Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems...
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Transcript of Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems...
![Page 1: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/1.jpg)
Chapter 13: Weighted Voting
• Banzhaf Power Index
• Shapley-Shubik Power Index
• Equivalent Systems
• Examples
![Page 2: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/2.jpg)
Chapter 13 - Lecture Part 1
• Vocabulary
• Assumptions
• Notation
• Examples of Weighted Voting
• The idea of power within a voting system
![Page 3: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/3.jpg)
Weighted Voting Systems – An Example
• The United States Electoral College is an example of a weighted voting system.
• We elect the President of the United States, not by a direct “one person-one vote” method, but by the Electoral College. When you vote for President, you are voting to have your state’s electoral votes cast in favor of your choice for President.
• This system provides different weights to each of the states (and the District of Columbia) based on population.
• Population size is assessed every 10 years through the census and electoral votes are re-distributed among the states. This gives the states differing voting weights.
• For example, California, with the highest population, currently has 55 votes in the Electoral College, while Florida has 27. Some states (and the District of Columbia) have only 3 electoral votes. The number of electoral votes a state has is the same as the total representatives that state has in Congress (House and Senate).
![Page 4: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/4.jpg)
The Electoral College – A Weighted Voting System
• There are a total of 538 votes divided among the states and a simple majority of the electoral votes (270 votes) is required to win the Presidency.
• A state’s electoral votes are awarded to the candidate with a plurality of votes in that state.
• There are some exceptions: Maine and Nebraska divide their electoral votes proportionally (according to the plurality winner of each district).
• Colorado recently rejected a proposal (in 2004) to divide electoral votes proportionally.
![Page 5: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/5.jpg)
Weighted Voting Systems – Vocabulary & Notation
• In the language of voting theory, we say that the number of electoral votes given to a state is that state’s voting weight.
• The number of electoral votes needed to elect a President, 270 in the U.S. Electoral College, is called the quota of the weighted voting system.
• In general, with n voters, we use the notation
[ q : w1, w2, w3, w4, …, wn ]
to represent a weighted voting system with quota q and weights w1, w2, w3, w4, …, wn .
• In the example of the Electoral College, q = 270, n = 51 (50 states + D.C.) and wi is the weight (number of electoral votes) for state i.
![Page 6: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/6.jpg)
Weighted Voting Systems - Vocabulary
• Some vocabulary:
– A coalition is a subset of the set of all voters within a voting system. A coalition may consist of some, all or none of the voters.
– Weighted voting systems decide measures. – A coalition’s weight is the sum of the voting weights of its
members.– A winning coalition consists of a set of voters with enough votes
to pass a measure. A losing coalition does not have enough votes to pass a measure.
– A coalition is winning if it has a weight that is greater than or equal to the quota of the system.
– A blocking coalition has enough votes to prevent a measure from passing.
![Page 7: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/7.jpg)
Weighted Voting Systems - Assumptions
• Some assumptions:
– A weighted voting system must be able to pass a measure. Therefore, the quota can not be greater than the total weight of all voters.
– We can not have opposing coalitions both win. Therefore, the quota must be more than half the total weight of all voters.
Symbolically, we have
nn wwwwqwwww ......2
1321321
![Page 8: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/8.jpg)
Weighted Voting Systems - Assumptions
• This statement represents a fundamental assumption about weighted voting systems:
nn wwwwqwwww ......2
1321321
nwwwww ...321
• Then the above statement simply becomes: wqw 2
1
• Let w represent the total weight of all voters …
• This translates into the statement: The quota of any weighted voting system must less than or equal to the total weight of the system and more than half the weight of the system.
![Page 9: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/9.jpg)
Weighted Voting Systems - Assumptions
• To answer this question, suppose we have a voting system with weight w and quota q satisfying the condition stated above.
• By definition, any coalition will need q votes to pass a measure. Also, any coalition can block passage of a measure if it can prevent any other coalition from collecting q votes.
• We’ll define the blocking quota of a system to be w – q + 1.
wqw 2
1
imply about the number of votes needed to block a measure ?
• What does the assumption about the quota of a weighted system
![Page 10: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/10.jpg)
Weighted Voting Systems - Assumptions
• We have w = total weight of the system
q = quota
w – q + 1 = blocking quota
• Suppose coalition X has q votes. Then any opposing coalition could collect at most the remaining w – q votes.
Coalition X Coalition Y
q w - q
• Assuming q is more than half the total weight w, then w – q is less than half the total weight.
• Coalition Y could block coalition X if it had more than w – q votes. In that way, coalition X could no longer reach the quota q. That is, Y can block if and only if it has the blocking quota w – q + 1.
![Page 11: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/11.jpg)
Two Fundamental Questions
Given any weighted voting system
– Can the blocking quota equal the quota ?
– Can the blocking quota be greater than the quota ?
![Page 12: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/12.jpg)
Two Fundamental Questions
1. Can the blocking quota equal the quota ?
The answer to this question is yes.
Suppose w – q + 1 = q. Then w + 1 = 2q and thus
q = (w + 1)/2.
Note that q is an integer whenever w is odd.
Note that q = (w +1)/2 will satisfy w/2 < q < w.
The blocking quota equals the quota when w is odd and
q = ( w + 1)/2.
It is possible that the quota and blocking quota are equal in a weighted voting system.
![Page 13: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/13.jpg)
Two Fundamental Questions
2. Can the blocking quota be greater than the quota ?
The answer to this question is no.
Suppose w – q + 1 > q. Then w + 1 > 2q and thus
q < (w + 1)/2. As in any weighted voting system, it must still be true that
w/2 < q. Consequently we have, w/2 < q < (w+1)/2. This implies w < 2q < w+1 which can never be true for integer
values of w and q.
We have shown deductively that w – q + 1 > q is impossible in a weighted voting system using a proof by contradiction.
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Weighted Voting Systems – An Example
Consider the weighted voting system [ 16 : 9, 9, 7, 3, 1, 1 ]. This weighted system this could represent shareholders in a company. Each shareholder has a different proportion of the vote when measures are considered.
The total weight of the system is 30. Thus 16 votes is a majority.
Because of the assumption w/2 < q < w , we deduce the quota must be more than 15 and less than or equal to 30. Thus, in this case16 < q < 30.
If q = 16, a majority of votes is required to pass a measure. If q = 30 then passage of a measure requires unanimous support.
Note that any coalition of voters with 15 or more votes is a blocking coalition in this system. Can you form a coalition that can block passage of any measure but is unable to pass a measure ?
![Page 15: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/15.jpg)
Weighted Voting Systems – An Example
Consider the weighted voting system [ 16 : 9, 9, 7, 3, 1, 1 ].
Here are some basic questions -
• How many voters do we have ?
• What is the quota ?
• What is the total weight of all voters combined ?
• What is the blocking quota ?
5. What is the weight of the coalition of Dr.’s Mansfield, Ide, Lambert and Edwards ?
6. Does the coalition mentioned above have sufficient votes to pass a measure ? Do they have sufficient votes to block a measure ?
![Page 16: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/16.jpg)
Weighted Voting Systems - Vocabulary
• More vocabulary:
– A dictator is a voter who can pass a measure even when all others oppose the measure.
– A voter has veto power when that voter can block a measure even when all others support it.
– A dummy voter is a voter who is never needed to win or block a measure.
![Page 17: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/17.jpg)
An Example of a Dummy
• In the weighted voting system [ 51: 26, 26, 26, 22 ] it seems as though voting weights are at least near to being equally divided.
• Let’s name these voters A, B, C, and D, in that order.
That is, A, B, C each have 26 votes and D has 22 votes.
• We might consider this fair – perhaps D is a new partner in a corporation, or is a new board member, perhaps D holds less stock in some company, or represents a state with a slightly smaller population.
• In this particular voting system, D is a dummy.
• We will find that D is never critical to any blocking or winning coalitions. That is, any blocking or winning coalition will remain as such with or without the support of D.
![Page 18: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/18.jpg)
An Example of a Dummy
• In the weighted voting system [ 51: 26, 26, 26, 22 ] it seems as though voting weights are at least close to equally divided.
• We have named these voters A, B, C, and D, in that order.
• Note that the total weight of this system is 100 and that the quota is 51. Also, note that the blocking quota is 50.
• Consider any coalition that includes D, the voter with 22 votes. D alone cannot pass or block a measure. If D joins with exactly one other voter, that coalition will still have insufficient votes (48 together) to pass or block a measure.
• A third voter is still required to pass or block a measure … and with a third voter, D is not needed. D is a dummy because D is never required by any coalition to pass or block a measure.
![Page 19: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/19.jpg)
An Example of a Dictator
• Consider the voting system [ 51: 60, 40 ].
• Suppose A is the voter with a voting weight of 60 and that B has weight equal to 40.
• In this example, A is a dictator. A can pass or block a measure with or without the support of B. The voting weight of A alone exceeds both the quota and blocking quota.
• In a system with a dictator, all other voters are dummies.
![Page 20: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/20.jpg)
An Example of Veto Power
• Consider the system [ 3 : 2, 1, 1 ] with voters A, B, and C, respectively.
• In this example, voter A has veto power because A can block any measure.
• The quota is 3 and the blocking quota is w – q + 1 = 4 – 3 + 1 = 2.
• Voter A ( a coalition of one ) has enough votes to block any measure alone and therefore has veto power in this system.
• Note that A is not a dictator and can not pass a measure alone because A does not have sufficient weight to pass a measure that all others oppose.
![Page 21: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/21.jpg)
Measuring Power
• Consider again the voting system [ 51: 26, 26, 26, 22 ] for voters A, B, C and D, respectively.
• How can we conceive of the power of individual voters within this system?
• The total weight w of all voters is 26 + 26 + 26 + 22 = 100.
• It seems reasonable to say each voter has some measure of power corresponding to that voter’s share of the total votes.
• Perhaps one might suggest that individual voter power be measured by the ratio
w
wi where wi is the weight of voter i and w is the total sum of the weights of all of the voters.
![Page 22: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/22.jpg)
Measuring Power
• In the voting system [ 51: 26, 26, 26, 22 ] with voters A, B, C and D, respectively, we consider the voting power of individual voters …
• We will call the ratio the nominal power for voter i.
• Does the nominal power truly represent voter D’s power?
• In this case 22/100 = .22 meaning we would say D has a nominal power of 22%. ( That is, D has 22% of voting weight of this system.)
• Remember that, in another sense, D really has no power in this particular voting system because we have already discovered that D is a dummy voter.
w
wi
![Page 23: Chapter 13: Weighted Voting Banzhaf Power Index Shapley-Shubik Power Index Equivalent Systems Examples.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649e6d5503460f94b6c217/html5/thumbnails/23.jpg)
Measuring Power
• In this chapter, we study two other methods for measuring power in a weighted voting system:
– The Banzhaf Power Index
– The Shapley-Shubik Power Index
• The goal is to provide a more reasonable measure of power for voters within a weighted voting system.
• We will consider when these methods are appropriate and when they may not be appropriate.