Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 35 / 5 / a

It is useful to prove the general core of the miners game calculation when $r$ miners carry one unit: $v(S)=\lfloor |S|/r\rfloor$, with $r\ge 2$. Individual rationality gives $x_i\ge 0$, and efficiency gives ${\sum }_i{x}_i=q:=\lfloor n/r\rfloor$.

If $n<r$, the grand coalition value is zero and the only core allocation is zero. If $n\ge r$, every $r$-person coalition must receive at least one. Sum these inequalities over all $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ such coalitions. Each player appears $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ times, soIf $r$ does not divide $n$, then $rq<n$, a contradiction. Thus the core of a cooperative game is empty in these cases.

If $n=r$, every proper coalition has value zero. The core is exactly the simplex of nonnegative allocations with total one. If $n$ is a multiple of $r$ with $n\ge 2r$, the averaged inequality is equality. Since each $r$-person coalition sum is at least one, every such sum equals one. For any two players $i,j$, choose $r-1$ other players and compare the two resulting $r$-person coalition sums. They imply $x_i=x_j$. Hence $x_i=1/r$ for all players. This vector belongs to the core because $|S|/r\ge \lfloor |S|/r\rfloor$ for every coalition.

For two miners per lump, therefore,In particular the even case has a small-group exception at $n=2$.

For three miners per lump, the same proof givesThus divisibility controls the existence of a stable allocation once enough miners are present, while exactly one possible lump leaves freedom to divide its value.