**Mathematics 483 ** (Game Theory)

Homework vii

prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, NY 11451

email: joeyc@cunyvm.cuny.edu

web page: http://www.york.cuny.edu/~malk

We will use the notation [Q; a, b, c, ...., z] to represent a weighted voting game with players 1, 2, 3, ...., where player 1 casts a votes, player 2, b votes, etc. In order for a "coalition" of players to act, the number of votes (weight) of the coalition must sum to Q or more.

A coalition whose weight is Q or more is called winning. A coalition C is called minimal winning if it is winning but no subset of C is also winning.

1. Given the voting game G = [5; 4, 3, 2] write down all the winning coalitions for G. Write down all the minimal winning coalitions for G. Which if any of the players in these games are "dummies?"

2. Given the voting game G = [6; 4, 3, 2] write down all the winning coalitions for G. Write down all the minimal winning coalitions for G. Which if any of the players in these games are "dummies?"

3. Given the voting game G = [7; 4, 3, 2] write down all the winning coalitions for G. Write down all the minimal winning coalitions for G. Which if any of the players in these games are "dummies?"

4. Given the voting game G = [8; 4, 3, 2] write down all the winning coalitions for G. Write down all the minimal winning coalitions for G. Which if any of the players in these games are "dummies?"

5. Given the voting games below, write down the minimal winning coalitions. Which if any of the players in these games are "dummies?"

a. [12; 6, 4, 3, 1]

b. [13; 7, 5, 4, 2]

c. [10; 7, 5, 4, 2]

d. [13; 9, 5, 4, 4]

e. [13; 9, 5, 4, 3]

6. Compute the Shapley, Coleman, and Banzhaf power for the players in the games in the previous exercises.