1. Excursions in Modern Mathematics Sixth Edition
Peter Tannenbaum

2. Chapter 2 Weighted Voting Systems
The Power Game

3. Weighted Voting Systems Outline/learning Objectives
Represent a weighted voting system using a mathematical model.
Use the Banzhaf and Shapley-Shubik indices to calculate the distribution of power in a weighted voting system.

4. Weighted Voting Systems

5. Weighted Voting Systems
The Players
The voters in a weighted voting system.
The Weights
That each player controls a certain number of votes.
The Quota
The minimum number of votes needed to pass a motion (yes-no votes)

6. Weighted Voting Systems
Dictator
The player’s weight is bigger than or equal to the quota.
Consider [11:12, 5, 4]
owns enough votes to carry a motion single handedly.

7. Weighted Voting Systems
Dummy
A player with no power.
Consider [30: 10, 10, 10, 9]
turns out to be a dummy! There is never going to be a time when is going to make a difference in the outcome of the voting.

8. Weighted Voting Systems
Veto Power
If a motion cannot pass unless player votes in favor of the motion.
Consider [12: 9, 5, 4, 2]
has the power to obstruct by preventing any motion from passing.

9. Weighted Voting Systems
2.2 The Banzhaf Power Index

10. Weighted Voting Systems
Coalitions
Any set of players that might join forces and vote the same way. The coalition consisting of all the players is called a grand coalition.

11. Weighted Voting Systems
Winning Coalitions
Some coalitions have enough votes to win and some don’t. We call the former winning coalitions and the latter losing coalitions.

12. Weighted Voting Systems
Critical players
In a winning coalition, a player is said to be a critical player for the coalition if the coalition must have that player’s votes to win. If and only if

13. Weighted Voting Systems
Computing a Banzhaf Power Distribution
Step 1. Make a list of all possible winning coalitions.

14. Weighted Voting Systems
Computing a Banzhaf Power Distribution
Step 2. Within each winning coalition determine which are the critical players. (To determine if a given player is critical or not in a given winning coalition, we subtract the player’s weight from the total number of votes in the coalition- if the difference drops below the quota q, then that player is critical. Otherwise, that player is not critical.

15. Weighted Voting Systems
Computing a Banzhaf Power Distribution
Step 3.Count the number of times that
is critical. Call this number Repeat for each of the other players to find

16. Weighted Voting Systems
Computing a Banzhaf Power Distribution
Step 4. Find the total number of times all players are critical. This total is given by

17. Weighted Voting Systems
Computing a Banzhaf Power Distribution
Step 5. Find the ratio 
This gives the Banzhaf power index of
Repeat for each of the other players to find
2, 3, …, N . The complete list of ’s gives the Banzhaf power distribution of the weighted voting system.

18. Weighted Voting Systems
2.3 Applications of Banzhaf Power

19. Weighted Voting Systems
Applications of Banzhaf Power
The Nassau County Board of Supervisors
John Banzhaf first introduced the concept
The United Nations Security Council
Classic example of a weighted voting system
The European Union (EU)
Relative Weight vs Banzhaf Power Index

20. Weighted Voting Systems
2.4 The Shapley-Shubik Power Index

21. Weighted Voting Systems
Three-Player Sequential Coalitions

22. Weighted Voting Systems
Shapley-Shubik- Pivotal Player
The player that contributes the votes that turn what was a losing coalition into a winning coalition.

23. Weighted Voting Systems
Computing a Shapley-Shubik Power Distribution
Step 1. Make a list of all possible sequential coalitions of the N players. Let T be the number of such coalitions.

24. Weighted Voting Systems
Computing a Shapley-Shubik Power Distribution
Step 2. In each sequential coalition determine the pivotal player.

25. Weighted Voting Systems
Computing a Shapley-Shubik Power Distribution
Step 3.Count the number of times that
is pivotal. Call this number
Repeat for each of the other players to find

26. Weighted Voting Systems
Computing a Shapley-Shubik Power Distribution
Step 4. Find the ratio 1 This gives the Shapley Shubik power index of
Repeat for each of the other players to find
 2,  3, …,  N . The complete list of  ’s gives the Shapley-Shubik power distribution of the weighted voting system.

27. Weighted Voting Systems
The Multiplication Rule
If there are m different ways to do X, and n different ways to do Y, then X and Y together can be done in m x n different ways.

28. Weighted Voting Systems
The Number of Sequential Coalitions
The number of sequential coalitions with N players is N! = N x (N-1) x…x 3 x 2 x 1.

29. Weighted Voting Systems
Applications of Shapley-Shubik Power
The Electoral College
There are 51! Sequential coalitions
The United Nations Security Council
Enormous difference between permanent and nonpermanent members
The European Union (EU)
Relative Weight vs Shapley-Shubik Power Index

30. Weighted Voting Systems Conclusion
The notion of power as it applies to weighted voting systems
How mathematical methods allow us to measure the power of an individual or group by means of an index.
We looked at two different kinds of power indexes– Banzhaf and Shapley-Shubik