Thread: Shieldbreaker vs Savage Mace - Mathcraft

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  1. #1
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    Shieldbreaker vs Savage Mace [Unexpected winner]

    ================================================== ===========

    Conclusion only applies for Armour>6 before debuff from shield breaker is applied.

    Pure DPS throughout the game


    DPS:Cost.



    Yes duck boots kick the **** out of everything for DPS:cost, but out of the two you care about


    Formula used
    a>6, (x+60)(1-((a-6)*0.06)/(1+0.06(a-6)))/(1-(a*0.06)/(1+0.06*a))=(x+123)*(1.05)

    a= armour
    x= damage before SB/SM
    1.05 is an approximation of how attack speed from mace affects DPS

    if you have over 200 agility+attackspeed this favours SM if not favours SB, for heroes with 100 agility+attackspeed it should be changed to 1.075 for more accurate readings.


    If you're interested in the comparison keep reading
    Last edited by ElementUser; 08-19-2011 at 11:51 AM.
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  2. #2
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    updated for v2.1.4. numbers used are for shieldbreaker level 3

    Introduction:

    The point of this post is to provide a comparative analysis between Savage Mace and Shieldbreaker WITHOUT MAKING ASSUMPTIONS.

    For most posts I see where people try to use math to prove a point, something is missing. Some people neglect speed factors. Some people assume a specific armor value (normally 0). Nearly everyone failed to recognize that this game has an economy, so damage needs to be normalized by gold costs.

    I will not make ANY assumptions. My calculations below factor in attack speed, armor of target, as well as the gold cost of the items (in this case 5400 vs 4300 is far from negligible).


    The Acronyms and Variables:

    SM = Savage Mace
    SMd = Added damage output, including all factors, that SM adds
    SMv = Value of SM = SMd / 5400
    SB = Shieldbreaker
    SBd = Added damage output, including all factors, that SB adds
    SBv = Value of SB = SBd / 4300
    D = Damage BEFORE purchasing any of the items or components
    A = Armor value of enemy target
    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)


    The Basics:

    SMd = (88+.35*100)/(1+.06A)+[15/(100+S)]*[(D+88+.35*100)/(1+06A)]
    SMv = {(88+.35*100)/(1+.06A)+[15/(100+S)]*[(D+88+.35*100)/(1+06A)]}/5400
    SBd = 70/[1+.06(A-6)]+D/[1+.06(A-6)]-D/(1+.06A)
    SBv = {70/[1+.06(A-6)]+D/[1+.06(A-6)]-D/(1+.06A)}/4300


    The Method:

    To solve the problem of which is better, we have to find the trend line where the values SMv and SBv are equal to each other. Since we have three variables (D, A, and S), we shall fix A at a given value then solve for S in terms of D. The method is repeated for varying values of A.

    Basically it boiled down to this:
    1) Set SMv = SBv
    2) Solve for S in terms of D and A
    3) Plot for fixed values of A


    The Results:

    v2.1.4


    v2.0.38:


    The Conclusion:

    As you can see from the graph, if your target has low armor, Shieldbreaker has greater effect. The numbers where damage, attack speed, and armor all come together are reasonable for in game scenarios.

    Intangibles will also factor largely into your decision:

    SM stops channeling and gives truestrike.
    SB reduces armor so your allies also benefit and bypasses HotBL block.


    Decision Tree:

    I put this together for those who like cookie cutter builds. Obviously you can use your judgement to vary from this!

    Instructions: Answer each question in order. If you answer yes, go with the item suggested. If the answer is no, go on to the next question!

    Do you already have an attack modifier? -> Savage Mace
    Does an allied hero already have Shieldbreaker? -> Savage Mace
    Does the enemy team have more than one annoying channeling spell? -> Savage Mace
    Do your or allied hero spells do mostly physical damage? -> Shieldbreaker
    Do you or an allied hero have spells that reduce enemy armor? -> Shieldbreaker
    Does an enemy hero have one annoying channeling spell? -> Savage Mace
    Does the enemy carry have a form of evasion? -> Savage Mace
    Is your current damage currently above 175? -> Shieldbreaker
    Did you answer "no" to all of the above? -> Savage Mace
    Last edited by MacroHard; 10-10-2011 at 11:42 AM.

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  3. #3
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    Your allies benefit from shieldbroken and its easier to build.

    Now, since you like maths, intergrate x factorial dx

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    Quote Originally Posted by TheSopranos View Post
    Your allies benefit from shieldbroken and its easier to build.

    Now, since you like maths, intergrate x factorial dx
    As its a tad difficult to do i hope this approximation of the intergral about x=0 is suffice.

    x - (gamma x^2)/2 + (1/36)*(6*gamma^2 + pi^2)*x^3 + (1/24)*(x^4)*( -gamma^3 - (gamma pi^2)/2 + psi^(2)(1)) + (1/120)*(x^5)*gamma^4 + gamma^2*pi^2 + (3 pi^4)/20 - 4*gamma*psi^(2)(1)) + (x^6*( -12 gamma^5 - 20 gamma^3*pi^2 - 9*gamma*pi^4 + 120*gamma^2*psi^(2)(1) + 20*pi^2*psi^(2)(1) + 12*psi^(4)(1)))/8640 + O(x^7) + constant

    -Where gamma is the Euler-Mascheroni constant
    -Where psi^(n)(x) is the nth derivative of the digamma function

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  5. #5
    ^ Great Odin's raven!

  6. #6
    Quote Originally Posted by yyr_ View Post
    Shieldbreaker vs Savagemace

    (x+60)y=0.81*(x+123)*(1.15), 50<x<120, 1<y<1.30

    X= Base damage
    Y= Damage increase due to the -6 armour.
    0.81 is the ratio of the cost of shield breaker to savage mace
    123 as 88+35 as it has a 35% chance to deal 100 damage

    Here is the question. As it gives +15 attack speed can i just multiply it with the rest of the damage on the right hand side as is the generation of attackspeed formed by attackspeed= 1.7*(agility increase + straight attackspeed increase) rather than 1.7*(agility increase)*(straight attackspeed increase).

    If it is the former what modifications are needed?

    Ignore the fact that shield breaker debuff affects teammates and lets assume it carries equal weight for purpose of this comparison.

    Maybe savage mace will be the best damage : gold item in the game...

    Note: Sorry mods if this isnt the correct forum, please shift if it nesc.
    Shieldbreaker actually isn't easy to model in this linear term.

    The reason being is that Shieldbreaker's effect doesn't increase your damage, it reduces your opponents effective HP by 36% of their Max HP, implicitly requiring you to do less damage to kill the same target.

    To model it properly you need to consider attacking and defending hero stats.

    For instance, on a hero with 1000 HP and 50 armor (giving him effectively 4000 HP, an attack that would normally strike a hero for 100 damage instead strikes him for 25, and what would've taken 10 attacks to kill a hero with 100 HP instead takes 40 attacks to kill him, resulting in him having survived the effective of 40 x 100 damage = 4000) Shieldbreaker's effect is equivalent to reducing his effective HP to 3636, meaning you would need only 36 attacks to kill the target, thereby increasing the value of each attack by the amount of effective health removed, thus reducing the time to kill this target by 4 attacks. In this case, it's as if you've done the equivalent of 4000 damage in 36 attacks, which is an increase of about 10 damage per attack equivalence.

    On a target with 2944 hp and 6 armor (still giving him 4000 effective HP) the Shieldbreaker's effect is equivalent to reducing his effective HP to his Max HP of 2944. Were you still dealing 100 damage per hit this would reduce the amount of time to kill this target by 10.5 attacks, such that you've done the equivalent of 4000 damage in 29.5 attacks which is an increase of about 35.8 damage per attack equivalence.

    For example's sake I mapped out a spreadsheet for Madman previously, take these hero stats:

    Attacker (Madman)
    Damage 135 BAT 1.7 Base AS 229 Modifier 1.28
    Base AS is his attack speed from Agility along with his ultimate active
    Modifier is to accomodate the increase in damage through his critical strike passive.

    Defender (Arbitrary hero)
    Target Max HP 2000 Target Armor X Target M. Armor 5.5
    Assuming at end game most heroes have around 2000 HP (often this is higher, the lower this number the less favourable Shieldbreaker is)

    With Shieldbreaker Madman attacks 1.94 times per second, with Savage Mace Madman attacks 2.02 times per second.

    At X=5, eHP = 2600 (Shieldbreaker reduces this to 1880 / by 720)
    Naked Madman takes 7.77 seconds to kill the target (uses 15 attacks @ 172.8 Damage per hit / 334.42 DPS)

    Shieldbreaker Madman takes 3.89 seconds to kill the target (uses 7.5 attacks @ 249.6 Damage per hit / 483.05 DPS), an improvement of 49.94% (1.14% improvement per 100g)

    Savage Mace Madman takes 4.02 seconds to kill the target (uses 8.1 attacks @ 320 Damage per hit / 647.53 DPS), an improvement of 48.35% (0.90% improvement per 100g)

    At X=10, eHP = 3200 (Shieldbreaker reduces this to 2480 / by 720)
    Naked Madman takes 9.57 seconds to kill the target (uses 18.5 attacks @ 172.8 Damage per hit / 334.42 DPS)

    Shieldbreaker Madman takes 5.13 seconds to kill the target (uses 9.9 attacks @ 249.6 Damage per hit / 483.05 DPS), an improvement of 46.35% (1.05% improvement per 100g)

    Savage Mace Madman takes 4.94 seconds to kill the target (uses 10 attacks @ 320 Damage per hit / 647.53 DPS), an improvement of 48.35% (0.90% improvement per 100g)

    At X=15, eHP = 3800 (Shieldbreaker reduces this to 3080 / by 720)
    Naked Madman takes 11.36 seconds to kill the target (uses 22 attacks @ 172.8 Damage per hit / 334.42 DPS)

    Shieldbreaker Madman takes 6.38 seconds to kill the target (uses 11.9 attacks @ 249.6 Damage per hit / 483.05 DPS), an improvement of 43.89% (1.00% improvement per 100g)

    Savage Mace Madman takes 5.84 seconds to kill the target (uses 12.3 attacks @ 320 Damage per hit / 647.53 DPS), an improvement of 48.35% (0.90% improvement per 100g)

    At X=20, eHP = 4400 (Shieldbreaker reduces this to 3680 / by 720)
    Naked Madman takes 13.16 seconds to kill the target (uses 25.5 attacks @ 172.8 Damage per hit / 334.42 DPS)

    Shieldbreaker Madman takes 7.62 seconds to kill the target (uses 14.7 attacks @ 249.6 Damage per hit / 483.05 DPS), an improvement of 42.10% (0.96% improvement per 100g)

    Savage Mace Madman takes 6.80 seconds to kill the target (uses 13.8 attacks @ 320 Damage per hit / 647.53 DPS), an improvement of 48.35% (0.90% improvement per 100g)

    At X=25, eHP = 5000 (Shieldbreaker reduces this to 4280 / by 720)
    Shieldbreaker gives an improvement of 40.74% (0.93% improvement per 100g)
    Savage Mace gives an improvement of 48.35% (0.90% improvement per 100g)

    At X=30, eHP = 5600 (Shieldbreaker reduces this to 4880 / by 720)
    Shieldbreaker gives an improvement of 39.67% (0.90% improvement per 100g)
    Savage Mace gives an improvement of 48.35% (0.90% improvement per 100g)

    At X=35, eHP = 6200 (Shieldbreaker reduces this to 5480 / by 720)
    Shieldbreaker gives an improvement of 38.81% (0.88% improvement per 100g)
    Savage Mace gives an improvement of 48.35% (0.90% improvement per 100g)

    So in short, Shieldbreaker's effect is a static 36% of the target's maxHP removed from their eHP, but as their eHP increases while maxHP stays the same the same (720 in this case) amount being removed becomes less and less significant.

    Meanwhile the Savage Mace's DPS gain is consistent and linear regardless of what your target armor levels or HP levels are, it will always provide a consistent return.

  7. #7
    in the end, the shieldbreaker reduction of EHP applies for all enemies attacking a unit where the savage mace's effectiveness is only for that single hero

    it is ALWAYS more beneficial to get a shieldbreaker over a savage mace for this reason

  8. #8
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    Quote Originally Posted by Urizen View Post
    it is ALWAYS more beneficial to get a shieldbreaker over a savage mace for this reason

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    Also consider that Savage Mace gives truestrike (good against Fayde, Night Hound, Wingbow) and a ministun (useful for heroes who are countered by TPs such as Arachna).

    Plus it can stack with lifesteal.

  10. #10
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    Quote Originally Posted by yyr_ View Post
    As its a tad difficult to do i hope this approximation of the intergral about x=0 is suffice.

    x - (gamma x^2)/2 + (1/36)*(6*gamma^2 + pi^2)*x^3 + (1/24)*(x^4)*( -gamma^3 - (gamma pi^2)/2 + psi^(2)(1)) + (1/120)*(x^5)*gamma^4 + gamma^2*pi^2 + (3 pi^4)/20 - 4*gamma*psi^(2)(1)) + (x^6*( -12 gamma^5 - 20 gamma^3*pi^2 - 9*gamma*pi^4 + 120*gamma^2*psi^(2)(1) + 20*pi^2*psi^(2)(1) + 12*psi^(4)(1)))/8640 + O(x^7) + constant

    -Where gamma is the Euler-Mascheroni constant
    -Where psi^(n)(x) is the nth derivative of the digamma function

    You just killed a kitten. I hope your happy.

  11. #11
    So if I'm reading this right, savage mace is good for killing a wingbow carry because of ministun and truestrike, while shieldbreaker is better for other targets because your allies get the effect too.

  12. #12
    Quote Originally Posted by RogerDodger2 View Post
    Also consider that Savage Mace gives truestrike (good against Fayde, Night Hound, Wingbow) and a ministun (useful for heroes who are countered by TPs such as Arachna).

    Plus it can stack with lifesteal.
    Good points. In these cases you would want to use Savage Mace.

    Otherwise, Sheild breaker is better. There have been many other threads that analyzed this topic.

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    This thread makes my brain hurt in places that don't exist.

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    wait there are smart people playing hon? I can do university calculas but I don't understand where those numbers are coming from lol i feel dumb

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    this thread is why the servers imploded today

  16. #16
    only get shieldbreaker on warbeast soulstealer pest and pred

  17. #17
    Quote Originally Posted by Autism_Prime View Post
    wait there are smart people playing hon? I can do university calculas but I don't understand where those numbers are coming from lol i feel dumb
    Well I know what a Gamma is, but I have no idea where he pulled that from.

  18. #18
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    I'm not that advanced in Mathematics, but I do recall that MKB vs. Buriza was done on the DA forums a lot. Can't find anyting that compares MKB vs. Desolator though.

    The reason being is that Shieldbreaker's effect doesn't increase your damage, it reduces your opponents effective HP by 36% of their Max HP, implicitly requiring you to do less damage to kill the same target.
    This is true if the opponent's armor is positive, but if the armor is negative then this is not necessarily true.

    For instance, assume the hero drops from 0 armor to -2 armor.

    Generally, EHP = HP/Damage Reduction

    EHP (0 Armor) = HP/(1-0) = HP (-0 is there because there is 0% damage reduction)

    -2 Armor gives 0.94^-(-2) - 1 = 0.94^2 - 1 = -0.1164 damage reduction

    EHP (-2 Armor) = HP/(1 - (-0.1164) = HP/(1.1164) = 0.89573629523468290935148692225009 ≈ 0.896 ≈ 89.6% EHP

    100% - 89.6% = 10.4% drop in EHP when you go from 0 to -2 armor, not a 12% drop like you would expect it to.

    (You can also use common sense and see that if the opponent has -17 armor, he would have 100 - 17*6 = -2% EHP, which would not make sense)

    Anyway, you're right about Shieldbreaker's DPS not being so easy to represent as it is a function of the enemy's EHP, which can differ depending on what damage reduction sources they have. Not to mention Shieldbreaker's damage coefficient varies with the enemy armor (see this: http://forums.heroesofnewerth.com/sh...ad.php?t=61190), while Savage Mace adds a static amount of damage & True Strike (means that Savage Mace would generally be better if the opponent has Evasion).

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  19. #19
    Quote Originally Posted by ElementUser View Post
    I'm not that advanced in Mathematics, but I do recall that MKB vs. Buriza was done on the DA forums a lot. Can't find anyting that compares MKB vs. Desolator though.

    This is true if the opponent's armor is positive, but if the armor is negative then this is not necessarily true.

    For instance, assume the hero drops from 0 armor to -2 armor.

    Generally, EHP = HP/Damage Reduction

    EHP (0 Armor) = HP/(1-0) = HP (-0 is there because there is 0% damage reduction)

    -2 Armor gives 0.94^-(-2) - 1 = 0.94^2 - 1 = -0.1164 damage reduction

    EHP (-2 Armor) = HP/(1 - (-0.1164) = HP/(1.1164) = 0.89573629523468290935148692225009 ≈ 0.896 ≈ 89.6% EHP

    100% - 89.6% = 10.4% drop in EHP when you go from 0 to -2 armor, not a 12% drop like you would expect it to.

    (You can also use common sense and see that if the opponent has -17 armor, he would have 100 - 17*6 = -2% EHP, which would not make sense)

    Anyway, you're right about Shieldbreaker's DPS not being so easy to represent as it is a function of the enemy's EHP, which can differ depending on what damage reduction sources they have. Not to mention Shieldbreaker's damage coefficient varies with the enemy armor (see this: http://forums.heroesofnewerth.com/sh...ad.php?t=61190), while Savage Mace adds a static amount of damage & True Strike (means that Savage Mace would generally be better if the opponent has Evasion).
    Aye this is a good amendment, negative armor values behave differently than positive ones and further alters how the shieldbreaker effect can be measured.

  20. #20
    Quote Originally Posted by yyr_ View Post
    As its a tad difficult to do i hope this approximation of the intergral about x=0 is suffice.

    x - (gamma x^2)/2 + (1/36)*(6*gamma^2 + pi^2)*x^3 + (1/24)*(x^4)*( -gamma^3 - (gamma pi^2)/2 + psi^(2)(1)) + (1/120)*(x^5)*gamma^4 + gamma^2*pi^2 + (3 pi^4)/20 - 4*gamma*psi^(2)(1)) + (x^6*( -12 gamma^5 - 20 gamma^3*pi^2 - 9*gamma*pi^4 + 120*gamma^2*psi^(2)(1) + 20*pi^2*psi^(2)(1) + 12*psi^(4)(1)))/8640 + O(x^7) + constant

    -Where gamma is the Euler-Mascheroni constant
    -Where psi^(n)(x) is the nth derivative of the digamma function

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