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treatment for the subgroup . The minimal effort required to achieve -level of outcome variable within
+
the subgroup is computed as follows:
+

Ψ ( ) = ∈ E[ ] ⩾ (11)
+
+
Then, for a certain outcome level , individual -equal effort is satisfied for individual if:

Ψ ( ) = Ψ ( ) (12)
+
−
Equality of effort can also be extended to identify discrimination at any sub-population level or system level,
when is extended to the entire group with = and denotes the entire group with = . To
+
−
−
+
distinguish individual -equal effort, is used to denote the first set, while denoted the second one. The
+
−
-equal effort is satisfied for a sub-population if:
(13)
Ψ ( ) = Ψ ( )
+
−
4.2.4. PC-Fairness
Path-specific Counterfactual Fairness (PC-fairness) [31] is used to denote a general fairness formalization for
representing various causality-based fairness notions. Given a factual condition O = o where O ∈ V and a
causal path set , a predictor achieves the PC-fairness if it satisfies the following expression:
ˆ

( ˆ → |o) ⩽ (14)
−
+
where ( ˆ → |o) = ( ˆ | , | ¯ |o) − ( ˆ |o) and is a predefined fairness threshold (typically, 0.05).
−
−
+
−
+
Intuitively, ( ˆ → |o) denotes when the value of the sensitive attribute changes from to , the causal
−
+
−
+
ˆ
effect of on through the causal path set and given the factual observation o.
PC-fairness matches different causality-based fairness notions by tuning its parameters. For example, if the
path set contains all causal paths and O = , PC-fairness corresponds to the total effects in Equation (3).
Apart from that, it also includes new types of fairness that have not been studied yet in the past. For example,
PC-fairness can detect individual indirect discrimination by letting O = V\{ } and the path set containing
all causal paths that pass through any redlining variables.
5. CAUSALITY-BASED FAIRNESS-ENHANCING METHODS
The need for causal models for detecting and eliminating discrimination is based on the intuition that the
same individuals experience different outcomes due to innate or acquired characteristics outside of their con-
trol (e.g., gender). Therefore, causal models are useful for investigating which characteristics cannot be con-
trolled by individuals and using the resulted understandings to identify and deal with discrimination. In other
words, understandingthestructureofrootcausesoftheproblemcanassistinidentifyingunfairnessandcauses.
Thus, there is a causal structure that must be considered rather than just the correlation between the sensitive
attribute and outcome. Because of these advantages, many recent studies introduce fairness-enhancing ap-
proachesfromtheperspectiveofcausality. Accordingtothestagesoftrainingthemachinelearningalgorithms,
pre-processing, in-processing, and post-processing mechanisms can be used to intervene in the algorithm to
achieve fairness. Therefore, causal-based methods can be divided into the above three categories. Figure 6
shows the general flow of different categorical causality-based approaches. This section provides an overview
of studies for these categories, and then the advantages and disadvantages of these three types of mechanisms
are summarized.