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us formalize the argumentation of the respondent by saying that a PD classification
rule
A
,
B
→
C
is an instance of a PND rule
D
,
B
→
C
when:
•
a transaction satisfying
A
in context
B
satisfies condition
D
as well, or, in sym-
bols,
con f
(
A
,
B
→
D
)
is close to 1;
•
and, the rule
D
,
B
→
C
holds at the same or higher confidence, or, in symbols,
;
A respondent argumenting against discriminatory allegations supported by a PD
rule
A
con f
(
D
,
B
→
C
)
≥
con f
(
A
,
B
→
C
)
C
,
and with
D
modeling a genuine occupational requirement. On the contrary, a com-
plainant or a control authority can prevent respondent's argumentation by showing
that the PD rule
A
,
B
→
C
must show that the rule is an instance of some PND rule
D
,
B
→
C
. In (Ruggieri
et al., 2010c), the concept of “instance” has been relaxed to the notion of
p
-instance,
requiring
con f
,
B
→
C
is not an instance of any PND rule
D
,
B
→
.Onthe
experimental side, the vast majority of discriminatory PD rules extracted from the
German credit dataset result (
p
-)instances of some PND rule, thus concluding that it
is (fortunately) extremely difficult to characterize
prima facie
evidence of discrim-
ination.
Another defence strategy of the respondent is to resort to the well-known Simp-
son's paradox. (Bickel, Hammel, & O'Connell, 1975) describes a real case of pos-
sible discrimination against women in university admission. Let us rephrase it using
our notation. Assume that the rule
SEX
=
FEMALE
(
A
,
B
→
D
)
≥
p
and
con f
(
D
,
B
→
C
)
≥
p
·
con f
(
A
,
B
→
C
)
ADMITTED
=
NO
has an high ex-
tended lift, so that a possible discrimination is raised. By examining each individual
department A of the university, however, it can happen that each rule
SEX
=
FEMALE
,
DEPT
=A
→
ADMITTED
=
NO
has a very low extended lift, denoting no discrimina-
tion at all. The paradox is that the discrimination observed at university level did
not actually occur in any department. If the examination commissions worked at
department level, then the department attribute is causal factor, and the standard ap-
proach (Pearl, 2009) is to condition probabilities and rules on it. As a consequence,
the rules at department level are the correct ones to be looked at, whilst the rule at
university level contains confounding factors (the commissions that took decisions).
→
5.6
Affirmative Actions
Affirmative actions (see ENAR, 2008; Sowell, 2005), sometimes called positive ac-
tions or reverse discrimination, are a range of policies to overcome and to com-
pensate for past and present discrimination by providing opportunities to those
traditionally denied for. Policies range from the mere encouragement of under-
represented groups to quotas in favor of those groups. For instance, US federal
contractors are required to identify and set goals for hiring under-utilized minori-
ties and women. Also, universities have voluntarily implemented admission policies
that give preferential treatment to women and minority candidates. Affirmative ac-
tion policies “shall in no case entail as a consequence the maintenance of unequal
or separate rights for different racial groups after the objectives for which they were
taken have been achieved” (United Nations Legislation, 2011, (a)). It is therefore
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