Sometimes the party wants to complete a multi-part task where the performance of one of them on one part can help or hinder the performance of another one on a different part. They have a common goal, but are involved in separate and discrete parts of the task.
For example, in one game I DM'd, an invisible party member entered in a busy public square and needed to attract the attention of another party member. They knew the second party member was close at hand, but did not know exactly where they were. I had them roll Intimidation(Charisma) for how loud they could shout, and had the other PC roll Perception(Wisdom) for how well they could pick out the shout against the background noise of the city. With the overall goal being to convey the message, the better the first PC performed on their part, the easier the part of the second PC should be.
Now, I could have just had the second PC roll Perception with advantage due to the assistance of the first PC, as in the rules for Working Together. However, this was unsatisfying to me, because it would have the first PC provide a 'static' bonus without accounting for how well they did on their part of the task. It also goes against the requirements of the 'Working Together' rules themselves, which state:
A character can only provide help if the task is one that he or she could attempt alone. For example, trying to open a lock requires proficiency with thieves' tools, so a character who lacks that proficiency can't help another character in that task.
Moreover, a character can help only when two or more individuals working together would actually be productive. Some tasks, such as threading a needle, are no easier with help.
Even though the PCs had a common and interdependent goal, the first PC can't really help the second one listen, and the second one can't really help the first one shout.
I could also call this a group check:
When a number of individuals are trying to accomplish something as a group, the DM might ask for a group ability check. In such a situation, the characters who are skilled at a particular task help cover those who aren't.
To make a group ability check, everyone in the group makes the ability check. If at least half the group succeeds, the whole group succeeds. Otherwise, the group fails. Group checks don't come up very often, and they're most useful when all the characters succeed or fail as a group.
While the PC's certainly are trying to accomplish something as a group and those who perform better are 'covering' those who do not, this misses the point by having everyone involved roll on the same skill when in my example they are using completely different skills to achieve different parts of a task.
At the time this event played out, I just had each of them roll, with the result of the first roll vaguely informing me of the DC for the second roll. I would like something a little more formal, however.
I am thinking that what I could do is set an 'overall' DC for the task and double it. The result of the first PC's roll would then reduce the DC of the task for the second PC.
For example, if my overall assessment of the DC for the task (hearing the shout) was 'Moderate' (DC15), then the combined DC would be 30. If the first PC rolled a 10 for their Intimidation attempt, that would reduce the DC by 10, such that a not-very-impressive shout meant that the second PC needed a DC20 Perception check to hear it.
I'm interested in an evaluation of this proposed mechanic. If you think this type of situation is covered in RAW, then please explain how. If you think it is not, please keep in mind good subjective as applied to homebrew review questions – a good answer should describe actual experience with trying to accomplish the same goal that I have, even if the mechanic to achieve it was different. Also, an objective assessment of the numerical consequences of such a mechanic (how it compares to a single skill check, to the 'working together' rules, to 'group checks', and how it is different from 'rolling to failure') would be welcome, even absent experience in implementing something like this.
Best Answer
I think this is elegant but very tough
My first observation is that when you multiply the base DC with the number of contributors1, having more helpers does not make the task easier. So this mechanic only applies to tasks that need all contributors, not to tasks that get easier the more contributors you have.
My second observation is that you are trying to achive two different things:
You can actually look at these separately:
Combining different skills
All of the book options assume characters use the same skill to help with the outcome of a DC challenge, there are no "mixed skill" DC checks in the game, even though there are challenges conceivable (as in your example) where it might seem plausible. I think such situations are not common, but the simplest way to handle them would be for the DM to allow using the specific needed skills that can reasonably contribute in the Help Action or the Group Check option, on a case-by-case basis. No new mechanism for combining them is needed.
Your new combination mechanism
For the following discussion, let us assume you allow combining different skills with the book mechanisms, so we can compare them to your method (as the book mechanisms do not allow combining different skills, there otherwise would be nothing to compare):
For example, imagine the DC is 15, and you have two characters A with +4 on intimidation, B with +2 on perception. Then with the Help action, A gets to roll with advantage, with 75% chance of success. In your system, the combined DC would be 30, and the expected chance to achieve this (if my anydice got this right) is only 38.25% -- which intuitively makes sense - the average for each of their rolls would need to be higher than the expected 10.5, for 2d20 + 2 + 4 to be higher than 29.
In our example, at DC 15, A has a 50% chance of success, B 40%, so the group check has a chance of 70% success, as it is sufficient for one of them to succeed (they fail if both fail, which happens 50% * 60% = 30% of cases). For your system, they need to get 30 or better with the sum of 2 rolls. Again, the chance for this is 38.25%, so again, your system is a a lot harder than a group check.
In our example, the chance of both characters succeeding on the check is 50% * 40% = 20%, which is much lower than the 38.25% from your mechanism.
Conclusion
The proposed mechanic is very elegant, but at the same time significantly more difficult than comparable mechanics the game already offers. You could try to address this by coming up with more complicated ways to set the "combined" DC, but I think this would quickly get clunky, and also may be hard to balance across DCs and numbers of participants.
In my mind, allowing different skills with the established mechanisms is more in line with the design spirit of 5e, than introducing a degrees-of-success mechanic for a corner case situation.
1 I am assuming you set a DC for a given task depending on difficulty using the normal difficulty tiers, which you then multiply with N, the number of characters involved, for the overall DC. Your example of N=2 is the case where only two characters work together.
2 By default, the game has no concept of degrees of success or failure for skill checks, it is binary, either you succeed on the check, or you don't. The DMG on p. 242 proposes "flourishes and approaches you can take when adjudicating success and failure to make things a little less black-and-white", including Success at a Cost ("Failing Forward") , Degrees of Failure, and Critical Success or Failure, but not something that in general allows degrees of success with increased rolls above the target DC. So your system is fundamentally different in this regard.