Savage Mace vs Charged Hammer vs Shieldbreaker vs Riftshards - Mathcraft
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
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...
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:
Originally Posted by xdvesper
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.
Last edited by MacroHard; 05-23-2013 at 05:10 PM.
When I was younger I had an imaginary friend.
Now, with Facebook, I have hundreds of imaginary friends.
-American proverb
im pretty sure sheildbreaker is worse then riftshards after 23 armor( i thought it was more around 19armor, but i could be wrong), your graph says the other way around. typo or ? (also, if im correct either way, then deso is much weaker then rift based on your graphs.
also, the graphs should really show much higher damage values. you go up to ridiculous armor values like 40, but stop at a fairly paltry 250 dmg.
you should toss in wingbow and hellflower if you have time
Last edited by LegoPirate; 10-11-2011 at 08:25 PM.
im pretty sure sheildbreaker is worse then riftshards after 23 armor( i thought it was more around 19armor, but i could be wrong), your graph says the other way around. typo or ? (also, if im correct either way, then deso is much weaker then rift based on your graphs.
also, the graphs should really show much higher damage values. you go up to ridiculous armor values like 40, but stop at a fairly paltry 250 dmg.
you should toss in wingbow and hellflower if you have time
You are half correct. I had a typo underneath the RS/SB graph that said higher armor favors Shieldbreaker. Thank you for catching this. The graph itself is correct; the higher the armor value the more area that favors Riftshards. Some people might find it peculiar that the graph of Riftshards vs Shieldbreaker changes with attack speed, seeing as neither item contains attack speed. The reason for this is that higher attack speed benefits from higher damage.
Since when was 40 armor ridiculous? Very high, yes, but no more uncommon than 3000 HP. As for the damage being paltry, consider that the damage listed on the graphs is BEFORE purchasing these items. Since most of these items are around 70 damage, that would bring you to ~320 damage after purchase. If you still think this is paltry I can re-scale the graphs to 300 damage or even higher.
I didn't add Wingbow or Hellflower because a significant portion of the gold cost goes to the other benefits (armor/evasion and mana/mana regen/silence). Since these graphs show pure dps divided by gold, these two items would be unfairly disadvantaged.
Last edited by MacroHard; 10-11-2011 at 08:44 PM.
When I was younger I had an imaginary friend.
Now, with Facebook, I have hundreds of imaginary friends.
-American proverb
showing that hellflower and wingbow are compareable (just in terms of dps, which they very much are) will go a long way in showing how really good they both are.
the upper end of the graph should end at 350 dmg imo.
the attack speed portion is redundant and obvious (more attack speed means that more damage will do more damage? derp?)
and the fact that desolator takes up an orb slot is insanely huge. means no fws/lifesteal
Last edited by LegoPirate; 10-11-2011 at 08:53 PM.
I've never seen anyone get Riftshards when they have less than 250 damage so your graphs are fairly biased towards your idea that Charged Hammer is better. I still don't get how the graphs work anyway though, considering you've just plotted damage against attack speed (what this is trying to prove is beyond me).
For any two items, set the value equations and solve for S in terms of D, Ar, and Ap.
This is also confusing since you haven't defined what Ar is.
I'd sing you a song, but
I'm just a little hoarse.
I've never seen anyone get Riftshards when they have less than 250 damage so your graphs are fairly biased towards your idea that Charged Hammer is better. I still don't get how the graphs work anyway though, considering you've just plotted damage against attack speed (what this is trying to prove is beyond me).
This is also confusing since you haven't defined what Ar is.
Typo. Meant Am.
Last edited by MacroHard; 10-11-2011 at 11:15 PM.
When I was younger I had an imaginary friend.
Now, with Facebook, I have hundreds of imaginary friends.
-American proverb
Agree, would be interesting to see Wingbow and Hellflower too.
Although clearly SM>Wingbow (pretty much SM > everything...) unless you have a very unusual combination of very high damage yet extremely low Attack Speed (both items are equally affected by armor).
I still don't get how the graphs work anyway though, considering you've just plotted damage against attack speed (what this is trying to prove is beyond me).
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 =)
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 =)
Thanks for that, makes a lot more sense now. I think MarcoHard should put this description in his original post.
Would be nice if he could protrude the graph out a bit though, since realistically some people will get beyond the areas shown way before buying a Riftshards or Charged Hammer.
Consider playing as The Dark Lady: before buying Riftshards you'd have at least Runed Axe, Ghost Marchers and Shrunken Head. Dark Lady has base damage 24 (average of 23-25), base agility 23 and agility gain of 2.9. At about level 20 when you'd be thinking about this next item purchase, she'd have (24+23+20*2.9+5*2) = 115 base damage. All bonuses including Dark Blades bring her attack damage up to (115*1.8+60+24+24) = 315 damage. With Charging Strike, she has (100+23+20*2.9+75) = 256 attack speed. It seems odd that he's graphed attack speed all the way up to 400, which is fairly unrealistic, while damage only goes up to 250 which is very frequently attained.
Other things to note: Charged Hammer is very inferior on Melee heroes because their capacity to deal out DPS is severely stunted by them often being unable to hit people -- in general more damage is superior to attack speed because the timing between each attack is needed to close distance. If your delay between each attack is smaller than the time needed to reach your target between attacks, any increase in attack speed has zero effect on total DPS. I would maintain that regardless of what the maths says, Charged Hammer is an extremely poor choice on any melee carry because of this. (That said, I wouldn't get Riftshards on many melee carries either.)
Last edited by Octavia; 10-12-2011 at 07:51 AM.
I'd sing you a song, but
I'm just a little hoarse.
It seems odd that he's graphed attack speed all the way up to 400, which is fairly unrealistic, while damage only goes up to 250 which is very frequently attained.
can get such quick attack speed (temporarily). I agree though that the dmg range could be increased.
Graphs now go to 350 damage, with the exception of Riftshards vs Savage Mace (because that graph has a single line that drops to negligible levels past 250 damage).
When I was younger I had an imaginary friend.
Now, with Facebook, I have hundreds of imaginary friends.
-American proverb
Can someone tell me what typical attack speeds and damages are for carries at various levels with common items? Otherwise, this graph does not mean much to me.