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## Chapter 2. Weighted Voting Systems

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**1. **Chapter 2. Weighted Voting Systems Goals
Study weighted voting systems
Coalitions
Dummies and dictators
Veto power
Study the Banzhaf power index
and Shapley-Shubik

**2. **Weighted Voting Systems In a weighted voting system, an individual voter may have more than one vote.
The number of votes that a voter controls is called the weight of the voter.
An example of a weighted voting system is the election of the U.S. President by the Electoral College.
_________________________
The voter with the largest weight is called the first voter, written P1.
The weight of the first voter is represented by W1.
The remaining voters and their weights are represented similarly, in order of decreasing weights.

**3. **Weighted Voting Systems, contd The weights of the voters are usually listed as a sequence of numbers between square brackets.
For example, the voting system in which Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11 is represented as
[12, 11, 9, 8].

**4. **Example 1 The voting system in which Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11 was represented as
[12, 11, 9, 8].
In this case, P1 = Roberta, P2 = Darrell, P3 = Angie, and P4 = Carlos.
Also, W1 = 12, W2 = 11, W3 = 9, and W4 = 8.

**5. **Weighted Voting Systems, contd A simple majority requirement means that a motion must receive more than half of the votes to pass.
A supermajority requirement means that the minimum number of votes required to pass a motion is set higher than half of the total weight.
A common supermajority is two-thirds of the total weight.

**6. **Weighted Voting Systems, contd The weight required to pass a motion is called the quota.
Example: A simple majority quota for the weighted voting system [12, 11, 9, 8] would be 21. Half of the total weight is
(12 + 11 + 9 + 8)/2 = 40/2 = 20.
More than half of the weight would be at least 21 Yes votes.
Half of the total weight is
(12 + 11 + 9 + 8)/2 = 40/2 = 20.
More than half of the weight would be at least 21 Yes votes.

**7. **Question: Given the weighted voting system
[10, 9, 8, 8, 5],
find the quota for a supermajority requirement of two-thirds of the total weight.
a. 27
b. 21
c. 26
d. 20 10+9+8+8+5=40
2/3*40=26.66 answer a10+9+8+8+5=40
2/3*40=26.66 answer a

**8. **Weighted Voting Systems, contd The quota for a weighted voting system is usually added to the list of weights.
Example: For the weighted voting system [12, 11, 9, 8] with a quota of 21 the complete notation is
[21 : 12, 11, 9, 8].

**9. **Example 2 Given the weighted voting system
[21 : 10, 8, 7, 7, 4, 4], suppose P1, P3, and P5 vote Yes on a motion.
Is the motion passed or defeated?

**10. **Example 3 Given the weighted voting system
[21 : 10, 8, 7, 7, 4, 4],
suppose P1, P5, and P6 vote Yes on a motion. Is the motion passed or defeated?
suppose P1, P4, and P6 vote Yes on a motion. Is the motion passed or defeated?
Solution:
The given voters have a combined weight of 10 + 4 + 4 = 18.
The quota is not met, so the motion is defeated.
Solution:
The given voters have a combined weight of 10 + 7 + 4 = 21.
The quota is met, so the motion passes.
Solution:
The given voters have a combined weight of 10 + 4 + 4 = 18.
The quota is not met, so the motion is defeated.
Solution:
The given voters have a combined weight of 10 + 7 + 4 = 21.
The quota is met, so the motion passes.

**11. **Coalitions Any nonempty subset of the voters in a weighted voting system is called a coalition.
If the total weight of the voters in a coalition is greater than or equal to the quota, it is called a winning coalition.
If the total weight of the voters in a coalition is less than the quota, it is called a losing coalition.

**12. **Question: Given the weighted voting system
[27: 10, 9, 8, 8, 5]
is the coalition {P1, P4, P5} a winning coalition or a losing coalition?
a. winning
b. losing

**13. **Example 4 For the weighted voting system
[8: 6, 5, 4], list all possible coalitions and determine whether each is a winning or losing coalition.

**14. **Example 4, contd Solution: Each coalition and its status is listed in the table below.

**15. **Coalitions, contd How many coalitions are possible in a weighted voting system with n voters?
n=1,
n=2,
n=3,
n=4
n=5
Formula for n voters
In a weighted voting system with n voters, exactly 2n - 1 coalitions are possible.
In a weighted voting system with n voters, exactly 2n - 1 coalitions are possible.

**16. **Example 5 The voting weights of EU members in a council in 2003 are shown in the table.

**17. **Example 5, contd
If resolutions must receive 71% of the votes to pass, what is the quota?
How many coalitions are possible? Solution:
There are 87 votes total. So the quota is 71% of 87, or approximately 62 votes.
There are n = 15 members, so there are 215 1 = 32,767 coalitions possibleSolution:
There are 87 votes total. So the quota is 71% of 87, or approximately 62 votes.
There are n = 15 members, so there are 215 1 = 32,767 coalitions possible

**18. **Dictators and Dummies A voter whose presence or absence in any coalition makes no difference in the outcome is called a dummy.
A voter whose presence or absence in any coalition completely determines the outcome is called a dictator.
When a weighted voting system has a dictator, the other voters in the system are automatically dummies.

**19. **Weighted Voting Systems Dummy
A player with no power.
Consider [30: 10, 10, 10, 9]
P4 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.

**20. **Veto Power In between the complete power of a dictator and the zero power of a dummy is a level of power called veto power.
A voter with veto power can defeat a motion by voting No but cannot necessarily pass a motion by voting Yes.
Any dictator has veto power, but a voter with veto power is not necessarily a dictator.

**21. **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.

**22. **Example 6 Consider the weighted voting system
[12: 7, 6, 4].
List all the coalitions and determine whether each is a winning or losing coalition.
Are there any dummies or dictators?
Are there any voters with veto power?

**23. **Example 6, contd Solution:
Each coalition and its status is listed in the table below.

**24. **Example 6, contd Solution, contd:
Removing the third voter from any coalition does not change the status of the coalition. P3 is a dummy.

**25. **Example 6, contd Solution, contd:
No voter has complete power to pass or defeat a motion. There is no dictator.

**26. **Example 6, contd Solution, contd:
If P1 is not in a coalition, then it is a losing coalition. P1 has veto power.

**27. **Question: In the weighted voting system
[27: 10, 9, 8, 8, 5], is P1 a:
a. dictator
b. dummy
c. voter with veto power
d. none of the above

**28. **Example 7 Consider the weighted voting system [10: 10, 5, 4].
Are there any dummies, dictators, or voters with veto power? Solution:
P1 has enough weight to pass a motion by voting Yes no matter how anyone else votes.
If P1 votes No, the motion will not pass no matter how anyone else votes.
P1 is a dictator and thus all other voters are dummies.
Solution:
P1 has enough weight to pass a motion by voting Yes no matter how anyone else votes.
If P1 votes No, the motion will not pass no matter how anyone else votes.
P1 is a dictator and thus all other voters are dummies.

**29. **Critical Voters If a voters weight is large enough so that the voter can change a particular winning coalition to a losing coalition by leaving the coalition, then that voter is called a critical voter in that winning coalition.

**30. **Question: Given the weighted voting system
[27: 10, 9, 8, 8, 5],
is the voter P4 a critical voter in the winning coalition {P1, P2, P4, P5}?
a. yes
b. no

**31. **Example 8 Consider the weighted voting system [21 : 10, 8, 7, 7, 4, 4].
Which voters in the coalition {P2, P3, P4, P5 } are critical voters in that coalition? Solution: The weight in the winning coalition is 26.
If P2 leaves, the weight goes down to
26 8 = 18 < quota.
If P3 leaves, the weight goes down to
26 7 = 19 < quota.
Solution contd:
If P4 leaves, the weight goes down to
26 7 = 19 < quota.
If P5 leaves, the weight goes down to
26 4 = 22 > quota.
The critical voters in this coalition are P2, P3, and P4.
Solution: The weight in the winning coalition is 26.
If P2 leaves, the weight goes down to
26 8 = 18 < quota.
If P3 leaves, the weight goes down to
26 7 = 19 < quota.
Solution contd:
If P4 leaves, the weight goes down to
26 7 = 19 < quota.
If P5 leaves, the weight goes down to
26 4 = 22 > quota.
The critical voters in this coalition are P2, P3, and P4.

**32. **The Banzhaf Power Index The more times a voter is a critical voter in a coalition, the more power that voter has in the system.
The Banzhaf power of a voter is the number of winning coalitions in which that voter is critical.

**33. **Banzhaf Power Index, contd The sum of the Banzhaf powers of all voters is called the total Banzhaf power in the weighted voting system.
An individual voters Banzhaf power index is the ratio of the voters Banzhaf power to the total Banzhaf power in the system.
The sum of the Banzhaf power indices of all voters is 100%.

**34. **Banzhaf Power Index, contd An individual voters Banzhaf power index is calculated using the following process:
Find all winning coalitions for the system.
Determine the critical voters for each winning coalition.
Calculate each voters Banzhaf power.
Find the total Banzhaf power in the system.
Divide each voters Banzhaf power by the total Banzhaf power.

**35. **Example 9 For the weighted voting system [18 : 12, 7, 6, 5], determine:
The total Banzhaf power in the system.
The Banzhaf power index of each voter.

**36. **Example 9, contd Solution Step 1: Find all the winning coalitions.

**37. **Example 9, contd Solution Step 2: Determine the critical voters for each winning coalition.
Remove each voter one at a time and check to see whether the resulting coalition is still a winning coalition.
This work is shown in the next slides.

**38. **Example 9, contd

**39. **Example 9, contd

**40. **Example 9, contd Solution Step 3: Count the number of times each voter is a critical voter:
P1: 5 times
P2: 3 times
P3: 3 times
P4: 1 time
Step 4: The total Banzhaf power in the system is 5 + 3 + 3 + 1 = 12

**41. **Example 9, contd Solution Step 5: Divide each voters Banzhaf power by the total Banzhaf power to find the Banzhaf power indices.

**42. **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

**43. **Weighted Voting Systems Three-Player Sequential Coalitions

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

**48. **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.

**49. **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

**50. **Exercise 2.28 Find the Shapely-Shubik index of following weighted voting systems
[6:4,3,2,1]
[7:4,3,2,1]
[8:4,3,2,1]
[9:4,3,2,1]
[10:4,3,2,1]

**51. **Four player coalitions <P1,P2,P3,P4> ; <P1,P2,P4,P3>;
<P1,P3,P2,P4>; <P1,P3,P4,P2>;
<P1,P4,P2,P3> ; <P1, P4,P3,P2>;
<P2,P1,P3,P4> ; <P2,P1,P4,P3>;
< P2,P3, P1,P4> ; < P2,P3,P4,P1>;
< P2,P4,P3, P1> ; < P2,P4, P1,P3 >;
<P3,P2,P1,P4> ; <P3,P2,P4,P1>;
<P3, P1,P2,P4> ; <P3, P1,P4,P2 >;
<P3,P4,P2,P3> ; <P3, P4,P3,P2 >;
<P4,P2,P1,P3> ; <P4,P2,P3,P1>;
<P4, P1,P2,P3> ; <P4, P1,P3,P2>;
<P4,P3,P2,P1> ; <P4, P3,P1,P2>;

**53. **Example: Let us consider the coalition {P1 ,P2,P3}. How many sequential coalitions contain these players? We have the following sequential coalitions:
<P1 , P2, P3 ? ? P1 , P3, P2 ?
? P2 , P1, P3 ? ? P2 , P1, P3 ? ? P3 , P1, P2 ? ? P3 , P2, P1 ?
We can see that there are a total of 6.(In the first sequential coalition what we are saying is that P1 started the coalition, then P2 joined who in turn was followed byP3.)

**54. **Example: [10: 6, 5, 4] We have already seen that the 6 possible sequential coalitions and 6 Pivotal Players
? P1 , P2, P3 ? P2 ? P1 , P3, P2 ? P3 ? P2 , P1, P3 ? P1 ? P2 , P1, P3 ? P1 ? P3 , P1, P2 ? P1 ? P3 , P2, P1 ? P1 P1 is pivotal four times.
P2 is pivotal one time.
P3 is pivotal one time.
Since there are a total of 6 sequential coalitions, under the Shapley-Shubik Power Index we have: P1 : 4/6 P2 : 1/6 P3 : 1/6
P1 is pivotal four times.
P2 is pivotal one time.
P3 is pivotal one time.
Since there are a total of 6 sequential coalitions, under the Shapley-Shubik Power Index we have: P1 : 4/6 P2 : 1/6 P3 : 1/6

**57. **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

**58. **Homework 2, 3, 9, 11,13,16, 17,23,24,33, 34, 42, 45, 49