1. Chapter 11: Weighted Voting Systems
MAT 105 Fall 2008
Chapter 11: Weighted Voting Systems

2. Unequal Voters
In many voting systems, the voters are not treated equally
Juries: If one voter votes “not guilty,” then the result is “not guilty”
Stockholders: If you have more shares of stock, then your vote is weighted more heavily
US Electoral College: Larger states get more votes
European Union: Larger member countries get more votes

3. Weighted Voting Systems
Each participant has a specified number of votes, called his or her weight
For simplicity, we will assume all elections are “yes” or “no”
There is a criterion for determining whether “yes” or “no” wins: the quota

4. Examples of Quotas
To pass a bill in the House of Representatives, you just need a majority of the votes to be “yes”
There are 435 total votes, so the quota is 218
To amend the US Constitution, 3/4 of the states must ratify the amendment
There are 50 states, so the quota is 38

5. The Electoral College
In US Presidential elections, plurality elections are held in each state
The candidate winning each state sends electors to vote for him or her in the Electoral College
The number of electors per state depends on the size of the state: bigger states have more electors

6. 2008 Electoral Map

7. Notation
When we talk about weighted voting systems, we use a compact notation to list the quota together with the various weights
[q; a, b, c, …]
q is the quota
a, b, c, etc. are the weights
So the electoral college system would be written [270; 55, 34, 31, 27, 21, 21, etc.]

8. Terminology
As we said before, to simplify things, our elections will always be deciding “yes” versus “no”
This isn’t so unreasonable, since the most common place to find weighted voting systems is in legislatures or other government bodies
We say that the voters cast their votes “in favor” or “against” a motion by voting yes or no

9. Properties of Weighted Voting Systems: Dictators
A dictator is a voter with all the power: a motion will pass only if the dictator votes in favor, and it doesn’t matter how the other participants vote
[51; 60, 40]
In this system, the weight-60 voter is a dictator
[20; 15, 10, 5]
In this system, none of the voters is a dictator

10. Properties of Weighted Voting Systems: Dummy Voters
A dummy voter is a voter whose vote does not matter. When voters form a coalition to vote in favor or against a motion, the dummy voter can be removed from the coalition without changing the result
[51; 26, 26, 26, 22]
The voter with weight 22 is not needed to win when two others combine to support a motion, and she does not have enough weight to pass a motion with only one other

11. Properties of Weighted Voting Systems: Veto Power
A voter whose vote is necessary to pass any motion is said to have veto power
[21; 20, 15, 5]
The voter with weight 20 has veto power: if that voter votes “no,” then the motion cannot pass
Jury: [12; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
Since all 12 voters must vote in favor to pass the motion, each voter has veto power

12. More Examples
[8; 5, 3, 1]
None of the voters is a dictator
The weight-1 voter is a dummy voter
The weight-5 and weight-3 voters have veto power
[9; 5, 3, 1]
Motions can only pass unanimously: none of the voters is a dictator, there are no dummies, and all of the voters have veto power

13. More Examples
[6; 5, 3, 1]
None of the voters is a dictator
None of the voters is a dummy
The weight-5 voter has veto power
[51; 49, 48, 3]
None of the voters has veto power

14. Looking Closer Consider the system [51; 49, 48, 3]
Even though the third participant only has a weight of 3, it has the same “power” as the other two
Any two of the three participants can combine to pass a motion, and none of the three can pass a motion alone

15. Looking Closer
In the system [51; 26, 26, 26, 22], the fourth participant has almost as much weight as the other three, but is a dummy voter
Dummies have no power to influence elections one way or the other

16. Measuring Power
We have seen that the power a voter wields is not necessarily directly related to the weight of the voter
We will want to measure the power of each voter, keeping in mind that:
dictators have all the power; when one voter is a dictator, all other voters are dummies
dummies have zero power