Mathematical Modeling: Weighted Voting: Practice Problems

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://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 this game 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 this game 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 this 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 this game 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.

7. Give some examples where it seems natural to use weighted voting games.

8. Given the weighted voting game: [Q; 4, 2, 1], write down the minimal winning coalitions as Q varies from 1 to 7. Are all of these games "different?" Do all of them seem reasonable?

9. Why might a value for the quota in a weighted voting game be set higher than the floor function of the sum of weights divided by two plus 1?

10. Why might two players in a weighted voting game who cast the same weight in the "real world" have different "power?"

"Researchy questions"

1. Suppose one has a weighted voting game G. One wants to construct a new weighted voting game G* which has one more player p and where the minimal winning coalitions in G* are the exactly those of G together with p. Is it always possible to modify the weights of the players in G to create the new weighted voting game G*?

2. How many inequivalent weighted voting games with n players? How many inequivalent weighted "majority" games are there with n players? (Here one wants the quota Q to be at least the floor function of the sum of the weights divided by 2 plus 1.

Shapley:

3 players

1 2 3

1 3 2

2 1 3

2 3 1

3 1 2

3 2 1

3 players

1 2 3

1 3 2

2 1 3

2 3 1

3 1 2

3 2 1

4 players:

1 2 3 4

1 2 4 3

1 3 2 4

1 3 4 2

1 4 2 3

1 4 3 2

2 1 3 4

2 1 4 3

2 3 1 4

2 3 4 1

2 4 1 3

2 4 3 1

3 1 2 4

3 1 4 2

3 2 1 4

3 2 4 1

3 1 2 4

3 1 4 2

3 4 1 2

3 4 2 1

4 1 2 3

4 1 3 2

4 2 1 3

4 2 3 1

4 3 1 2

4 3 2 1
4 players:

1 2 3 4

1 2 4 3

1 3 2 4

1 3 4 2

1 4 2 3

1 4 3 2

2 1 3 4

2 1 4 3

2 3 1 4

2 3 4 1

2 4 1 3

2 4 3 1

3 1 2 4

3 1 4 2

3 2 1 4

3 2 4 1

3 1 2 4

3 1 4 2

3 4 1 2

3 4 2 1

4 1 2 3

4 1 3 2

4 2 1 3

4 2 3 1

4 3 1 2

4 3 2 1

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