Members of legislature decide about proposed bills, which are either passed or rejected. Consider the following scenarios:

i) Each member of the legislature has to vote for a bill to pass it, otherwise the bill is rejected.

ii) Three key members of the legislature have to vote for a bill to pass it, otherwise the bill is rejected.

iii) The chairman of the legislature can pass any bill by himself, but other (groups of) legislators cannot pass a bill.

a) Depict these situations as coalitional games by assigning the value of 1 to each coalition that can pass a bill, and by assigning the value of zero to coalitions otherwise. Find the core of each game.

b) Let each coalition with value 1 be called a winning coalition. A player who belongs to each winning coalition is called a veto player. Find the veto players for each of the above three scenarios.