Author Topic: Optimization by the Numbers  (Read 12583 times)

Offline Agita

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Optimization by the Numbers
« on: February 19, 2012, 05:24:28 PM »
Repost of a repost of a
Originally written by Tleilaxu_Ghola. This is a repost; no text has been altered. Formatting has been fixed to account for the boards, and sum signs that were lost on the old BG boards have been reinserted in the proper places. This post will not be edited further; any additions will instead go in separate posts.
If you spot broken formatting or missing symbols, or want to discuss the data in this thread, post in the discussion thread.
New table formatted by Maat Mons.

Caveats: The results of this analysis may not fit your personal idea of optimization.  I fully realize that many builds exhibit unquantifiable benefits which do not fit in the analysis I provide below.  That said, this strictly numeric analysis contains some interesting results.
Introduction:
The entire premise of this thread will be to analyze characters based on two things: the probability that a monster will succeed in resisting an attack or the probability that a character will succeed in resisting a monster's attack.  From here it should be obvious that one desires the former probability to be as low as possible and the latter to be as high as possible.  This thread will not assess any character traits that cannot be easily described by a probability.  Essentially this means I'm limiting myself to opposed rolls and DC checks.  I realize this is not a comprehensive analysis, and I will never claim that it is. Nevertheless, it should be a good analysis for the area it attempts to assess.

The Data:
This thread concept was previously impossible and the realm of speculation because of a lack of proper data.  This has changed, two individuals (CubeKnight & Dielzen) have performed an excellent job of data mining to bring us what follows:

(click to show/hide)

Average and Max Values per CR:
The above data are extracted from all monsters in the SRD; this work was done by Dielzen.

The next data is similar, but contains numerous "CO sources" such as MMI through MMIV, stormwrack, sandstorm, and others:
>>LINK<<

The above link has a great deal more functionality than Dielzen's table, so I strongly recommend using it.  The tool is simply wonderful.

Data Analysis: There's a lot of data here and making sense of it is perhaps more of a chore than collecting it.  The first thing I will do is establish a rubric for character analysis that will make use of this data.  Once that is done, analysis of any particular character will involve nothing more than looking up values in a table and clicking away at your calculator.  The next thing I will do is go through the data and extract whatever interesting and salient pieces of information that I can.

Character Rubric:
  • Step 1 (Set up the problem): Figure out what level you would like to analyze the character.  This will determine the CR band that your character might be dealing with.  I will use the following notation for single opponents:
  • Worthless: CR = Level -5 to Level -4
  • Trash: CR = Level -3 to Level -2
  • Average: CR = Level -1 to Level +1
  • Boss: CR = Level +2 to Level +3
  • Impossible: CR = Level +4 to Level +5
One doesn't always fight single monsters, luckily there's a good resource for assessing groups: DMG Pg 49.  Simply look up your level and you can see how many trash mobs to expect per encounter.  For example, a 20th level character can expect up to 4 16 CR creatures in one encounter and up to 12 CR 13s.  Dealing with groups is a little trickier as far as assessment goes, I'll discuss this further in step 6.
  • Step 2 (Offensive Capabilities): Find the salient attack forms of the character.  Is it a spell slinger?  A crowd controller?  A melee type?  An archer?  Whatever it does, it's probably dependent on some sort of opposed check to attack, be it saving throws or AC.  The next few sub-steps will walk you through the probability calculations.  It's really not that hard, once you get a hang of it.  Trust me. First, determine the mechanic of the offensive capability.  Is it an opposed roll (like trips, skill checks, etc), or is it a DC check (like spell saves)?  There are two fundamentally different things, the former being the most complex.

    • Opposed Rolls:
      (click to show/hide)
      Formulas (for d20 system... N = 20).
      If your mod is bigger:
      Probability of Success = 1 - (1/2)N-2(N-(X-Z)-1)(N-X+Z), where X = your modifier, Z = monster modifier

      If your mod is smaller:
      1 - (1/2)N-2[((N-(Z-X))(N-(Z-X)+1) - 2(N-Z+X)], where X = your modifier, Z = monster modifier

      Note that if X-Z >= N, we always win.  If Z-X >= N+1 we always lose. So these form the bounds for X and Z with respect to N.
    • Roll versus DC:
      (click to show/hide)
      Formulas:
      If your modifier is smaller than the DC: Probability of success = N-1[N-(Z-X)+1]
      If your modifier is equal to or greater than the DC: 100% success, unless 1 fails, then 95%.
      If your modifier is N+1 less than the DC: 0% success, unless 20 is auto-success, then 5%.
  • Step 4: Weighted Damage (if applicable): If your character deals damage, then it needs to be weighted by its probability of success to determine just how good it is.  One can say, "assuming all hits" and come up with a huge damage figure, but it means nothing if you have only a 1% chance of landing all hits.  Calculating weighted damage is as simple as multiplying the probability of each attack by the damage that it deals.
  • Probability Grading: If you want to "certify" your character or give it some "grade" of performance, I suggest the following:
    • Passable: 50% probability of success for binary attack forms (like save or die spells).  10% of total monster HP in weighted damage dealt.  With 4 passable party members, 10% weighted damage means that the monster will go down in a little over 2 rounds.  Provided the monster is also passable, this should be alright.  These probabilities are calculated from the AVERAGE values for CR = level +1.
    • Optimal: 75% probability of success for binary attack forms (like save or die spells).  20% of total monster HP in weighted damage dealt.  With 4 passable party members, 20% weighted damage means that the monster will go down in a little over 1 round.  Provided the monster is also optimal, this should be alright.  These probabilities are calculated from the AVERAGE values for CR = level +2 OR the max values of CR = level.
    • Overpowered: 95% probability of success for binary attack forms (like save or die spells).  50% of total monster HP in weighted damage dealt.  With 4 passable party members, 50% weighted damage means that the monster will go down in under 1 round.  Provided the monster is also overpowered this will be the only way to avoid PK.  These probabilities are calculated from the AVERAGE values for CR = level +3 OR the max values of CR = level +1.
    • Twink: 95% probability of success for binary attack forms (like save or die spells).  100% of total monster HP in weighted damage dealt.  With 4 passable party members, 100% weighted damage means that the monster will go down in a little over 1 rounds. Provided the monster is also ridiculous, you basically need to one-shot it or it will one-shot you.  These probabilities are calculated from the AVERAGE values for CR = level +5 OR the max values of CR = level +2.
  • Step 3 (Defensive Capabilities): This is pretty simple.  Just calculate the offensive capabilities of your monster, like you calculated yours, find his probability of success.  Your probability of successfully defending yourself is 1 - monster's probability of success.  Some considerations:
    • AC versus multiple opponents: The most optimal attack method for a group of monsters fighting a group of characters of higher level is to focus fire on the most damaging or threatening individuals.  Essentially you need to determine if you would be a threatening target to the monsters.  Estimate how many of them you would attract fire from.  If you plan on being the party "tank" assume that all will attack you, as a worst case scenario analysis.  Calculate the probability that each one will hit you (as with the single enemy case) then do the following:
    • The probability that N-N (zero) monsters of N total monsters will hit you is equal to the product of your probability of success against each individual monster.
    • The probability that N - m, where m < N, will hit you is equal to the product your probability of success for the m that miss you, multiplied by the product of your AC's probability of failure for the N- m monsters that hit you.  Your probability of failure = 100% - Probability of Success.
  • Miss Chances: For miss chances you simply multiply the probability of the attacker's success by the percentage and then use this modified offensive success rate to calculate your defense rate (as discussed above).  If multiple miss chances are present take the product of them all and then multiply that by the attacker's success rate.
Interesting Observations on the Data:
This will be an expanding list of observations that I make on the data. There is no real overarching theme for these observations as of yet. Just interesting trends.
Hit point Trends:
  • Average HP from CR 0-15: HP scale very linearly in this region at about 12.5 HP per CR.
  • Average HP from CR 15-66: HP scale linearly (I can't tell you the chi-squared of the fit without additional data) at about a rate of 49.6 HP per CR minus 647.
This tells me that damaging characters need to linearly increase their damage output with level and at about level 14 they need to receive a significant damage boost to enter the next linear regime.

Average Saving Throws:
Saving throws increase linearly in the region of 1-30 CR, which is the most interesting area.  Fort saves are generally the highest, with will saves being the lowest.  Interestingly though, ref saves increase at a decreased rate compared to will and fort, such that near 17-23 CR there's a region where Ref saves are lowest on average.  That's good to know for AoE mages.

20th level Average Saves:
Avg saves are: 22.5 fort, 16 ref, 19.12 will.  Lets assume a character with 18 starting int/cha/wis, +5 levels, +5 tome, +6 item and +2 DC from feats casting a 9th level spell with binary save.  Spell DC is 19+12+2 = 33. Against fort save that gives the monster about a 50% chance of survival, against ref its 20%, and against will its about a 35%.  Assuming the spell slays the monster outright, it's equivalent of dealing 100% avg HP dmg (weighted with the probability of monster's failure).  At 20 CR this is 364 dmg.  Best case is ref, and against a single creature it deals 291.2 weighted damage.  Of course most ref-saves are area effects and thus would apply to more than one creature.  Say it was against 4 lvl 16 creatures (a EL 20 encounter), slaying them outright with one ref-save spell of 9th level would yield 778 weighted damage.
« Last Edit: February 19, 2012, 07:17:35 PM by Agita »
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Offline Agita

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Re: Optimization by the Numbers
« Reply #1 on: February 19, 2012, 05:25:42 PM »
Compiled into a .pdf file for readability on 17.02.2011. Download here: http://www.mediafire.com/?ibe8q042m88o362
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Offline Agita

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Re: Optimization by the Numbers
« Reply #2 on: February 19, 2012, 07:13:00 PM »
A better table for those that don't want to download the pdf follows below. Thanks to Maat Mons for formatting it.

(click to show/hide)
« Last Edit: February 19, 2012, 07:17:45 PM by Agita »
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