Core of the miners game ($v(S)=\lfloor |S|/r\rfloor$)

For $r\ge 2$, the core is zero when $n<r$, the nonnegative unit simplex when $n=r$, the single allocation $x_i=1/r$ when $n\ge 2r$ is divisible by $r$, and empty for all other $n\ge r$. To prove the obstruction, every $r$-person coalition requires payoff at least one. Summing these constraints counts each player $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ times and gives $r\lfloor n/r\rfloor /n\ge 1$, impossible when $r\nmid n$. For a divisible $n$, all such coalition sums must equal one. Comparing coalitions with the same $r-1$ members makes all individual payoffs equal when $n>r$. The uniform allocation satisfies every coalition constraint because $|S|/r\ge \lfloor |S|/r\rfloor$. At $n=r$ all proper coalition values are zero, giving the whole simplex instead.