A graph game extends a simple game with a list of edges,

L = [(i, j), (j, k), …]

Banzhaf (1965) value

Shapey value using permutations

Shapey value using Harsanyi dividends

Convert a graph game to a simple game based on Myerson (1977). Namely, assume v(S) > 0 iff S is connected subgraph.

Breadth first test of convexity of a game,

For all S, T ∈ P(G): v(S ∪ T) + v(S ∩ T) ≥ v(S) + v(T)

Check if a solution x is in the core of G. Namely,

- ∑ x < v(N)
- ∑ₛ x < v(S) ∀ S

Maps v to every possible coalition P(G)

Harsanyi dividends for S ⊂ N

Myerson solution, it is equivalent to the Shapley value of the Myerson game.