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→ Income → Loan.

Similar to no unresolved discrimination, no proxy discrimination [28] also focuses on indirect discrimination.
Given a causal graph, if this criterion is satisfied, the effects of the sensitive attribute on the output cannot
be transmitted through any proxy variable (which is also denoted as redlining variable). A proxy variable is
a descendant of sensitive attribute and the ancestor of decision attribute . It is labeled as a proxy because
it is exploited to capture the information of . The outcome of an automated decision making exhibits no
proxy discrimination if the equality of the following equation is valid for all potential proxies :

( | ( = 0 )) = ( | ( = 1 )) ∀ 0 , 1 ∈ ( ) (6)

In other words, this notion implies that changing the value of should not have any impact on the prediction.
A simple example is shown in Figure 5. ZipCode is a redlining variable due to it reflects the information of
the sensitive attribute Race. There is no proxy discrimination in causal graph shown in Figure 5(c), since the
causal path Race → ZipCode → Loan has been blocked by intervening ZipCode.

No unresolved discrimination is a flawed definition of fairness. Specifically, no unresolved discrimination
criterion is unable to identify some counterfactual unfair scenarios where some attributes are deemed as the
resolved attributes. On the other hand, policy makers and domain professionals should carefully examine the
relevance between sensitive variables and other endogenous variables so as to discover all resolving attributes
and potential proxies that may lead to discrimination spread.

4.2. Individual causality­based fairness notions
Different from group fairness notions that measure the differences in the outcome of decision models between
advantaged groups and disadvantaged ones, individual fairness notions aim to examine whether the outcome
of decision models is fair to each individual in the population. Some representative group causality-based
fairness notions are discussed here.

4.2.1. Counterfactual fairness
An outcome achieves counterfactual fairness towards an individual (i.e., O = o) if the probability of the
outcome = for such individual is the same as the probability of = for the same individual whose value
of sensitive attribute changing to another one. Formally, counterfactual fairness can be expressed as follows
for any O = o:

−
| ( |O = o, = ) − ( |O = o, = )| ≤ (7)
−
+
−
where O ⊆ V \ { , } is the subset of endogenous variables except sensitive variables and decision variables.
Any context O = o represents a certain sub-group of the population, specifically, when O = V \ { , }, it
representsaspecificindividual. AccordingtoEquation(7), thedecisionmodelachievescounterfactualfairness
if, for every possible individual (O = o, = ) of the entire population, the probability distribution of the
−
outcome is the same in both the actual ( = ) and counterfactual ( = ) worlds.
+
−
Counterfactual fairness was proposed by Kusner et al. [11] . They empirically tested whether the automated
decision making systems are counterfactual fairness by generating the samples given the observed sensitive at-
tribute value and their counterfactual sensitive value; then, they fitted decision models to both the original and
counterfactual sampled data and examined the differences in the prediction distribution of predictor between
the original and the counterfactual data. If an outcome is fair, the predictor is expected that the predicted
results of actual and counterfactual distributions lie exactly on top of each other.