Fairness (machine learning)

We say the random variables ${\textstyle (R,A)}$ satisfy independence if the sensitive characteristics ${\textstyle A}$ are statistically independent of the prediction ${\textstyle R}$, and we write

$${\displaystyle R\bot A.}$$

We can also express this notion with the following formula:

$${\displaystyle P(R=r\ |\ A=a)=P(R=r\ |\ A=b)\quad \forall r\in R\quad \forall a,b\in A}$$

This means that the classification rate for each target classes is equal for people belonging to different groups with respect to sensitive characteristics ${\displaystyle A}$.

Yet another equivalent expression for independence can be given using the concept of mutual information between random variables, defined as

$${\displaystyle I(X,Y)=H(X)+H(Y)-H(X,Y)}$$

In this formula, ${\textstyle H(X)}$ is the entropy of the random variable ${\displaystyle X}$. Then ${\textstyle (R,A)}$ satisfy independence if ${\textstyle I(R,A)=0}$.

A possible relaxation of the independence definition include introducing a positive slack ${\textstyle \epsilon >0}$ and is given by the formula:

$${\displaystyle P(R=r\ |\ A=a)\geq P(R=r\ |\ A=b)-\epsilon \quad \forall r\in R\quad \forall a,b\in A}$$

Finally, another possible relaxation is to require ${\textstyle I(R,A)\leq \epsilon }$.

We say the random variables ${\textstyle (R,A,Y)}$ satisfy separation if the sensitive characteristics ${\textstyle A}$ are statistically independent of the prediction ${\textstyle R}$ given the target value ${\textstyle Y}$, and we write

$${\displaystyle R\bot A\ |\ Y.}$$

We can also express this notion with the following formula:

$${\displaystyle P(R=r\ |\ Y=q,A=a)=P(R=r\ |\ Y=q,A=b)\quad \forall r\in R\quad q\in Y\quad \forall a,b\in A}$$

This means that all the dependence of the decision ${\displaystyle R}$ on the sensitive attribute ${\displaystyle A}$ must be justified by the actual dependence of the true target variable ${\displaystyle Y}$.

Another equivalent expression, in the case of a binary target rate, is that the true positive rate and the false positive rate are equal (and therefore the false negative rate and the true negative rate are equal) for every value of the sensitive characteristics:

$${\displaystyle P(R=1\ |\ Y=1,A=a)=P(R=1\ |\ Y=1,A=b)\quad \forall a,b\in A}$$

$${\displaystyle P(R=1\ |\ Y=0,A=a)=P(R=1\ |\ Y=0,A=b)\quad \forall a,b\in A}$$

A possible relaxation of the given definitions is to allow the value for the difference between rates to be a positive number lower than a given slack ${\textstyle \epsilon >0}$, rather than equal to zero.

In some fields separation (separation coefficient) in a confusion matrix is a measure of the distance (at a given level of the probability score) between the predicted cumulative percent negative and predicted cumulative percent positive.

The greater this separation coefficient is at a given score value, the more effective the model is at differentiating between the set of positives and negatives at a particular probability cut-off. According to Mayes:[24] "It is often observed in the credit industry that the selection of validation measures depends on the modeling approach. For example, if modeling procedure is parametric or semi-parametric, the two-sample K-S test is often used. If the model is derived by heuristic or iterative search methods, the measure of model performance is usually divergence. A third option is the coefficient of separation...The coefficient of separation, compared to the other two methods, seems to be most reasonable as a measure for model performance because it reflects the separation pattern of a model."

We say the random variables ${\textstyle (R,A,Y)}$ satisfy sufficiency if the sensitive characteristics ${\textstyle A}$ are statistically independent of the target value ${\textstyle Y}$ given the prediction ${\textstyle R}$, and we write

$${\displaystyle Y\bot A\ |\ R.}$$

We can also express this notion with the following formula:

$${\displaystyle P(Y=q\ |\ R=r,A=a)=P(Y=q\ |\ R=r,A=b)\quad \forall q\in Y\quad r\in R\quad \forall a,b\in A}$$

This means that the probability of actually being in each of the groups is equal for two individuals with different sensitive characteristics given that they were predicted to belong to the same group.

Finally, we sum up some of the main results that relate the three definitions given above:

• Assuming ${\textstyle Y}$ is binary, if ${\textstyle A}$ and ${\textstyle Y}$ are not statistically independent, and ${\textstyle R}$ and ${\textstyle Y}$ are not statistically independent either, then independence and separation cannot both hold except for rhetorical cases.
• If ${\textstyle (R,A,Y)}$ as a joint distribution has positive probability for all its possible values and ${\textstyle A}$ and ${\textstyle Y}$ are not statistically independent, then separation and sufficiency cannot both hold except for rhetorical cases.

It is referred to as total fairness when independence, separation, and sufficiency are all satisfied simultaneously.[25] However, total fairness is not possible to achieve except in specific rhetorical cases.[26]

Most statistical measures of fairness rely on different metrics, so we will start by defining them. When working with a binary classifier, both the predicted and the actual classes can take two values: positive and negative. Now let us start explaining the different possible relations between predicted and actual outcome:[27]

Confusion matrix

• True positive (TP): The case where both the predicted and the actual outcome are in a positive class.
• True negative (TN): The case where both the predicted outcome and the actual outcome are assigned to the negative class.
• False positive (FP): A case predicted to befall into a positive class assigned in the actual outcome is to the negative one.
• False negative (FN): A case predicted to be in the negative class with an actual outcome is in the positive one.

These relations can be easily represented with a confusion matrix, a table that describes the accuracy of a classification model. In this matrix, columns and rows represent instances of the predicted and the actual cases, respectively.

By using these relations, we can define multiple metrics which can be later used to measure the fairness of an algorithm:

• Positive predicted value (PPV): the fraction of positive cases which were correctly predicted out of all the positive predictions. It is usually referred to as precision, and represents the probability of a correct positive prediction. It is given by the following formula:
  $${\displaystyle PPV=P(actual=+\ |\ prediction=+)={\frac {TP}{TP+FP}}}$$

  
• False discovery rate (FDR): the fraction of positive predictions which were actually negative out of all the positive predictions. It represents the probability of an erroneous positive prediction, and it is given by the following formula:
  $${\displaystyle FDR=P(actual=-\ |\ prediction=+)={\frac {FP}{TP+FP}}}$$

  
• Negative predicted value (NPV): the fraction of negative cases which were correctly predicted out of all the negative predictions. It represents the probability of a correct negative prediction, and it is given by the following formula:
  $${\displaystyle NPV=P(actual=-\ |\ prediction=-)={\frac {TN}{TN+FN}}}$$

  
• False omission rate (FOR): the fraction of negative predictions which were actually positive out of all the negative predictions. It represents the probability of an erroneous negative prediction, and it is given by the following formula:
  $${\displaystyle FOR=P(actual=+\ |\ prediction=-)={\frac {FN}{TN+FN}}}$$

  
• True positive rate (TPR): the fraction of positive cases which were correctly predicted out of all the positive cases. It is usually referred to as sensitivity or recall, and it represents the probability of the positive subjects to be classified correctly as such. It is given by the formula:
  $${\displaystyle TPR=P(prediction=+\ |\ actual=+)={\frac {TP}{TP+FN}}}$$

  
• False negative rate (FNR): the fraction of positive cases which were incorrectly predicted to be negative out of all the positive cases. It represents the probability of the positive subjects to be classified incorrectly as negative ones, and it is given by the formula:
  $${\displaystyle FNR=P(prediction=-\ |\ actual=+)={\frac {FN}{TP+FN}}}$$

  
• True negative rate (TNR): the fraction of negative cases which were correctly predicted out of all the negative cases. It represents the probability of the negative subjects to be classified correctly as such, and it is given by the formula:
  $${\displaystyle TNR=P(prediction=-\ |\ actual=-)={\frac {TN}{TN+FP}}}$$

  
• False positive rate (FPR): the fraction of negative cases which were incorrectly predicted to be positive out of all the negative cases. It represents the probability of the negative subjects to be classified incorrectly as positive ones, and it is given by the formula:
  $${\displaystyle FPR=P(prediction=+\ |\ actual=-)={\frac {FP}{TN+FP}}}$$

  

Relationship between fairness criteria as shown in Barocas et al.[23]

The following criteria can be understood as measures of the three general definitions given at the beginning of this section, namely Independence, Separation and Sufficiency. In the table[23] to the right, we can see the relationships between them.

To define these measures specifically, we will divide them into three big groups as done in Verma et al.:[27] definitions based on a predicted outcome, on predicted and actual outcomes, and definitions based on predicted probabilities and the actual outcome.

We will be working with a binary classifier and the following notation: ${\textstyle S}$ refers to the score given by the classifier, which is the probability of a certain subject to be in the positive or the negative class. ${\textstyle R}$ represents the final classification predicted by the algorithm, and its value is usually derived from ${\textstyle S}$, for example will be positive when ${\textstyle S}$ is above a certain threshold. ${\textstyle Y}$ represents the actual outcome, that is, the real classification of the individual and, finally, ${\textstyle A}$ denotes the sensitive attributes of the subjects.

The definitions in this section focus on a predicted outcome ${\textstyle R}$ for various distributions of subjects. They are the simplest and most intuitive notions of fairness.

• Demographic parity, also referred to as statistical parity, acceptance rate parity and benchmarking. A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal probability of being assigned to the positive predicted class. This is, if the following formula is satisfied:
  $${\displaystyle P(R=+\ |\ A=a)=P(R=+\ |\ A=b)\quad \forall a,b\in A}$$

  
• Conditional statistical parity. Basically consists in the definition above, but restricted only to a subset of the instances. In mathematical notation this would be:
  $${\displaystyle P(R=+\ |\ L=l,A=a)=P(R=+\ |\ L=l,A=b)\quad \forall a,b\in A\quad \forall l\in L}$$

  

These definitions not only considers the predicted outcome ${\textstyle R}$ but also compare it to the actual outcome ${\textstyle Y}$.

• Predictive parity, also referred to as outcome test. A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal PPV. This is, if the following formula is satisfied:
  $${\displaystyle P(Y=+\ |\ R=+,A=a)=P(Y=+\ |\ R=+,A=b)\quad \forall a,b\in A}$$

  

Mathematically, if a classifier has equal PPV for both groups, it will also have equal FDR, satisfying the formula:

$${\displaystyle P(Y=-\ |\ R=+,A=a)=P(Y=-\ |\ R=+,A=b)\quad \forall a,b\in A}$$

• False positive error rate balance, also referred to as predictive equality. A classifier satisfies this definition if the subjects in the protected and unprotected groups have aqual FPR. This is, if the following formula is satisfied:
  $${\displaystyle P(R=+\ |\ Y=-,A=a)=P(R=+\ |\ Y=-,A=b)\quad \forall a,b\in A}$$

  

Mathematically, if a classifier has equal FPR for both groups, it will also have equal TNR, satisfying the formula:

$${\displaystyle P(R=-\ |\ Y=-,A=a)=P(R=-\ |\ Y=-,A=b)\quad \forall a,b\in A}$$

• False negative error rate balance, also referred to as equal opportunity. A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal FNR. This is, if the following formula is satisfied:
  $${\displaystyle P(R=-\ |\ Y=+,A=a)=P(R=-\ |\ Y=+,A=b)\quad \forall a,b\in A}$$

  

Mathematically, if a classifier has equal FNR for both groups, it will also have equal TPR, satisfying the formula:

$${\displaystyle P(R=+\ |\ Y=+,A=a)=P(R=+\ |\ Y=+,A=b)\quad \forall a,b\in A}$$

• Equalized odds, also referred to as conditional procedure accuracy equality and disparate mistreatment. A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal TPR and equal FPR, satisfying the formula:
  $${\displaystyle P(R=+\ |\ Y=y,A=a)=P(R=+\ |\ Y=y,A=b)\quad y\in \{+,-\}\quad \forall a,b\in A}$$

  
• Conditional use accuracy equality. A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal PPV and equal NPV, satisfying the formula:
  $${\displaystyle P(Y=y\ |\ R=y,A=a)=P(Y=y\ |\ R=y,A=b)\quad y\in \{+,-\}\quad \forall a,b\in A}$$

  
• Overall accuracy equality. A classifier satisfies this definition if the subject in the protected and unprotected groups have equal prediction accuracy, that is, the probability of a subject from one class to be assigned to it. This is, if it satisfies the following formula:
  $${\displaystyle P(R=Y\ |\ A=a)=P(R=Y\ |\ A=b)\quad \forall a,b\in A}$$

  
• Treatment equality. A classifier satisfies this definition if the subjects in the protected and unprotected groups have an equal ratio of FN and FP, satisfying the formula:
  $${\displaystyle {\frac {FN_{A=a}}{FP_{A=a}}}={\frac {FN_{A=b}}{FP_{A=b}}}}$$

  

These definitions are based in the actual outcome ${\textstyle Y}$ and the predicted probability score ${\textstyle S}$.

• Test-fairness, also known as calibration or matching conditional frequencies. A classifier satisfies this definition if individuals with the same predicted probability score ${\textstyle S}$ have the same probability of being classified in the positive class when they belong to either the protected or the unprotected group:
  $${\displaystyle P(Y=+\ |\ S=s,A=a)=P(Y=+\ |\ S=s,A=b)\quad \forall s\in S\quad \forall a,b\in A}$$

  
• Well-calibration is an extension of the previous definition. It states that when individuals inside or outside the protected group have the same predicted probability score ${\textstyle S}$ they must have the same probability of being classified in the positive class, and this probability must be equal to ${\textstyle S}$:
  $${\displaystyle P(Y=+\ |\ S=s,A=a)=P(Y=+\ |\ S=s,A=b)=s\quad \forall s\in S\quad \forall a,b\in A}$$

  
• Balance for positive class. A classifier satisfies this definition if the subjects constituting the positive class from both protected and unprotected groups have equal average predicted probability score ${\textstyle S}$. This means that the expected value of probability score for the protected and unprotected groups with positive actual outcome ${\textstyle Y}$ is the same, satisfying the formula:
  $${\displaystyle E(S\ |\ Y=+,A=a)=E(S\ |\ Y=+,A=b)\quad \forall a,b\in A}$$

  
• Balance for negative class. A classifier satisfies this definition if the subjects constituting the negative class from both protected and unprotected groups have equal average predicted probability score ${\textstyle S}$. This means that the expected value of probability score for the protected and unprotected groups with negative actual outcome ${\textstyle Y}$ is the same, satisfying the formula:
  $${\displaystyle E(S\ |\ Y=-,A=a)=E(S\ |\ Y=-,A=b)\quad \forall a,b\in A}$$

  

With respect to confusion matrices, independence, separation, and sufficiency require the respective quantities listed below to not have statistically significant difference across sensitive characteristics.[26]

• Independence: (TP + FP) / (TP + FP + FN + TN)
• Separation: TN / (TN + FP) and TP / (TP + FN) (i.e., specificity and recall).
• Sufficiency: TP / (TP +FP) and TN / (TN + FN) (i.e., precision and negative predictive value), and

The distribution of the confusion matrix is known when the values of separation and sufficiency are given. As a result, any measure based on confusion matrices, including independence, may also be computed. Therefore, confusion matrices cover all three criteria and any other fairness metric based on TP, FP, TN, and FN.

The notion of equal confusion fairness[28] desires the confusion matrices of a given decision system to have the same distributions across all sensitive characteristics. Equal confusion fairness test to identify any unfair behavior, confusion parity error to quantify the extent of unfairness, and a post hoc analysis method to identify the impacted groups are available as a open-source software.[29]

Some scholars have proposed defining algorithmic fairness in terms of a social welfare function. They argue that using a social welfare function enables an algorithm designer to consider fairness and predictive accuracy in terms of their benefits to the people affected by the algorithm. It also allows the designer to trade off efficiency and equity in a principled way.[30] Sendhil Mullainathan has stated that algorithm designers should use social welfare functions in order to recognize absolute gains for disadvantaged groups. For example, a study found that using a decision-making algorithm in pretrial detention rather than pure human judgment reduced the detention rates for Blacks, Hispanics, and racial minorities overall, even while keeping the crime rate constant.[31]

Usually, the classifier is not the only problem; the dataset is also biased. The discrimination of a dataset ${\textstyle D}$ with respect to the group ${\textstyle A=a}$ can be defined as follows:

$${\displaystyle disc_{A=a}(D)={\frac {|\{X\in D|X(A)\neq a,X(Y)=+\}|}{|\{X\in D|X(A)\neq a\}|}}-{\frac {|\{X\in D|X(A)=a,X(Y)=+\}|}{|\{X\in D|X(A)=a\}|}}}$$

That is, an approximation to the difference between the probabilities of belonging in the positive class given that the subject has a protected characteristic different from ${\textstyle a}$ and equal to ${\textstyle a}$.

Algorithms correcting bias at preprocessing remove information about dataset variables which might result in unfair decisions, while trying to alter as little as possible. This is not as simple as just removing the sensitive variable, because other attributes can be correlated to the protected one.

A way to do this is to map each individual in the initial dataset to an intermediate representation in which it is impossible to identify whether it belongs to a particular protected group while maintaining as much information as possible. Then, the new representation of the data is adjusted to get the maximum accuracy in the algorithm.

This way, individuals are mapped into a new multivariable representation where the probability of any member of a protected group to be mapped to a certain value in the new representation is the same as the probability of an individual which doesn't belong to the protected group. Then, this representation is used to obtain the prediction for the individual, instead of the initial data. As the intermediate representation is constructed giving the same probability to individuals inside or outside the protected group, this attribute is hidden to the classificator.

An example is explained in Zemel et al.[40] where a multinomial random variable is used as an intermediate representation. In the process, the system is encouraged to preserve all information except that which can lead to biased decisions, and to obtain a prediction as accurate as possible.

On the one hand, this procedure has the advantage that the preprocessed data can be used for any machine learning task. Furthermore, the classifier does not need to be modified, as the correction is applied to the dataset before processing. On the other hand, the other methods obtain better results in accuracy and fairness.[41]

Reweighing is an example of a preprocessing algorithm. The idea is to assign a weight to each dataset point such that the weighted discrimination is 0 with respect to the designated group.[42]

If the dataset ${\textstyle D}$ was unbiased the sensitive variable ${\textstyle A}$ and the target variable ${\textstyle Y}$ would be statistically independent and the probability of the joint distribution would be the product of the probabilities as follows:

$${\displaystyle P_{exp}(A=a\wedge Y=+)=P(A=a)\times P(Y=+)={\frac {|\{X\in D|X(A)=a\}|}{|D|}}\times {\frac {|\{X\in D|X(Y)=+\}|}{|D|}}}$$

In reality, however, the dataset is not unbiased and the variables are not statistically independent so the observed probability is:

$${\displaystyle P_{obs}(A=a\wedge Y=+)={\frac {|\{X\in D|X(A)=a\wedge X(Y)=+\}|}{|D|}}}$$

To compensate for the bias, the software adds a weight, lower for favored objects and higher for unfavored objects. For each ${\textstyle X\in D}$ we get:

$${\displaystyle W(X)={\frac {P_{exp}(A=X(A)\wedge Y=X(Y))}{P_{obs}(A=X(A)\wedge Y=X(Y))}}}$$

When we have for each ${\textstyle X}$ a weight associated ${\textstyle W(X)}$ we compute the weighted discrimination with respect to group ${\textstyle A=a}$ as follows:

$${\displaystyle disc_{A=a}(D)={\frac {\sum W(X)X\in \{X\in D|X(A)\neq a,X(Y)=+\}}{\sum W(X)X\in \{X\in D|X(A)\neq a\}}}-{\frac {\sum W(X)X\in \{X\in D|X(A)=a,X(Y)=+\}}{\sum W(X)X\in \{X\in D|X(A)=a\}}}}$$

It can be shown that after reweighting this weighted discrimination is 0.

Another approach is to correct the bias at training time. This can be done by adding constraints to the optimization objective of the algorithm.[43] These constraints force the algorithm to improve fairness, by keeping the same rates of certain measures for the protected group and the rest of individuals. For example, we can add to the objective of the algorithm the condition that the false positive rate is the same for individuals in the protected group and the ones outside the protected group.

The main measures used in this approach are false positive rate, false negative rate, and overall misclassification rate. It is possible to add just one or several of these constraints to the objective of the algorithm. Note that the equality of false negative rates implies the equality of true positive rates so this implies the equality of opportunity. After adding the restrictions to the problem it may turn intractable, so a relaxation on them may be needed.

This technique obtains good results in improving fairness while keeping high accuracy and lets the programmer choose the fairness measures to improve. However, each machine learning task may need a different method to be applied and the code in the classifier needs to be modified, which is not always possible.[41]

We train two classifiers at the same time through some gradient-based method (f.e.: gradient descent). The first one, the predictor tries to accomplish the task of predicting ${\textstyle Y}$, the target variable, given ${\textstyle X}$, the input, by modifying its weights ${\textstyle W}$ to minimize some loss function ${\textstyle L_{P}({\hat {y}},y)}$. The second one, the adversary tries to accomplish the task of predicting ${\textstyle A}$, the sensitive variable, given ${\textstyle {\hat {Y}}}$ by modifying its weights ${\textstyle U}$ to minimize some loss function ${\textstyle L_{A}({\hat {a}},a)}$.[44][45]

An important point here is that, in order to propagate correctly, ${\textstyle {\hat {Y}}}$ above must refer to the raw output of the classifier, not the discrete prediction; for example, with an artificial neural network and a classification problem, ${\textstyle {\hat {Y}}}$ could refer to the output of the softmax layer.

Then we update ${\textstyle U}$ to minimize ${\textstyle L_{A}}$ at each training step according to the gradient ${\textstyle \nabla _{U}L_{A}}$ and we modify ${\textstyle W}$ according to the expression:

$${\displaystyle \nabla _{W}L_{P}-proj_{\nabla _{W}L_{A}}\nabla _{W}L_{P}-\alpha \nabla _{W}L_{A}}$$

where $\alpha$ is a tuneable hyperparameter that can vary at each time step.

Graphic representation of the vectors used in adversarial debiasing as shown in Zhang et al.[44]

The intuitive idea is that we want the predictor to try to minimize ${\textstyle L_{P}}$ (therefore the term ${\textstyle \nabla _{W}L_{P}}$) while, at the same time, maximize ${\textstyle L_{A}}$ (therefore the term ${\textstyle -\alpha \nabla _{W}L_{A}}$), so that the adversary fails at predicting the sensitive variable from ${\textstyle {\hat {Y}}}$.

The term ${\textstyle -proj_{\nabla _{W}L_{A}}\nabla _{W}L_{P}}$ prevents the predictor from moving in a direction that helps the adversary decrease its loss function.

It can be shown that training a predictor classification model with this algorithm improves demographic parity with respect to training it without the adversary.

The final method tries to correct the results of a classifier to achieve fairness. In this method, we have a classifier that returns a score for each individual and we need to do a binary prediction for them. High scores are likely to get a positive outcome, while low scores are likely to get a negative one, but we can adjust the threshold to determine when to answer yes as desired. Note that variations in the threshold value affect the trade-off between the rates for true positives and true negatives.

If the score function is fair in the sense that it is independent of the protected attribute, then any choice of the threshold will also be fair, but classifiers of this type tend to be biased, so a different threshold may be required for each protected group to achieve fairness.[46] A way to do this is plotting the true positive rate against the false negative rate at various threshold settings (this is called ROC curve) and find a threshold where the rates for the protected group and other individuals are equal.[46]

The advantages of postprocessing include that the technique can be applied after any classifiers, without modifying it, and has a good performance in fairness measures. The cons are the need to access to the protected attribute in test time and the lack of choice in the balance between accuracy and fairness.[41]

Given a classifier let ${\textstyle P(+|X)}$ be the probability computed by the classifiers as the probability that the instance ${\textstyle X}$ belongs to the positive class +. When ${\textstyle P(+|X)}$ is close to 1 or to 0, the instance ${\textstyle X}$ is specified with high degree of certainty to belong to class + or - respectively. However, when ${\textstyle P(+|X)}$ is closer to 0.5 the classification is more unclear.[47]

We say ${\textstyle X}$ is a "rejected instance" if ${\textstyle max(P(+|X),1-P(+|X))\leq \theta }$ with a certain ${\textstyle \theta }$ such that ${\textstyle 0.5<\theta <1}$.

The algorithm of "ROC" consists on classifying the non-rejected instances following the rule above and the rejected instances as follows: if the instance is an example of a deprived group (${\displaystyle X(A)=a}$) then label it as positive, otherwise, label it as negative.

We can optimize different measures of discrimination (link) as functions of ${\textstyle \theta }$ to find the optimal ${\textstyle \theta }$ for each problem and avoid becoming discriminatory against the privileged group.[47]