A committee consists of 7 people (6 regular members and a chairman) and uses simple majority rule to make decisions. Each of the 6 regular members have one vote each, and the chair itself has two votes plus one extra vote if there is a tie.

a) Define the corresponding coalitional game. That is, specify the set of players and the worth of every coalition. Is it a simple game?

b) Define the core of this game. Is the core of this game empty? If yes, prove it. If no, describe the core exactly.

c) Calculate the Shapely value of this game. Is the Shapely value in the core?

d) Is this game a weighted majority game? If yes, find two possible representations. If no, prove it.