Game Theory Question variant of 3player majority game Find t

Game Theory Question: (variant of 3-player majority game) Find the core of the variant of the three player majority game in which player 1 has 3 votes (and players 1 and 3 have one vote, as in the original game).

How would you set this question up and solve it?

Solution

Let the core allocation for player 1 be A1

and that for player 2 and 3 be A2 and A3 respectively

Player 1 has three votes so he can win any vote and allocate the whole Game to himself.

Players 2 and 3 together do not have enough votes to win and they can only get zero together.

Therefore any location in the core must satisfy the following restrictions :

A1 >= 1

A2 >= 0

A3 >= 0

A1 + A2 >= 1

A1 + A3 >= 1

A2 + A3 >= 0

There\'s just one condition which satisfy the above inequalities and that is :

A1 = 1 , A2 = 0 and A3 = 0