)/(1+.06A)+X/(1+.06M)) - H/((1-X)/(1+.06A)+X/(1+.06M)))/4.4
Value of A = H/((1-X)/(1+.06(A+1))+X/(1+.06M)) - H/((1-X)/(1+.06A))+X/(1+.06M))
Setting these equal yields the following solution for H in terms of A:
HP = (3AX-3M(X-1)+50)((3A+3)X-3M(X-1)+50))/((1-X)/(1+.06A)+X/(1+.06M))/4.4/(3M+50)^2/(1-X)/.06/110
Graph
http://i42.tinypic.com/aexmb8.png
Example
You have 2000 HP and 10.5 magic armor. Based on the enemy team composition, you expect half the damage you receive will be magical (X = 0.5).
Using the X = 0.5, M = 10.5 line, you should have about 18 to 19 physical armor for an optimal balance between health and armor.
Health/Armor Balance Quick Reference
Let's face it. Looking at multiple graphs, tables, and charts is silly and impractical while in a game. While it makes for an interesting thought experiment while cruising the forums, nobody will ever think "okay, if I'm going to take X amount of magic damage, my next item purchase should be...".
To address this, I put together a "Hybrid" method - a single simple graph that combines the essence of everything you've learned up until this point. The result is a continuous line that simply lets you know if you're in the right ballpark for efficient EHP gain. If you frequently stray from this line perhaps you have a tendency to invest too much in Health or too much in Armor. Remember, balance is key.
WARNING! Hard numbers and math end here. What follows is purely subjective!
What is This?
If you look at the optimization lines, how likely is it that you will take 1500 magic damage at a point in the game where you are still sitting on 750 health? Zero. The following table and graph is essentially a smoothed line based on the fact that magic is more prevalent early game.
Table
http://i39.tinypic.com/2a5zomc.png
Graph
http://i40.tinypic.com/35k1kzo.png
Example
You have 2000 HP. The enemy team is more physical based than usual, but there are still some magic sources.
Using the graph and estimating between the "Heave Physical" and "Typical" lines, you should have approximately 20 physical armor for an optimal balance between health and armor.
Conclusion
Because magic damage is more prominent early game, armor is mostly not necessary very early game until your health reaches around 1000. However, after that point, armor's economic effectiveness skyrockets. Even with expecting to take a whopping 1500 magic damage after mitigation, you should have about 23 armor by the time you hit 2500 HP. Keep this in mind next time you contemplate building your hero to have 3500+ health. More often then not you are better off buying some simple armor items if your goal truly is to be as tanky as possible.
The benefits of Armor are completely linear and there is no upper bounds for armor
Armor is often the least expensive way to increase EHP
High Health / Low Armor has the advantage of providing better protection against magical attacks
High Armor / Low Health makes better use of health regeneration items and spells
Preface - What and why
Damage Reduction and Damage Factor - How armor reduces incoming damage
Offensive Damage Amplification - The "sweet spot" for armor reduction
Effective Health and the Linearity of Armor - How the benefits of armor are linear (not diminishing)
"Diminishing" Returns of EHP - The illusion of diminishing returns
Incremental Benefits of Health and Armor - The benefits of 1 extra HP or armor
Lines of Equivalency - How much health a single point of armor is equal to
Optimization and Adjustments for Fixed Magic Damage - How to optimize your gold for maximum EHP for fixed incoming magic damage
Optimization and Adjustments for Scaling Magic Damage - How to optimize your gold for maximum EHP for incoming damage as a percentage of total damage
Health/Armor Balance Quick Reference - Easy to read rule of thumb that quickly summarizes all of the above
Conclusion - Meaningless generalities
Item EHP Tables - 60 individual tables itemizing the EHP and EHP/gold for every item in HoN
FAQ - Your questions answered
Preface
Armor is one of, if not the most, under appreciated and underrated attributes in HoN.
After searching through the threads, stickied posts, and google searches, I still could not find a guide truly dedicated to unraveling the mysteries of EHP specific to HoN. The closest guide that met my standards, by Malle, was mostly focused on comparing armor to shield blocking. While brilliant, it still didn't quite get the message out on the true value of armor. Malle's guide can be read here: http://forums.heroesofnewerth.com/showthread.php?p=444209#post=444209.
I am aware that these formulas are commonly available and have been hashed out many, many times before. The first few sections of this guide are just for ease of reference and overall consolodation. Please continue to the end of this guide for newer revelations.
Damage Reduction and Damage Factor
for positive armor:
DR = 0.06*A/(1 + .06*A)
DF = 1/(1+0.06*A)
for negative armor:
DR = 0.94^(-A) - 1
DF = 2 - 0.94^(-A)
These are the basic formulas that are the building block of effective health. All of these formulas are available on various threads and guides on the mechanics forum and honwiki. They are repeated here for your convenience.
Definitions
Health HP
Health is the in-game attribute that determines your capacity for life. When health drops to zero your hero dies. Maximum health increases with strength (19 health for every 1 strength), can be purchased directly, or enhanced by skills/spells.
Armor A
Armor is the in-game attribute that is used to calculate Damage Reduction. Higher armor corresponds to more damage resistance. Armor increases with agility (.14 armor for every 1 agility), can be purchased directly, or enhanced by skills/spells.
Damage Reduction DR
DR is how much, by percentage, incoming damage is reduced. For example, a DR value of 25% reduces one-fourth of damage. DR increases with armor. Both physical and magical armor work the same way, but against different damage types.
Damage Factor DF
DF is how much damage an attack actually does as a proportion of the listed damage. For example, for DF = 0.60, a 200 damage attack will only remove 120 health from the target. DF = 1 - DR.
Damage Reduction Formulas, Table, and Graph
Damage Reduction DR
DR = 0.06*A/(1 + .06*A), A > 0
DR = 0.94^(-A) - 1, A < 0
note: For this equation, since A < 0, -A will be positive
Table:
http://i53.tinypic.com/259it92.jpg
Graph:
http://i56.tinypic.com/256wpyq.jpg
Damage Factor Formulas, Table, and Graph
Damage Factor DF
DF = 1/(1+0.06*A), A > 0
DF = 2 - 0.94^(-A), A < 0
note: For this equation, since A < 0, -A will be positive
Table:
http://i56.tinypic.com/2gt8zrp.jpg
Graph:
http://i51.tinypic.com/2ep4yo8.jpg
Offensive Damage Amplification
When talking about DPS, all you care about is how fast you can eliminate the opponent. If the enemy has 20 armor, how much faster can I kill him with Shieldbreaker Level 2? If the enemy has -5 armor because of Pesti ultimate, is it really worth buying Sol's Bulwark? These questions can be answered by looking at when armor makes the enemy most vulnerable to physical damage.
This section identifies the "sweet spot" where armor reducing items and spells have the biggest effect. This sweet spot is typically between 3 and 5 armor. You can also see that armor reducing items and spells still maintain usefulness even into the negative armor range.
Definition and Example
Damage Amplification
Damage Amplification will be defined, for the sake of this guide, the rate of change of the Damage Factor with respect to Armor, divided by the original Damage Factor. It is essentially the increase of damage on a percentage basis.
For example, the enemy has an armor value of 15 and you intend to decrease it by 5.
DF(15) = 1/(1+.06*15) = 0.526 = 52.6%
DF(10) = 1/(1+.06*10) = 0.625 = 62.5%
The increase in Damage Factor is 9.9%.
However, the Damage Amplification is (0.625-0.526)/(0.526) = 18.8%.
By lowering the enemies' armor from 15 to 10 you will kill them 18.8% faster.
Formula
For negative armor,
DF = 2-0.94^(-A)
-dDF/dA = 2489*2^(A-1)*5^(2*A)/(20113*47^A)
-dDF/dA / DF = 2489*2^(A-1)*5^(2*A)/(20113*47^A)/(2-0.94^(-A))
For positive armor,
DF = 1/(1+.06*A)
-dDF/dA = 150/(9*A^2+300*A+2500)
-dDF/dA / DF = 150/(9*A^2+300*A+2500)*(1+.06*A)
Table, General Case
http://i52.tinypic.com/2cr8uc5.jpg
Graph, General Case
http://i54.tinypic.com/2cgld04.jpg
Table, Item Specific
This shows the total Damage Amplification from Shieldbreaker and Sol's Bulwark.
http://i51.tinypic.com/2ciiatt.jpg
Graph, Item Specific
http://i51.tinypic.com/dxjrqp.jpg
Effective Health and the Linearity of Armor
Effective Health, or EHP, is another commonly referred to factor along with Damage Reduction. EHP is more meaningful to reality however, as explained in the definition below. Once again these formulas are repeated here for your convenience.
Definition
Effective Health EHP
EHP is the amount of unmitigated damage you can take before dying. For example, it takes 20 attacks of 200 listed damage to kill a target with 4000 EHP. Due to this direct relation to unmitigated incoming damage EHP is the single best measurement of survive-ability. Both Health and Armor contribute to the calculation of EHP.
Effective Health Formula
EHP = HP/(1 - DR) = HP/DF = HP*(1 + .06*A), A > 0
EHP = HP/(1 - DR) = HP/DF = HP/[2 - 0.94^(-A)], A < 0
note: For this equation, since A < 0, -A will be positive
Tables
http://i53.tinypic.com/fw6ps0.jpg
http://i51.tinypic.com/2vnm7gy.jpg
Graphs
http://i51.tinypic.com/2wf30gw.jpg
http://i54.tinypic.com/2s6rgua.jpg
Linearity
As shown in the graphs, the benefit of armor is purely linear for positive armor even though damage reduction is diminishing.
Every additional point of armor gives the same benefit as the point before.
In other words, going from 5 to 6 armor has the same benefit as going from 45 to 46 armor. Not only is there no functional armor cap, there is not even an upper bound where armor becomes less effective.
To prove this with the formulas, let's take the example above with 2000 health:
2000 HP and 5 armor gives 2600 EHP
2000 HP and 6 armor gives 2720 EHP... a gain of 120 EHP
2000 HP and 45 armor gives 7400 EHP
2000 HP and 46 armor gives 7520 EHP... also a gain of 120 EHP
"Diminishing" Returns of EHP
While total EHP increases linearly with armor, the percentage of EHP gain compared to current EHP naturally decreases. The same applies for every attribute in HoN. For example, increasing Health (or EHP, or mana, etc) from 1000 to 1100 is a gain of 100 or 10%. However, increasing Health from 2000 to 2100 is still a gain of 100 but now only 5%.
For example, if you already have 20 armor, 1 additional point of armor will increase your EHP by 2.7%, NOT by 6%.
Calculation
%EHP gain is calculated by taking the dirivative of the EHP formula and dividing the result by the EHP formula.
For negative armor,
EHP = 1/(2-0.94^(-A))
dEHP/dA = 2489*2^A*47^A*5^(2*A)/(20113*5^(4*A)*2^(2*A+1)-20113*47^A*5^(2*A)*2^(A+3)+160904*47^(2*A))
%EHP = dEHP/da / EHP = [2489*2^A*47^A*5^(2*A)/(20113*5^(4*A)*2^(2*A+1)-20113*47^A*5^(2*A)*2^(A+3)+160904*47^(2*A))]*(2-0.94^(-A))
For positive armor,
EHP = (1+.06*A)
dEHP/dA = 0.06 (somewhat easier than negative formula!)
%EHP = dEHP/dA / EHP = 0.06/(1+.06*A)
Table
http://i55.tinypic.com/11qs400.jpg
Graph
http://i51.tinypic.com/2w1v6nq.jpg
Incremental Benefits of Health and Armor
This is the first step towards understanding how to maximize EHP. In order to optimizing the balance between Health and Armor, we must first identify exactly the benefit, in terms of EHP, from one additional unit of Health or Armor.
HP's incremental effect on EHP: (increase in EHP) = (additional HP)*(1 + .06*A)
(original EHP) = (original HP)*(1 + .06*A)
(new EHP) = (original HP + additional HP)*(1 + .06*A)
(increase in EHP) = (new EHP) - (original EHP) = (additional HP)*(1 + .06*A)
Armor's incremental effect on EHP: (increase in EHP) = .06*HP*(additional A)
(original EHP) = HP*[1+.06*(original A)]
(new EHP) = HP*{1 + .06*[(original A)+(additional A)]}
(increase in EHP) = (new EHP) - (original EHP) = .06*HP*(additional A)
Lines of Equivalency
For every given Health and Armor combination, there is a precise ratio at which X Health is equivalent to Y Armor.
Each "line of equivalency" is:
(additional HP)/(additional A) = .06*HP/(1 + .06*A)
Explanation
Set incremental EHP gain from Health equal to incremental EHP gain from Armor:
(additional HP)*(1 + .06*A) = .06*HP*(additional A)
(additional HP)/(additional A) = .06*HP/(1 + .06*A)
For example, if you currently have 15 armor and 2000 health, increasing armor by 1 is equivalent to increasing health by 63.2. To put it in more game-real quantities, increasing armor by 5 is equivalent to increasing health by 316.
Table
http://i52.tinypic.com/2u5x7jt.jpg
Graph
http://i51.tinypic.com/24q2uqt.jpg
Optimization and Adjustments for Fixed Magic Damage
By assigning in game currency, gold, to each incremental point in Health or Armor, we can calculate a single line of Health/Armor optimization.
It is important to note that this gold assignment is subjective. My calculations and graphs are based on 1 HP = 4.4 gold and 1 Armor = 110 gold.
Using this assumption, the optimal balance of Health and Armor is:
A = 0.04*[HP-(Anticipated Magic Damage Received)] - 16.67
Assumptions
Fortified Bracelet = 510 gold, 114 HP -> 4.47 gold/HP
Beastheart = 1100 gold, 250 HP -> 4.40 gold/HP
Ringmail = 550 gold, 5 A -> 110 gold/A
Although there are many items that give armor and/or HP, I am assuming that the portion of the cost of these items directly responsible for health and/or armor fall within these same ratios.
For instance, Frostfield Plate costs 4700 gold and gives 15 armor. The cost per armor is not 313. Rather, I am assuming that the cost associated with 15 armor is only 1650 gold. The remaining 3050 gold for this item go towards the other benefits for that item (int, mana, active, etc).
Calculation
(additional HP)/(additional A) = .06*HP/(1 + .06*A)
(1/4.4)/(1/110) = .06*HP/(1 + .06*A)
25 = .06*HP/(1 + .06*A)
A = 0.04*HP - 16.67
This works fine for when 100% of the damage you take is physical. However, there is a lot of magic in HoN. Therefore you must account for the post-mitigated magic damage you anticipate to absorb in a fight. This damage deducts from your health such that the formula becomes:
A = 0.04*[HP-(Anticipated Magic Damage Received)] - 16.67
Graph
http://i56.tinypic.com/2u70do2.jpg
Example
You have 2000 HP. Based on the enemy team composition, you expect to receive 1000 damage after reductions in a particular teamfight.
Using the 1000 damage line, you should have about 23 physical armor for an optimal balance between health and armor.
Optimization and Adjustments for Scaling Magic Damage
Similar to above, this section uses the same gold assumptions of 1 HP = 4.4 gold and 1 Armor = 110 gold.
This time, rather than a fixed amount of magic damage, the balance is calculated using the percentage of incoming damage that is magical.
Calculations
Let X = percentage of incoming damage that is magical (.25 = 25% of unmitigated damage received is magic)
Let M = magic armor
Amount of Magic Damage = X/DF = X/(1+.06M)
Amount of Physical Damage = (1-X)/DF = (1-X)/(1+.06A)
Effective Combined Damage Factor = X/(1+.06M) + (1-X)/(1+.06A)
Effective Combined EHP = HP/DF = HP/((1-X)/(1+.06A)+X/(1+.06M))
Value of HP = ((H+1)/((1-X)/(1+.06A)+X/(1+.06M)) - H/((1-X)/(1+.06A)+X/(1+.06M)))/4.4
Value of A = H/((1-X)/(1+.06(A+1))+X/(1+.06M)) - H/((1-X)/(1+.06A))+X/(1+.06M))
Setting these equal yields the following solution for H in terms of A:
HP = (3AX-3M(X-1)+50)((3A+3)X-3M(X-1)+50))/((1-X)/(1+.06A)+X/(1+.06M))/4.4/(3M+50)^2/(1-X)/.06/110
Graph
http://i42.tinypic.com/aexmb8.png
Example
You have 2000 HP and 10.5 magic armor. Based on the enemy team composition, you expect half the damage you receive will be magical (X = 0.5).
Using the X = 0.5, M = 10.5 line, you should have about 18 to 19 physical armor for an optimal balance between health and armor.
Health/Armor Balance Quick Reference
Let's face it. Looking at multiple graphs, tables, and charts is silly and impractical while in a game. While it makes for an interesting thought experiment while cruising the forums, nobody will ever think "okay, if I'm going to take X amount of magic damage, my next item purchase should be...".
To address this, I put together a "Hybrid" method - a single simple graph that combines the essence of everything you've learned up until this point. The result is a continuous line that simply lets you know if you're in the right ballpark for efficient EHP gain. If you frequently stray from this line perhaps you have a tendency to invest too much in Health or too much in Armor. Remember, balance is key.
WARNING! Hard numbers and math end here. What follows is purely subjective!
What is This?
If you look at the optimization lines, how likely is it that you will take 1500 magic damage at a point in the game where you are still sitting on 750 health? Zero. The following table and graph is essentially a smoothed line based on the fact that magic is more prevalent early game.
Table
http://i39.tinypic.com/2a5zomc.png
Graph
http://i40.tinypic.com/35k1kzo.png
Example
You have 2000 HP. The enemy team is more physical based than usual, but there are still some magic sources.
Using the graph and estimating between the "Heave Physical" and "Typical" lines, you should have approximately 20 physical armor for an optimal balance between health and armor.
Conclusion
Because magic damage is more prominent early game, armor is mostly not necessary very early game until your health reaches around 1000. However, after that point, armor's economic effectiveness skyrockets. Even with expecting to take a whopping 1500 magic damage after mitigation, you should have about 23 armor by the time you hit 2500 HP. Keep this in mind next time you contemplate building your hero to have 3500+ health. More often then not you are better off buying some simple armor items if your goal truly is to be as tanky as possible.
The benefits of Armor are completely linear and there is no upper bounds for armor
Armor is often the least expensive way to increase EHP
High Health / Low Armor has the advantage of providing better protection against magical attacks
High Armor / Low Health makes better use of health regeneration items and spells