# Maximal fairness

MaryBeth Defrance  University of GhentGhentBelgium9000
<EMAIL_ADDRESS>and  Tijl De Bie  University of
GhentGhentBelgium9000<EMAIL_ADDRESS>
###### Abstract.

Fairness in AI has garnered quite some attention in research, and increasingly
also in society. The so-called ”Impossibility Theorem” has been one of the
more striking research results with both theoretical and practical
consequences, as it states that satisfying a certain combination of fairness
measures is impossible. To date, this negative result has not yet been
complemented with a positive one: a characterization of which combinations of
fairness notions _are_ possible. This work aims to fill this gap by
identifying maximal sets of commonly used fairness measures that can be
simultaneously satisfied. The fairness measures used are demographic parity,
equal opportunity, false positive parity, predictive parity, predictive
equality, overall accuracy equality and treatment equality. We conclude that
in total 12 maximal sets of these fairness measures are possible, among which
seven combinations of two measures, and five combinations of three measures.
Our work raises interest questions regarding the practical relevance of each
of these 12 _maximal fairness_ notions in various scenarios.

Fairness, Fairness in AI, Fairness definitions, confusion tables, Combining
Fairness definitions

††ccs: Computing methodologies Artificial intelligence

##  1\. Introduction

The ProPublica article about the risk assessment system COMPAS generated quite
some buzz in the field of fairness of AI (Angwin et al., 2016). It started
raising questions on what characteristics an AI system needs to satisfy for it
to be fair in a practical, intuitive, or legal sense. One of the main
criticisms from the authors of the article were the differing false positive
and false negative rates for black people as compared to for white people. In
response, the creators of the COMPAS-tool defended their system by arguing
that it was properly calibrated, meaning that the predicted probability of
recidivism was accurate for both demographic groups, and that this made the
system fair. One might argue that ideally these properties would both be
satisfied. However, the works of Chouldechova et al. (Chouldechova, 2016),
Hardt et al. (Hardt et al., 2016), and Kleinberg et al. (Kleinberg et al.,
2016) all led to the same conclusion, colloquially referred to as the
”Impossibility Theorem”. It states that it is mathematically _impossible_ to
have calibration, equal false positive and equal false negative rates all at
the same time, except in a practically irrelevant degenerate case. (See Sec. 2
for some further details.)

##### Contributions

To the best of our knowledge, this negative result has never been complemented
with a positive one, namely the characterisation of maximal sets of fairness
notions that can be simultaneously satisfied. Thus, in this paper, we
investigate which and how many of the more frequently used fairness measures
can be combined, leading to a set of maximal notions of fairness. We do this
for the simplest but commonly studied case of binary classification. Out of a
range of seven commonly used fairness measures, we identify a total of 12
maximal combinations, including seven maximal combinations of two and five
maximal combinations of three fairness measures. Our results trigger
intriguing questions on the practical relevance and interpretation of each of
these maximal fairness notions.

##  2\. Related Work

Most of the conceptual research about fairness in machine learning is about
the introduction of new fairness measures, about the interpretation of
previously proposed fairness notions, about techniques to satisfy fairness
measures, or about so-called impossibility theorems that show that certain
combinations of fairness measures are impossible to achieve. Here we survey
the work most directly related to the present paper.

###  2.1. Surveys on Fairness definitions

Verma S. and Rubin J. (Verma and Rubin, 2018) discuss different fairness
measures for classification with a focus on creating a human-understandable
interpretation of them and applying them to a specific use case. A more in-
depth discussion about the design of a predictive model concerning pitfalls
and biases that occur in the process can be found in Mitchell et al. (Mitchell
et al., 2021). This paper also covers different fairness measures and relates
the choice between these definitions back to the discussion of design choices,
and it lightly touches on the impossibilities of combining certain fairness
measures when working with scores instead of binary decisions. The survey of
N. Mehrabi et al. contains an extensive list of possible biases and fairness
measures (Mehrabi et al., 2021). The fairness measures are grouped by type,
group, subgroup or individual. It also discusses methods to satisfy them. Ruf
and Detyniecki (Ruf and Detyniecki, 2021) strive to create a type of Fairness
Compass which would help practitioners decide what fairness measure is best
suited in a given problem. Throughout the work, different fairness measures
are clearly explained through the use of confusion matrices and with a
theoretical use case.

###  2.2. Prior work on combining Fairness definitions

Most research related to combining fairness measures focusses on the
impossibility of certain combinations. Chouldechova A. (Chouldechova, 2016)
discuss how adherence to a certain criterion can still lead to a considerable
disparate impact between groups. It states that it is impossible to achieve
equality between groups for false positive rate (FPR), positive predictive
value (PPV) and false negative rate (FNR) when the base rates differ between
groups. Kleinberg et al. (Kleinberg et al., 2016) discuss combining three
different fairness measures in the context of risk scoring. It is focussed on
scoring instead of binary classification. They come to the conclusion that
combining those three measures is only possible under unique circumstances,
namely if the groups have equal base rates or if the model is capable of
perfect prediction. Berk et al. (Berk et al., 2021) focus on the trade-off
that would occur when using multiple incompatible fairness measures such as
discussed in the work of Chouldechova A. (Chouldechova, 2016). They also
include different techniques that can be used to achieve this goal.

##  3\. Maximal Combinations of Fairness Definitions

Common fairness measures can be expressed by referring to the confusion
matrices (as in Table 1) for the different protected demographic groups. They
are defined by choosing a statistic computed on a confusion matrix, for
example the false negative rate or some other statistic, and requiring this
statistic to be equal across the demographic groups. We will demonstrate this
for seven common fairness measures in Sec. 3.1. This approach makes combining
fairness metrics easy: simply by combining the constraints they impose on the
contingency matrices. In this paper we assume there are two demographic
groups, and we will refer to the statistics computed on the respective
confusion matrix using subscripts a and b. Note that in their most common
definition, confusion matrices contain counts. However, because this
simplifies our derivations somewhat, in this paper we normalize them by
dividing each of the four cells by their sum.

Table 1. A confusion matrix, showing the relation between ground truth labels, predicted labels, true positives, true negatives, false positives, and false negatives. Commonly used fairness notions require a particular statistic computed on the confusion matrices for different demographic groups to be equal to each other. | Predicted
---|---
| Positive | Negative
True | Positive | TP | FN
Negative | FP | TN

By definition, the (normalized) confusion matrices satisfy certain constraints
of their own. These constraints are expressed in the system of equations
depicted in Figure 1. Here, the variables $1-x$ and $1-y$ denote the base
rates of group a and group b respectively. The base rate is the fraction of
positive samples in the data set for a given group. The inequalities will be
left out of subsequent derivations in order to simplify the notations.

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ 0<=TP_{a}<=1-x,0<=TP_{b}<=1-y\\\
0<=TN_{a}<=x,0<=TN_{b}<=y\\\ 0<=FP_{a}<=x,0<=FP_{b}<=y\\\
0<=FN_{a}<=1-x,0<=FN_{b}<=1-y\\\ \end{cases}$

Figure 1. Constraints always present in the system of equations, due to the
properties of the confusion matrix.

The constraints that come from the properties of the confusion matrix

Before investigating which fairness measures can be combined, we must be
precise about what we mean by that. To do this, let us first point out that
any combination of fairness measures can be satisfied if the base rates are
equal between both demographic groups, i.e. if $x=y$. Indeed, in those cases
it is possible for the matrices to be exact copies of each other. However as
base rates are a property of the data and hence cannot be controlled, this
possibility must not lead us to judge that any set of fairness definitions is
compatible. Another possibility we will see in our derivations below is that
certain combinations are possible only if some of the variables in the
confusion table are equal to a fixed (and often trivial) value. Again,
although this constraint can be satisfied in principle, it leaves very little
freedom to the classifier, effectively preventing it from being useful. Thus,
we argue that also such combinations cannot be deemed ‘possible’. The
following definition determines what properties are necessary to deem a
combination of fairness measures as ‘possible’.

###### Definition 1 (Possibility of combining fairness measures).

A combination of fairness measures is deemed _possible_ if after combining the
constraints, all elements in both confusion matrices can still take on values
within a subset of $[0,1]$ with non-zero Lebesgue measure, and this for all
pairs of base rates $(x,y)$ from a subset of $[0,1]^{2}$ with non-zero
2-dimensional Lebesgue measure. Or informally speaking, the elements of the
confusion matrices should be able to take on values within a non-trivial
range, and this for all base rate pairs from within a non-trivial
2-dimensional range.

###  3.1. Implemented Fairness measures

The fairness measures used in this paper are demographic parity, equal
opportunity, false positive parity, predictive parity, predictive equality,
overall accuracy equality and treatment equality. More detailed information
about these measures can be found in Table 2. This table includes two
equivalent definitions: one expressed in terms of probabilities, and another
in terms of a constraint on the confusion matrices. The table also mentions
the statistical property it requires to be equal between groups (if it has a
standard name). Finally, it mentions the ‘orientation’ of the constraint on
the confusion matrices, which may offer insight into the possible combinations
of fairness notions. Other frequently used fairness measures exist,
particularly in other settings than binary classification, but in the paper
our focus is on measures for this setting only. Note that each of these
measures only require one constraint to be satisfied, and by combining them
the number of constraints is growing.

Below we discuss each of the considered fairness measures in some greater
detail. Figure 2 also illustrates them more visually, highlighting the
orientations on the confusion matrix, and in this way also showing their
differences and similarities.

Table 2. The list of fairness measures considered in this paper, with their definition expressed in probabilities (second column) as well as by means of an expression evaluated on the confusion matrices (third column), and the statistical property it requires to be equal (fourth column). The fourth column mentions the orientation of the constraints on the confusion matrix. Name | In probabilities | In the confusion matrix | Statistic | Orientation
---|---|---|---|---
Demographic Parity | $P(\hat{Y}|A=0)=P(\hat{Y}|A=1)$ | $\frac{FP_{A}+TP_{A}}{FP_{A}+TP_{A}+FN_{A}+TN_{A}}=\frac{FP_{B}+TP_{B}}{FP_{B}+TP_{B}+FN_{B}+TN_{B}}$ | Positive Rate (PR) | $B_{v}$ Board
Equal Opportunity | $P(\hat{Y}=1|A=0,Y=1)=P(\hat{Y}=1|A=1,Y=1)$ | $\frac{TP_{A}}{TP_{A}+FN_{A}}=\frac{TP_{B}}{TP_{B}+FN_{B}}$ | False Negative Rate (FNR) | H
Horizontal
False positive parity | $P(\hat{Y}=1|A=0,Y=0)=P(\hat{Y}=1|A=1,Y=0)$ | $\frac{FP_{A}}{FP_{A}+TN_{A}}=\frac{FP_{B}}{FP_{B}+TN_{B}}$ | False Positive Rate (FPR) | H
Horizontal
Predictive Parity | $P(Y=1|\hat{Y}=1,A=0)=P(Y=1|\hat{Y}=1,A=1)$ | $\frac{TP_{A}}{TP_{A}+FP_{A}}=\frac{TP_{B}}{TP_{B}+FP_{B}}$ | Positive Prediction Value (PPV) | V
Vertical
Predictive Equality | $P(Y=1|\hat{Y}=0,A=0)=P(Y=1|\hat{Y}=0,A=1)$ | $\frac{FN_{A}}{FN_{A}+TN_{A}}=\frac{FN_{B}}{FN_{B}+TN_{B}}$ | False Omission Rate (FOR) | V
Vertical
Overall accuracy Equality | $P(\hat{Y}=Y|A=0)=P(\hat{Y}=Y|A=1)$ | $\frac{TP_{A}+TN_{A}}{TP_{A}+FP_{A}+TN_{A}+FN_{A}}=\frac{TP_{B}+TN_{B}}{TP_{B}+FP_{B}+TN_{B}+FN_{B}}$ | Accuracy (ACC) | $B_{d}$ Board
Treatment Equality | $-$ | $\frac{FN_{A}}{FP_{A}}=\frac{FN_{B}}{FP_{B}}$ |  | D
Diagonal

###### Definition 2 (Demographic Parity (Dwork et al., 2012)).

Demographic parity is a fairness measure which requires that both groups
proportionally receive the positive outcome (TP + FP) equally regardless of
the ground truth and thus also regardless of the base rates. This independence
makes demographic parity unusual. Because demographic parity is independent of
the ground truth it often tends to result in lower accuracies when enforced.
Note that a similar notion is conditional statistical parity (Corbett-Davies
et al., 2017), which is similar except that it requires equality between
sensitive groups when restricted to people that share a certain attribute.
Conditional statistical parity will not be discussed in this paper.

###### Definition 3 (Equal Opportunity (Hardt et al., 2016)).

Equal opportunity is both a measure of itself and also one of two conditions
that is required for equalised odds. Equal opportunity means that people have
the same probability regardless of their sensitive attribute of receiving a
negative prediction given that they belong in the positive category.

###### Definition 4 (False positive parity (Hardt et al., 2016)).

False positive parity is the second condition for equalised odds, alongside
equal opportunity. It requires that the probability of receiving a positive
prediction when the actual class is negative is equal across sensitive groups.
If both false positive parity and equal opportunity are satisfied then
equalised odds is satisfied.

###### Definition 5 (Predictive Parity (Chouldechova, 2016)).

Predictive parity differs from equal opportunity in that it is concerned with
the probabilities of the actual class and not of the predicted class. In other
words, rather than horizontal ratios in the confusion matrices for equal
opportunity and false positive parity, predictive parity considers a vertical
ratio. In order for predictive parity to be satisfied, the probability that a
positive prediction is correct must be equal across sensitive groups.

###### Definition 6 (Predictive Equality (Corbett-Davies et al., 2017)).

Predictive equality is very analogous to predictive parity. It requires that
the probability for a negative prediction to be correct is equal across
sensitive groups. The combination of satisfying both predictive parity and
predictive equality is also called conditional use accuracy equality.

###### Definition 7 (Overall accuracy Equality (Berk et al., 2021)).

Overall accuracy equality requires that the accuracy of the classifier is
equal between sensitive groups. While straightforward and intuitive, this
definition on its own is often not sufficient to guarantee intuitive fairness.
Indeed, it implicitly assumes an equal impact of positive and negative
predictions, which is often unjustified in practice.

###### Definition 8 (Treatment Equality (Berk et al., 2021)).

Treatment equality is an atypical fairness measure as it cannot be expressed
as a probability nor a commonly used statistical property. It is a constraint
on the false positives and false negatives, requiring their ratio to be equal
between sensitive groups.

Figure 2. Orientations of the fairness measures in the confusion matrix.

A visual representations of what areas of the contingency table are effected
by the constraint of the fairness measures.

###  3.2. Combining two Fairness measures

We begin with a result on the _pairwise_ combinations of the fairness measures
discussed in Sec. 3.1.

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\
\frac{TP_{a}}{TP_{a}+FN_{a}}=\frac{TP_{b}}{TP_{b}+FN_{b}}\\\
\frac{TP_{a}+FP_{a}}{TP_{a}+FP_{a}+FN_{a}+TN_{a}}=\frac{TP_{b}+FP_{b}}{TP_{b}+FP_{b}+FN_{b}+TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
\frac{TP_{a}}{1-x}=\frac{TP_{b}}{1-y}\\\
TP_{a}+FP_{a}=TP_{b}+FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=TP_{b}+FP_{b}-\frac{1-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=(1-\frac{1-x}{1-y})TP_{b}+FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{x-y}{1-y}TP_{b}-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\
TN_{b}=y-FP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\end{cases}$

Figure 3. Combining the fairness measures of Demographic Parity with Equal
Opportunity.

A system of equations where the constraint of the base rates on the variables
are denoted, but also the range in which each variable needs to lie.

$\begin{cases}0\leq 1-x-\frac{1-x}{1-y}TP_{b}\\\ 1-x-\frac{1-x}{1-y}TP_{b}\leq
1-x\\\ 0\leq x-\frac{x-y}{1-y}TP_{b}-FP_{b}\\\
x-\frac{x-y}{1-y}TP_{b}-FP_{b}\leq x\\\ 0\leq\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{x-y}{1-y}TP_{b}+FP_{b}\leq
x\end{cases}\Longleftrightarrow\begin{cases}\frac{1}{1-y}TP_{b}\leq 1\\\
\frac{1-x}{1-y}TP_{b}\geq 0\\\ FP_{b}\leq x+\frac{-x+y}{1-y}TP_{b}\\\
\frac{-x+y}{1-y}TP_{b}\leq FP_{b}\\\ \frac{-x+y}{1-y}TP_{b}\leq FP_{b}\\\
FP_{b}\leq x+\frac{-x+y}{1-y}TP_{b}\end{cases}$

Figure 4. Shortened version of calculating the constraints on x and y when
combining Demographic Parity with Equal Opportunity, full system of
inequalities can be found in Appendix A.1.1.

Calculating the constraints on x and y when combining Demographic Parity with
Equal Opportunity. Most of the constraints are just properties of the
variables and thus insignificant. Four of the constraints however do constrain
the value of x, given y and the value of the free variables

Figure 5. Resulting constraints when combining pairs of fairness measures.

17 square boxes in which the resulting constraints are denoted in order to
satisfy the two fairness measures denoted on the top of the box. In the top
right corner of each box a number is added which denotes the kind of group in
which a combined measure lies.

######  Proposition 3.0.

All pairwise combinations of demographic parity, equal opportunity, false
positive parity, predictive parity, predictive equality, overall accuracy
equality and treatment equality are possible according to Definition 1.

######  Proof Outline 3.1.

To prove this for a given pair of fairness measures, we start with the
constraints that originate from the confusion matrices from Fig. 1, and add
the respective fairness constraints to yield a combined system of equations.
We manipulate this system of equations to express a subset of the variables in
terms of the other variables, which remain free. To ensure that the base rates
are unconstrained (as required for Definition 1), they must be among these
other free variables.

An example of calculating the constraints for combining two fairness measures
can be found in Figure 3, for the case of combining demographic parity with
equal opportunity. Using the inequalities from Figure 1 a range can be found
for the free variables and the base rates, as is done in Figure 4. This system
of inequalities shows a specific range for $FP_{b}$, the other free variable
$TP_{b}$, and both base rates $x$ and $y$ are not constrained further. The
resulting range for $FP_{b}$ is
$[max(0,\frac{-x+y}{1-y}),min(x+\frac{-x+y}{1-y},y)]$ which simplifies to
$[0,x+\frac{-x+y}{1-y}]$ if $x\geq y$, resulting in a non-zero Lebesgue
measure, or $[\frac{-x+y}{1-y},y]$ if $y>x$ then $TP_{b}$ will have a non-zero
Lebesgue measure if $x>y^{2}$. Thus the Lebesgue measure is non-zero for each
free variable for a set of base rates with a non-zero 2-dimensional Lebesgue
measure, meaning that the combination of demographic parity and equal
opportunity is possible.

The proofs for the other pairwise combinations of fairness measures follow a
similar structure. The calculations are in Appendix A. $\Box$

Note that Proposition 9 indicates that none of the fairness measures are
contradictory to each other. It is perhaps unsurprising that what is fair
according to one definition is not considered impossibly fair according to
another. In fact, two sets of three pairwise combinations are even equivalent
with each other, such that they are essentially already combinations of three
fairness measures: EOP, PP, and Tr. Eq. form one such group, and FPP, PE, and
Tr. Eq. forms the other one.

Two constraints in each system of equations are derived from the constraints
of the fairness measures and are unique to them. These two resulting
constraints for each possible pairwise combination can be found in Figure 5.
It is clear that certain combinations result in constraints that are more
easily interpreted. An exception was made for predictive parity and predictive
equality as the result becomes fairly convoluted. The constraint can be found
in the calculations of combining predictive parity, predictive equality and
overall accuracy equality in Section B.25.

Some structure can be found between these two resulting constraints. This
structure leads to defining seven types of constraint combinations. These
seven types are denoted in the top right corner of the combination in Figure
5. They serve as a kind of check for the calculations, where the same
orientations of the combined fairness definitions would lead to the same type.
A summary of the structures and associated measure orientations can be found
in Table 3. The elements in the Factor term are denoted by its subscript.

Table 3. Table showing the structure of the seven different constraint types
identified and the combination of orientations associated with the type.

This table contains the seven different constraint types and the associated
structure of these constraints. The orientations of the fairness measures
who’s combination of constraints fall into the type are also added.  Nr.
Orientation of the combined measures Constraint structure (1) $B_{v}H$
$X_{a}=Factor_{BR}*X_{b}$ $Y_{b}=Factor_{BR}*X_{b}+Y_{b}$ (2) $B_{v}V$,
$B_{d}D$ $X_{a}=X_{b}$ $Y_{a}=Y_{b}$ (3) $HH,HV,HD,VD$
$X_{a}=Factor_{BR}*X_{a}$ $Y_{a}=Factor_{BR}*Y_{b}$ (4) $B_{v}D$, $VB_{d}$
$X_{a}=Factor_{BR,X_{b},Y_{b}}*X_{b}$ $Y_{a}=Factor_{BR,X_{b},Y_{b}}*Y_{b}$
(5) $B_{v}B_{d}$ $X_{a}=X_{b}+Factor_{BR}$ $Y_{a}=Y_{b}+Factor_{BR}$ (6) $VV$
$X_{a}=Factor_{BR,Y_{a},Y_{b}}*Y_{b}+Factor_{BR,Y_{a},Y_{b}}*Y_{a}$
$X_{b}=Factor_{BR,Y_{a},Y_{b}}*Y_{b}+Factor_{BR,Y_{a},Y_{b}}*Y_{a}$ (7)
$HB_{d}$ $X_{a}=Factor_{BR}*(Y_{b}-Y_{a})$ $X_{b}=Factor_{BR}*(Y_{b}-Y_{a})$

###  3.3. Combining three Fairness measures

We now proceed to characterize the set of possible combinations of three
fairness measures.

######  Proposition 3.0.

For combining three measures out of demographic parity, equal opportunity,
false positive parity, predictive parity, predictive equality, overall
accuracy equality, and treatment equality only five out of 35 different
combinations are possible (according to Definition 1).

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x}{y}FP_{b}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{1-x}{1-y}TP_{b}=\frac{y-x}{y}FP_{b}+TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x-y}{y}FP_{b}=\frac{x-y}{1-y}TP_{b}\\\
\frac{1-x-1+y}{1-y}TP_{b}=\frac{y-x}{y}FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x}{1-y}TP_{b}\\\ FP_{b}=\frac{y}{1-y}TP_{b}\\\
TP_{b}=TP_{b}\end{cases}\lor\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\\\ 0=0\end{cases}$

Figure 6. Combining demographic parity, equal opportunity and false positive
parity resulting in a solution that has a free variable.

The calculations made in order for combining demographic parity, equal
opportunity and false positive parity, this can also be found in a formula in
the appendix

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\ FP_{a}=FP_{b}\\\ TN_{a}=TN_{b}\\\
FN_{a}=FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x-FP_{a}=y-FP_{b}\\\
1-x-TP_{a}=1-y-TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x-FP_{a}=y-FP_{a}\\\
-x-TP_{a}=-y-TP_{a}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\\\ x=y\end{cases}$

Figure 7. Combining demographic parity, predictive parity and predictive
equality resulting in a solution that requires equal base rates

The calculations made in order for combining demographic parity, predictive
parity and predict equality, this can also be found in calculations in the
appendix

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ \frac{x-y}{1-y}TP_{b}=0\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{a}\\\
TN_{a}=x-FP_{a}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ x=y\\\
TP_{a}=TP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x\\\ TN_{a}=x-FP_{a}\\\
FN_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ TP_{b}=0\\\ TP_{a}=0\\\ TP_{a}=0\\\
FP_{a}=FP_{b}\end{cases}$

Figure 8. Combining demographic parity, equal opportunity and predictive
parity

A shortened version of the calculations made in order for combining
demographic parity, equal opportunity and predictive parity, the full length
calculations can be found in the appendix.

######  Proof Outline 3.2.

In Sec. 3.2 we saw that some pairwise combinations are equivalent with each
other, in this way already giving rise to two combinations with three fairness
measures. Besides these, combining three fairness measures can be done most
conveniently by combining the constraints of two pairwise combinations with
one fairness measure in common. As an example, Figure 6 shows the calculations
for combining demographic parity, equal opportunity, and false positive
parity. The resulting constraints show that one variable, $TP_{b}$, is free,
the base rates are unconstrained, and none of the variables are fixed to a
particular value. This shows that the combination of demographic parity, equal
opportunity and false positive parity is possible.

However, not all combinations of three fairness measures are possible. An
example is the combination of demographic parity, predictive parity, and
predictive equality. The calculations can be found in Figure 7. The final set
of constraints requires $x=y$, which violates the condition of the base rates
having a non-zero 2-dimensional Lebesgue measure. It means that these three
fairness measures can only be achieved simultaneously when the base rates
happen to be equal to each other, which is entirely data-dependent and not
within the control of the algorithm. This means that the combination of
demographic parity, predictive parity and predictive equality is not possible.

Another example of how a combination can be impossible is demonstrated in the
combination of demographic parity, equal opportunity and predictive parity.
The calculations thereof are shown in Figure 8. The solution either requires
equal base rates, which violates the requirements to be deemed possible, or it
requires for $TP_{b}$ and $TP_{a}$ to equal zero. The result is that four out
of eight variables in the confusion matrices have a Lesbesgue measure of zero,
violating the requirements for being a possible combination.

The final possible kind of outcome, when combining three fairness measures
from the ones used in this paper, arises when combining predictive parity,
predictive equality and overall accuracy equality. In this case, the following
second degree polynomial needs to be satisfied:

(1)
$(-2x+1)TN_{b}^{2}+(xy-y+x+2yTN_{a}-x^{2}+2xTN_{a}-2TN_{a})TN_{b}+(xy-x+TN_{a}-2yTN_{a}+y-y^{2})TN_{a}=0.$

As this equation is non-transparent, we used a numerical approach to solve it.
We then investigated the results in plots as in Figure 9(b). Using this
approach it is clear that a free variable, e.g. $TN_{a}$, exists. It is
solvable for a range of base rates, although interestingly not for _all_ base
rates. However, this range of base rates consists of non-trivial real
intervals, such that it satisfies Definition 1. On the plots it is evident
that none of the variables have a Lebesgue measure of zero, such that the
combination of predictive parity, predictive equality and overall accuracy
equality is possible.

The calculations for all other combinations of three fairness measures can be
found in Appendix B, and the results are summarized in Table 4. $\Box$

(a) Relation true negative a versus true negative b

(b) Relation true negative a versus true positive of a and b

Figure 9. Relation between the free variable $TN_{a}$ and
$TN_{b},TP_{a},TP_{b}$ when combining predictive parity with predictive
equality and overall accuracy equality.

Two figures showing the program used to calculate the results when combining
predictive parity with predictive equality and overall accuracy equality. The
left figure shows the two base rate sliding bars designed for input and the
graph shows the true negative value for group A related to the true negative
group B values. It shows that at a certain point two lines can be drawn
relating the two. The right-hand figure shows true negative of group a plotted
against the true positive value of b and the true positive value of a. This
graph again shows the double possibility of them being related.

Table 4. Table containing whether satisfying a certain set of three fairness
measures is possible or impossible.

This table separates all the possible triplet combinations of fairness
definitions into a possible or an impossible combinations. The first column
denotes possible fairness measures combinations with the final three
containing impossible combinations of fairness measures.  . Possible
combinations Impossible combinations DP $|$ EOP $|$ FPP DP $|$ PP $|$ PE DP
$|$ EOP $|$ PP DP $|$ PE $|$ Tr. Eq. FPP $|$ PP $|$ PE EOP $|$ FPP $|$ OaE DP
$|$ PP $|$ OaE DP $|$ EOP $|$ PE DP $|$ OaE $|$ Tr. Eq. FPP $|$ PP $|$ OaE EOP
$|$ PP $|$ Tr. Eq. DP $|$ PE $|$ OaE DP $|$ EOP $|$ OaE EOP $|$ FPP $|$ PP FPP
$|$ OaE $|$ Tr. Eq. FPP $|$ PE $|$ Tr. Eq. DP $|$ PP $|$ Tr. Eq. DP $|$ FPP
$|$ Tr. Eq. EOP $|$ FPP $|$ Tr. Eq. FPP $|$ OaE $|$ PE PP $|$ PE $|$ OaE EOP
$|$ PE $|$ Tr. Eq. DP $|$ FPP $|$ PP EOP $|$ FPP $|$ PE FPP $|$ PP $|$ Tr. Eq.
EOP $|$ OaE $|$ PP DP $|$ FPP $|$ PE EOP $|$ PE $|$ OaE PP $|$ PE $|$ Tr. Eq.
EOP $|$ OaE $|$ Tr. Eq. DP $|$ FPP $|$ PE EOP $|$ PE $|$ PP PP $|$ OaE $|$ Tr.
Eq. PE $|$ OaE $|$ Tr. Eq. DP $|$ FPP $|$ Tr. Eq.

###  3.4. Combining four fairness measures

As there are only four free variables in the two confusion matrices, due to
inherent constraints as in Fig. 1, combinations of four _independent_ fairness
notions will not be possible. In other words: for a combination of four
fairness measures to be possible, at least one of the constraints should
mathematically follow from the combination of other constraints in order for
the values in the confusion matrix to have a non-zero Lebesgue measure. This
is not the case for the fairness measures considered in this paper.

######  Proposition 3.0.

Out of the fairness measures demographic parity, equal opportunity, false
positive parity, predictive parity, predictive equality, overall accuracy
equality and treatment equality no sets of four fairness measures are possible
(according to Definition 1).

###### Proof.

If a set of four fairness measures were possible, then all subsets of size
three of these four measures would be possible. However, based on Table 4, it
is easy to exhaustively verify that this is not the case: no set of size four
exists for which all its size three subsets are possible. Thus, no set of four
fairness measures is possible. ∎

###  3.5. Integration of the results

Combining the propositions above, we can now state the main result of this
paper.

###### Theorem 12.

There are twelve sets that form a maximal possible combinations of fairness
measures that are possible according to Definition 1, out of demographic
parity, equal opportunity, false positive parity, predictive parity,
predictive equality, overall accuracy equality and treatment equality. We call
these twelve combinations the _maximal fairness_ notions. Seven of these sets
are pairwise combinations and five are a combination of three fairness
measures. Of course, all subsets from these twelve sets are also possible.

###### Proof.

From Propositions 9, 10, and 10 Figure 10 is constructed, where each element
is a possible combination of fairness measures. It also shows the subset
relationships between the possible sets of fairness measures. The 12 red
coloured elements indicate that they are maximal sets with regard to possible
fairness measure combinations. ∎

Figure 10. All possible combinations of fairness measures.

On this figure all possible combinations of fairness measures are combined and
represented in a bubble. This means that when combining these measures there
is still some freedom for the predictor to work within. The bubbles are
connected where a smaller set of measures is connected to a larger set of
measures if that smaller set is a subset of the larger set. Some of the
bubbles are coloured red denoted that they are a leaf node, meaning that the
set cannot be expanded with a new measure without violating the definition of
possible.

Next we try to get some insight into the combinations that are possible. To
this end, Table 5 shows the number of times each fairness measure can be
combined with two others, and the number of times it is impossible because it
requires a trivial 2-dimensional range for base rates, or because it requires
a trivial range for variables or a trivial 2-dimensional range for base rates.
The table reveals an interesting symmetry between equal opportunity and false
positive parity and also between predictive parity and predictive equality.
This symmetry is due to their mathematical symmetry, as formalized by the
following proposition.

######  Proposition 3.0.

Given any possible combination of fairness measures, another possible
combination can be obtained by swapping the semantics of the classes, i.e. by
renaming the positive class into the negative and vice versa.

Note that intuitively, the swapping of the positive and negative classes can
be visualized as a point inversion of the contingency tables through its
center.

###### Proof.

In Fig. 11 the point inversions of all possible combinations of three fairness
measures are illustrated, showing the symmetry relations between them.

A formal proof of correctness follows from the symmetry relations between the
_individual_ fairness measures. If the positives are switched with negatives
in the constraint of equal opportunity, it becomes the constraint for false
positive parity, and vice versa. The same relation holds between predictive
parity and predictive equality. The demographic parity constraint, however,
changes into a constraint that is equivalent to the original one, and the
measures of overall accuracy equality and treatment equality remain
unaffected, by switching the positive and negative classes. ∎

Proposition 13 also provides an explanation for the symmetry between equal
opportunity and false positive parity and between predictive parity and
predictive equality. If for every possible combination their inversion is also
a possible combination, then it means that for every possible combination with
predictive parity there will be a possible combination with predictive
equality. Something analogous holds for equal opportunity and false positive
parity. This explains why in Table 5 they have the same number of occurrences
in each of the categories.

Note that Fig. 11 shows that among the five maximal combinations of three
fairness measures, three treat the positive and negative classes equally: they
remain unaltered by swapping the role of the positive and negative class.
These are (DP, EOP, FPP), (EOP, PP, AoE), and (PP, PE, AoE). The other two
possible combinations of three fairness measures do treat the classes
differently, and they represents each other’s symmetrical analogue. These are
(EOP, PP, Tr. Eq.) and (FPP, PE, Tr. Eq.).

Table 5. Table containing the prevalence of each fairness measure for a
characteristic when combined with two others.

This table contains the prevalence of each fairness measure in different
categories when combining three fairness measures. The first category requires
that the constraints still allow for one variable to be completely free. The
second category requires that the base rates are equal otherwise it is not
possible to satisfy all definitions. The third category has constraints where
at least one of the variables has only occurs in a trivial range or the base
rates are in a trivial 2-dimensional range. Demographic parity has the least
occurrences in the first group with one. Equal opportunity has the exact same
distribution as false positive parity and together are highly prevalent in the
first class. Predictive parity and predictive equality also have the same
distribution as each other. Overall accuracy equality is the most prevalent in
the first class out of all fairness measures.  Measure Nr. of times in
possible combinations Nr. of times requiring equal base rates Nr. of times
trivial range for variables or trivial 2-dim. range for base rates Demographic
Parity 1 3 11 Equal Opportunity 3 2 10 False positive parity 3 2 10 Predictive
Parity 2 6 7 Predictive Equality 2 6 7 Overall accuracy Equality 2 2 11
Treatment Equality 2 3 10

Figure 11. Point inversion of the possible combinations of three fairness
measures.

This figure shows the combinations of three measures that still leave room for
some freedom by drawing their orientations and effects on the confusion
matrix. Related to that a point of version is done in the middle of the square
signifying the confusion matrix and the resulting orientations are drawn from
this. This results in 8 squares as three out of 5 combinations when applying
the point inversion remain the same combination.

Finally, note that both Figure 10 and Table 5 indicate that possible
combinations with demographic parity are most rare. Demographic parity is a
fairness definition that does not relate to the base rates of the groups due
to it being independent from the ground truth. This is most likely the reason
why it can only be combined with equal opportunity and false positive parity
as these definitions are directly related to the base rates.

##  4\. Conclusion

We investigated which and how many fairness notions can be combined and
imposed simultaneously, for the simple setting of binary classification with
two demographic groups. We find that out of seven fairness measures, namely
demographic parity, equal opportunity, false positive parity, predictive
parity, predictive equality, overall accuracy equality and treatment equality,
in total twelve maximal combinations are possible. Five of these maximal
combinations consist of three fairness measures, while seven of them consist
of just two fairness measures. Each of these sets could be regarded as
_maximal fairness_ notions, in that it is not possible to impose any further
fairness constraints without making the problem infeasible. Of course, all
subsets of these twelve maximal sets are also possible combinations of
fairness definitions. An overview of the possible combinations, maximal as
well as non-maximal, and the subset relations between them, is given in Figure
10. Our results confirm and extend related work, such as the theorem
colloquially referred to as the fairness impossibility theorem.

We consider the five maximal sets of three fairness measures as particularly
interesting. Of these five, three treat the positive and negative classes
symmetrically, in that they remain unaltered after swapping the class labels.
The remaining two are each other’s symmetrical analogue: one of these two sets
of fairness measures is equivalent to the other set after swapping the class
labels. This property might provide some initial guidance and insight to
determine which maximal fairness notion is most appropriate in a given
application: if the costs of different kinds of misclassifications are very
different, one of the maximal fairness notions that are not invariant with
respect to a label swap might be most appropriate.

An important next step building on these results would be to develop methods
that are capable of imposing these (maximal) sets of fairness measures.
Subsequently, it would be interesting to investigate the impact of imposing
(maximal) sets of fairness measures on the accuracy and learning capabilities
(e.g. learning rate and convergence rate) of machine learning methods.

A separate line of further work would be towards including other fairness
measures, in addition to the seven measures included in the present paper.
Moreover, we consider extensions towards other problems besides binary
classification and just two demographic groups as fruitful avenues for further
research.

Finally, and perhaps most importantly, the meaning of the maximal fairness
notions derived in the present paper needs to be better understood from a
practical, ethical and/or legal point of view. This is an important aspect of
research in fairness in AI as this translation to real world concepts is a
pre-requisite for wide-spread adoption.

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##  Appendix A Combining two definitions

###  A.1. Equal Opportunity and Demographic Parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\
\frac{TP_{a}}{TP_{a}+FN_{a}}=\frac{TP_{b}}{TP_{b}+FN_{b}}\\\
\frac{TP_{a}+FP_{a}}{TP_{a}+FP_{a}+FN_{a}+TN_{a}}=\frac{TP_{b}+FP_{b}}{TP_{b}+FP_{b}+FN_{b}+TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
\frac{TP_{a}}{1-x}=\frac{TP_{b}}{1-y}\\\
TP_{a}+FP_{a}=TP_{b}+FP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=TP_{b}+FP_{b}-\frac{1-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=(1-\frac{1-x}{1-y})TP_{b}+FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{x-y}{1-y}TP_{b}-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\
TN_{b}=y-FP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\end{cases}$

####  A.1.1. Applying inequalities

$\begin{cases}0\leq 1-x-\frac{1-x}{1-y}TP_{b}\\\ 1-x-\frac{1-x}{1-y}TP_{b}\leq
1-x\\\ 0\leq x-\frac{x-y}{1-y}TP_{b}-FP_{b}\\\
x-\frac{x-y}{1-y}TP_{b}-FP_{b}\leq x\\\ 0\leq 1-y-TP_{b}\\\ 1-y-TP_{b}\leq
1-y\\\ 0\leq y-FP_{b}\\\ y-FP_{b}\leq y\\\ 0\leq\frac{1-x}{1-y}TP_{b}\\\
\frac{1-x}{1-y}TP_{b}\leq 1-x\\\ 0\leq\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{x-y}{1-y}TP_{b}+FP_{b}\leq
x\end{cases}\Longleftrightarrow\begin{cases}\frac{1}{1-y}TP_{b}\leq 1\\\
\frac{1-x}{1-y}TP_{b}\geq 0\\\ FP_{b}\leq x+\frac{-x+y}{1-y}TP_{b}\\\
\frac{-x+y}{1-y}TP_{b}\leq FP_{b}\\\ TP_{b}\leq 1-y\\\ TP_{b}\geq 0\\\
FP_{b}\leq y\\\ FP_{b}\geq 0\\\ 0\leq\frac{1-x}{1-y}\\\
\frac{1}{1-y}TP_{b}\leq 1\\\ \frac{y-x}{1-y}\leq\frac{FP_{b}}{TP_{b}}\\\
FP_{b}+\frac{x-y}{1-y}TP_{b}\leq
x\end{cases}\Longleftrightarrow\begin{cases}\frac{y-x}{1-y}\leq\frac{FP_{b}}{TP_{b}}\\\
FP_{b}+\frac{x-y}{1-y}TP_{b}\leq x\end{cases}$

###  A.2. Demographic Parity and False positive parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}+FP_{a}=TP_{b}+FP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{a}=TP_{b}+(1-\frac{x}{y})FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{y-x}{y}FP_{b}-TP_{b}\\\
TN_{a}=x-\frac{x}{y}FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\end{cases}$

####  A.2.1. Added constraints through inequalities

$\begin{cases}\frac{x-y}{y}\leq\frac{TP_{b}}{FP_{b}}\\\
1-x\geq\frac{y-x}{y}FP_{b}+TP_{b}\\\ \end{cases}$

###  A.3. Demographic Parity and Predictive Parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}*FP_{b}=TP_{b}*FP_{a}\\\
TP_{a}+FP_{a}=TP_{b}+FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
\frac{TP_{b}*FP_{a}}{FP_{b}}+FP_{a}=TP_{b}+FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
\frac{FP_{b}+TP_{b}}{FP_{b}}*FP_{a}=TP_{b}+FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
FP_{a}=FP_{b}\lor
FP_{b}+TP_{b}=0\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x-FP_{a}\\\
TN_{a}=x-FP_{a}\\\ FN_{b}=1-y\\\ TN_{b}=y\\\ TP_{a}=FP_{a}\\\
FP_{b}=TP_{b}=0\end{cases}$

####  A.3.1. Added constraints through inequalities

$\begin{cases}1-x\geq TP_{b}\\\ x\geq FP_{b}\end{cases}$

###  A.4. Demographic parity and predictive equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FN_{a}*TN_{b}=FN_{b}*TN_{a}\\\
TP_{a}+FP_{a}=TP_{b}+FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
1-x-FN_{a}+x-TN_{a}=1-y-FN_{b}+y-TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
FN_{a}+TN_{a}=FN_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
\frac{FN_{b}*TN_{a}}{TN_{b}}+TN_{a}=FN_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
\frac{FN_{b}+TN_{b}}{TN_{b}}*TN_{a}=FN_{b}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=TN_{b}\lor FN_{b}=TN_{b}=0\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{a}\\\
FP_{a}=x-FN_{a}\\\ TP_{b}=1-y\\\ FP_{b}=y\\\ FN_{b}=TN_{b}=0\\\
FN_{a}=TN_{a}\end{cases}\lor\begin{cases}TP_{a}=1-x-FN_{b}\\\
FP_{a}=x-TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\ TN_{a}=TN_{b}\\\
FN_{a}=FN_{b}\end{cases}$

####  A.4.1. Added constraints through inequalities

$\begin{cases}FN_{b}\leq 1-x\\\ TN_{b}\leq x\end{cases}$

###  A.5. Demographic Parity and Overall accuracy Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}+FP_{a}=TP_{b}+FP_{b}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}-TP_{b}=FP_{b}-FP_{a}\\\ TP_{a}-TP_{b}=TN_{b}-TN_{a}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{b}-TN_{a}=FP_{b}-FP_{a}\\\
TP_{a}-TP_{b}=FP_{b}-FP_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
y-FP_{b}-x+FP_{a}=FP_{b}-FP_{a}\\\ TP_{a}-TP_{b}=FP_{b}-FP_{a}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ y-x=2FP_{b}-2FP_{a}\\\
TP_{a}-TP_{b}=FP_{b}-FP_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{b}-FP_{a}=\frac{y-x}{2}\\\
TP_{a}-TP_{b}=\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}-\frac{y-x}{2}\\\
TN_{a}=x-FP_{b}-\frac{x-y}{2}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\ TP_{a}=TP_{b}+\frac{y-x}{2}\end{cases}$

####  A.5.1. Added constraints through inequalities

$\begin{cases}TP_{b}\leq 1-\frac{x+y}{2}\\\ TP_{b}\geq\frac{x-y}{2}\\\
FP_{b}\leq\frac{x+y}{2}\\\ FP_{b}\geq\frac{y-x}{2}\\\ \end{cases}$

###  A.6. Demographic parity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}+FP_{a}=TP_{b}+FP_{b}\\\
\frac{FN_{a}}{FP_{a}}=\frac{FN_{b}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=\frac{FN_{a}*FP_{b}}{FN_{b}}\\\
1-x-FN_{a}+\frac{FN_{a}*FP_{b}}{FN_{b}}=1-y-FN_{b}+FP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=\frac{FN_{a}*FP_{b}}{FN_{b}}\\\
\frac{FP_{b}-FN_{b}}{FN_{b}}FN_{a}=x-y+FP_{b}-FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
TN_{a}=x-FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\ TP_{b}=1-y-FN_{b}\\\
TN_{b}=y-FP_{b}\\\ FN_{a}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\lor\begin{cases}TP_{a}=1-x-FN_{a}\\\
TN_{a}=x-FN_{a}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FN_{b}\\\ FP_{b}=FN_{b}\\\
x=y\\\ FP_{a}=FN_{a}\end{cases}$

####  A.6.1. Added constraints through inequalities

$\begin{cases}\frac{x-y}{FP_{b}-FN_{b}}\geq-1\\\ 1-x\geq FN_{b}\\\ x\geq
FP_{b}\end{cases}$

###  A.7. Equalised Odds

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{FP_{a}}{FP_{a}+TN_{a}}=\frac{FP_{b}}{FP_{b}+TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{FP_{a}}{x}=\frac{FP_{b}}{y}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{x}{y}FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\end{cases}$

###  A.8. Equal Opportunity and Predictive Parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{TP_{a}}{TP_{a}+FP_{a}}=\frac{TP_{b}}{TP_{b}+FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TP_{a}*FP_{b}=TP_{b}*FP_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{1-x}{1-y}*TP_{b}*FP_{b}=TP_{b}*FP_{a}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{1-x}{1-y}FP_{b}=FP_{a}\lor
TP_{b}=0\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x\\\
TN_{a}=x-FP_{a}\\\ FN_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=0\\\
TP_{b}=0\end{cases}$

####  A.8.1. Added constraint through inequalities

$FP_{b}\leq x\frac{1-y}{1-x}FP_{b}$

###  A.9. Equal Opportunity and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{FN_{a}}{FN_{a}+TN_{a}}=\frac{FN_{b}}{FN_{b}+TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FN_{a}*TN_{b}=FN_{b}*TN_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
1-x-FN_{a}=\frac{1-x}{1-y}(1-y-FN_{b})\\\
FN_{a}*TN_{b}=FN_{b}*TN_{a}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
\frac{1-x}{1-y}FN_{b}*TN_{b}=FN_{b}*TN_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ \frac{1-x}{1-y}TN_{b}=TN_{a}\lor
FN_{b}=0\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=x-\frac{1-x}{1-y}TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
FP_{a}=x-TN_{a}\\\ TP_{b}=1-y\\\ FP_{b}=y-TN_{b}\\\ FN_{a}=0\\\
FN_{b}=0\end{cases}$

####  A.9.1. Added constraint through inequalities

$TN_{b}\leq\frac{1-y}{1-x}*x$

###  A.10. Equal opportunity and Overall accuracy equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{1-x}{1-y}TP_{b}-TP_{b}=TN_{b}-TN_{a}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{y-x}{1-y}TP_{b}=TN_{b}-TN_{a}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{y-x}(TN_{b}-TN_{a})\\\
FP_{a}=x-TN_{a}\\\ FN_{b}=1-y-\frac{1-y}{y-x}(TN_{b}-TN_{a})\\\
FP_{b}=y-TN_{b}\\\ TP_{a}=\frac{1-x}{y-x}(TN_{b}-TN_{a})\\\
TP_{b}=\frac{1-y}{y-x}(TN_{b}-TN_{a})\end{cases}$

####  A.10.1. Added constraints through inequalities

$\begin{cases}y>x\\\ TN_{a}\leq TN_{b}\\\ y-x\geq TN_{b}-TN_{a}\\\
\end{cases}\lor\begin{cases}x>y\\\ TN_{b}\leq TN_{b}\\\ y-x\leq
TN_{b}-TN_{a}\\\ \end{cases}$

###  A.11. Equal opportunity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{FN_{a}}{FP_{a}}=\frac{FN_{b}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
\frac{1-x}{1-y}\frac{FN_{b}}{FP_{a}}=\frac{FN_{b}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=\frac{1-x}{1-y}FP_{b}\lor
FN_{b}=0\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
TN_{a}=x-FP_{a}\\\ TP_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ FN_{a}=0\\\
FN_{b}=0\end{cases}$

####  A.11.1. Added constraints through inequalities

$\frac{1-y}{1-x}x\geq FP_{b}$

###  A.12. False positive parity and Predictive Parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{a}*FP_{b}=TP_{b}*FP_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{a}*FP_{b}=TP_{b}*\frac{x}{y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{a}\\\
TN_{a}=x-\frac{x}{y}FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ TP_{a}=\frac{x}{y}TP_{b}\lor FP_{b}=0\end{cases}$

####  A.12.1. Added constraints through inequalities

$\frac{y}{x}(1-x)\geq TP_{b}$

###  A.13. False positive parity and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FN_{a}*TN_{b}=FN_{b}*TN_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
x-TN_{a}=\frac{x}{y}(y-TN_{b})\\\ FN_{a}*TN_{b}=FN_{b}*TN_{a}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
FN_{a}*TN_{b}=FN_{b}*\frac{x}{y}*TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{a}\\\
FP_{a}=x-\frac{x}{y}TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ FN_{a}=\frac{x}{y}FN_{b}\lor TN_{b}=0\end{cases}$

####  A.13.1. Added constraints through inequalities

$\frac{y}{x}(1-x)\geq FN_{b}$

###  A.14. False positive parity and Overall accuracy Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=\frac{x}{y}TN_{b}\\\
TP_{a}+\frac{x}{y}TN_{b}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ \frac{x-y}{y}TN_{b}=TP_{b}-TP_{a}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{b}=\frac{y}{x-y}(TP_{b}-TPa)\\\
TN_{a}=\frac{x}{y}\frac{y}{x-y}(TP_{b}-TP_{a})\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{a}\\\
FP_{a}=x-\frac{x}{x-y}(TP_{b}-TP_{a})\\\ FN_{b}=1-y-TP_{b}\\\
FP_{b}=y-\frac{y}{x-y}(TP_{b}-TPa)\\\ TN_{b}=\frac{y}{x-y}(TP_{b}-TPa)\\\
TN_{a}=\frac{x}{x-y}(TP_{b}-TP_{a})\end{cases}$

####  A.14.1. Added constraints through inequalities

$\begin{cases}x>y\\\ TP_{a}\leq TP_{b}\\\ x-y\geq TP_{b}-TP_{a}\\\
\end{cases}\lor\begin{cases}x<y\\\ TP_{a}\geq TP_{b}\\\ x-y\leq
TP_{b}-TP_{a}\\\ \end{cases}$

###  A.15. False positive parity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{FN_{a}}{FP_{a}}=\frac{FN_{b}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=\frac{x}{y}FP_{b}\\\
FN_{a}*FP_{b}=FN_{b}*\frac{x}{y}*FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{a}\\\
TN_{a}=x-\frac{x}{y}FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ FN_{a}=\frac{x}{y}*FN_{b}\lor FP_{b}=0\end{cases}$

####  A.15.1. Added constraints through inequalities

$\frac{y}{x}(1-x)\geq FN_{b}$

###  A.16. Predictive Parity and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{TP_{b}FP_{a}}{FP_{b}}\\\
FN_{a}=\frac{FN_{b}TN_{a}}{TN_{b}}\end{cases}$

###  A.17. Predictive Parity and Overall accuracy Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}*FP_{b}=TP_{b}*FP_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
\frac{TP_{b}*FP_{a}}{FP_{b}}+x-FP_{a}=TP_{b}+y-FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
\frac{FP_{a}-FP_{b}}{FP_{b}}TP_{b}=y-x+FP_{a}-FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
TP_{b}=FP_{b}\frac{y-x}{FP_{a}-FP_{b}}+FP_{b}\lor(FP_{a}=FP_{b}\land
x=y)\end{cases}$
$\Longleftrightarrow\begin{cases}FN_{a}=1-x-FP_{a}\frac{y-x}{FP_{a}-FP_{b}}-FP_{a}\\\
TN_{a}=x-FP_{a}\\\ FN_{b}=1-y-FP_{b}\frac{y-x}{FP_{a}-FP_{b}}-FP_{b}\\\
TN_{b}=y-FP_{b}\\\ TP_{b}=FP_{b}\frac{y-x}{FP_{a}-FP_{b}}+FP_{b}\\\
TP_{a}=FP_{a}\frac{y-x}{FP_{a}-FP_{b}}+FP_{a}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\land x=y\end{cases}$

####  A.17.1. Added constraints through inequalities

$\begin{cases}1-y\geq 1-x\\\ FP_{a}\geq FP_{b}\\\
\frac{1-y-FP_{a}}{1-x-FP_{a}}\geq\frac{FP_{b}}{FP_{a}}\\\
\frac{1-y+FP_{b}}{1-x+FP_{b}}\geq\frac{FP_{b}}{FP_{a}}\end{cases}\lor\begin{cases}1-y\leq
1-x\\\ FP_{a}\leq FP_{b}\\\
\frac{1-y-FP_{a}}{1-x-FP_{a}}\leq\frac{FP_{b}}{FP_{a}}\\\
\frac{1-y+FP_{b}}{1-x+FP_{b}}\leq\frac{FP_{b}}{FP_{a}}\end{cases}$

###  A.18. Predictive Parity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}*FP_{b}=TP_{b}*FP_{a}\\\
FN_{a}*FP_{b}=FN_{b}*FP_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\\\
(1-x-\frac{FN_{b}*FP_{a}}{FP_{b}})*FP_{b}=(1-y-FN_{b})*FP_{a}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\\\
FP_{b}-xFP_{b}-FN_{b}*FP_{a}=FP_{a}-yFP_{a}-FN_{b}*FP_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\\\
FP_{b}-xFP_{b}=FP_{a}-yFP_{a}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}$

####  A.18.1. Added constraints through inequalities

$\frac{1-y}{1-x}x\geq FP_{b}$

###  A.19. Predictive Equality and Overall accuracy Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FN_{a}*TN_{b}=FN_{b}*TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
1-x-FN_{a}+TN_{a}=1-y-FN_{b}+TN_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
\frac{TN_{b}-FN_{b}}{TN_{b}}TN_{a}=x-y+TN_{b}-FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=TN_{b}\frac{x-y}{TN_{b}-FN_{b}}+TN_{b}\lor(TN_{b}=FN_{b}\land x=y)\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\frac{x-y}{TN_{b}-FN_{b}}-FN_{b}\\\
FP_{a}=x-TN_{b}\frac{x-y}{TN_{b}-FN_{b}}-TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\
FP_{b}=y-TN_{b}\\\ TN_{a}=TN_{b}\frac{x-y}{TN_{b}-FN_{b}}+TN_{b}\\\
FN_{a}=FN_{b}\frac{x-y}{TN_{b}-FN_{b}}+FN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x-TN_{a}\\\
FP_{a}=x-TN_{a}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-FN_{b}\\\ TN_{b}=FN_{b}\land
x=y\\\ FN_{a}=TN_{a}\end{cases}$

####  A.19.1. Added constraints through inequalities

$\begin{cases}TN_{b}\geq FN_{b}\\\ y\geq x\\\
\frac{x-TN_{b}}{y-TN_{b}}\leq\frac{TN_{b}}{FN_{b}}\\\
\frac{1-x-FN_{b}}{1-y-FN_{b}}\geq\frac{FN_{b}}{TN_{b}}\end{cases}\lor\begin{cases}TN_{b}\leq
FN_{b}\\\ y\leq x\\\ \frac{x-TN_{b}}{y-TN_{b}}\geq\frac{TN_{b}}{FN_{b}}\\\
\frac{1-x-FN_{b}}{1-y-FN_{b}}\leq\frac{FN_{b}}{TN_{b}}\end{cases}$

###  A.20. Predictive Equality and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FN_{a}*TN_{b}=FN_{b}*TN_{a}\\\
\frac{FN_{a}}{FP_{a}}=\frac{FN_{b}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
\frac{FN_{b}*FP_{a}}{FP_{b}}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}*FP_{b}=TN_{b}*FP_{a}\lor
FN_{b}=0\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}(y-TN_{b})=TN_{b}(x-TN_{a})\lor FN_{b}=0\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\lor
FN_{b}=0\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{x}{y}FN_{b}\\\
FP_{a}=x-\frac{x}{y}TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\
TN_{a}=\frac{x}{y}TN_{b}\\\
FN_{a}=\frac{x}{y}FN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
TN_{a}=x-FP_{a}\\\ TP_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ FN_{b}=0\\\
FN_{a}=0\end{cases}$

####  A.20.1. Added constraints through inequalities

$\frac{y}{x}(1-x)\geq TN_{b}$

###  A.21. Overall accuracy Equality and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}+FN_{a}=FP_{b}+FN_{b}\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}+\frac{FN_{b}*FP_{a}}{FP_{b}}=FP_{b}+FN_{b}\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}\frac{FP_{b}+FN_{b}}{FP_{b}}=FP_{b}+FN_{b}\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\lor FP_{b}=FN_{b}=0\\\
FN_{a}=\frac{FN_{b}*FP_{a}}{FP_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\\\
TN_{a}=x-FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=FP_{b}\\\
FN_{a}=FN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\ TN_{a}=x-FP_{a}\\\
TP_{b}=1-y\\\ TN_{b}=y\\\ FP_{b}=FN_{b}=0\\\ FN_{a}=0\end{cases}$

####  A.21.1. Added constraints through inequalities

$\begin{cases}x\geq FP_{b}\\\ 1-x\geq FN_{b}\end{cases}$

##  Appendix B Combining three definitions

###  B.1. Demographic Parity and Equalised Odds

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x}{y}FP_{b}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{1-x}{1-y}TP_{b}=\frac{y-x}{y}FP_{b}+TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x-y}{y}FP_{b}=\frac{x-y}{1-y}TP_{b}\\\
\frac{1-x-1+y}{1-y}TP_{b}=\frac{y-x}{y}FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ FP_{b}=\frac{y}{1-y}TP_{b}\lor x=y\\\
\frac{y-x}{1-y}TP_{b}=\frac{y-x}{y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}=\frac{y}{1-y}TP_{b}\\\
\frac{y-x}{1-y}TP_{b}=\frac{y-x}{y}\frac{y}{1-y}TP_{b}\end{cases}\lor\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\\\ 0=0\end{cases}$
$\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{x}{1-y}TP_{b}\\\ FN_{b}=1-y-TP_{b}\\\
TN_{b}=y-\frac{y}{1-y}TP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x}{1-y}TP_{b}\\\ FP_{b}=\frac{y}{1-y}TP_{b}\\\
TP_{b}=TP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\\\ 0=0\end{cases}$

###  B.2. Demographic Parity, Equal Opportunity and Predictive Parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{a}\\\
TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ \frac{x-y}{1-y}TP_{b}=0\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
x-y=0\lor TP_{b}=0\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ x=y\\\
TP_{a}=TP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x\\\ TN_{a}=x-FP_{b}\\\
FN_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ TP_{b}=0\\\ TP_{a}=0\\\ TP_{a}=0\\\
FP_{a}=FP_{b}\end{cases}$

###  B.3. Demographic Parity, Equal Opportunity and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\ TN_{a}=TN_{b}\\\
FN_{a}=FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
-FP_{a}=-x+y-FP_{b}\\\
-TP_{a}=-1+x+1-y-TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ TP_{a}=-x+y+TP_{b}\\\
-x+y+TP_{b}=\frac{1-x}{1-y}TP_{b}\\\
x-y+FP_{b}=\frac{x-y}{1-y}TP_{b}+FP_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=x-y+FP_{b}\\\
TP_{a}=-x+y+TP_{b}\\\ -x+y=\frac{y-x}{1-y}TP_{b}\\\
x-y=\frac{x-y}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ TP_{a}=-x+y+TP_{b}\\\ 1=\frac{1}{1-y}TP_{b}\lor x=y\\\
1=\frac{1}{1-y}TP_{b}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=0\\\ TN_{a}=y-FP_{b}\\\
FN_{b}=0\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=x-y+FP_{b}\\\ TP_{a}=1-x\\\
TP_{b}=1-y\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\ TN_{a}=x-FP_{b}\\\
FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=FP_{b}\\\ TP_{a}=TP_{b}\\\
x=y\end{cases}$

###  B.4. Demographic Parity, Equal Opportunity and Overall accuracy Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\ FP_{a}=FP_{b}+\frac{x-y}{2}\\\
TP_{a}=TP_{b}+\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{x-y}{1-y}TP_{b}+FP_{b}=FP_{b}+\frac{x-y}{2}\\\
\frac{1-x}{1-y}TP_{b}=TP_{b}+\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{x-y}{1-y}TP_{b}=\frac{x-y}{2}\\\
\frac{y-x}{1-y}TP_{b}=\frac{y-x}{2}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
\frac{x-y}{1-y}TP_{b}=\frac{x-y}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
TP_{b}=\frac{1-y}{2}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=\frac{1-x}{2}\\\
TN_{a}=x-\frac{x-y}{2}-FP_{b}\\\ FN_{b}=\frac{1-y}{2}\\\ TN_{b}=y-FP_{b}\\\
TP_{a}=\frac{1-x}{2}\\\ FP_{a}=\frac{x-y}{2}+FP_{b}\\\
TP_{b}=\frac{1-y}{2}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\end{cases}$

###  B.5. Demographic Parity, Equal Opportunity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{x-y}{1-y}TP_{b}+FP_{b}\\\
FN_{a}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=\frac{x-y}{1-y}(1-y-FN_{b})+FP_{b}\\\
FN_{a}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\
\frac{1-x}{1-y}FN_{b}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\
\frac{1-x}{1-y}=(\frac{x-y}{FP_{b}-FN_{b}}+1)\lor FN_{b}=0\\\
x-y-\frac{x-y}{1-y}FN_{b}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}})\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\
-\frac{1}{1-y}=\frac{1}{FP_{b}-FN_{b}}\lor FN_{b}=0\lor x=y\\\
(x-y)(1-\frac{1}{1-y}FN_{b})=FP_{b}\frac{x-y}{FP_{b}-FN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\
-1+y=FP_{b}-FN_{b}\lor FN_{b}=0\lor x=y\\\
1-\frac{1}{1-y}FN_{b}=\frac{FP_{b}}{FP_{b}-FN_{b}}\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\ -1+y=FP_{b}-FN_{b}\lor FN_{b}=0\lor
x=y\\\ FP_{b}-FN_{b}-\frac{1}{1-y}(FP_{b}-FN_{b})FN_{b}=FP_{b}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\
-1+y=FP_{b}-FN_{b}\lor FN_{b}=0\lor x=y\\\
-1-\frac{1}{1-y}(FP_{b}-FN_{b})=0\lor x=y\lor FN_{b}=0\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\ -1+y=FP_{b}-FN_{b}\lor FN_{b}=0\lor
x=y\\\ FP_{b}-FN_{b}=-1+y\lor x=y\lor
FN_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}+FP_{b}\\\
FP_{b}=-1+y+FN_{b}\lor FN_{b}=0\lor x=y\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=y+\frac{x-y}{1-y}FN_{b}\\\ FN_{b}=1-y\\\ TN_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=x-y-\frac{x-y}{1-y}FN_{b}\\\
FP_{b}=-TP_{b}=0\end{cases}\lor\begin{cases}TP_{a}=1-x\\\ TN_{a}=y+FP_{b}\\\
TP_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ FN_{a}=0\\\ FP_{a}=x-y-FP_{b}\\\
FN_{b}=0\end{cases}\lor\begin{cases}TP_{a}=1-x-FN_{b}\\\ TN_{a}=x-FP_{b}\\\
TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ FN_{a}=FN_{b}\\\ FP_{a}=FP_{b}\\\
x=y\end{cases}$

###  B.6. Demographic Parity, False positive parity and Predictive Parity

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{b}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ FP_{b}=\frac{x}{y}FP_{b}\\\
TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
0=\frac{y-x}{y}FP_{b}\\\ 1=\frac{x}{y}\lor FP_{b}=0\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ x=y\lor FP_{b}=0\\\ x=y\lor
FP_{b}=0\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ x=y\\\
TP_{a}=TP_{b}\\\ FP_{a}=FP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y\\\ FP_{b}=0\\\ TP_{a}=TP_{b}\\\
FP_{a}=0\end{cases}$

###  B.7. Demographic Parity, False positive parity and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ TN_{a}=TN_{b}\\\
FN_{a}=FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ TN_{b}=\frac{x}{y}TN_{b}\\\
TN_{a}=TN_{b}\\\
-TP_{a}=-1+x+1-y-TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ 1=\frac{x}{y}\lor TN_{b}=0\\\
TN_{a}=TN_{b}\\\ TP_{a}=-x+y+TP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
-x+y+TP_{b}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ x=y\lor TN_{b}=0\\\ TN_{a}=TN_{b}\\\
TP_{a}=-x+y+TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
-x+y=\frac{y-x}{y}FP_{b}\\\ x=y\lor TN_{b}=0\\\ TN_{a}=TN_{b}\\\
TP_{a}=-x+y+TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
1=\frac{1}{y}FP_{b}\lor x=y\\\ x=y\lor TN_{b}=0\\\ TN_{a}=TN_{b}\\\
TP_{a}=-x+y+TP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ y=FP_{b}\lor x=y\\\ x=y\lor
TN_{b}=0\\\ TN_{a}=TN_{b}\\\
TP_{a}=-x+y+TP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
FP_{a}=x-TN_{b}\\\ FN_{b}=1-y-TP_{b}\\\ FP_{b}=y-TN_{b}\\\ x=y\\\ x=y\\\
TN_{a}=TN_{b}\\\ TP_{a}=TP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-y-TP_{b}\\\
FP_{a}=x\\\ FN_{b}=1-y-TP_{b}\\\ 0=0\\\ FP_{b}=y\\\ TN_{b}=0\\\ TN_{a}=0\\\
TP_{a}=-x+y+TP_{b}\end{cases}$

###  B.8. Demographic Parity, False positive parity and Overall accuracy
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ TP_{a}=TP_{b}+\frac{y-x}{2}\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{y-x}{y}FP_{b}+TP_{b}=TP_{b}+\frac{y-x}{2}\\\
\frac{x}{y}FP_{b}=FP_{b}+\frac{x-y}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{y-x}{y}FP_{b}=\frac{y-x}{2}\\\
\frac{x-y}{y}FP_{b}=\frac{x-y}{2}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ \frac{1}{y}FP_{b}=\frac{1}{2}\lor x=y\\\
\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}=\frac{y}{2}\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{y-x}{2}-TP_{b}\\\
TN_{a}=\frac{x}{2}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=\frac{y}{2}\\\
TP_{a}=\frac{y-x}{2}+TP_{b}\\\ FP_{a}=\frac{x}{2}\\\
FP_{b}=\frac{y}{2}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\end{cases}$

###  B.9. Demographic Parity, False positive parity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{y-x}{y}FP_{b}+TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ FN_{a}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
-FN_{a}=-1+x+1-y-FN_{b}+\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x}{y}FP_{b}=FP_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FN_{a}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x}{y}=(\frac{x-y}{FP_{b}-FN_{b}}+1)\lor FP_{b}=0\\\
y-x+FN_{b}-\frac{y-x}{y}FP_{b}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{x-y}{y}=\frac{x-y}{FP_{b}-FN_{b}}\lor FP_{b}=0\\\
y-x-\frac{y-x}{y}FP_{b}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}})\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}-FN_{b}=y\lor FP_{b}=0\lor x-y=0\\\
1-\frac{1}{y}FP_{b}=-FN_{b}*\frac{1}{FP_{b}-FN_{b}}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}-FN_{b}=y\lor FP_{b}=0\lor x-y=0\\\
y-FP_{b}=-yFN_{b}*\frac{1}{FP_{b}-FN_{b}}\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}-FN_{b}=y\lor FP_{b}=0\lor x-y=0\\\
yFP_{b}-yFN_{b}-FP_{b}^{2}+FN_{b}FP_{b}=-yFN_{b}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}-FN_{b}=y\lor FP_{b}=0\lor x-y=0\\\
yFP_{b}-FP_{b}^{2}+FN_{b}FP_{b}=0\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}-FN_{b}=y\lor FP_{b}=0\lor x=y\\\ y-FP_{b}+FN_{b}=0\lor x=y\lor
FP_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{b}-FN_{b}=y\lor FP_{b}=0\lor x=y\\\ FP_{b}-FN_{b}=y\lor x=y\lor
FP_{b}=0\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FN_{b}=FP_{b}-y\lor FP_{b}=0\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{a}\\\
TN_{a}=x-FP_{a}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ FN_{b}=FP_{b}-y\\\
FN_{a}=-x(1-\frac{1}{y}FP_{b})\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\end{cases}\lor\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{y}FP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FP_{a}=0\lor FP_{a}=FP_{b}\\\ \end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}=1-x\\\ TN_{a}=0\\\ TP_{b}=1-y\\\
TN_{b}=0\\\ FN_{b}=0\\\ FN_{a}=0,FP_{b}=y\\\ FP_{a}=x\\\
\end{cases}\lor\begin{cases}TP_{a}=1-y-FN_{b}\\\ TN_{a}=x\\\
TP_{b}=1-y-FN_{b}\\\ TN_{b}=y\\\ FP_{b}=0\\\ FN_{a}=y-x+FN_{b}\\\ FP_{a}=0\\\
\end{cases}\lor\begin{cases}TP_{a}=1-x-FN_{b}\\\ TN_{a}=x-FP_{b}\\\
TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ x=y\\\ FN_{a}=FN_{b}\\\
FP_{a}=FP_{b}\\\ \end{cases}$

###  B.10. Demographic Parity, Predictive Parity and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\ FP_{a}=FP_{b}\\\ TN_{a}=TN_{b}\\\
FN_{a}=FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x-FP_{a}=y-FP_{b}\\\
1-x-TP_{a}=1-y-TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x-FP_{a}=y-FP_{a}\\\
-x-TP_{a}=-y-TP_{a}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\\\ x=y\end{cases}$

###  B.11. Demographic Parity, Predictive Parity and Overall Accuracy equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\ FP_{a}=FP_{b}\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\
TP_{a}=TP_{b}+\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ FP_{a}=FP_{a}+\frac{x-y}{2}\\\
TP_{a}=TP_{a}+\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ 0=\frac{x-y}{2}\\\
0=\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\\\ x=y\end{cases}$

###  B.12. Demographic Parity, Predictive Parity and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=TP_{b}\\\ FP_{a}=FP_{b}\\\
FN_{a}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
1-x-FN_{a}=1-y-FN_{b}\\\ FP_{a}=FP_{b}\\\
FN_{a}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{b}=FP_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=y-x+FN_{b}\\\ FP_{a}=FP_{b}\\\
y-x+FN_{b}=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
1=\frac{x-y}{FP_{b}-FN_{b}}+1\lor FP_{b}=0\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=y-x+FN_{b}\\\ FP_{a}=FP_{b}\\\
y-x=FN_{b}*(\frac{x-y}{FP_{b}-FN_{b}})\\\ x=y\lor
FP_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\\\
TN_{a}=x-FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ FN_{a}=FN_{b}\\\
FP_{a}=FP_{b}\\\ 0=0\\\ x=y\end{cases}\lor\begin{cases}TP_{a}=1-y-FN_{b}\\\
TN_{a}=x\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y\\\ FN_{a}=y-x+FN_{b}\\\
FP_{a}=FP_{b}=0\\\ y-x=-(x-y)\\\ FP_{b}=0\end{cases}$

###  B.13. Demographic Parity, Predictive Equality and Overall accuracy
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TN_{a}=TN_{b}\\\ FN_{a}=FN_{b}\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\
TP_{a}=TP_{b}+\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
x-FP_{a}=y-FP_{b}\\\ 1-x-TP_{a}=1-y-TP_{b}\\\ FP_{a}=FP_{b}+\frac{x-y}{2}\\\
TP_{a}=TP_{b}+\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ TP_{a}=y-x+TP_{b}\\\ x-y+FP_{b}=FP_{b}+\frac{x-y}{2}\\\
y-x+TP_{b}=TP_{b}+\frac{y-x}{2}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=x-y+FP_{b}\\\
TP_{a}=y-x+TP_{b}\\\ x-y=\frac{x-y}{2}\\\
y-x=\frac{y-x}{2}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ TP_{a}=y-x+TP_{b}\\\ x=y\\\
x=y\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=FP_{b}\\\
TP_{a}=TP_{b}\\\ x=y\end{cases}$

###  B.14. Demographic Parity, Predictive Equality and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=x-y+FP_{b}\\\ FN_{a}=FN_{b}\\\
FN_{a}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ FN_{a}=FN_{b}\\\
FN_{b}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
x-y+FP_{b}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ FN_{a}=FN_{b}\\\ 1=\frac{x-y}{FP_{b}-FN_{b}}+1\lor
FN_{b}=0\\\ x-y=FP_{b}\frac{x-y}{FP_{b}-FN_{b}}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=x-y+FP_{b}\\\ FN_{a}=FN_{b}\\\
0=\frac{x-y}{FP_{b}-FN_{b}}\lor FN_{b}=0\\\ 1=\frac{FP_{b}}{FP_{b}-FN_{b}}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ FN_{a}=FN_{b}\\\ x=y\lor FN_{b}=0\\\
FP_{b}-FN_{b}=FP_{b}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=x-y+FP_{b}\\\ FN_{a}=FN_{b}\\\ x=y\lor FN_{b}=0\\\ FN_{b}=0\lor
x=y\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\\\
TN_{a}=x-FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=FP_{b}\\\
FN_{a}=FN_{b}\\\ x=y\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
TN_{a}=y-FP_{b}\\\ TP_{b}=1-y\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=x-y+FP_{b}\\\
FN_{a}=0\\\ FN_{b}=0\end{cases}$

###  B.15. Demographic Parity, Overall accuracy Equality and Treatment
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}+\frac{x-y}{2}\\\
TP_{a}=TP_{b}+\frac{y-x}{2}\\\ FN_{a}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{a}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\ FN_{a}=y-x+FN_{b}-\frac{y-x}{2}\\\
y-x+FN_{b}-\frac{y-x}{2}=FN_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\\\
FP_{b}+\frac{x-y}{2}=FP_{b}(\frac{x-y}{FP_{b}-FN_{b}}+1)\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}+\frac{x-y}{2}\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{2}\\\
\frac{y-x}{2}=FN_{b}\frac{x-y}{FP_{b}-FN_{b}}\\\
\frac{x-y}{2}=FP_{b}\frac{x-y}{FP_{b}-FN_{b}}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\ FN_{a}=y-x+FN_{b}-\frac{y-x}{2}\\\
-\frac{1}{2}=FN_{b}\frac{1}{FP_{b}-FN_{b}}\lor x=y\\\
\frac{1}{2}=FP_{b}\frac{1}{FP_{b}-FN_{b}}\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}+\frac{x-y}{2}\\\
FN_{a}=y-x+FN_{b}-\frac{y-x}{2}\\\ \frac{-FP_{b}+FN_{b}}{2}=FN_{b}\lor x=y\\\
\frac{FP_{b}-FN_{b}}{2}=FP_{b}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\ FN_{a}=FN_{b}+\frac{y-x}{2}\\\
\frac{-FP_{b}}{2}=\frac{FN_{b}}{2}\lor x=y\\\
\frac{-FN_{b}}{2}=\frac{FP_{b}}{2}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=FP_{b}+\frac{x-y}{2}\\\ FN_{a}=FN_{b}+\frac{y-x}{2}\\\
-FP_{b}=FN_{b}=0\lor x=y\\\ -FN_{b}=FP_{b}=0\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{y-x}{2}\\\
TN_{a}=x-\frac{x-y}{2}\\\ TP_{b}=1-y\\\ TN_{b}=y\\\ FP_{a}=\frac{x-y}{2}\\\
FN_{a}=\frac{y-x}{2}\\\
-FP_{b}=FN_{b}=0\end{cases}\lor\begin{cases}TP_{a}=1-x-FN_{a}\\\
TN_{a}=x-FP_{a}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=FP_{b}\\\
FN_{a}=FN_{b}\\\ x=y\end{cases}$

###  B.16. Equalised odds and Predictive Parity/Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ \frac{1-x}{1-y}FP_{b}=\frac{x}{y}FP_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ \frac{1-x}{1-y}=\frac{x}{y}\lor FP_{b}=0\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
y-xy=x-xy\lor FP_{b}=0\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ y=x\lor FP_{b}=0\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
y=x\\\
FP_{a}=FP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x-\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{b}=0\\\ FP_{a}=0\end{cases}$

###  B.17. Equalised odds and Predictive Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
\frac{x}{y}TN_{b}=\frac{1-x}{1-y}TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
\frac{x}{y}=\frac{1-x}{1-y}\lor TN_{b}=0\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ x=y\lor
TN_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\\\
FP_{a}=x-TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\ FN_{a}=FN_{b}\\\
TN_{a}=TN_{b}\\\
x=y\end{cases}\lor\begin{cases}TP_{a}=1-x-\frac{1-x}{1-y}FN_{b}\\\ FP_{a}=x\\\
TP_{b}=1-y-FN_{b}\\\ FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=0\\\
TN_{b}=0\end{cases}$

###  B.18. Equalised odds and Overall accuracy Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
\frac{y-x}{1-y}TP_{b}=TN_{b}-TN_{a}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
\frac{1-x}{1-y}TP_{b}=\frac{1-x}{1-y}TP_{b}\\\
\frac{y-x}{1-y}TP_{b}=\frac{y-x}{y}TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{b}=\frac{1-y}{y}TN_{b}\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}FN_{a}=1-x-\frac{1-x}{y}TN_{b}\\\
TN_{a}=x-\frac{x}{y}FP_{b}\\\ FN_{b}=1-y-\frac{1-y}{y}TN_{b}\\\
TN_{b}=y-FP_{b}\\\ TP_{a}=\frac{1-x}{y}TN_{b}\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
TP_{b}=\frac{1-y}{y}TN_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\
TN_{a}=x-FP_{b}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
FP_{a}=FP_{b}\\\ x=y\end{cases}$

###  B.19. Equal Opportunity, Predictive Equality and Overall accuracy
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=TN_{b}-\frac{y-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
\frac{1-x}{1-y}TN_{b}=TN_{b}-\frac{y-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
\frac{1-x-1+y}{1-y}TN_{b}=-\frac{y-x}{1-y}TP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
\frac{y-x}{1-y}TN_{b}=-\frac{y-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{a}\\\
FP_{a}=x-TN_{a}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
TN_{b}=-TP_{b}\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=x-\frac{1-x}{1-y}TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\ TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
TN_{b}=-TP_{b}=0\lor x=y\end{cases}$

###  B.20. Equal Opportunity, Predictive Equality and Predictive
Parity/Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ x-TN_{a}=\frac{1-x}{1-y}(y-TN_{b})\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
x-\frac{1-x}{1-y}(y-TN_{b})=\frac{1-x}{1-y}TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
x=\frac{1-x}{1-y}y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
x=y\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{b}\\\
FP_{a}=x-TN_{b}\\\ FN_{b}=1-y-TP_{b}\\\ FP_{b}=y-TN_{b}\\\ TP_{a}=TP_{b}\\\
TN_{a}=TN_{b}\\\ x=y\end{cases}$

###  B.21. Equal Opportunity, Overall accuracy Equality and Predictive
Parity/Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=TN_{b}-\frac{y-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ x-TN_{a}=\frac{1-x}{1-y}(y-TN_{b})\\\
TN_{a}=TN_{b}-\frac{y-x}{1-y}TP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}=TN_{b}-\frac{y-x}{1-y}TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
-TN_{b}+\frac{1-x}{1-y}TN_{b}=-\frac{y-x}{1-y}TP_{b}-x+\frac{1-x}{1-y}y\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
\frac{y-x}{1-y}TN_{b}=-\frac{y-x}{1-y}TP_{b}-x+\frac{1-x}{1-y}y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
TN_{b}=-TP_{b}-\frac{1-y}{y-x}x+\frac{1-x}{1-y}\frac{1-y}{y-x}y\lor
x=y\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
TN_{b}=-TP_{b}-\frac{1-y}{y-x}x+\frac{1-x}{y-x}y\lor
x=y\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}(y-TN_{b})\\\
TN_{b}=-TP_{b}+\frac{1}{y-x}(-x+y)\lor x=y\end{cases}$
$\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{a}\\\ TN_{a}=x-FP_{a}\\\
FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
TN_{a}=1-\frac{1-x}{1-y}TP_{b}\\\
TN_{b}=1-TP_{b}\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{a}\\\
TN_{a}=x-FP_{a}\\\ FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\
TN_{a}=TN_{b}\\\ x=y\end{cases}$ $\Longleftrightarrow\begin{cases}FN_{a}=0\\\
FP_{a}=0\\\ FN_{b}=0\\\ TN_{b}=0\\\ TP_{a}=1-x\\\ TN_{a}=x\\\ TN_{b}=y\\\
TP_{b}=1-y\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\ TN_{a}=x-FP_{b}\\\
FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ TP_{a}=TP_{b}\\\ TN_{a}=TN_{b}\\\
x=y\end{cases}$

###  B.22. False positive parity, Predictive Parity and Predictive
Equality/Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=\frac{x}{y}FP_{b}\\\ TP_{a}=\frac{x}{y}TP_{b}\\\
FP_{a}=\frac{x}{y}FP_{b}\\\
FN_{a}=\frac{x}{y}FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ FN_{a}=\frac{x}{y}FN_{b}\\\
1-x-FN_{a}=\frac{x}{y}(1-y-FN_{b})\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=\frac{x}{y}FP_{b}\\\
FN_{a}=\frac{x}{y}FN_{b}\\\
1-x-\frac{x}{y}FN_{b}=\frac{x}{y}-x-\frac{x}{y}FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
FP_{a}=\frac{x}{y}FP_{b}\\\ FN_{a}=\frac{x}{y}FN_{b}\\\
1=\frac{x}{y}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\\\
TN_{a}=x-FP_{b}\\\ TP_{b}=1-y-FN_{b}\\\ TN_{b}=y-FP_{b}\\\ y=x\\\
FP_{a}=FP_{b}\\\ FN_{a}=FN_{b}\end{cases}$

###  B.23. False positive parity, Predictive Parity and Overall accuracy
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\\\ TP_{a}=\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{x-y}(TP_{b}-TP_{a})\\\
TN_{b}=\frac{y}{x-y}(TP_{b}-TP_{a})\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ TP_{a}=\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{x-y}(TP_{b}-\frac{x}{y}TP_{b})\\\
TN_{b}=\frac{y}{x-y}(TP_{b}-\frac{x}{y}TP_{b})\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ TP_{a}=\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{x-y}(\frac{y-x}{y}TP_{b})\\\
TN_{b}=\frac{y}{x-y}(\frac{y-x}{y}TP_{b})\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
TP_{a}=\frac{x}{y}TP_{b}\\\ TN_{a}=-\frac{x}{y}TP_{b}\\\
TN_{b}=-TP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{x}{y}TP_{b}\\\ TN_{a}=-\frac{x}{y}TP_{b}\\\ TN_{b}=-TP_{b}\\\
-\frac{x}{y}TP_{b}=-\frac{x}{y}TP_{b}\\\
\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x\\\ FP_{a}=x\\\
FN_{b}=1-y\\\ FP_{b}=y\\\ TP_{a}=\frac{x}{y}TP_{b}=0\\\
TN_{a}=-\frac{x}{y}TP_{b}=0\\\ TN_{b}=-TP_{b}=0\\\ 1=1\\\ \end{cases}$

###  B.24. False positive parity, Overall accuracy Equality and Predictive
Equality/Treatment equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\\\ FN_{a}=\frac{x}{y}FN_{b}\\\
TN_{a}=\frac{x}{x-y}(TP_{b}-TP_{a})\\\
TN_{b}=\frac{y}{x-y}(TP_{b}-TP_{a})\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ TP_{a}=1-\frac{x}{y}+\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{x-y}(TP_{b}-1+\frac{x}{y}-\frac{x}{y}TP_{b})\\\
TN_{b}=\frac{y}{x-y}(TP_{b}-1+\frac{x}{y}-\frac{x}{y}TP_{b})\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
TP_{a}=1-\frac{x}{y}+\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{x-y}(\frac{x-y}{y}+\frac{y-x}{y}TP_{b})\\\
TN_{b}=\frac{y}{x-y}(\frac{x-y}{y}+\frac{y-x}{y}TP_{b})\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=\frac{x}{y}TN_{b}\\\ TP_{a}=1-\frac{x}{y}+\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{y}-\frac{x}{y}TP_{b}\\\ TN_{b}=1-TP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=1-\frac{x}{y}+\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{y}-\frac{x}{y}TP_{b}\\\ TN_{b}=1-TP_{b}\\\
\frac{x}{y}-\frac{x}{y}TP_{b}=\frac{x}{y}(1-TP_{b})\\\
\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=1-x-TP_{a}\\\
FP_{a}=x-TN_{a}\\\ FN_{b}=1-y-TP_{b}\\\ FP_{b}=y-TN_{b}\\\
TP_{a}=1-\frac{x}{y}+\frac{x}{y}TP_{b}\\\
TN_{a}=\frac{x}{y}-\frac{x}{y}TP_{b}\\\ TN_{b}=1-TP_{b}\\\ 1=1\\\
\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=0\\\ FP_{a}=0\\\ FN_{b}=0\\\
FP_{b}=0\\\ TP_{a}=1-x\\\ TN_{a}=x\\\ TN_{b}=y\\\ TP_{b}=1-y\\\ 1=1\\\
\end{cases}$

###  B.25. Predictive Parity, Predictive Equality and Overall accuracy
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{TP_{b}*FP_{a}}{FP_{b}}\\\
FN_{a}=\frac{FN_{b}*TN_{a}}{TN_{b}}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}*(y-TN_{b})=TP_{b}*(x-TN_{a})\\\
(1-x-TP_{a})*TN_{b}=(1-y-TP_{b})*TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=TP_{b}*\frac{x-TN_{a}}{y-TN_{b}}\\\
(1-x-TP_{b}*\frac{x-TN_{a}}{y-TN_{b}})*TN_{b}=(1-y-TP_{b})*TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=TP_{b}*\frac{x-TN_{a}}{y-TN_{b}}\\\
-TP_{b}TN_{b}\frac{x-TN_{a}}{y-TN_{b}}+TP_{b}TN_{a}=-TN_{b}+xTN_{b}+TN_{a}-yTN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=TP_{b}*\frac{x-TN_{a}}{y-TN_{b}}\\\
TP_{b}(TN_{a}-TN_{b}*\frac{x-TN_{a}}{y-TN_{b}})=(x-1)TN_{b}+(1-y)TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=TP_{b}*\frac{x-TN_{a}}{y-TN_{b}}\\\
TP_{b}(\frac{yTN_{a}-TN_{a}TN_{b}-xTN_{b}+TN_{b}TN_{a}}{y-TN_{b}})=(x-1)TN_{b}+(1-y)TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=TP_{b}*\frac{x-TN_{a}}{y-TN_{b}}\\\
TP_{b}(\frac{yTN_{a}-xTN_{b}}{y-TN_{b}})=(x-1)TN_{b}+(1-y)TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{(x-1)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{a}\\\
TP_{b}=\frac{(x-1)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{a}\\\
TP_{a}+TN_{a}=TP_{b}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{(x-1)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{a}\\\
TP_{b}=\frac{(x-1)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{a}\\\
\frac{(x-1)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{a}+TN_{a}=\frac{(x-1)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{a}+TN_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{(1-x)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{a}\\\
TP_{b}=\frac{(1-x)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{a}\\\
(x^{2}-xTN_{a}-x+TN_{a})TN_{b}+(x-xy-
TN_{a}+yTN_{a})TN_{a}+(yTN_{a}-xTN_{b})TN_{a}=\\\ (xy-
xTN_{b}-y+<tn-b)TN_{b}+(y-TN_{b}-y^{2}+yTN_{b})TN_{a}+(yTN_{a}-xTN_{b})TN_{b}\par\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{(1-x)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{a}\\\
TP_{b}=\frac{(1-x)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{a}\\\
(xy-2xTN_{b}-y+TN_{b}+yTN_{a}-x^{2}+xTN_{a}+x-TN_{a})TN_{b}=\\\ (x-xy-
TN_{a}+2yTN_{a}-xTN_{b}-y+TN_{b}+y^{2}-yTN_{b})TN_{a}\par\end{cases}$

$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TP_{a}=\frac{(1-x)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(x-TN_{a})}{yTN_{a}-xTN_{b}}TN_{a}\\\
TP_{b}=\frac{(1-x)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{b}+\frac{(1-y)(y-TN_{b})}{yTN_{a}-xTN_{b}}TN_{a}\\\
(-2x+1)TN_{b}^{2}+(xy-y+x+2yTN_{a}-x^{2}+2xTN_{a}-2TN_{a})TN_{b}+(xy-x+TN_{a}-2yTN_{a}+y-y^{2})TN_{a}=0\par\end{cases}$

###  B.26. Predictive Parity, Predictive Equality and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ TP_{a}=\frac{TP_{b}FP_{a}}{FP_{b}}\\\
FN_{a}=\frac{FN_{b}TN_{a}}{TN_{b}}\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
\frac{1-x}{1-y}TP_{b}=\frac{TP_{b}FP_{a}}{FP_{b}}\\\
\frac{1-x}{1-y}FN_{b}=\frac{FN_{b}TN_{a}}{TN_{b}}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
\frac{1-x}{1-y}FP_{b}=FP_{a}\lor TP_{b}=0\\\ TN_{a}=\frac{1-x}{1-y}TN_{b}\lor
FN_{b}=0\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=\frac{1-x}{1-y}TN_{b}\lor
FN_{b}=0\\\ FN_{a}=\frac{1-x}{1-y}FN_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
FP_{a}=x-TN_{a}\\\ TP_{b}=1-y\\\ FP_{b}=y-TN_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}\\\ FN_{b}=0\\\
FN_{a}=0\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
\frac{1-x}{1-y}TN_{b}=x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}\\\
TN_{a}=\frac{1-x}{1-y}TN_{b}\\\
FN_{a}=\frac{1-x}{1-y}FN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
FP_{a}=x-TN_{a}\\\ TP_{b}=1-y\\\ FP_{b}=y-TN_{b}\\\
TN_{a}=x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}\\\ FN_{b}=0\\\
FN_{a}=0\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}=1-x-FN_{b}\\\
FP_{a}=x-TN_{b}\\\ TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\ x=y\\\
TN_{a}=TN_{b}\\\ FN_{a}=FN_{b}\end{cases}\lor\begin{cases}TP_{a}=1-x\\\
FP_{a}=\frac{1-x}{1-y}y-\frac{1-x}{1-y}TN_{b}\\\ TP_{b}=1-y\\\
FP_{b}=y-TN_{b}\\\ TN_{a}=x-\frac{1-x}{1-y}y+\frac{1-x}{1-y}TN_{b}\\\
FN_{b}=0\\\ FN_{a}=0\end{cases}$

###  B.27. Predictive Parity, Overall accuracy Equality and Treatment Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\\\ FN_{a}=FN_{b}\\\
TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{a}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\\\
1-x-TP_{a}=1-y-TP_{b}\\\ TP_{a}=\frac{1-x}{1-y}TP_{b}\\\
FP_{b}=\frac{1-x}{1-y}FP_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\\\
TP_{a}=y-x+TP_{b}\\\ y-x-TP_{b}=\frac{1-x}{1-y}TP_{b}\\\ x=y\lor
FP_{b}=0\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\\\
TP_{a}=y-x+TP_{b}\\\ y-x=\frac{y-x}{1-y}TP_{b}\\\ x=y\lor
FP_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\\\
TP_{a}=y-x+TP_{b}\\\ TP_{b}=1-y\lor x=y\\\ x=y\lor
FP_{b}=0\end{cases}\Longleftrightarrow\begin{cases}FN_{a}=0\\\ TN_{a}=x\\\
FN_{b}=0\\\ TN_{b}=y\\\ FP_{a}=0\\\ TP_{a}=1-x\\\ TP_{b}=1-y\\\
FP_{b}=0\end{cases}\lor\begin{cases}FN_{a}=1-x-TP_{b}\\\ TN_{a}=x-FP_{b}\\\
FN_{b}=1-y-TP_{b}\\\ TN_{b}=y-FP_{b}\\\ FP_{a}=FP_{b}\\\ TP_{a}=TP_{b}\\\
x=y\end{cases}$

###  B.28. Predictive Equality, Overall accuracy Equality and Treatment
Equality

$\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\
TN_{b}+FP_{b}=y\\\ FP_{a}=FP_{b}\\\ FN_{a}=FN_{b}\\\
TN_{a}=\frac{x}{y}TN_{b}\\\
FN_{a}=\frac{x}{y}FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
x-TN_{a}=y-TN_{b}\\\ FN_{a}=FN_{b}\\\ TN_{a}=\frac{x}{y}TN_{b}\\\
FN_{b}=\frac{x}{y}FN_{b}\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=x-y+TN_{b}\\\ FN_{a}=FN_{b}\\\ x-y+TN_{b}=\frac{x}{y}TN_{b}\\\
1=\frac{x}{y}\lor FN_{b}=0\end{cases}$
$\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\ TN_{a}+FP_{a}=x\\\
TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\ TN_{a}=x-y+TN_{b}\\\ FN_{a}=FN_{b}\\\
x-y=\frac{x-y}{y}TN_{b}\\\ x=y\lor
FN_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=x-y+TN_{b}\\\ FN_{a}=FN_{b}\\\ 1=\frac{1}{y}TN_{b}\lor x=y\\\ x=y\lor
FN_{b}=0\end{cases}\Longleftrightarrow\begin{cases}TP_{a}+FN_{a}=1-x\\\
TN_{a}+FP_{a}=x\\\ TP_{b}+FN_{b}=1-y\\\ TN_{b}+FP_{b}=y\\\
TN_{a}=x-y+TN_{b}\\\ FN_{a}=FN_{b}\\\ TN_{b}=y\lor x=y\\\ x=y\lor
FN_{b}=0\end{cases}$ $\Longleftrightarrow\begin{cases}TP_{a}=1-x\\\
FP_{a}=0\\\ TP_{b}=1-y\\\ FP_{b}=0\\\ TN_{a}=x\\\ FN_{a}=0\\\ TN_{b}=y\\\
FN_{b}=0\end{cases}\lor\begin{cases}TP_{a}=1-x-FN_{b}\\\ FP_{a}=x-TN_{b}\\\
TP_{b}=1-y-FN_{b}\\\ FP_{b}=y-TN_{b}\\\ TN_{a}=TN_{b}\\\ FN_{a}=FN_{b}\\\
x=y\end{cases}$