= Core of the miners game
{title2=$v(S)=\lfloor|S|/r\rfloor$}
For $r\geq2$, the <core> is zero when $n<r$, the nonnegative unit simplex when $n=r$, the single allocation $x_i=1/r$ when $n\geq2r$ is divisible by $r$, and empty for all other $n\geq r$. To prove the obstruction, every $r$-person <coalition> requires payoff at least one. Summing these constraints counts each player $\binom{n-1}{r-1}$ times and gives $r\lfloor n/r\rfloor/n\geq1$, 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 $|S|/r\geq\lfloor|S|/r\rfloor$. At $n=r$ all proper coalition values are zero, giving the whole simplex instead.
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