Voting (2) A voting committee consists of five members: members 1 and 2 belong to party I, while members 3, 4, and 5 belong to party II. In order to pass a decision at least a weak majority of each...

Voting (2)

A voting committee consists of five members: members 1 and 2 belong to party I, while members 3, 4, and 5 belong to party II. In order to pass a decision at least a weak majority of each party (that is, at least 50 % of the votes of each party) is required. A coalition that has a weak majority of both parties is called winning. We model this situation as a so-called simple game: winning coalitions obtain worth 1, all other coalitions worth 0.

(a) Show that there are six winning coalitions of minimal size: list all of them.

(b) Use your answer to (a) to give a concise description of the game (i.e., without listing all 32 coalitions).

(c) Compute the Shapley value of this game. According to the Shapley value, which players (members) are most powerful?

(d) Compute the nucleolus of this game. According to the nucleolus, which players (members) are most powerful?

(e) Compute the core of this game.