Let us consider a game with two actors named A and B. Let we have the following pay-offs depending on the coalitions:
- (empty coalition) = $0$, (A) = $0$, (B) = $200$, (AB) = $300$.
The calculation shows that the Shapley value is (50, 250) and that the core consists of vectors $(x,300-x)$ for $0<x<100$.
I claim that any vector in the core satisfies the four conditions of the Shapley value:
symmetry: trivial, because there is no equivalent actors,
linearity: trivial, because there is only one game (more precisely, no other game is presented),
zero player: trivial, because there is no zero player.
So, any vector in the core is Shapley value. But Shapley's theorem says that the Shapley values is unique. Where is my mistake? Have I got a misunderstanding of the linearity condition?