Prove that the symmetry relation between two players is

Question

Players i, j are symmetric players if for every coalition S that does not include any one of them,

v(S ∪ {i}) = v(S ∪ {j}).

(a) Prove that the symmetry relation between two players is transitive: if i and j are symmetric players, and j and k are symmetric players, then i and k are symmetric players.

(b) Show that if the core is nonempty, then there exists an imputation x in the core that grants every pair of symmetric players the same payoff, i.e., x_i = x_j for every pair of symmetric players i, j .