Let us define $\gamma([n],v)$ and $(\gamma'([n],v')$ as the two cooperative games in coalition form. Both games have the same set of players. Let this hold for every non-empty coalition $S$: $v(S) > v'(S)$.
Does this lead to the coclusion that $\phi_i(v) > \phi_i(v') $, whereby we define $\phi$ as Shapley value.
My attempt: $$\phi_i(v) > \phi_i(v') $$ according to the fourth aksiom (additivity): $$\phi_i(v - v') > 0 $$ So I am guessing that $v - v'$ shall be greater than 0.
This result is due to H. P. Young (1985): Monotonic solutions of cooperative games, International Journal of Game Theory, 14, 65-72. The associated Theorem states that if a solution satisfies strong monotonicity, the equal treatment property and efficiency, then it is the Shapley value.