Bins in a Circle

Alice and Bob are playing a game. In this game, there are initially $N$ bins in a circle, conveniently labeled $1$ through $N$ in clockwise order — for each valid $i$, bin $i+1$ is the next bin clockwise from bin $i$, and bin $1$ is the next bin clockwise from bin $N$. Also, there is a ball in bin $1$. The game is played with fixed parametres $R$ and $K$. Initially, the players designate one of t