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Thread: Hand of Salvation Analysis (8/21/08)

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    Blessing of Salvation, the paladin's current form of reducing threat provides a flat, 30% threat reduction on all generated threat (without affecting the threat accumulated before that). It is being removed in the WotLK and being replaced with a spell called Hand of Salvation (HoS).

    **** As a disclaimer, this is still Beta, so Hand of Salvation is subject to change.

    The current implementation of Hand of Salvation:
    Hand of Salvation
    6% of base mana
    30 yd range
    Instant cast
    2 min cooldown
    Places a Hand on the party or raid member, reducing their total threat by 2% every 1 sec. for 10 sec. Players may only have one Hand on them per Paladin at any one time.
    Now at first glance, one might think "oh 2% every second for 10 seconds, so that is 20% threat reduction". Well that isn't the case. It's incremental so the effects of the last threat reduction affect the effects of the next one. Further more, once the buff runs out it no longer affects threat generated after it. So the idea behind this Hand spell is a bit different than the old Blessing of Salvation that we are all familiar with.

    Now Blessing of Salvation reduces threat by 30%, which can be modeled by the equation:
    Code:
    Threat_With_BoSalv = (1-percent_reduction)*Threat_Without_BoSalv
    Rearranging that gives:
    Code:
                            Threat_With_BoSalv
    percent_reduction = 1 - ---------------------
                            Threat_Without_BoSalv
    Now one my think it silly to even model BoSalv since we already know that it is a 30% reduction, but I wanted to set up the stage to show how we are going to tackle HoS. HoS will be a different beast because it reduces threat in a recursive manner. Not to mention, it affects all the previous threat you have already generated before the buff was placed (unlike BoSalv).

    Still, I am going to try to emulate the same model to get an equivalent percent_reduction value for HoS. Using this value, you can look at your threat and get a good general idea of how it will look after HoS is done. The new equation will look like:

    Code:
                            Threat_With_HoSalv
    percent_reduction = 1 - ---------------------
                            Threat_Without_HoSalv
    Now we just need to calculate the quantities Threat_With_HoSalv and Threat_Without_HoSalv.

    Let's look at Threat_With_HoSalv first since it is the harder part.

    Say you have some amount of threat you have generated. We will call it AT for Accumulated Threat. We will assume at least some type of semi stable tps over the 10 second period of Hand of Salvation. Hand of Salvation will work like this:

    +1 seconds: Threat = (AT + TPS)*0.98
    +2 seconds: Threat = ((AT + TPS)*0.98 + TPS)*0.98
    +3 seconds: Threat = (((AT + TPS)*0.98 + TPS)*0.98 + TPS)*0.98

    and so on down to +10 seconds.

    If you run the numbers you will find that it isn't going to come out to 0.8*(AT+10TPS), which is what you would expect from a flat 20% reduction after 10 seconds.

    Actually what you end up getting is more like:

    (0.98^10)*AT + TPS*[0.98 + 0.98^2 + 0.98^3 + ... + 0.98^10]

    One thing to note from this equation is that if AT is sufficiently big enough (I.E. A huge threat lead or perhaps a really really long time), then it will swamp out the second part, leaving only (0.98^10)*AT = 0.817073*AT, or an 18.2927% reduction in threat (well while this number is true, using this equation to get it as is is a bit wrong, but I will go into that later). This is also true if the person affected by Hand of Salvation stops generating threat while it is active (so TPS=0).

    ************************************************** *******

    ASIDE: [0.98 + 0.98^2 + 0.98^3 + ... + 0.98^10] is a geometric series, which reduces down to the equation:

    Code:
    (0.98 - 0.98^11)
    ----------------
       (1 - 0.98)
    I won't go into how this is calculated unless someone really really wants to know.

    ************************************************** *******

    So the equation we have thus far is:

    Code:
                                  (0.98 - 0.98^11)
    Threat = (0.98^10)*AT + TPS * ----------------
                                     (1 - 0.98)
    Now that we have Threat_With_HoSalv ready to go, let's look at Threat_Without_HoSalv:

    Threat_With_HoSalv = AT + TPS*10

    Pretty simple. That is the threat you will generate if HoSalv wasn't there.

    To get the actual threat reduction, we need to simply divide the two:

    Code:
    AT*(0.98^10)+TPS*(0.98 - 0.98^11)/(1 - 0.98)
    --------------------------------------------
                     AT + 10*TPS
    This leaves a function of two variables, AT and TPS. If we assume (for a moment) that TPS was constant the whole time, then we can say that AT = TPS*time and the equation simplifies down to one variable, time:
    Code:
    time*(0.98^10)+(0.98 - 0.98^11)/(1 - 0.98)
    --------------------------------------------
                     time + 10
    The characteristic for this is



    There is something else to consider: We really don't care about the spikes/degradations of TPS during AT as it happens before we use Hand of Salvation (HoS from now on). During HoS, however, it is important to watch the TPS. The TPS generated might not be the same as the average TPS used to generate AT. That might be due to fight mechanics, threat dumps, etc., but for whatever reason it is possible that the DPS'er won't be doing the same TPS when HoS is up versus their average TPS up until AT was accumulated.

    In order to combat this, I simply multiplied the 10 second HoS portion of both the numerator and denominator by a value m, which is simply a relative multiplier (I.E. is the TPS during HoS twice the TPS before, m=2, or is it the same as before, m=1, etc.).

    The equation will now look like this:
    Code:
    time*(0.98^10) + m*(0.98 - 0.98^11)/(1 - 0.98)
    -----------------------------------------------
                     time + 10*m
    So now it is back to being a two variable equation, using time to represent the various combinations of accumulated threat and m to identify how the TPS during HoS looks like compared to when the initial threat was accumulated (is it more, less, or the same).

    There are two ways to view this data in a 2D graph. You can hold either m or time constant and plot against the other variable. Here is how it looks in both formats:



    Here you can see some trends:
    1. Increasing relative TPS during the duration of HoS gives a lower threat reduction in terms of percentage. This doesn't mean your total threat is lower, just that the comparison of your threat with HoS to threat without HoS is lower as m gets larger.
    2. Increasing time gives a higher threat reduction in terms of percentage. The longer you wait to use HoS, the better the threat reduction percentage is.
    3. The characteristics all converge to a TPS value of 18.2927% as time increases towards infinity. Consequently, if m=0 (dps is stopped during HoS), then the characteristic is a constant value of 18.2927% as well.
    Last edited by jere; 08-23-2008 at 04:13 PM.

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    Unless I'm reading your post incorrectly, your conclusion is completely wrong. I think your analysis gets drawn too much into a few unnecessary variables. There is no need to introduce m, t, or z. You spent too much time getting confused by all the graphs and the formulas and the relationships.

    The percentage reduction in threat matters not at all. What matters is the actual, numerical reduction in threat.

    If you think about it qualitatively, this is obviously true--any permanent, percentage-based threat reduction mechanism would be best used as late as possible to take advantage of a larger base upon which to reduce a percentage. Theoretically, you'd want to use it near the middle of the fight, to allow the DPS to make up the difference and ride neck and neck with the tank when the boss is about to die.

    Here's a quick spreadsheet that shows it better:



    No matter how you DPS after getting Salved, you still reduce more of your potential threat if you give it more to calculate that 2% off.

    2. The longer the time waited before using HoS, the less impact your relative TPS has on the outcome. I.E. at infinity, the threat reduction goes to 18.2927% threat reduction regarless of how much extra or less TPS one does during the duration of HoS.
    All you're saying here is "the longer the fight's been going on, the smaller the percentage of the total threat ten more seconds of DPS is going to be."

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    Doing 0 TPS during the duration of HoS gives the most benefit in terms of relative threat reduction.
    I'll be honest, I didn't dig into the math of your post. Are you trying to say here that doing no TPS during HoS will result in a larger numerical threat reduction from HoS than continuing to DPS? Or are you saying that doing no TPS will result in a larger total numerical threat loss over those 10 seconds?

    Either way, I'm not really sure what the point of the conclusion is - optimal salv usage might be important if it had a long CD, but in its current state you can fire it off several times per fight. This kind of makes the whole effectiveness vs time argument a little academic, don't you think?

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    Last edited by jere; 08-23-2008 at 04:53 AM.

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    jere, the "threat" column in my char is really "threat you would have had doing the moves you're doing if you didn't get BoS'd" but for space concerns I just shortened it to "threat" to contrast with "HoS threat." It is simply adding either 1k TPS or 500 TPS or 0 TPS depending on which action sequence you go with.

    More importantly, what I mean by "actual, numerical reduction in threat" is simply that--the actual number your threat got reduced by. This is the best measure of how much benefit you got out of HoS. The whole point of aggro-reducing moves, provided they are not done reactively but proactively (i.e., not because you pulled aggro but so you won't pull aggro in the future) is to reduce your threat by as much as possible so you can go balls out for a longer period of time.

    Percentage reduction does not matter. At all.

    If there were a move called "hand of lolvation" usable every 60 min that reduced your threat by 50% and then an additional 50 flat, then by your logic, we would get a better benefit out of it by popping it as soon as possible in the fight. Someone who hit the boss once and got 1000 threat, then got HoL'd, would have his threat reduced by 50%, then 50, resulting in 450 threat. That's a 65% reduction in threat! Someone who has been DPSing for a while and has hit 20k threat would get it reduced to 9950,
    or 50.25%.

    But which one actually reduced his threat more? Which one is more useful? Obviously 10050 > 550, and would result in 9500 more TPS worth of damage being doable to the boss without pulling aggro.

    Your chart is valid if the person always starts at 10000 previous threat, but is invalid otherwise.
    Incorrect. Percentages always favor stop-DPSing because the value by which you're dividing is not increasing. But, again, the only useful measure is the scalar quantity, which favors DPSing through the HoS, and HoSing later. I have the spreadsheet here. Starting threat matters not at all.

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    Last edited by jere; 08-23-2008 at 04:54 AM.

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    Just to chime in:

    Try to keep it calm and civil guys. I don't want this great post turning into a closed thread.

    On Topic Though:

    Doesn't the start of the fight matter most? That's when most tanks lose their threat in fights imo. Once I get my rotation going and have a stream of rage going, I rarely ever lose threat then. Of course I'm sure the DPS are holding back so then another HoS later one would be great, but to say that Beginning threat doesn't matter concerns me a bit.

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    Quote Originally Posted by Kazeyonoma View Post
    Just to chime in:

    Try to keep it calm and civil guys. I don't want this great post turning into a closed thread.
    I am confused. Where have we not been civil or calm? I know you have been having trouble with lots of other threads, but I don't think we have any harsh feelings in this one at all. EDIT: unless I have upset Forklift in some way, for which I am sorry. I am happy to drop it if you feel we are being rude or mean. I can erase the whole post even if it makes things easier. I don't want to start trouble.

    Quote Originally Posted by Kazeyonoma View Post
    On Topic Though:

    Doesn't the start of the fight matter most? That's when most tanks lose their threat in fights imo. Once I get my rotation going and have a stream of rage going, I rarely ever lose threat then. Of course I'm sure the DPS are holding back so then another HoS later one would be great, but to say that Beginning threat doesn't matter concerns me a bit.
    The threat at the beginning does matter for sure. The thing is though, taking a larger threat reduction percentage of a smaller value of total (actual) threat may not have much of an impact at the start of the fight. It really depends on the values (initial threat, dps'ers TPS, etc.). It could allow the dpser to throw 5 more shadow bolts without care or it could only remove the last one. It all depends. But I agree with you, the start of the fight does indeed have great importance.
    Last edited by jere; 08-22-2008 at 03:06 PM.

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    Agreed. And I wasn't saying you guys weren't civil, you guys have been and it's great. I just don't want it to escalate. I've found letting people know that I'm watching keeps people more civil then letting it get bad, then having to step in and regulate post uncivil activity =]

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    Well like I said, I will be happy to delete my post if it becomes a problem. I don't want to cause issues at all.

    Just let me know if it becomes a problem and I will delete the post so it won't be a bother.

    Maybe I should just keep stuff like this in a blog instead so I don't mess yall's site up so much.
    Last edited by jere; 08-23-2008 at 04:52 AM.

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    no no no, we love this stuff. Thank you very much for it. It'll benefit many many more people by making it a thread. If I can get a paladin author or two to authorize the math in this especially once wotlk goes live I'll probably move this to the guides.

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    /agree with Kaze, this is definitely a worthwhile post. Even if the information itself isn't earth-shattering stuff, the ensuing discussion has been very interesting.

    I show that using HoS early can provide a much larger threat reduction % if the DPS'er stops DPS'ing
    I'm really not convinced on this point. I'll lay out my generalization without arguing the math, and if you still disagree I'll try to prove myself.

    BoS aggro reduction relies on the initial threat, plus the threat delta across those ten seconds. If all TPS were to stop the moment salv was dropped, the total threat reduction would be 0.183 * threat.

    However, if you factor in the delta you get that 0.183 * threat reduction plus 0.183 times whatever the delta threat is.

    This means that in any case, the more threat someone can put out over the course of a salv, the more threat the salv will remove.

    Edit: for clarity...
    And isn't that the point of salv, to allow people do DPS more? If I was worried about pulling aggro at the beginning of the fight, I think it would make more sense to not DPS in the first place than to DPS normally and stop DPSing while BoS ticks away.
    Last edited by shiz98; 08-22-2008 at 05:33 PM.

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    jere, ain't nothing wrong with our discussion. I don't think anyone has gotten the least bit upset. Kazey is just hyper vigilant, that's his job.

    Quote Originally Posted by shiz98 View Post
    /agree with Kaze, this is definitely a worthwhile post. Even if the information itself isn't earth-shattering stuff, the ensuing discussion has been very interesting.


    I'm really not convinced on this point. I'll lay out my generalization without arguing the math, and if you still disagree I'll try to prove myself.

    BoS aggro reduction relies on the initial threat, plus the threat delta across those ten seconds. If all TPS were to stop the moment salv was dropped, the total threat reduction would be 0.183 * threat.

    However, if you factor in the delta you get that 0.183 * threat reduction plus 0.183 times whatever the delta threat is.

    This means that in any case, the more threat someone can put out over the course of a salv, the more threat the salv will remove.

    Edit: for clarity...
    And isn't that the point of salv, to allow people do DPS more? If I was worried about pulling aggro at the beginning of the fight, I think it would make more sense to not DPS in the first place than to DPS normally and stop DPSing while BoS ticks away.
    Bingo. Both points shiz makes are right on. jere calculates the percentage reduction as a percentage of actual damage done rather than as a percentage of potential damage done. So yeah, if you stop DPSing, you stop dividing by an increasingly larger number each 1s interval, so it looks like you're reducing your aggro by a greater percentage. In reality, that percentage doesn't matter. It's meaningless. It's the actual numerical drop that matters--this is the measurement of the difference between how many threat points behind the tank you are because of BoS and how many threat points behind the tank you would have been without BoS.

    The second point answers Kazey's point--if your DPS isn't threat-capped, then this entire argument is irrelevant to you. HoS in general ain't much use to you. If you pull aggro, HoS will let you die in all likelihood. The point of HoS is to allow someone who is threatcapped to DPS above where they ought to be able to and not pull aggro. So if you're threat-capped, chilling for a few seconds at the beginning doesn't do anything to you since you're going to catch up to the tank anyway. And if you're not threat-capped, pulling aggro at the beginning of the fight will still kill you.

    But

    "actual number your threat got reduced by" = threat_before_reduction*Percentage_Reduction.

    They are one in the same. Perhaps we are using the same term for different quantities and getting confused on what each other are saying?
    Nope. That equation holds true if you don't DPS during HoS's 10s, but if you do, then you can't simply multiply 10k threat by percentage reduction to get actual threat reduction. I mean you know this. Your geometric series shows this clearly--that threat(t) depends on threat(t-1), in addition to whatever threat was created by attacking during that interval.

    The whole point is moot, really, because there is no need to break up "numerical threat reduction" into "threat before reduction" and "percentage reduction" for the whole thing, since percentage reduction is irrelevant. You need it for each step of the calculation--each 1 second interval--but at the end, to figure out when you should use it, it's ultimately unimportant.

    Can you run your spread sheet exactly the same except change your initial threat to 40000 (instead of 10000)? Keep the DPS as 1000 and all the other factors the same. Could you post a pic of those results for comparison?
    Sure:



    As you can see, the stop-DPS strategy will allow HoS to reduce a greater percentage of your threat, but the full-DPS strategy will still, in the end, allow you to reduce more actual threat points.

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    *** Removed
    Last edited by jere; 08-23-2008 at 04:51 AM.

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    Quote Originally Posted by Klimpen View Post
    The threat% reduction talked about in this thread is one of the silliest metrics I've come across in a long time.
    Ouch....

    My apologies then, I will remove it. I am sorry I embarrassed you.
    Last edited by jere; 08-23-2008 at 10:28 AM.

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    No need to do that! If there is indeed a difference between the percentage of threat reduced and the numerical amount of threat reduced by BoS, it's worth analyzing and discussing. At the very least, showing that the percentage of threat reduction was 18.3% and not 20% was very useful, and worthy of a post.

    If you still have those maths, I'd like to take a look at them. It doesn't really jive with my sense of intuition that the larger percentage can end up with the smaller result, so I'd like to make sure nothing was missing from your calculations. Truth be told I didn't fully understand what you were arguing before your second last (and now deleted) post.
    Last edited by shiz98; 08-23-2008 at 09:30 AM.

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    Well I won't go back and repost the whole original post cause I was already getting remarks about it and I don't want to get everyone riled up again

    Well the basic math was something like this:

    AAT = Already_Accumulated_Threat

    Without Hand of Salv:
    AAT + TPS + TPS + ... + TPS = AAT + 10*TPS

    With Hand of Salv:
    (...(((AAT + TPS)*0.98 + TPS)*0.98 + TPS)*0.98 + ... + TPS)*0.98
    or
    AAT*0.98^10 + TPS*(0.98^10+0.98^9+...+0.98^1)

    (0.98^10+0.98^9+...+0.98^1) is equivalent to:
    Code:
    (0.98 - 0.98^11)
    ----------------
       (1 - 0.98)
    So Threat with HoS up was:
    Code:
                                   (0.98 - 0.98^11)
    Threat = (0.98^10)*AAT + TPS * ----------------
                                      (1 - 0.98)
    I just took that equation (the threat with HoS up) and divided it by AAT + 10*TPS (the threat generated without HoS up). The same way you would calculate BoSalv (threat with salv/threat without salv = 0.7).

    I used different variables in the OP though, x, Y, etc.
    Last edited by jere; 08-23-2008 at 10:29 AM.

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    Jere- this was a useful thread.

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    Jere! This was a very useful thread! Please if you still have it can you repost it? Don't let a few remarks discourage you!

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    Quote Originally Posted by Turelliax View Post
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    http://i.imgur.com/3vbQi.gif

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