Induced subgraph game with self-loops ($v(S)={\sum }_{i\in S}{\ell }_i+{\sum }_{\{i,j\}\subseteq S,i\ne j}{w}_{ij}$)

An induced subgraph game extended by one self-loop weight ${\ell }_i$ per player. Each loop is counted once, so $v(\{i\})={\ell }_i$ andSingleton and pair coalition values determine all weights; larger coalition values must then agree with their sums. Signed weights are allowed. For example, two loops of weight $1$ joined by an edge of weight $-1$ represent the simple cooperative game in which any coalition meeting those two players wins.