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Sunic_Flames
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« on: May 26, 2008, 02:36:01 PM » |
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I take no credit for any of this. Everything below is the writing of Tleilaxu_Ghola, not mine. I did not contribute to this in any way, shape, or form barring this copy paste. And some minor formatting changes to adjust it to these boards. Either way, nothing special. 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: 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.
[list=1] - Opposed Rolls:
[SBLOCK=Explaination/Derviation]Opposed rolls all work the same way. Each side rolls a d20 and adds a modifier to it. Whoever has the highest modifier wins the tie. Here's how I think about it (may not be the best way of solving it, but it works):
Treat the monster's roll as a probability and yours as a fixed variable. Examine it case by case. Each of your rolls has a probability of 1/20, calculate the probability that the monster could beat your roll with the modifiers in place. Multiply that probability by 1/20 and sum over all twenty events. The total result cannot be greater than 100%. To put it in generalized terms:
N-1?Pr(X+n<Z+Y), where X is your modifier to the opposed roll, the sum index is n (ranging from 1 to N), Z is the monster's modifier, and Y an RV representing his roll (uniform discrete from 1 to N).
Basically the only chore is to figure out how many possible rolls there are such that the monster succeeds, ie where X+n < Z + Y is satisfied. We restrict outselves to the case where X-Z <N. If Z-X >= N+1, then we always lose. Take a concrete example, Z = +15, X = +10, and you roll a 6. The monster needs to roll a 2 or better, leaving 19 possible rolls in which he could succeed. For any roll of 5 or lower on your part, the monster always succeeds, contributing a total of 25% to his total probability of success. In variables, that's (N+[Z-(X+n)]). Plug in N = 20, Z = 15, X = 10, and n = 6, and you'll see it gives 19, that is 19 possible rolls to succeed. Going into this we knew that we needed to roll better than (Z-X), since Z > X. Now we have a general formula for Z > X.
N-2?(N+[Z-(X+n)]) + N-1(Z-X), where n ranges from (Z-X+1) to N.
Now lets evaluate the sum... this will require a few tricks. If you don't follow... I'm sorry. I'll write out each step.
N-2?(N+[Z-(X+n)]) = N-2?(N+Z-X)-N-2?(n), both sums over the original range.
N-2(N+Z-X)(N-(Z-X))-N-2N(N+1)/2 + N-2(Z-X)(Z-X+1)/2
Bringing the total to:
N-2(N+Z-X)(N-(Z-X))-N-2N(N+1)/2 + N-2(Z-X)(Z-X+1)/2+N-1(Z-X)
Just to check, lets try the original case, plugging in N = 20, X = 10, Z = 15.
(1/400)(25)(15)-210/400+15/400+100/400= 280/400 = 70%. Meaning you would only have a 30% chance of beating a monster in this opposed check.
Now lets look at the case where Z<X. We will restrict ourselves to the case X-Z <N, because if X-Z >= N, we'd always win. Again, let’s use a concrete example. Z = 10, X = 15, and you roll a 6. The monster has to roll a 12 to beat you, giving him 9 possible rolls to succeed. So N-([X+n]-Z) gives the number of possible rolls. The monster always fails for rolls less than 7, so the best he can get is a 70% success rate (he automatically fails 6/20 times). In variables, that's (X+1)-Z automatic failures. We, as the player, needn't roll any higher than N-(X-Z) to ensure success on our part. I translate this to mean that the monster automatically fails for n > N-(X-Z). We're now in position to calculate the general formula for X > Z.
N-2?(N-[(X+n)-Z]), where n ranges from 1 to N-(X-Z)-1.
Evaluation time... again standard manipulations to evaluate the sum
N-2?(N-[(X+n)-Z]) = N-2?(N-X+Z]) - N-2?(n), both sums over the original range.
N-2(N-(X-Z)-1)(N-X+Z) - N-2(N-(X-Z))(N-(X-Z)-1)/2
Bringing the total to:
(1/2)N-2(N-(X-Z)-1)(N-X+Z)
Again, lets check with the variables from the original scenario to make sure I didn't screw up badly:
1-(1/2)(1/400)(14)(15) = 1-105/400 = .7375.
Note the above derivation assumes that the offender wins the tie, this is actually wrong in retrospect, so rather than redoing all the sums, I'm just going to calculate the tie probability. Then we'll subtract it from the "offender".
Same deal as before, we need to find all the possible rolls such that the condition X + n = Z + Y. In other words X-Z = Y-n. Obviously if Z-X > N, then the two cannot ever be equal. Likewise, if X-Z > N, then the two cannot ever be equal. These form the bounds for our problem. The interesting thing is that for all the possible values of X + n (ranging from X + 1 to X + N), there is only one combination Z + Y that will satisfy the tie condition. The only difficulty then becomes discovering the range for n. Basically we're going to have two scenarios: your mod is bigger or the monster's mod is bigger. If your mod is bigger then the range of n in the sum is from 1 to N - (X-Z). If your mod is smaller, the sum ranges from Z-X+1 to N. So basically we come up with these to formulas:
If your mod is bigger: Tie probability = N-2*(N-X+Z)
If your mod is smaller: Tie probability = N-2*(N-Z+X)
Wanna check? Take my mod to be 15 and the monster's to be 10. Check that link (below) and you'll see that the tie probability is 3.8%. My formula gives: (20-15+10)/400 = 15/400 = 3.75 %. The other one is the exact same thing, just with the modifiers reversed in role. So now our final formulas come to:
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
In a moment of insecurity, I decided to check these numbers against what other people have said. I googled a bit and came to this site: >>LINK<<
His numbers agree with mine (rounded to one decimal place). Just add the tie probability to the win probability to see. To check the case Z > X, add the tie probability to the lose and you get the 70/30 split I calculated.
[/SBLOCK]
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:
[SBLOCK=Explaination/Derivation]
Whew, this guy is waaaay easier to do. For any roll versus a fixed DC where X = your modifier, Y is the random variable, and Z = the DC, we have the following situations:
Pr(Y+X>=Z) = 1-Pr(Y+X<Z) = 1-Pr(Y<Z-X)
Obviously for X >= Z, we always win, since Y cannot be less than zero. If Z-X>N, we always lose, since Y cannot be larger than N. For anything in between, we have the probability N-1[N-(Z-X)+1]. For example say we were trying to make a DC 15 check with a +5 modifier. We'd need to roll 10 or better to make it. That means there's 11 different rolls which would mean success. Hence the equation. Now, if a 20 automatically succeeds, as with saves, then the minimum probability of success is 5%. If 1 always fails, as with saves, then the maximum probability of success is 95%.
[/SBLOCK]
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. |
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« Last Edit: May 27, 2008, 07:26:32 AM by Sunic_Flames »
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Sunic_Flames
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« Reply #1 on: May 26, 2008, 02:46:59 PM » |
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Ok, finished making the necessary adjustments to adapt it to this forum.
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AfterCrescent
Honorary Moderator
Organ Grinder
Posts: 4220
Here After
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« Reply #2 on: May 26, 2008, 08:15:37 PM » |
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I suggest taking the Code out of the spoiler. It should keep its format decently enough.
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Tshern
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« Reply #3 on: May 26, 2008, 08:29:20 PM » |
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Did you actually ask for his permission before you copied it? Not sure if you actually are obligated to do so for some reason, but I'd do that anyway.
Thanks for moving this here.
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Ja vuosia myöhemmin
kalvas kaksikko lattialla motellin tihrustelee,
kun sama keiju katossa leijailee.
Kyselevät: "Mikä päivä nyt on?
Tiedätkö sen?"
Kuiskaten laulaa keiju:
"Tämän elämän viimeinen."
Handy Links |
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Sunic_Flames
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« Reply #5 on: May 26, 2008, 09:46:38 PM » |
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Did you actually ask for his permission before you copied it? Not sure if you actually are obligated to do so for some reason, but I'd do that anyway.
Thanks for moving this here.
I was not aware that was part of the policy. When I asked about finding it because I needed it for something, and someone mentioned that should be copied over I took it upon myself to contribute by doing so. If he objects for some reason I'll remove it. I didn't think it would be a problem as long as credit was given where it was due. No offense was intended. |
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Sunic_Flames
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« Reply #6 on: May 26, 2008, 09:52:11 PM » |
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I suggest taking the Code out of the spoiler. It should keep its format decently enough.
I tested this. The alignment broke pretty badly, rendering it mostly unreadable. So it has to stay. Link fixed. |
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blargney the second
Ring-Tailed Lemur
Posts: 45
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« Reply #7 on: May 27, 2008, 03:46:27 AM » |
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The link doesn't quite work - the " now" at the end got copied over as well.
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Tshern
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« Reply #8 on: May 27, 2008, 06:32:11 AM » |
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Did you actually ask for his permission before you copied it? Not sure if you actually are obligated to do so for some reason, but I'd do that anyway.
Thanks for moving this here.
I was not aware that was part of the policy. When I asked about finding it because I needed it for something, and someone mentioned that should be copied over I took it upon myself to contribute by doing so. If he objects for some reason I'll remove it. I didn't think it would be a problem as long as credit was given where it was due. No offense was intended. I am not sure if there is a policy about this, but to me it seems like a smart move. He's a smart guy, so he most probably will not object and definitely knows that no-one intended to offend him... |
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Ja vuosia myöhemmin
kalvas kaksikko lattialla motellin tihrustelee,
kun sama keiju katossa leijailee.
Kyselevät: "Mikä päivä nyt on?
Tiedätkö sen?"
Kuiskaten laulaa keiju:
"Tämän elämän viimeinen."
Handy Links |
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Sunic_Flames
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« Reply #9 on: May 27, 2008, 07:27:43 AM » |
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The link doesn't quite work - the " now" at the end got copied over as well.
 ...And this kids, is why you should always get a restful night's sleep before posting on the forums. |
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Bauglir
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« Reply #10 on: September 06, 2009, 10:15:14 PM » |
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Hi, my name is Bauglir and I'm a thread necromancer. I'm so very sorry.
The link to the superior table is dead, looks like cuberocks is down. I'd try and find another source myself, but I'm a lazy bum and I wanted to see if anybody knows about one already.
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So you end up stuck in an endless loop, unable to act, forever.
In retrospect, much like Keanu Reeves.
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Endarire
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« Reply #11 on: November 13, 2010, 02:57:04 AM » |
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I agree that the link to that table is down. Also, this board supports SPOILER but not SBLOCK.
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Hood - My first answer to all your build questions; past, present, and future. Speaking of which: Don't even need TO for this. Any decent Hood build, especially one with Celerity, one-rounds [Azathoth, the most powerful greater deity from d20 Cthulu]. Does it bug anyone else that we've reached the point where characters who can obliterate a greater deity in one round are considered "decent?"
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awaken DM golem
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« Reply #12 on: November 13, 2010, 02:18:36 PM » |
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In the Interesting Observations part ... the monster's Hitpoints and Saves are linear.
The exception is the hitpoints trending up around level 14.
Ok
This looks a lot like 4e to my eye. At least what(-ever) they were trying to do.
The problem on the low end of both 3e and 4e, is the math breaks down a little.
Over at the den, FrankT's recent thread has an idea, I wholeheartedly agree with:
add a few "levels" at the start. That's it.
You get rid of the whoopsy zero problem.
Everything works closer to real linear attempt.
Of course this says nothing about what powers monster have or should have.
NPC builds are even further off, into weird mathy nonsense.
4e CO guys are running +5 levels during Epic, at this point.
Expect that number to increase, as more becomes known about the game.
That's not so bad of a linear vs. geometric semi-fixable CO smackdown.
3e gets off the linear, on the PCs side, rather early. But "we" know this.
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Agita
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« Reply #13 on: February 16, 2011, 08:25:09 PM » |
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*cue Thriller* So I was looking at this yesterday and today, and thinking that the table in the OP isn't very readable due to the code tags making the font so darn small, and that the boards don't like the sum signs in the mathematical formulas. So I decided to prettify the whole thing a bit. Like Sunic, I did not alter the content in any way, though I did take some liberties with the formatting. Everything else was purely copy/pasted, including Step 3 coming after Step 4 for some reason. Download here. |
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Brainpiercing
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« Reply #14 on: February 17, 2011, 08:07:39 AM » |
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Did someone by any chance archive the "critter filter"? On Waybackmachine the scripts don't seem to work.
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awaken DM golem
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« Reply #15 on: February 17, 2011, 05:57:56 PM » |
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The critter filter may have had some stuff on it, like what wotc wanted yanked off other sites.  |
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juton
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« Reply #16 on: February 17, 2011, 06:24:03 PM » |
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*cue Thriller* So I was looking at this yesterday and today, and thinking that the table in the OP isn't very readable due to the code tags making the font so darn small, and that the boards don't like the sum signs in the mathematical formulas. So I decided to prettify the whole thing a bit. Like Sunic, I did not alter the content in any way, though I did take some liberties with the formatting. Everything else was purely copy/pasted, including Step 3 coming after Step 4 for some reason. Download here. This is a lot more readable, thanks. By the way, did you use LateX to make this pdf? |
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Agita
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« Reply #17 on: February 17, 2011, 06:26:58 PM » |
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*cue Thriller* So I was looking at this yesterday and today, and thinking that the table in the OP isn't very readable due to the code tags making the font so darn small, and that the boards don't like the sum signs in the mathematical formulas. So I decided to prettify the whole thing a bit. Like Sunic, I did not alter the content in any way, though I did take some liberties with the formatting. Everything else was purely copy/pasted, including Step 3 coming after Step 4 for some reason. Download here. This is a lot more readable, thanks. By the way, did you use LateX to make this pdf? Ayup. It's become pretty much my go-to if I want anything well formatted. Doubly so if it contains any number of formulas more complicated than X+Y=Z. |
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Esgath
Barbary Macaque at the Rock of Gibraltar
Posts: 131
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« Reply #18 on: March 01, 2011, 07:14:14 AM » |
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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. Link doesn't work anymore. Did anyone out there compile a complete statlist? |
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Sunic_Flames
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« Reply #19 on: March 01, 2011, 07:38:33 AM » |
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If the link is down, I have no idea where a mirror would be.
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