1. Weighted Voting, Algorithms and Voting Power

2. Student Council
A small high school has 110 students. The school's student council is composed of a single representative from each class, sophomore, junior and senior.
The class populations are 50 sophomores, 30 juniors and 30 seniors.
Each of the three members is given a number of votes proportionate to the size of the class represented.

3. Student Council (cont'd)
Therefore, the sophomore class has five votes, and the junior and senior classes each have three.
In order for any issue to pass the council, a simple majority of six votes is required.
This procedure is an example of weighted voting.
WEIGHTED VOTING occurs whenever some members of the voting body have more votes than others do.

4. Ways of Passing an Issue
Issues facing this board can be passed by having either the sophomore and junior (or senior) votes, or by having the junior and senior votes, together.
Listed are all the possible voting combinations:
{;0} {So; 5} {Jr; 3} {Sr; 3} {So, Jr; 8} {So, Sr; 8} {Jr, Sr; 6} {So, Jr, Sr; 11}
Each of these collections is known as a coalition.
Those that are successful are known as winning coalitions.

5. Coalitions
A coalition is any group of voters and a winning coalition is any group of voters that has enough votes to pass an issue.
Of the winning coalitions, {So, Jr, Sr; 11} is different in one very important way: If any one of the members decides to vote differently, the coalition sill still win. None of the members is essential to the coalition.

6. Minimal Winning Coalitions
When all of the members are essential to a winning coalition, it is called a minimal winning coalition.
The minimal winning coalitions in this example are: {So, Jr; 8} {So, Sr; 8} {Jr, Sr; 6}
A winning coalition where, it will become a losing coalition if any of the members are removed is called a minimal winning coalition.

7. Interesting Results
Notice that the sophomore representative is essential to two of the winning coalitions.
This is also true of the Senior and Junior representative.
In other words, in about the same number of times, each of the representatives can expect to cast a key vote in passing an issue.

8. A Paradox
Although the votes have been distributed to give greater power to the sophomores, the actual outcome is that all the members have the same amount of power!

9. Distribution of Power
Since distributing the number of votes in this way population does not reflect the fair distribution of power, mathematical procedures can be used to find a way to measure actual power when weighted voting is used.

10. Power Index
A measure of the power of a member of a voting body is called a power index.
The power index is the number of winning coalitions to which each member of that body is essential.
In our example, each member is essential to 2 winning coalitions and thereby has a power index of 2.

11. Algorithms An algorithm is a procedure for solving a problem.
They are usually presented in numbered steps to make them easy to apply.
Here is an example of a power index algorithm:

12. Power Index Algorithm
List all coalitions of voters that are winning coalitions.
Select any voter and record a 0 for that voters power index.
From the list in Step 1 select any coalition of which the voter is a member. Subtract from the coalition's vote total the number of votes that this voter has. If the result is less than the number of votes necessary to pass an issue, add 1 to the voter's power index.

13. Power Index Algorithm(cont'd)
Repeat step until you have checked all the coalitions for which the voter is a member.
Repeat Steps 2 through 4 until you have checked all the voters.

14. Consider this!
We have a group of three members in which A has seven 7 votes, B has 3 and C has 3.
List all the coalitions and then divide out the winning coalitions:

15. Winning Coalitions All coalitions:
{ ; 0} {A; 7} {B; 3} {C; 3} {A, B; 10}
{A, C; 10} {B, C; 6} {A, B, C; 13}
Winning coalitions:
{A; 7} {A, B; 10} {A, C; 10} {A, B, C; 13}

16. Power Holder In this situation, A is the only one with any power.
All four of the winning coalitions will become losing coalitions if A is removed.
B and C have no power because every winning coalition will remain winning is removed.

17. Power Holder (cont’d)
If one member has over half of the votes, that member alone is essential to the winning coalitions and so is known as a dictator.
When a member is essential to no winning coalitions, that member is known as a dummy.
In the last example, A is the dictator and B and C are dummies.

18. Practice Problems
Consider a situation in which A, B and C have 3, 2 and 1 votes, respectively in which 4 votes are needed to pass an issue.
List all possible coalitions, all winning coalitions and all minimal winning coalitions.
Determine the power index for each voter.
Suppose that the number of votes needed to pass an issue is increased from 4 to 5. Determine the power index of each.

19. More Practice
In the student council example in this section, can the votes be distributed so that the members’ power indices will follow the ratio of the class sizes?
In the student council example, suppose that the representatives of the junior and senior classes always differ on issues and never vote alike. Does this make any practical difference in the power of the three representatives?

20. More Practice
4. One way to determine all winning coalitions and all minimal winning coalitions in a weighted voting situation is to work from a list of all possible coalitions. Use A, B, C and D to represent all possible coalitions.

21. More Practice
Consider a situation in which A has 4 votes, B has 5, C has 7 and D has 9. If 13 votes are needed to pass an issue, determine:
All possible coalitions, all winning coalitions and all minimal winning coalitions.
Determine the power index of each voter
Suppose the number of votes need to pass an issue increased from 12 to 15. How would this affect the power index of each voter?