1. Chapter 11. 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 P 1.  The weight of the first voter is represented by W 1.  The remaining voters and their weights are represented similarly, in order of decreasing weights.

3. Weighted Voting Systems, cont’d  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, P 1 = Roberta, P 2 = Darrell, P 3 = Angie, and P 4 = Carlos.  Also, W 1 = 12, W 2 = 11, W 3 = 9, and W 4 = 8.

5. Weighted Voting Systems, cont’d  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, cont’d  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.

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

8. Weighted Voting Systems, cont’d  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 P 1, P 3, and P 5 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 P 1, P 5, and P 6 vote ‘Yes’ on a motion. Is the motion passed or defeated?  suppose P 1, P 4, and P 6 vote ‘Yes’ on a motion. Is the motion passed or defeated?

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 {P 1, P 4, P 5 } 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, cont’d  Solution: Each coalition and its status is listed in the table below.

15. Coalitions, cont’d  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

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

17. Example 5, cont’d a) If resolutions must receive 71% of the votes to pass, what is the quota? b) How many coalitions are 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] P 4 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]. a) List all the coalitions and determine whether each is a winning or losing coalition. b) Are there any dummies or dictators? c) Are there any voters with veto power?

23. Example 6, cont’d  Solution: a) Each coalition and its status is listed in the table below.

24. Example 6, cont’d  Solution, cont’d: b) Removing the third voter from any coalition does not change the status of the coalition. P 3 is a dummy.

25. Example 6, cont’d  Solution, cont’d: b) No voter has complete power to pass or defeat a motion. There is no dictator.

26. Example 6, cont’d  Solution, cont’d: c) If P 1 is not in a coalition, then it is a losing coalition. P 1 has veto power.

27. Question: In the weighted voting system [27: 10, 9, 8, 8, 5], is P 1 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?

29. Critical Voters  If a voter’s 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 P 4 a critical voter in the winning coalition {P 1, P 2, P 4, P 5 }? a. yes b. no

31. Example 8  Consider the weighted voting system [21 : 10, 8, 7, 7, 4, 4].  Which voters in the coalition {P 2, P 3, P 4, P 5 } are critical voters in that coalition?

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, cont’d  The sum of the Banzhaf powers of all voters is called the total Banzhaf power in the weighted voting system.  An individual voter’s Banzhaf power index is the ratio of the voter’s 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, cont’d  An individual voter’s Banzhaf power index is calculated using the following process: 1) Find all winning coalitions for the system. 2) Determine the critical voters for each winning coalition. 3) Calculate each voter’s Banzhaf power. 4) Find the total Banzhaf power in the system. 5) Divide each voter’s 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, cont’d  Solution Step 1: Find all the winning coalitions.

37. Example 9, cont’d  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, cont’d

40.  Solution Step 3: Count the number of times each voter is a critical voter: P 1 : 5 times P 2 : 3 times P 3 : 3 times P 4 : 1 time  Step 4: The total Banzhaf power in the system is 5 + 3 + 3 + 1 = 12

41. Example 9, cont’d  Solution Step 5: Divide each voter’s 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.

45. The Shapley-Shubik Power Index  In each (winning) sequential coalition there is a pivotal player--a player whose joining causes the coalition to change from a losing coalition to a winning coalition.  We will use the concept of the pivotal player to define the Shapley-Shubik Power Index. The Shapley-Shubik  Power Index concerns itself with sequential coalitions--coalitions in which the order that players join matters

46. The Shapley-Shubik Power Index  The Shapley-Shubik Power Index concerns itself with sequential coalitions-- coalitions in which the order that players join matters.  In general, the number of sequential coalitions with N players is:  N ! = (N)(N - 1)...(2)(1)

47. The Shapley-Shubik Power Index Finding the Shapley-Shubik Power Index of Player P : Finding the Shapley-Shubik Power Index of Player P : Step 1. Make a list of all sequential coalitions containing all N players. Step 2. In each sequential coalition determine the pivotal player. Step 3. Count the number of times P is pivotal--call this number S. The Shapley-Shubik Power Index for the player P is the fraction S/(N !).

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  ; ; ; ;  ; ; ; ;  ; ; ; ;  ; ; ; ;

52. The Shapley-Shubik Power Index  When discussing power of a coalition in terms of the Banzhaf Index we did not care about the order in which player’s cast their votes.  In other words, in Banzhaf index {P 1,P 2 } and {P 2,P 1 } to be the same coalition.

53. Example: Example: Let us consider the coalition {P 1,P 2,P 3 }. 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

55. The Shapley-Shubik Power Index  The list of all of the Shapley-Shubik Power Indices for a given election is the Shapley-Shubik power distribution of the weighted voting system.

56. Example: Example: The European Union (revisited). There are a total of 15! = 1,307,674,368,000 possible sequential coalitions (and 2 15 - 1 = 32,767 ‘normal’ coalitions) to consider. CountryVotesBanzhaf PowerShapley-Shubik Power France, Germany, Italy, UK 101849/16,565 ≈ 11.16%11.67% Spain81531/16,565 ≈ 9.24%9.55% Belgium, Greece, Netherlands, Portugal 5973/16,565 ≈ 5.87%5.52% Austria, Sweden4793/16,565 ≈ 4.79%4.54% Denmark, Finland, Ireland 3595/16,565 ≈ 3.59%3.53% Luxembourg2375/16,565 ≈ 2.26%2.07%

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