Savage Mace vs Charged Hammer vs Shieldbreaker vs Riftshards - Mathcraft

[[S2]MacroHard]

Updated as of v3.1.1

Introduction:

The calculations and resulting charts factor in attack speed, damage, physical armor, and magic armor. All graphs and comparisons are also normalized by gold cost. It does not include incidentals such as synergy with lifesteal, splash, ally attacks and spells, occupation of attack modifier, other proc items, or ease of buildup.

The Acronyms and Variables:

TC = additional damage per second from Thunderclaw
CH = additional damage per second from Charged Hammer
RS = additional damage per second from Riftshards
SB = additional damage per second from Shieldbreaker
SM = additional damage per second from Savage Mace
HB = additional damage per second from Harkon's Blade

TCv = value of TC = TC/2900
CHv = value of CH = CH/5400
RSv = value of RS = RS/5400
SBv = value of SB = SB/4600
SMv = value of SM = SM/5400
HBv = value of HB = HB/4775

D = Damage BEFORE purchasing any of the items or components
Ap = Physical Armor of the target you are attacking
Am = Magic Armor of the target you are attacking
S = Attack speed bonus BEFORE purchasing any of the items or components
(includes everything... so if you have 80 agility and +30 attack speed worth of items, S = 110)

P = Damage Factor from Physical Armor = 1/(1+.06Ap)
M = Damage Factor from Magic Armor = 1/(1+.06Am)
Q = Damage Factor from Physical Armor after Shieldbreaker debuff = 1/(1+.06(Ap-6)), Ap > 6

The Formulas:

General Formula = [(D*P + itemdamage*P + magicproc*M + physicalproc*P)*(100+S+attackspeed)-D*P*(100+S)]/itemcost

TCv = (2760P+3450M+24PS+30MS+15PD)/2900
CHv = (5100P+6800M+30PS+40MS+70PD)/5400
RSv = (10240P+28PD+102.4PS+0.28PDS)/5400
SBv = (100QD+7000Q+QSD+70QS-100PD-PDS)/4600
SMv = (12300P+123PS)/5400
HBv = (110MD+MDS+2970M+27MS-100PD-PDS)/4775 (for Agi/Str heroes)
HBv = (110MD+MDS+6820M+62MS-100PD-PDS)/4775 (for Int heroes)

The Method:

For any two items*, equate the value equations and solve for S in terms of D, Ap, and Am.
*For Harkon's solve for Ap in terms of D, S, and Am
The result is the line where both items have equivalent DPS per gold.
Most graphs have multiple lines with each line representing fixed values for the remaining variables.

The Results:

Thunderclaw vs Charged Hammer

Charged Hammer is obviously more DPS. What this shows is that Charged Hammer is also more efficient than Thunderclaw.

Charged Hammer vs Shieldbreaker

5.5 magic armor-

10.5 magic armor-

15.5 magic armor-

Generally favors Charged Hammer. Very high attack speed combined with low enemy physical armor and high magic armor favors Shieldbreaker.

Charged Hammer vs Riftshards

Generally favors Charged Hammer. High attack speed combined with low enemy physical armor and high magic armor favors Riftshards.

Charged Hammer vs Savage Mace

Generally favors Charged Hammer. Low damage and high attack speed, combined with low enemy physical armor, favors Savage Mace.

Shieldbreaker vs Savage Mace

Generally favors Shieldbreaker. Low damage and high enemy physical armor favors Savage Mace.

Riftshards vs Savage Mace

No graph required!
Since neither of these items factor in attack speed nor armor...

Savage Mace adds 123 damage
Riftshards adds 102.4 + .28D damage

Riftshards out-damages Savage Mace when D > 73.6

Riftshards will always exceed Savage Mace for pure DPS.

Riftshards vs Shieldbreaker

Generally favors Shieldbreaker. Enemy armor higher than ~14 favors Riftshards.

Harkon's Blade vs Charged Hammer

For Agi/Stri Heroes, generally favors Harkon's Blade when Physical Armor is more than 18 higher than Magic Armor (uncommon).
For Int Heroes, generally favors Harkon's Blade when Physical Armor is more than 7 higher than Magic Armor (common).

Harkon's Blade vs Savage Mace

For Agi/Stri Heroes, generally favors Harkon's Blade when Physical Armor is more than 12 higher than Magic Armor.
For Int Heroes, generally favors Harkon's Blade when Physical Armor is more than 6 higher than Magic Armor.

Explanation of 2-Dimensional Graphing Involving Three or More Variables:

Since Macrohard didn't explain it maybe I'll take a stab for him. Maths and Stats graph time!

Basically graphs only plot the relationship between 2 variables. For example, how (1) Total Damage scales with (2) Armor reduction. After all, the graph only has 2 axes, and can therefore only show 2 things.

In some situations, like this, Macro is trying to show the relationship of (1) Total Damage against... several things. Basically (2) Current Hero Damage before buying item, (3) Current Hero Atk Speed before buying item, (4) Enemy Physical Armor and (5) Enemy Magic Armor.

If we really wanted to plot this, we would need a 5 dimensional graph, which would make your head explode. We need to get rid of 3 dimensions to bring it back to something we can represent on a 2 dimensional chart.

The first two dimensions (physical armor and magic armor) are easy. All he did was plot the same thing multiple times - so there are many lines of different colors to show the various graphs. The problem with this is that it's not exact - you can only pick and choose several values, rather than have a graph that shows you clearly what the relationship between the dimensions are and where the cross-over points are.

The way Macrohard got rid of one more dimension is a clever bit of work that is pretty standard in statistics - instead of plotting the relationship between two factors that are the axes on the graph, Macro instead plotted the line where the damage contribution from both items being compared is exactly the same. Each line therefore shows all the combinations of Attack Speed and Damage for which both items being compared are exactly the same - and from this contour, we know any points lying above the line favour one item, while points lying below the line favour the other item.

And that's how to compress a 5 dimensional graph down to a 2 dimensional graph =)

Incidentals:

+ Positive Side Effects

• Charged Hammer has an activatable that punishes attackers
• Charged Hammer also hits nearby enemies with lightning
• Charged Hammer has a guaranteed proc on first attack between fights
• Riftshards can lifesteal from critical strikes
• Riftshards critical strikes apply to splash attacks
• Shieldbreaker lowers enemy armor such that allied attacks and physical spells also benefit
• Shieldbreaker and Riftshards have easier build up
• Savage Mace removes all chances to miss
• Savage Mace interrupts enemy attacks and stops channeling abilities
• Harkon's Blade lowers enemy magic armor such that allied spells also benefit

- Negative Side Effects

• Charged Hammer loses effectiveness against Magic Immunity
• Shieldbreaker occupies an attack modifier which may be needed for slow or lifesteal
• Harkon's Blade occupies an attack modifier which may be needed for slow or lifesteal
• Harkon's Blade requires mana to use

Ignorant Generalization:

Notwithstanding reality, when it comes to pure DPS per gold cost for most Damage and Attack Speeds...
Harkon's Blade (int) > Charged Hammer > Riftshards > Harkon's Blade (agi/str) > Shieldbreaker > Savage Mace

My Recommendation:

Screw math. Do what feels right.