Sunday, April 17, 2016

Scaling Effects & Expanded Advantage

Ahoy, dear readers. It feels like it's been awhile since I assaulted you with spreadsheets. Brace yourself!



Over here:

http://autarch.co/forums/general-forums/general-discussion/monsters-saves

Nimas made a good point about poison and such - shouldn't larger creatures need more?

Broadly, some of that is taken into account via their saving throws, but specifically, the poisoner is still using a single dose to instantiate the effect.

I answered a bit lower, thinking about multiple doses and what that may entail.

Assume, then, that poison, as a chemical/biological process, works much like digestion. "The dose makes the poison", as they say. If that's the case, we already know that larger creatures require more supply. We could also go by a more generic "size" measurement, in how many creatures it takes to make a full unit in Domains At War:

Herbivore/OmnivoreCarnivoreUnit Sizing
SizeSupply CostMultiplierSupply CostMultiplier# of MenMultiplier
Man-Sized0.521
Large4816822
Huge12348363
Gigantic4841924244
Colossal240596051205


Unit sizing makes the scope of the issue a lot smaller, but we'll look into both for the heck of it.

Let's look at a default case. We take a creature and dose it from 1 to 4 times; any failure dooms it. (you may notice that due to rounding, some things may not add right up to 100%)

Fighter LevelSave vs PoisonSuccess - 1DFailure - 1DSuccess - 2DFailure - 2DSuccess - 3DFailure - 3DSuccess - 4DFailure - 4D
20-211+100%0%100.0%0.0%100.0%0.0%100.0%0.0%
192+95%5%90.3%9.8%85.7%14.3%81.5%18.5%
17-183+90%10%81.0%19.0%72.9%27.1%65.6%34.4%
154+85%15%72.3%27.8%61.4%38.6%52.2%47.8%
14-155+80%20%64.0%36.0%51.2%48.8%41.0%59.0%
136+75%25%56.3%43.8%42.2%57.8%31.6%68.4%
11-127+70%30%49.0%51.0%34.3%65.7%24.0%76.0%
108+65%35%42.3%57.8%27.5%72.5%17.9%82.1%
8-99+60%40%36.0%64.0%21.6%78.4%13.0%87.0%
710+55%45%30.3%69.8%16.6%83.4%9.2%90.8%
5-611+50%50%25.0%75.0%12.5%87.5%6.2%93.8%
412+45%55%20.3%79.8%9.1%90.9%4.1%95.9%
2-313+40%60%16.0%84.0%6.4%93.6%2.6%97.4%
114+35%65%12.3%87.8%4.3%95.7%1.5%98.5%
015+30%70%9.0%91.0%2.7%97.3%0.8%99.2%

Note the 2 Dose columns are essentially 5E's Advantage/Disadvantage mechanic, presented slightly differently - check the numbers in bold.

If you need a 12+ to succeed, then the 20.3% chance is your chance to succeed with a Disadvantage; the 69.8% in the next column is your chance to succeed with Advantage. I realize that doesn't seem intuitive, but, that's the fault of how I did the table - you're looking to match the Success and Failure percentages from the 1 Dose column to find the row for Advantage or Disadvantage. Keep in mind I'm not defining failure as Disadvantage - taking the lowest of 2d20 - I'm defining failure as neither of the d20 rolls being 12+.

Anyway. A graph may be more illustrative:


This looks more like the Advantage/Disadvantage chart you see floating around; though I've presented it a bit differently again. You see the curves, though - middle values get more benefit. It's a bit truncated, since I stopped our calculation at the highest save available vs Poison; the Normal Man save of 15+.

Now, what we can do next is collapse those multiple-die percentages into a single roll. Random is random, after all. We're going to lose some resolution; up to a few percent here and there, but it'd make the whole process a bunch easier - if not, we're looking at having to roll dozens or hundreds if d20s.

Fighter LevelSave vs PoisonSuccess - 1DFailure - 1DPenalty - 2DPenalty - 3DPenalty - 4D
20-211+100%0%000
192+95%5%-1-2-3
17-183+90%10%-2-3-5
154+85%15%-3-5-7
14-155+80%20%-3-6-8
136+75%25%-4-7-9
11-127+70%30%-4-7-9
108+65%35%-5-8-9
8-99+60%40%-5-8-9
710+55%45%-5-8-9
5-611+50%50%-5-8-9
412+45%55%-5-7-8
2-313+40%60%-5-7-7
114+35%65%-5-6-7
015+30%70%-4-5-6

We take the difference in chance of success and convert that to a straight die modifier - one point per ~5%. For example, our 45% chance of a 12+ is modified by -5 at two dice - there's about a ~25% difference in the chances to save from 1 dose to 2 doses.

Conversely, if we underdose, we take the chance of "succeeding at failing", which is 55%, and invert our bonuses - if 4 doses is a "full dose", with no modifier, then underdosing by one, two, or three doses gives us a +5, +8, and +9 bonus to our single save, respectively.

One hole we have in this is that the 20HD+ creature isn't taking a penalty, so we'll add that in manually later on here.

Rolling this back around, we've got two possible choices for what our "dosing" multipliers are - we're either basing it on raw size - D@W unit sizing, or on dietary requirements - supply cost.

Dosing Multipliers
Supply Cost
SizeMultiplierMan DosesLarge DosesHuge DosesGigantic DosesColossal Dose
Man-Sized1112.50%4.17%1.04%0.21%
Large88133%8.33%1.67%
Huge3243125.00%5.00%
Gigantic496124120%
Colossal5480602051
Unit Sizing
SizeMultiplierMan DosesLarge DosesHuge DosesGigantic DosesColossal Dose
Man-Sized1150%17%4.17%0.83%
Large22133%8.33%1.67%
Huge363125%5.00%
Gigantic424124120%
Colossal5120602051

Fortunately for us, there's a built in break on this - the multiple-die percentages asymptotically approach the point of 100% success or failure pretty quickly - asking a normal man to roll 480d20 to see if he makes it is unnecessary. Like so:

Man-Sized Creature SavingSupply CostUnit Sizing
Fighter LevelSave vs PoisonSuccess - 1Dvs Large 8Dvs Huge 24Dvs Gigantic 96Dvs Large 2Dvs Huge 6Dvs Gigantic 24Dvs Colossal 120D
20-211+100%1111111
192+95%66.34%29.20%0.73%90.25%73.51%29.20%0.21%
17-183+90%43.05%7.98%0.00%81.00%53.14%7.98%0.00%
154+85%27.25%2.02%0.00%72.25%37.71%2.02%0.00%
14-155+80%16.78%0.47%0.00%64.00%26.21%0.47%0.00%
136+75%10.01%0.10%0.00%56.25%17.80%0.10%0.00%
11-127+70%5.76%0.02%0.00%49.00%11.76%0.02%0.00%
108+65%3.19%0.00%0.00%42.25%7.54%0.00%0.00%
8-99+60%1.68%0.00%0.00%36.00%4.67%0.00%0.00%
710+55%0.84%0.00%0.00%30.25%2.77%0.00%0.00%
5-611+50%0.39%0.00%0.00%25.00%1.56%0.00%0.00%
412+45%0.17%0.00%0.00%20.25%0.83%0.00%0.00%
2-313+40%0.07%0.00%0.00%16.00%0.41%0.00%0.00%
114+35%0.02%0.00%0.00%12.25%0.18%0.00%0.00%
015+30%0.01%0.00%0.00%9.00%0.07%0.00%0.00%

I stopped where it made sense to do so - no real need to show a row full of 0%.

If we swing that over to  bonuses, we get something like this:
Man-Sized Creature SavingSupply CostUnit Sizing
Fighter LevelSave vs PoisonSuccess - 1Dvs Large 8Dvs Huge 24Dvs Gigantic 96Dvs Large 2Dvs Huge 6Dvs Gigantic 24Dvs Colossal 120D
20-211+97.5%-3-9-180-2-9-19
192+95%-6-13-19-1-4-13-19
17-183+90%-9-16-18-2-7-16-18
154+85%-12-17-17-3-9-17-17
14-155+80%-13-16-16-3-11-16-16
136+75%-13-15-15-4-11-15-15
11-127+70%-13-14-14-4-12-14-14
108+65%-12-13-13-5-11-13-13
8-99+60%-12-12-12-5-11-12-12
710+55%-11-11-11-5-10-11-11
5-611+50%-10-10-10-5-10-10-10
412+45%-9-9-9-5-9-9-9
2-313+40%-8-8-8-5-8-8-8
114+35%-7-7-7-5-7-7-7
015+30%-6-6-6-4-6-6-6
The numbers in orange are the spots where success is impossible without additional positive modifiers to one's save. Also note I've reduced the 100% chance of success of the F20 save to a 97.5% - thereby giving some penalty to the roll of whatever creatures are that high (in ACKS default, that's just the most ancient of dragons). In theory I'd work out some sort of approach to 100% that includes higher HD save values, if any ever end up existing.

So, let's proceed. Here's Man-Sized again; I've made a couple adjustments to the 1+ save:

Man-Sized Creature
Unit Scale SizingSupply Cost Sizing
Save vs Poisonvs Large 2Dvs Huge 6Dvs Gigantic 24Dvs Colossal 120Dvs Large 8Dvs Huge 24Dvs Gigantic 96Dvs Colossal 480D
1+0-2-9
Automatic Failure
-4-10-18
Automatic Failure
2+-1-4-13-6-13-19
3+-2-7-16-9-16-18
4+-3-9-17-12-17-17
5+-3-11-16-13-16-16
6+-4-11-15-13-15-15
7+-4-12-14-13-14-14
8+-5-11-13-12-13-13
9+-5-11-12-12-12-12
10+-5-10-11-11-11-11
11+-5-10-10-10-10-10
12+-5-9-9-9-9-9
13+-5-8-8-8-8-8
14+-5-7-7-7-7-7
15+-4-6-6-6-6-6

...which ended up autofailing vs Colossal Sized poison via both methods. 

Anything in red is an automatic failure - we've approached the limit, and the penalty doesn't allow for a success even with a roll of 20. You can see here perhaps a little better what I was talking about with the asymptotic approach - the jump from the penalty on Large 2D to Large 8D is much larger than from Gigantic 24D to Gigantic 96D. And, while it's not shown, the two Colossal values are the same.

Let's move up to larger creatures. I'm assuming here that a Large Dose causes a Large creature to save with a single die, where larger or smaller doses cause it to save with more die or less die. I'm not doing anything more focused than that - I don't care if we do, say 2 Huge doses vs a Gigantic creature.

Large Creature
Unit Scale SizingSupply Cost Sizing
Save vs Poisonvs Man-Sized -1Dvs Huge 3Dvs Gigantic 12Dvs Colossal 60Dvs Man-Sized -8Dvs Huge 3Dvs Gigantic 12Dvs Colossal 60D
1+1-1-6-161-1-6-16
2+1-2-8-181-2-8-18
3+2-3-12-182-3-12-18
4+3-5-14-173-5-14-17
5+3-6-15-164-6-15-16
6+4-7-14-155-7-14-15
7+4-7-14-146-7-14-14
8+5-8-13-137-8-13-13
9+5-8-12-128-8-12-12
10+5-8-11-119-8-11-11
11+5-8-10-1010-8-10-10
12+5-7-9-911-7-9-9
13+5-7-8-812-7-8-8
14+5-6-7-712-6-7-7
15+4-5-6-613-5-6-6

The only difference between Unit Scale sizing and Supply Cost sizing is that initial jump from Man-Sized to Large, so here we see another good example of scaling. Everything in green is an auto-success.

Huge Creature
Unit Scale SizingSupply Cost Sizing
Save vs Poisonvs Man-Sized -6Dvs Large -2Dvs Gigantic 4Dvs Colossal 20Dvs Man-Sized -24Dvs Large -3Dvs Gigantic 4Dvs Colossal 20D
1+00-1-7
Automatic Success
0-1-7
2+11-3-121-3-12
3+22-5-162-5-16
4+33-7-163-7-16
5+44-8-164-8-16
6+55-9-155-9-15
7+65-9-145-9-14
8+76-9-136-9-13
9+87-9-127-9-12
10+97-9-117-9-11
11+108-9-108-9-10
12+108-8-98-8-9
13+118-7-88-7-8
14+118-7-78-7-7
15+125-6-65-6-6

And now it starts to get a little boring:
Gigantic Creature
Unit Scale SizingSupply Cost Sizing
Save vs Poisonvs Man-Sized -24Dvs Large -12Dvs Huge -4Dvs Colossal 5Dvs Man-Sized -96Dvs Large -12Dvs Huge -4Dvs Colossal 5D
1+
Automatic Success
0-2
Automatic Success
0-2
2+1-41-4
3+2-62-6
4+3-83-8
5+4-94-9
6+5-105-10
7+6-116-11
8+7-117-11
9+7-107-10
10+8-108-10
11+9-99-9
12+9-99-9
13+9-89-8
14+9-79-7
15+9-69-6

Colossal Creature
Unit Scale SizingSupply Cost Sizing
Save vs Poisonvs Man-Sized -120Dvs Large -60Dvs Huge -20Dvs Gigantic -5Dvs Man-Sized -480Dvs Large -60Dvs Huge -20Dvs Gigantic -5D
1+
Automatic Success
0
Automatic Success
0
2+11
3+22
4+33
5+44
6+55
7+66
8+77
9+88
10+99
11+99
12+1010
13+1010
14+1111
15+1111


So, what's the point?

Something we may have accomplished here is to "size" tasks, something that 5E brushes up against with it's Advantage mechanic.

1) Let's say Iocaine Powder is a "Large" poison. Vizzini, a 5th level Thief, is saving versus that at a 11+, with a -5 penalty. Good luck, Vizzini.

Our Westley, however, spent two years building up a tolerance to iocaine. He's a 7th level Fighter saving at 10+, but, he took the proficiency "Single Poison Tolerance", so he saves as a Huge creature versus iocaine. He gets a +7 to his roll.

Much better odds, still a gamble though.

2) Thief Al and Thief Bob are both attempting to pick a standard lock; both are  4th level, so it's a 15+ to do it.

Thief Bob took "Lockpicking", though, and picks locks as a Large creature. He gets a +4 to his roll. When Thief Bob is 10th level (rolling 4+), he'll auto-succeed.

Conversely, Thief Chuck, at level 1, is rolling an 18+ to open a standard lock; he'll never get a Large Lock, until next level, when he's rolling 17+ on a -3 penalty (he'll need a 20).

3) In a dungeon, there's a 60'x60' room. The whole room is a trap, set to drop the floor into a 100' pit when a majority of the weight gets to the middle. Or it drops the ceiling, or whatever. Big area of effect, is my point.

A 60' area is a Colossal area, as defined from D@W:Battles, since 1 colossal creature takes up a 60' area. 

Everyone automatically fails their save versus the trap; except Thief Derek, 14th level, who saves as a Gigantic creature versus traps, but still has a very large penalty to the roll. Assuming someone has adopted this idea and given thieves that ability, of course.


You could take this any number of ways; large-area dragon breath, stuff like that. Anything that can resolve on a pass/fail and could benefit from some scaling - especially if you want that scaling to taper off when a character is either really good or really bad at something.

Plus, it's already a generally accepted mechanic - 5E's Advantage - just taken another couple steps further (though we do nothing with Disadvantage here).