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Voting Blocs in Academic Divisions of CLU: A Mathematical Explanation of Faculty Power

Voting Blocs in Academic Divisions of CLU: A Mathematical Explanation of Faculty Power. Andrea Katz April 29, 2004 Advisor: Dr. Karrolyne Fogel. Definitions. Coalition : Any collection of voters in a yes-no voting system

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Voting Blocs in Academic Divisions of CLU: A Mathematical Explanation of Faculty Power

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

  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 We need 51 votes to pass Suppose Dr. Wyels casts 50 votes, Dr. Fogel casts 49, and Andrea casts 1 vote. The six possible orderings for the system are: Joining 1st 2nd 3rd W F A W A F F W A F A W AW 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%

  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 faculty Percentage of Power

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

  7. N pivots What if Two Divisions Form a Bloc?For example, N and H H votes before N H votes after N N = 1 H = 1 C = 16 S = 19 N Other H H1-37 N votes before H N votes after H H H Pivots Other N N axis 1-37

  8. Some Results for HCNS N does not organize N organizes N H N H H does not organize H organizes

  9. 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 What if Three Divisions Form a Bloc?HCNSThe Power Polynomial Consider C = 16 p = probability an event occurs. Let the event be that a voter votes yes (p – 1) = probability event does not occur 23 of these

  10. 3 Blocs Cont’d = 0.2760256410

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

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

  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. 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 coalitionany more. For H: 5 times C: 1 timeN: 3 times S: 3 times 12 The Banzhaf Index of Power

  15. The Business, Education, CaS Example B = 14, E = 20, CaS = 76 Education does not org. Education organizes. B E B E Business does not org.Business org. E is better off organizing when B does, however, B should not organize when E does

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