= Solution
\b[No representation exists, even allowing arbitrary signed weights and <self-loops>.] Every singleton <coalition> loses, so each loop weight must be $\ell_i=v(\{i\})=0$. Every two-player <coalition> wins, which then forces
$$
w_{12}=w_{13}=w_{23}=1.
$$
The represented grand-<coalition> value would consequently be
$$
v(\{1,2,3\})=\ell_1+\ell_2+\ell_3+w_{12}+w_{13}+w_{23}=3,
$$
contradicting its required value $1$. The obstruction is the additive <edge> representation, not a restriction that weights be nonnegative: the singleton and pair values already determine every relevant weight.
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