Transcript Slide 1

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
• 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.
The Shapley-Shubik Index of Power
• Defined as:
thenumber of occurancesthata player p is pivotal
the totalnumber of possible orderingsof the votingsystem
• 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
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
W
F
F
A
A
F
A
W
A
W
F
A
F
A
W
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%
A More Familiar Example:
The College of Arts and Sciences
Number of voting faculty
•
•
•
•
Humanities
(H)
Social Sciences (S)
Creative Arts (C)
Natural Sciences (N)
23
19
16
20
Percentage of Power
30%
24%
20%
26%
If no division forms a bloc
For a total of 78 voters
Need 40 votes to pass
What if One Division Forms a Bloc?
• Index of Power for the
bloc is given by:
• The other groups in the
system have power
b
b

n  b 1 m
g  (n  2b  2)  (m  2)!
m!
b  size of bloc
n  number of vot ers
in t hesyst em
g = the size of the group
as a whole
m  n-b  1
 voters bloc
What if Two Divisions Form a Bloc?
For example, N and H
H votes before N
N = 1 H = 1 C = 16 S = 19
N pivots
Other
H votes after N
N
H
N votes before H
H Pivots
H
1-37
H
N
Other
N votes after H
N axis 1-37
Some Results for HCNS
N does not organize
N
H
N organizes
N
H
H does not
organize
(26, 30)
(34, 26)
H organizes
(21, 41)
(22, 30)
What if Three Divisions Form a Bloc? HCNS
The Power Polynomial
Consider C = 16
23 of these
C
N S
H
[16 : 20, 19, 1, … , 1]
0
1
1
p = probability an event occurs.
Let the event be that a voter
votes yes
(p – 1) = probability event does
not occur
1 need 5 – 20 1s p 6 (1  p)19 or p 7 (1  p)18 or...
0 need 4 – 19 1s p 5 (1  p) 20 or ...
2
23
p
(
1

p
)
1 need 0 1s
3 Blocs Cont’d
19 23
 23 j 1
  i 1
24  j
24 i
2
23





p
(
1

p
)


p
(
1

p
)

p
(
1

p
)


j 
i 
j5 
i4


20
1
21
4
2
23
20
5
17
8
1771
p
(
1

p
)

p
(
1

p
)

17710
p
(
1

p
)

490314
p
(
1

p
)

0
 201894p18 (1  p) 7  2288102p14 (1  p )11  67298p19 (1  p ) 6
 490314p 8 (1  p)17  8855p 5 (1  p) 20  1634380p15 (1  p )10
 980628p16 (1  p) 9  67298p 6 (1  p)19  2704156p12 (1  p )13
 2288132p11 (1  p)14  2704156p13 (1  p)12  201894p 7 (1  p )18
 980628p 9 (1  p)16  1634380p10 (1  p)15 dp
= 0.2760256410
Analysis of 3 Blocs on the Hypercube!
HCNS
(42,8,25,25)
HCNS
HCNS
HCNS
HCNS
HCNS
(10,28,32,30)
HCNS
(41,17,21,20)
HCNS
HCNS
HCNS
(x,x,28,21)
HCNS
(30,x,22,x)
HCNS
(26,18,23,32)
HCNS
HCNS
(26,18,34,22)
HCNS
(30,21,26,24)
HCNS
(28,25,24,23)
HCNS
(10,28,32,30)
A Slice of the Cube
?
HCNS
( x, x,28,21)
HCNS
(26,18,23,32)
HCNS
(26,18,34,22)
When N organizes,
S should not, for it
loses 1%. When S
organizes N should
definitely organize!
HCNS
(30,21,26,24)
HCNS
(41,17,21,20)
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
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
5
 42%
12
1
 8%
12
3
 25%
12
3
 25%
12
The Business, Education, CaS Example
B = 14, E = 20, CaS = 76
Education does not org.
B
E
Education organizes.
B
E
Business
does not org.
(13, 18)
(13, 19)
Business org.
(14, 16)
(10, 36)
E is better off organizing when B does, however, B should not organize when E does