plasmid

Giving it a shot at writing with different notation
 

Having to actually write it out in such a way that others can read and follow it does wonders for imposing clarity...
 

Only if they're imaginary stonenibblers.
 

Maybe something less ultimate
 

I agree with Flamebirde's answer. But I would add a bit more to it.
 

Pretty sure it's one of these
 

Optimizing Captain Ed’s answer a little more
 

Ah, the hospital's rule. Of course. I vaguely remember that from high school.
More importantly, I also found out that my memory of integration by parts was faulty, and that seems to have been the real issue. From the top:
 

This might count as an ah-hah if it's true.

The four laws of card shifting

Here's what I came up with
Edit: an aside
Edit2: even further aside
 

Ah, this is probably what you have in mind.
 

Not fully calculated because the math gets extremely messy and complicated pretty fast, but a description of how to go about it
Although a better answer might be that if there are 10 people playing a game with at least $10 entry fee and $100 payout with a non-zero chance that no one will win and the house will keep everything if everyone ends up picking a number in common with at least one other player, then your best move is to not play.

You are 
or more seriously
 

The picture in the spoiler above is an explanation with formulas that wouldn't look very nice if I tried to just write them out in text.

I don't have a full solution, but this might help get things started.
After that point, it gets too abstract for me to continue comprehending.

Question #2
 

For the sake of kicking this off, and without proof of optimality, I'll say this is the best I can come up with for now.