Apportionment Game

The appointment game is studied, and the normal form is defined. It is proved that the game satisfy superadditivety, monotonic, and non-convexity. It is proved that the core of the game is non-empty when n≤3. It is proved the necessary and sufficient conditions of the core are non-empty. In addition, it is proved that the bargaining set and the stable set coincides with the imputation set. It is proved that the necessary and sufficient conditions of the imputation set belong to the nuclear set. It is proved that the nuclear set is empty when n≤3. It pointed out that Davis's conclusions on kernel set is no longer satisfy in this case, and Schmeidler's conclusions on the nucleolus is no longer establishment in this case, too. Finally, it is discussed that the Hamilton method is reasonable from the point of nuclear.