Weighted Voting Systems

1. Weighted Voting Systems Brian Carrico

2. What is a weighted voting system? • A weighted voting system is a decision making procedure in which the participants have varying numbers of votes. • Examples: • Shareholder elections • Some legislative bodies • Electoral College

3. Key Terms and Notation • Weight • Quota • Shorthand notation: • [q: w1, w2, …, wn]

4. Coalition Building • Rarely will one voter have enough votes to meet the quota so coalitions are necessary to pass any measure • Types of coalitions • Winning Coalition • Losing Coalition • Blocking Coalition • Dummy voters

5. Coalition Illustration • On the right is a table of the weights of shareholders of a company. • A simple majority (16 votes) is needed for any measure. • Ide, Lambert, and Edwards are all Dummy Voters as any winning coalition including any subset of those three would be a winning coalition without them.

6. How do we Measure an individual’s power? • Critical Voter • Banzhaf Power Index • Developed by John F Banzhaf III • 1965- “Weighted Voting Doesn’t Work” • The number of winning or blocking coalitions in which a participant is the critical voter

7. Critical Voter Illustration • Consider a committee of three members • The voting system follows this pattern: • [3: 2, 1, 1] • For ease, we’ll refer to the members as A, B, and C

8. Extra Votes • A helpful concept in calculating Banzhaf Power Index • A winning coalition with w votes has w-q extra votes • Any voter with more votes than the extra votes in the coalition is a critical voter

9. Calculating Banzhaf Index • In Winning Coalitions; A is a critical voter three times, B and C are critical voters once • In Blocking Coaltions; A is a critical voter three times, B and C are critical voters once • Banzhaf Index of this system: (6,2,2)

10. Notice a Pattern there? • Each voter is a critical voter in the same number of winning coalitions as blocking coalition • When a voter defects from a winning coalition they become the critical voter in a corresponding blocking coalition • [A, B, C]=>[A] • [A, B]=>[A, C] • [A, C]=>[A, B]

11. How does this help? • Because these numbers are identical, we can calculate the Banzhaf Power Index by finding the number of winning coalitions in which a voter is the critical voter and double it • Can make computations easier in systems with many voters

12. [51: 40, 30, 20, 10]

13. Banzhaf Index • From the table above we can see that in winning coalitions, • A is a critical vote 5 times • B and C are critical votes 3 times each • D is a critical vote once • So, their Banzhaf Index is twice that, • A=10, B=6, C=6, and D=2 • Their voting power is • A=10/24 B=6/24 C=6/24 D=2/24

14. The Electoral College

15. Shapley-Shubik Power Index • For coalitions built one voter at a time • The voter whose vote turns a losing coalition into a winning coalition is the most important voter • Shapley-Shubik uses permutations to calculate how often a voter serves as the pivotal voter • This index takes into account commitment to an issue

16. How do we find the pivotal voter? • The first voter in a permutation of voters whose vote would make a the coalition a winning coalition is the pivotal voter • The Shapley-Shubik power index is the fraction of the permutations in which that voter is pivotal • Formula: • (number times the voter is pivotal) • (number of permutations of voters)

17. What does this overlook?

18. Example • Permutations Weights • Shapley-Shubik indexes: • A=4/6 B=1/6 C=1/6

19. For a larger corporation

20. Larger Corporation (cont) • This is the same corporation we looked at earlier distributed as [51: 40, 30, 20, 10] • The Shapley-Shubik Index for the four people in the corporation is: • A=10/24 B=6/24 C=6/24 D=2/24 • So here, the Banzhaf and Shapley Shubik indexes agree, but is this always true?

21. Comparing the Indexes • The Banzhaf index assumes all votes are cast with the same probability • Shapley-Shubik index allows for a wide spectrum of opinions on an issue • Shapley-Shubik index takes commitment to an issue into account

22. An illustration of the difference • Consider a corporation of 9001 shareholders • Such a large corporation can only be analyzed if nearly all of the voters have the same power • So, we will consider a corporation with 1 shareholder owning 1000 shares and 9000 shareholders each owning one share, and assume a simple majority

23. Under Shapley-Shubik • The big voter will be the critical voter in any permutation that positions at least 4001 of the small voters before him, but no more than 5000 • We can group the permutations into 9001 equal groups based on the location of the big shareholder

24. Shapley-Shubik (cont) • We can see that the big shareholder is the pivotal voter in all permutations in groups 4002 through 5001 • So, the big shareholder has a Shapley-Shubik index of 1000/9001 • The remaining 8001/9001 power goes equally to the 9000 small voters

25. Under Banzhaf • We can estimate the big shareholder’s Banzhaf Power Index can be estimated assuming a each small shareholder decides his vote by a coin toss • The big shareholder will be a critical voter unless his coalition is joined by fewer than 4001 small shareholders or at least 5001 small shareholders

26. Banzhaf (cont) • When the 9000 small shareholders toss their coins, the expected number of heads is ½ * 9000 = 4500 • The standard deviation is roughly 50 • By the 68-95-99.7 rule we can see that there is a • 68% chance of 4450-4550 heads • 95% chance of 4400-4600 heads • 99.7% chance of 4350-4650 heads • You can see that the big shareholder’s Banzhaf Index is nearly 100%

27. Which seems fairer? • The Shapley-Shubik Index gave the big shareholder roughly 11% of the power while the Banzhaf Index gave him nearly 100% of the power • The big shareholder has roughly 11% of the votes • Which index seems more realistic? • Why are the indexes so different when earlier they came out the same?

28. Homework