= Solution
\b[No. Two positive <self-loops> and one negative <edge> give a <simple cooperative game> with an empty <core of a cooperative game>.] Give players $1,2$ loop weights $\ell_1=\ell_2=1$, put $w_{12}=-1$, and give every other loop and <edge> weight zero. For a <coalition> $S$,
$$
v(S)=\mathbf1_{\{1\in S\}}+\mathbf1_{\{2\in S\}}
-\mathbf1_{\{\{1,2\}\subseteq S\}}
=\mathbf1_{\{S\cap\{1,2\}\ne\varnothing\}}.
$$
Thus a <coalition> wins exactly when it contains player $1$ or player $2$. This is monotone, has values in $\{0,1\}$, and satisfies $v(\varnothing)=0,v(N)=1$. Extra players for any $n\ge3$ do not alter any <coalition> value.
The <core> singleton inequalities would require $x_1\ge1$, $x_2\ge1$ and $x_i\ge0$ for every other player, while efficiency requires $\sum_i x_i=1$. These requirements are inconsistent. Equivalently, the two disjoint winning singleton <coalitions> leave no <veto player>. Signed weights are essential to this particular representation and are permitted in the <induced subgraph game> model.
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