= Solution
\b[The <core> is the singleton $\{(1/2,1/2,1/2,1/2)\}$.] Here the <characteristic function of a coalitional game> is $v(C)=\lfloor |C|/2\rfloor$, the two-person instance of the <core of the miners game> calculation.
Let $x$ be in the <core>. Efficiency requires $x_1+x_2+x_3+x_4=2$, and every pair <coalition> requires $x_i+x_j\ge1$. Summing the six pair inequalities counts each coordinate three times, giving $3\sum_i x_i\ge6$. Efficiency makes this an equality; since every individual slack is nonnegative, all six pair inequalities must be equalities. In particular $x_1+x_2=x_1+x_3=x_1+x_4=1$ forces $x_2=x_3=x_4$, and comparison with $x_2+x_3=1$ gives every $x_i=1/2$.
Conversely, for the uniform allocation every <coalition> satisfies
$$
\sum_{i\in C}x_i=\frac{|C|}{2}\ge\left\lfloor\frac{|C|}{2}\right\rfloor=v(C),
$$
and its total is $2=v(N)$. It is therefore a <core> member, completing both existence and uniqueness. In particular, a three-player <coalition> receives $3/2\ge1$; the pair constraints are the tight ones.
Back to article page
