Extension of owen value for the game with a coalition structure under the limited feasible coalition

In a cooperative game with a coalition structure, it is assumed that all the profits of feasible coalitions can be obtained. Before the cooperation relationship was formed, the profit for a single priori union (i.e., firm) could be obtained based on the former cooperation history. However, it is difficult to get the feasible coalition worth among multiple firms. Hence, a cooperative game with a coalition structure under the limited feasible coalition (IFCS game) is defined. For the solution of IFCS game, two different methods are used. Firstly, a Limited Owen value is defined by known values, and a generalized extension of Limited Owen value (i.e., q-probabilistic value) is given by distribution matrices of players. Some axioms are used to characterize Limited Owen and q-probabilistic value, such as I-Efficiency, I-Symmetric within coalitions, I-Symmetric across coalitions, I-Null player property, I-Linearity, I-Null coalition property and I-Proportionality. In the second solution, unknown coalition values are also estimated to make Owen value usable. By defining the lower and upper limit values of IFCS game, IFCS game is transformed into the coalition structures cooperative games. An interval Owen value is represented by the lower and upper limit values of IFCS game, which is a range of all the possible Owen values. Finally, the relationship between two methods is discussed