A solution set for fine games

Bumb and Hoede have shown that a cooperative game can be split into two games, {\it the reward game} and {\it the fine game}, by considering the sign of quantities $c_S^v$ in the c-diagram of the game. One can then define a solution $x$ for the original game as $x=x_{r}-x_{f}$, where $x_{r}$ is a solution for the reward game and $x_{f}$ is a solution for the fine game. Due to the distinction of cooperation rewards and fines, for allocating the fines one may use another solution concept than for the rewards.