Game-theoretic analysis of Smallest Unique Integer game

What blew my mind when I played around with this in college was the fact that it had infinite support (easy to prove) but also doubly-exponential in the tail.

I could see the concave/convex behavior when I explored it computationally, but I couldn't figure out the theoretical/equational basis for it. I was working with a fixed number of players. I'm pleasantly surprised to see the variable-player Poisson version is simpler. I definitely wish I knew more stats, because the fixed-number version stumped me when I tried to solve it directly. (I don't know if it's possible, except for N = 3.)