We consider a scenario where a group of agents needs to elect a committee to lead them in accomplishing a project. They elect a committee to maximize social welfare, and the question is how to distribute the total value of the project to every agent. This scenario encodes a cooperative game setting where the reward of the chosen coalition must be distributed fairly. First, we establish the axiomatic foundation of solution concepts in this cooperative committee election game. We show that a natural extension of Shapley value to this game does not meet the classical axioms when the values of different coalitions are binary. We then propose a value distribution rule that satisfies all the desired properties. Furthermore, we prove that this rule is unique in meeting these properties and also satisfies an additional monotonicity property. When the values of the coalitions can take any general values, we decompose the game into a linear combination of simple games. This decomposition is unique, allowing us to extend our value distribution rule to solve this general class of games.