D .S. Felsenthal, M. Machover Mathematical Social Sciences 37 1999 25 –37
27
the difference between equal suffrage ‘one person, one vote’, briefly: ‘OPOV’ and majoritarianism, which have sometimes been conflated with each other.
2. Sensitivity and mean majority deficit
In this section we state some definitions and results from the general theory of a priori voting power.
Definition 1. i By an SVG we mean a set 0 of subsets of a finite set N such that [ [
⁄ 0; N [ 0; and whenever X Y N and X [ 0 then also Y [ 0.
We refer to N as the assembly, to its members as voters, and to its subsets as coalitions [of 0]. The winning coalitions are just those belonging to 0; the rest are
losing coalitions. Voter a is critical in coalition S if S [ 0 but S 2
haj [⁄ 0; a is critical outside S if
S [ ⁄ 0 but S
haj [ 0; a is critical for S if it is critical in or outside S. We say that voter a agrees with the outcome of coalition S [in 0] if a [ S [ 0 or
1
a [ ⁄ S [
⁄ 0.
We shall take uNu, the number of members of N, to be n. While generally N can be any
finite set, we shall use the set I 5 h1, . . . , nj as a ‘canonical’ assembly of size n.
n
Definition 2. Let q be a positive real and let w , . . . , w be nonnegative reals such that
1 n
n
0 , q o
w . In this connection we refer to q as the quota and to the w as weights.
i 51 i
i
Further, if S I we put wS 5 o
w and refer to wS as the weight of S.
n i [S
i
We denote by ‘[q; w , . . . , w ]’ the SVG with assembly I 5 h1, . . . , nj whose win-
1 n
n
ning coalitions are just those S I with wS q.
n 2
Any SVG 0 isomorphic to [q; w , . . . , w ] is called a weighted voting game WVG.
1 n
It ‘inherits’, via the isomorphism, a quota and weights. The WVG n 1 1
]]; 1, 1, . . . , 1 , 1
F G
2
n times
is denoted by ‘} ’; any WVG isomorphic to it is called a majority SVG.
n
Definition 3. For any finite set N we define the Bernoulli space of N, denoted by ‘B ’,
N
to be the probability space whose points are just the subsets of N, each of which is
n
assigned the same probability, 1 2 .
Comment 1. i If each subset S of N is replaced by its characteristic function, then B
N
becomes the well-known space of n Bernoulli trials, with equal probability, 1 2, for success and failure.
ii In what follows, we shall denote by ‘P’ and ‘E’ the probability measure and
expected value operator, respectively, on the space B , where N is the assembly of some
N
1
This is short for saying that in a division in which S is the set of ‘yes’ voters, the decision goes according to the way a votes.
2
We take the notion of isomorphism of SVGs to be self-explanatory.
28 D
.S. Felsenthal, M. Machover Mathematical Social Sciences 37 1999 25 –37
SVG. We shall usually not need to specify which SVG is being referred to, because this will be clear from the context.
Definition 4. 0 is an SVG with assembly N and a is any of its voters, we put
9
b [0]5 P
hX N : a is critical for X in 0j. 2
a def
9
We refer to b [0] as the Bz power of a in 0, and to the function b 9 itself as the Bz
a
measure [of voting power].
Comment 2. i In the literature b 9 is often called the absolute Banzhaf index as distinct from the ordinary, or relative, Banzhaf index b, which is obtained from b 9 by
normalization so that o
b [0]51.
x [N x
ii Our probabilistic definition of b 9 is easily seen to be equivalent to the definition more commonly given in the literature.
The measure proposed by Penrose 1946, p. 53 was in fact b 9 2. In stating his definition, he asserts as an obvious fact the following result, whose proof is indeed quite
3
simple.
Lemma 1. If 0 is an SVG and a is any of its voters , then
9
1 1 b 0
a
]]] P
hX N : a agrees with the outcome of X in 0j 5 ,
3 2
where N is the assembly of 0. j
Definition 5. For any SVG 0 we put
9
S[0]5
O
b [0], 4
def x
x [N
where N is the assembly of 0. We refer to S[0] as the sensitivity of 0. Further, we put S 5
S[} ]; so that S is the sensitivity of any majority SVG with
n def
n n
exactly n voters.
Comment 3. By an easy combinatorial argument one obtains:
n m
]] S 5
, 5
S D
n n 21
m 2
where m 5[n 2]11 is the least integer greater than n 2. Hence, a routine calculation, using Stirling’s well-known approximation formula for n, yields
] 2n
] S |
, 6
n
œ
p where ‘|’ means that the ratio of the two sides tends to 1 as n increases.
The sensitivity of an SVG has an interesting interpretation, as the expected value of an important random variable which we now proceed to define.
3
For a proof see, for example, Dubey and Shapley 1979, pp. 124–125.
D .S. Felsenthal, M. Machover Mathematical Social Sciences 37 1999 25 –37
29
Definition 6. For any SVG 0 with assembly N, we define the random variable Z
[0] on
the Bernoulli space B by stipulating that the value of
Z [0] at any coalition S N
N
equals the number of voters who agree with the outcome of S in 0 minus the number of those who do not agree with the outcome of S in 0.
Further, if 0 is the majority SVG with assembly N, we denote Z
[0] by ‘ M
’, and
N
refer to it as the margin [in N].
Comment 4. Note that the value of M
at any coalition S N equals the absolute value
N
of the difference between uSu and uN2Su, that is, the excess of the size of the majority
camp over that of the minority camp. If the voters are evenly divided, then the value of M
is of course 0.
N
Theorem 1. Let 0 be an SVG . Then
E Z
[0] 5 S[0]. 7
Proof. For each voter x and coalition S, let Z
S equal 1 or 21 according as x agrees or
x
does not agree with the outcome of S. From Lemma 1 it follows by an easy calculation
9
that E Z
5b [0]. But clearly Z
[0]5 o
Z . Hence
x x
x [N x
9
E Z
[0] 5
O
b [0] 5 S[0], 8
x x [N
as claimed. j
Corollary 1. E M
5S . j
N n
Definition 7. For any SVG 0 with assembly N, we define the random variable D
[0],
called the majority deficit [of 0], on the Bernoulli space B by stipulating that the value
N
of D
[0] at any coalition S N equals the size of the majority camp minus the number of voters who agree with the outcome of S in 0. Thus, if the voters who agree with the
outcome of S in 0 are the majority, or if the voters are evenly divided, then the value of D
[0] at S is 0; but if the voters who agree with the outcome of S in 0 are the minority then the value of
D [0] at S equals that of the margin
M .
N
Further, we put D[0]5 E
D [0]. We call D[0] the mean majority deficit MMD
def
[of 0]. Clearly, the MMD is a measure of the average absolute deviation of an SVG from
majority rule.
Theorem 2. For any SVG 0 with exactly n voters,
S 2 S[0]
n
]]]] D[0] 5
. 9
2
Proof. By Comment 4, the value of M
always equals the margin by which the size of
N
the majority camp exceeds that of the minority camp. Hence the size of the majority camp itself is given by the value of
30 D
.S. Felsenthal, M. Machover Mathematical Social Sciences 37 1999 25 –37
n 1 M
N
]]. 10
2 Also, by Definition 6 the number of voters who agree with the outcome is always given
by the value of n 1
Z [0]
]]]. 11
2 Thus by Definition 7 we have:
n 1 M
M 2
Z [0]
n 1 Z
[0]
N N
]] ]]]
]]]] D
[0] 5 2
5 .
12 2
2 2
We now apply the expected value operator E to both sides. By Definition 7 we obtain on the left-hand side D[0]; and by Corollary 1 and Theorem 1 we obtain on the right-hand
side
S 2 S[0]
n
]]]], 13
2 as claimed.
j
Comment 5. From Theorem 2 it is clear that, for a given assembly N, an SVG 0 maximizes S[0] iff it minimizes D[0].
Now, since the random variable D
[0] has no negative values, it follows that its expected value D[0] is minimized—in fact, vanishes—iff
D [0] vanishes everywhere in
the space B . By Definition 7 this means that all coalitions of size .n 2 must win and
N
all those of size ,n 2 must lose. For odd n there is just one such SVG: the majority SVG with assembly N. For even n there are other such SVGs, all having sensitivity S ,
n
since the vanishing of D
[0] imposes no condition on any coalition of size n 2: it may be winning or losing.
This characterization of the SVGs that maximize S[0] is proved by Dubey and Shapley 1979, pp. 106–107 using a very different argument. Below we shall use a
somewhat modified form of their argument to prove Theorem 3.
3. The second square-root rule
