A solution defined by fine vectors

Bumb and Hoede have shown that a cooperative game can be split into two games, the reward game and the fine game, by considering the sign of quantities $c_v^S$ 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. In this paper, a fine vector is introduced and a solution is defined by fine vectors. The structure and properties of this solution are studied. And the solution is characterized as the unique solution having efficiency and f-potential property (resp. f-balanced contributions property).