Calculating the core of a game In a coalitional game n miners find equal blocks of gold. Two can carry one piece home. The payoff of a coalition S is $\nu(S)=|S|/2$ if $|S|$ even and $(|S|-1)/2$ for $|S|$ odd.

Determine the core if n is even, that is the set of imputations $x$ such that $\sum_{i\in S}x_i\ge \nu(S)$ for all S.

Could someone show how to do this systematically? What if three miners can carry one lump?
If the cardinal of the grand coalition is odd, the core is empty because it is impossible to share an amount $v(N)$ such that $\forall i \in N$, the sum of the payoff of the players belonging to $v(N\setminus i)$ receive more than $v(N)=v(N\setminus i)$.

If the cardinal of the grand coalition is even, then it is easy to see that $x_i=\frac{1}{2}$ $\forall i \in N$ belongs to the core and it is the only one possible payoff.
If the cardinal of the grand coalition is odd, the core is empty because it is impossible to share an amount $v(N)$ such that $\forall i \in N$, the sum of the payoff of the players belonging to $v(N\setminus i)$ receive more than $v(N)=v(N\setminus i)$.

If the cardinal of the grand coalition is even, then it is easy to see that $x_i=\frac{1}{2}$ $\forall i \in N$ belongs to the core and it is the only one possible payoff.