1. Andrea Katz April 29, 2004 Advisor: Dr. Karrolyne Fogel Voting Blocs in Academic Divisions of CLU: A Mathematical Explanation of Faculty Power

2. Definitions Coalition: Any collection of voters in a yes-no voting system Voting Bloc: An organized group of voters (unit) all casting the same vote in a yes-no system. Coalitions exist within voting blocs. Pivotal Player: A voter in the system that, by joining a coalition, turns it from a losing coalition to a winning coalition.

3. The Shapley-Shubik Index of Power Defined as: Yields a player’s probability of being pivotal Being pivotal tells us the chance of a voter has to make a difference of swaying the outcome of the vote

4. Example Suppose Dr. Wyels casts 50 votes, Dr. Fogel casts 49, and Andrea casts 1 vote. The six possible orderings for the system are: Joining 1 st 2 nd 3 rd W F A W A F F W A F A W A W F A F W Dr. Wyels is pivotal 4/6 of the time = 67% Dr. Fogel is pivotal 1/6 of the time = 17% Andrea is pivotal 1/6 of the time = 17% We need 51 votes to pass

5. Humanities (H) 23 Social Sciences (S) 19 Creative Arts (C) 16 Natural Sciences (N) 20 For a total of 78 voters Need 40 votes to pass 30% 24% 20% 26% If no division forms a bloc A More Familiar Example: The College of Arts and Sciences Number of voting facultyPercentage of Power

6. What if One Division Forms a Bloc? Index of Power for the bloc is given by: The other groups in the system have power g = the size of the group

7. What if Two Divisions Form a Bloc? For example, N and H N pivots H votes before N H votes after N H Pivots N votes before H N votes after H Other N H H N N axis 1-37 H 1-37 N = 1 H = 1 C = 16 S = 19

8. Some Results for HCNS N does not organize N organizes N H N H H does not organize H organizes (26, 30)(34, 26) (21, 41)(22, 30)

9. What if Three Divisions Form a Bloc? HCNS The Power Polynomial C N S H [ 16 : 20, 19, 1, …, 1] 0 1 need 5 – 20 1s 1 0 need 4 – 19 1s 1 1 need 0 1s 23 of these p = probability an event occurs. Let the event be that a voter votes yes ( p – 1) = probability event does not occur Consider C = 16

10. 3 Blocs Cont’d = 0.2760256410

11. Analysis of 3 Blocs on the Hypercube! H C N S H C N S (42,8,25,25) H C N S H C N S (30,x,22,x) H C N S H C N S (41,17,21,20) H C N S (10,28,32,30 ) H C N S H C N S (26,18,34,22) H C N S (x,x,28,21 ) H C N S (30,21,26,24) H C N S (26,18,23,32) H C N SH C N S (28,25,24,23)

12. A Slice of the Cube H C N S ( x, x,28,21) H C N S (26,18,34,22) H C N S (26,18,23,32) H C N S (30,21,26,24) When N organizes, S should not, for it loses 1%. When S organizes N should definitely organize! H C N S (41,17,21,20) H C N S (10,28,32,30) ?

13. Sources Straffin, Philip D. Game Theory and Strategy. Washington. The Mathematical Association of America. 1993 Straffin, Philip D. The Power of Voting Blocs: An Example. Mathematics Magazine 50.1 1977 Straffin, Philip D. Measuring Voting Power. Applications of Calculus. Vol. 3. The Mathematical Association of America 1997 Taylor, Alan D. Mathematics and Politics – Strategy, Voting, Power and Proof. New York. Springer- Verlag 1995

14. The Banzhaf Index of Power 1.List all possible winning coalitions {HCNS} {HCN} {HCS} {CNS} {HNS} {HN} {HS} for a total of 7 2. Count the number of occurrences such that when a player is removed from a winning coalition, the coalition is not a winning coalition any more. For H: 5 times C: 1 time N: 3 times S: 3 times 12

15. The Business, Education, CaS Example B = 14, E = 20, CaS = 76 Education does not org. Education organizes. (13, 18) (13, 19) (14, 16)(10, 36) Business does not org. Business org. B E E is better off organizing when B does, however, B should not organize when E does