Interval value of a cooperative game with both a communication graph and a level structure: Considering possible values of infeasible coalitions

In a cooperative game with both a communication graph and a level structure (CLS game), some infeasible coalitions' values are unknown. In this paper, two values (i.e., the interval graph-level value and the interval level-graph value) are introduced by considering the lower and upper bounds of these unknown values. We further introduce degree weights to capture players' bridging (intergroup degree) and coordination (intragroup degree) roles. The interval graph-level value first applies a communication graph restriction and then a level structure, whereas the interval level-graph value reverses this order. As two solutions for CLS games, the interval weighted graph-level value and the interval weighted level-graph value provide alternative allocation ranges, reflecting the uncertainty arising from unknown values of infeasible coalitions under dual restrictions.