D&D 5e has few guidelines on fixing a Difficulty Classes (DCs). Basically, it boils down to this:
The other problem is that many rolls in 5e are contests (opposed rolls) - they rely on your foe's stats and rolls, not on a fixed DC. The simple answer is advantage/disadvantage - but what if I want to add some degrees of EPIC craziness? Say, for example, I have disadvantage if my enemy is in a "Hard" situation - what if I'm on a "Nearly impossible" situation?
It seems to me that, if the DC is that different between "Hard" and "Nearly impossible", there should also be some distinction when you're NOT using DCs.
Well, you can always adopt a +10 modifier instead of ad/disad for extreme circumstances. There is at least one good supplement - Dungeon Grappling - that does that.
There is no easy answer to all situations, but I use a simple rule that works for many circumstances, provided the challenge can be objectively "measured" somehow - in feet, pounds, number of creatures, minutes, etc.
It goes like this: you can double the effect of a roll by rolling two dice, triple it by rolling three dice, quadruple it by rolling four dice, etc. So, a "double challenge" would require two dice, and so on.
Let us say, for example, that you want to grapple or push four goblins at once with your shield. The GM thinks your idea is both plausible (you have Strength 18 and are proficient in Athletics) and cool, so she allows it - although she thinks pushing four goblins should be harder than pushing two or three.
Just roll four dice and pick the worst - if you succeed, all four goblins are affected.
Likewise, a Warlock could use Dark Delirium against three creatures instead of one - just roll three dice for their saving throws, and if the highest one succeeds, all three make their saves.
Or if you want to use a paladin's Abjure Enemy within 120 feet instead of 60 feet, to stop a skeleton. Technically it should be impossible, but why not allow it - specially for for a high level paladin against low level foes? Just roll two dice and pick the worst (since you doubled the distance).
This is not only for dealing with multiple foes. As you can see, you can double distances, do things three times faster, etc.
This assumes, of course, you must roll to hit and have both a chance of success and a chance of failure (no matter how minimal). However, you can also use this idea with powers or situations that require NO die roll - just assume a natural 1 is a critical failure, a natural 2 is a failure, and everything else succeeds.
This adds a lot of flexibility to the whole system. Say, if you have a power that can automatically provide food for six people every day, what happens if you're travelling with a dozen people? Or if you're in the a dry land with little food? Just roll a couple of dice and you're good to go.
I know, I know, creating a "dice pool" with disadvantages is verboten in 5e - but modifiers also are, as a general rule. In any case, if you prefer modifiers and dislike dice pools, just use the guidelines here. Or DOUBLE the number to get the modifier: -4 for two creatures, -6 for tripling the distance, -8 for acting four times faster, etc.
What is the point?
I added this rule to my RPG (Days of the Damned) to quickly adjust DCs in various circumstances. In 5e, I think it is useful for another reason: it allows high level characters to be more flexible and impressive against low level foes.
Because of bounded accuracy and the action economy, some PCs - specially Fighters, for example - have few options when fighting multiple weak creatures at once (and vice-versa - some high level creatures can be outclassed by a group of low level PCs).
This is deliberate, from what I understand about 5e's design goals - but not to everyone's tastes.
I, for one, think that there should be a greater gap between, say, levels 6 and 16. While I appreciate 5e's more "grounded" heroes, high level characters (specially fighters, barbarians etc.) feel a bit underwhelming.
In short, I like what 5e did - I just think they went too far.
A 12th level barbarian (according to the PHB, someone that deals with threats to whole regions or continents!) should have an easy time against a dozen of goblins archers, and not be completely unable to move if four kobolds ever manage to grapple him! I don't think it is too much to ask - at 12th level, a wizard can cast Mass Suggestion against a dozen foes, and even the fighter will survive a 100 feet fall with no serious injuries... So why not kick a few goblins away in a single round?
This little rule, by itself, is not enough to make high level characters more "epic", but it might be a good start.
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