On Grappling and Shoving While Raging in 5.5e

One way to determine this might be to just work through a couple of examples and try to keep the probability of success as close as possible to what it was in 2014.

Example 1:  Low Level Barbarian with +3 Strength and proficient (+2) in Athletics VS Commoner with +0 in both Strength and Dex and no proficiencies.

To get the exact probability we'd probably want to create a simulator and create a large sample size of data.  But we can estimate by seeing that on average the Commoner will roll 10.5 on the D20 even after modifiers.  So, on average the Barbarian will have to roll a 6 (+5) = 11 on the D20 to succeed.  That's a 75% success rate with a normal roll, but a 93.75% success rate with advantage.  Note also that in 2014 he must exceed the Commoner's roll -- a tie in a contest keeps the status quo which is "not grappled".  Conversely, it's the same contest for the Commoner to attempt to break free later on, but in that case the status quo is "grappled".

Example 2:  Low Level Barbarian with +3 Strength and proficient (+2) in Athletics VS Ogre with +4 in Strength but no proficiencies.

We can estimate the chance for success here by seeing that on average the Ogre will roll 10.5 + 4 = 14.5 on the D20 after modifiers.  So, on average the Barbarian will have to roll a 10 (+5) = 15 on the D20 to succeed.  That's a 55% success rate with a normal roll, but a 79.75% success rate with advantage.  Again, slightly different for initiating the grapple vs subsequently maintaining the grapple due to the rules for a tie.

Proposed 2024 Homebrew Solutions:

Solution A:  Increase the DC by 5.

Example 1:  Same Barbarian VS Commoner.

Normally, the Saving Throw DC to avoid being grappled = 8 + STRMOD + PB = 13.  Homebrew increases this to 18.  For the Saving Throw mechanic we use "meets it beats it" to resolve a tie, in favor of the one that is making the save.  In this particular example our success rate is now only 85%.  However, if our Barbarian was slightly more advanced to the point where he ends up creating a DC of 21 then we would actually create a 100% success rate, which was never realistically possible with the 2014 contest mechanic and therefore might not be a desirable solution depending on what we are trying to accomplish with this Homebrew.

Example 2:  Same Barbarian VS Ogre.

Again, the DC is increased to 18.  The Ogre gets +4 to the save.  He must roll a 14 or better.  Our success rate here is 65%.  Again, lower than in the corresponding 2014 contest.

Solution B:  The target creature gets disadvantage on their Saving Throw.

Example 1:  Same Barbarian VS Commoner.

Here the DC is 13.  The Commoner must roll a 13 or better on the D20, but at disadvantage.  Our success rate here is 84%.  This is similar to solution A, but note the more gradual increase of the curve from here -- it is much more difficult here to achieve 100% success, which we presumably would like to avoid if we are trying to replicate the results of the 2014 mechanic.

Example 2:  Same Barbarian VS Ogre.

Again, the DC is 13.  The Ogre gets +4 to the save.  He must roll 9 or better.  Our success rate here is 64%.  Again, lower than in the 2014 contest.  Also, again we approach 100% success more slowly as the Barbarian becomes more powerful as compared to Solution A.

Overall, it looks like Solution B (disadvantage on the save) is a better model for creating a similar curve to the 2014 system for how the chance for success improves as the Barbarian becomes more powerful.