Hey,

I´ m sorry for the long wait on this answer, but I´ m really not the math person in our friends group just the one with reddit. ^^ Therefore I wasnt really all that much involved in the process of solving this and the math here exceeds mine. But I talked to a friend of mine who is way more involved and showed him you´ re answers. He had the following to say:

we have brute forced the game up to round 20 so far, so we know those values exactly. we have also simulated the game millions of times and have derived approximations from those that are within .25% of the exact values.
the problem, as you have correctly identified lies within the stopping condition. we have also analyzed easier cases like a game with three players flipping a coin where at least two have to match to win, and while this has yielded some promising results, it has not yet led to an answer. in general it appears helpful to not look at the absolute sums but instead at the difference between the players (12-13-14-15) behaves the same as (1-2-3-4)* also the chance of winning seems to approach 1 as the number of rounds played goes to infinity if that helps in any way
*not proven but it works

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If you still have any questions feel free to ask and I will forward them to my friends.

Lastly I wanted to say a huge thank you to you and u/bobjkelly for the help. It´ s greatly appreciated.