$n$ balls are thrown randomly into $k$ bins, where the bins have a limited capacity $c$. If a ball would land in a full bin, it "bounces" randomly into a different bin, until it finds a non-full bin.
How many bins are expected to be full after all balls have been thrown? A solution for $c = 2$ for $n < 2k$ is specifically what I'm after. (So in this case, finding the number of empty bins is just as good.)
I'm having trouble dealing with the fact that the number of eligible bins changes as the balls are thrown, depending where they have landed so far. I've looked at other ball-and-bin problems here, but I can't find any that have this feature. Edit: This question also has this feature but has no answer. One commenter says "Unfortunately this is, as far as I'm aware, a rather intractable problem." But perhaps something can be done for $c=2$.
If you want to know, I'm trying to calculate equilibria for a population simulation where creatures (players) may or may not run into each other. If a creature is lucky enough to not run into another creature (like a single ball in a bin), it gets free food. If two creatures run into each other (two balls in a bin), they play a round of hawk-dove. The sim assumes a creature can tell when two creatures are already meeting at a location, and then stays away.
I could just numerically find the specific results I need, but I would love to find a general solution (for any $n<2k$, even with $c=2$).