Transcript Part 2 - Banzhaf Index

Chapter 13 – Weighted Voting
Lecture Part 2
Chapter 13 – Lecture Part 2
• The Banzhaf Power Index
– Counting the number of subsets of a set
– Listing winning coalitions and blocking coalitions
– Listing a coalition’s critical voters
Counting Subsets – Finding Banzhaf Power Index
• A set of n elements will have 2n subsets and will have 2n – 1 proper
subsets.
• Thus, for a weighted voting system with n voters, there are 2n
possible coalitions. We count both the set of all voters and the empty
set as coalitions. The set of all voters is sometimes called the grand
coalition. The subset consisting of no voters is called the empty
coalition.
• To determine the Banzhaf power index for a given system we list all
of a voting system’s coalitions (or sometimes to save time we list
only the winning coalitions.)
Banzhaf Power Index – Finding Winning Coalitions
• We determine the weight of each of the winning coalitions and the
number of extra votes that coalition has above the quota. That is,
we determine the number of votes the coalition could lose and still
be a winning coalition.
• A particular voter is critical in a given coalition if it’s weight is greater
than the extra votes of that coalition. A certain voter would not be
critical if its weight was equal or less than the extra votes of the
coalition.
Banzhaf Power Index – An Example
• Consider the voting system [ 51: 26, 26, 26, 22 ].
• There are 24 = 16 different coalitions including both the grand and
empty coalitions. We are interested in those coalitions that are
winning (and the ones that are blocking).
• We will create a list of all coalitions (later we might try to be quicker
by listing only winning coalitions).
• We list the coalition of all voters, coalitions of 3 voters , coalitions of
2 voters, coalitions of only 1 voter, and the empty coalition.
A = 26, B = 26
C = 26, D = 22
Banzhaf Power Index – An Example
q = 51
coalition
coalition weight (w)
extra votes ( w – q )
winning ?
critical voters
A, B, C, D
100
49
yes
none
A, B, C
78
27
yes
none
A, B, D
74
23
yes
A, B
A, C, D
74
23
yes
A, C
B, C, D
74
23
yes
B, C
A, B
52
1
yes
A, B
A, C
52
1
yes
A, C
A, D
48
B, C
52
B, D
48
no
C, D
48
no
A
26
no
B
26
no
C
26
no
D
22
no
empty
0
no
no
1
yes
B, C
q = 51
bq = 50
Banzhaf Power Index – An Example
A = 26, B = 26
C = 26, D = 22
coalition
weight (w)
extra votes: blocking (w – bq)
blocking ?
critical
voters
A, B, C, D
100
50
yes
none
A, B, C
78
28
yes
none
A, B, D
74
24
yes
A, B
A, C, D
74
24
yes
A, C
B, C, D
74
24
yes
B, C
A, B
52
2
yes
A, B
A, C
52
2
yes
A, C
A, D
48
B, C
52
B, D
48
no
C, D
48
no
A
26
no
B
26
no
C
26
no
D
22
no
empty
0
no
no
2
yes
B, C
A = 26, B = 26
C = 26, D = 22
Banzhaf Power Index – An Example
q = 51
coalition
coalition weight (w)
extra votes ( w – q )
winning ?
critical voters
A, B, C, D
100
49
yes
none
A, B, C
78
27
yes
none
A, B, D
74
23
yes
A, B
A, C, D
74
23
yes
A, C
B, C, D
74
23
yes
B, C
A, B
52
1
yes
A, B
A, C
52
1
yes
A, C
B, C
52
1
yes
B, C
Here we list only the winning coalitions.
Note that A is critical in 4 of the winning coalitions, B is also critical in 4
winning and likewise, C is critical in exactly 4 winning coalitions.
Notice that D is not critical in any winning coalitions.
q = 51
bq = 50
Banzhaf Power Index – An Example
A = 26, B = 26
C = 26, D = 22
coalition
weight (w)
extra votes: blocking (w – bq)
blocking ?
critical
voters
A, B, C, D
100
50
yes
none
A, B, C
78
28
yes
none
A, B, D
74
24
yes
A, B
A, C, D
74
24
yes
A, C
B, C, D
74
24
yes
B, C
A, B
52
2
yes
A, B
A, C
52
2
yes
A, C
B, C
52
2
yes
B, C
Here we list only the blocking coalitions.
Notice that A, B and C are each critical in exactly 4 blocking coalitions.
Also, notice that D is not critical to any blocking coalition.
The Banzhaf Power Index – An Example
• A is critical in 4 winning and blocking coalitions. Thus, we say the
Banzhaf Power of voter A is 8. This is the same for voters B and C.
• We found that D was critical in 0 winning and 0 blocking coalitions.
D’s Banzhaf Power is 0.
• The Banzhaf Power Index for the system [ 51: 26, 26, 26, 22 ] is
given by the numbers ( 8, 8, 8, 0 ).
• Sometimes the Banzhaf Power Index is written so that each term
represents the fraction of times each voter is critical out of the total
number of times all voters are critical. In this case, the total number
of times all voters are critical is 8 + 8 + 8 = 24. Thus, we could write
the Banzhaf index as ( 8/24, 8/24, 8/24, 0/24 ) or as, after reducing,
( 1/3, 1/3, 1/3, 0 ). In a sense, voters A, B and C each share 1/3 of
the power while D really has no power in this system.
Banzhaf Power Index - Shortcuts
• The number of blocking coalitions to which a voter is critical equals
the number of winning coalitions to which that voter is critical.
– If a voting member is critical to a winning coalition, that member is
therefore also critical to the opposing coalition – that is, to the coalition
of voters who’d hope to block that measure.
– And likewise, for any blocking coalition to which a voter is critical, that
voter is also critical to the opposing coalition – that is, to the coalition of
voters who’d hope to pass that measure.
– For each coalition that a voter is critical there is exactly one opposing
coalition to which that voter is also critical. Thus, once a voter is critical
to a coalition, that voter is critical to an equal number of winning and
blocking coalitions.
Banzhaf Power Index - Shortcuts
• The number of blocking coalitions to which a voter is critical equals
the number of winning coalitions to which that voter is critical.
• To determine the Banzhaf power index for a weighted voting system,
we list winning coalitions, count the number of times each voter is
critical to each winning coalition and double that number.
• We can begin by considering all coalitions of each possible size and
determine which of those are winning.
Banzhaf Power Index – An example
• Let’s return to the voting system given by [ 16 : 9, 9, 7, 3, 1, 1 ].
There are 26 = 64 coalitions,
however not all of them are
winning coalitions.
There are
C
6
C
6
4
2
= 15 coalitions of 4
= 15 coalitions of 2
C
C
6
C
6
6
5
= 1 coalition of 6
= 6 coalitions of 5
6
3
C
= 20 coalitions of 3
6
1
= 6 coalitions of 1
Banzhaf Power Index – An example
• Coalitions of 1: none are
winning
• Coalitions of 2:
s1, s2
18
2
s1, s2
s1, m
16
0
s1, m
s2, m
16
0
s2, m
Let’s refer to the Smith’s as s1 and
s2 and the others by the first letter of
their last name.
• Coalitions of 3:
s1, s2, m
25
9
none
s2, m, I
19
3
s2, m
s1, s2, I
21
5
s1, s2
s2, m, L
17
1
s2, m
s1, s2, L
19
3
s1, s2
s2, m, e
17
1
s2, m
s1, s2, e
19
3
s1, s2
s1, m, I
19
3
s1, m
s1, m, L
17
1
s1, m
s1, m, e
17
1
s1, m
Note: tables list winning coalitions,
weight, extra votes, and critical
voters, in that order, from left to right.
Banzhaf Power Index – An example
• Coalitions of 4:
s2, m, L, e
18
2
s2, m
s2, m, I, e
20
4
s2, m
s2, m, I, L
20
4
s2, m
s1, m, L, e
18
2
s1, m
s1, m, I, e
20
4
s1, m
s1, m, I, L
20
4
s1, m
s1, s2, L, e
20
4
s1, s2
s1, s2, I, e
22
6
s1, s2
s1, s2, I, L
22
6
s1, s2
s1, s2, m, e
26
10
none
s1 ,s2, m, L
26
10
none
s1, s2, m, I
28
12
none
Banzhaf Power Index – An example
• Coalitions of 5:
s2, m, I, L, e
21
5
s2, m
s1, m, I, L, e
21
5
s1, m
s1,s2, I, L, e
23
7
s1, s2
s1,s2, m, L, e
27
11
none
s1,s2, m, I, e
29
13
none
s1,s2, m, I, L
29
13
none
• Coalitions of 6: There is only one (all voters) and no
voter is critical to that coalition.
Banzhaf Power Index – An example
s1, s2
18
2
s1, s2
s2, m, L, e
18
2
s2, m
s1, m
16
0
s1, m
s2, m, I, e
20
4
s2, m
s2, m
16
0
s2, m
s2, m, I, L
20
4
s2, m
s1, m, L, e
18
2
s1, m
s1, s2, m
25
9
none
s1, m, I, e
20
4
s1, m
s1, s2, I
21
5
s1, s2
s1, m, I, L
20
4
s1, m
s1, s2, L
19
3
s1, s2
s1, s2, L, e
20
4
s1, s2
s1, s2, e
19
3
s1, s2
s1, s2, I, e
22
6
s1, s2
s1, m, I
19
3
s1, m
s1, s2, I, L
22
6
s1, s2
s1, m, L
17
1
s1, m
s1, s2, m, e
26
10
none
s1, m, e
17
1
s1, m
s1 ,s2, m, L
26
10
none
s2, m, I
19
3
s2, m
s1, s2, m, I
28
12
none
s2, m, L
17
1
s2, m
s2, m, e
17
1
s2, m
For each voter, we count the
number of times that voter is critical
to some coalition ….
s2, m, I, L, e
21
5
s2, m
s1, m, I, L, e
21
5
s1, m
s1,s2, I, L, e
23
7
s1, s2
s1,s2, m, L, e
27
11
none
s1,s2, m, I, e
29
13
none
s1,s2, m, I, L
29
13
none
Banzhaf Power Index – An example
• Results: S1 is critical 16 times, S2 is critical 16 times, and M is
critical 16 times. Neither I, L or E are ever critical.
• Conclusion: For the voting system given by [ 16 : 9, 9, 7, 3, 1, 1 ]
we have determined that the Banzhaf power index for this system is
given by ( 32, 32, 32, 0, 0, 0 ).
• Significance: This implies that Dr.’s Ruth
and Ralph Smith and Dr. Mansfield share
power equally in this system while neither
of Dr.’s Ide, Lambert or Edwards have any
power. Even though the nominal power of
Dr. Mansfield is less than the Smiths, he
has equal Banzhaf power in this system.
Dr. Ide, Dr. Lambert and Dr. Edwards are
dummy voters ( their vote is of no
significance.)