Information Technology Reference
In-Depth Information
2 Formal Analysis of SP
2.1 Conditions of SP
2
We begin with an analysis of the paradox in response to question (ii), “what
are the conditions in which the paradox arises?” Consider two groups, [A, B],
taken to be mutually exclusive and jointly exhaustive. The overall rates for each
group are [
ʱ
,
ʲ
] respectively. Each group is partitioned into categories [1, 2] and
the rates within each partition are [
A
1
,
A
2
,
B
1
,
B
2
]. Let's assume that
f
1
=the
number of females accepted in
D
1
,
F
1
= the total number of females applied
to
D
1
,
m
1
= the number of males accepted in
D
1
,
M
1
= the total number of
males applied to
D
1
.Then
A
1
=
f
1
/F
1
,and
B
1
=
m
1
/M
1
. Defining
f
2
,
F
2
,
m
2
and
M
2
in a similar way, we get
A
2
=
f
2
/F
2
and
B
2
=
m
2
/M
2
. Likewise, we
could understand
ʱ
and
ʲ
as representing the overall rates for females and males,
respectively. So the terms
ʱ
=(
f
1
+
f
2
)
/
(
F
1
+
F
2
)and
ʲ
=(
m
1
+
m
2
)
/
(
M
1
+
M
2
).
To help conceptualize these notations in terms of Table 1, we provide their
corresponding numerical values:
A
1
= 180
/
200 = 90%,
A
2
= 100
/
300 = 33%,
B
1
= 480
/
600 = 80%,
B
2
=10
/
100 = 10%,
ʱ
= 280
/
500 = 56%, and finally
ʲ
= 490
/
700 = 70%. Since
ʱ
,
ʲ
,
A
1
,
A
2
,
B
1
,and
B
2
are rates of some form,
they will range between 0 and 1 inclusive. We further stipulate the following
definitions where, “
≡
” means “is defined as”.
C
1
≡ A
1
≥ B
1
.
C
2
≡
A
2
≥
B
2
.
C
3
≡
ʲ
≥
ʱ.
(
C
1
&
C
2
&
C
3
)
.
In terms of Table 1, these definitions become
C
1
: 90%
>
80%,
C
2
: 33%
>
10%,
C
3
: 70%
>
56% and thus C is satisfied. But C alone is not a sucient condition
for SP. We could have a case where
A
1
=
B
1
,
A
2
=
B
2
and
ʲ
=
ʱ
resulting in
no paradox, yet C being satisfied. Hence, we stipulate another definition:
C
≡
C
4
≡
ʸ>
0
.
where,
ʸ
=(
A
1
−
B
1
)+(
A
2
−
B
2
)+(
ʲ
−
ʱ
)
.
For the data in Table 1,
ʸ
equals 10% + 23% + 14%. Again,
C
4
alone is
not sucient for SP since we could have a case where
A
1
>B
1
,
B
2
>A
2
and
ʲ>ʱ
resulting in no paradox (C is violated) and yet
C
4
being satisfied.
3
Hence,
2
Some parts of this section are based on our previous work [1,2].
3
As a heuristic rule we take
A
1
to be that sub-group ratio which is the greater of
the two ratios and
B
1
as that which is the lesser of the two. In table 1, the ratio
of women admitted to department 1 is greater than that of men. Hence, the former
will be taken as
A
1
and the latter will be taken as
B
1
. Similarly, since the ratio of
women admitted to department 2 is greater than that of men, the former is taken
as
A
2
and the latter as
B
2
. This avoids the complexity of taking the absolute value
of their difference in the calculation of
ʸ
.
