# Fairness in Algorithmic Decision-Making: Applications in Multi-Winner Voting, Machine Learning, and Recommender Systems

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## Abstract

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## 1. Introduction

## 2. Multi-Winner Voting

**Dichotomous preference.**Each voter classifies candidates into two classes, namely, the approved candidates and the disapproved candidates. In particular, all approved candidates are preferred to all disapproved candidates, and candidates inside each class are equally preferred.**Linear preference.**Each voter ranks all candidates in a linear order ≻, from the best to the worst. For two candidates, a and b, $a\succ b$ means that the corresponding voter strictly prefers a to b.

#### 2.1. Voter Fairness in Ranking-Based Voting

**q-Proportionality for solid coalition (q-PSC) [16].**For a rational number, q, a k-committee $w\subseteq C$ satisfies q-PSC if for every positive integer ℓ and for every solid coalition $U\subseteq V$ supporting some ${C}^{\prime}\subseteq C$ such that $\left|U\right|\ge \ell q$, it holds that $|w\cap {C}^{\prime}|\ge min\{\ell ,|{C}^{\prime}\left|\right\}$.

**Weak q-PSC. [16].**A committee $w\subseteq C$ satisfies weak q-PSC if the following holds, for every positive integer ℓ, every ${C}^{\prime}\subseteq C$ such that $|{C}^{\prime}|\le \ell $, and every ${C}^{\prime}$-solid coalition U of size at least $\ell q$, it holds that ${C}^{\prime}\subseteq w$.

$\tau $-Computing | |

Input: | An election $(C,V)$ and a positive integer $k\le \left|C\right|$. |

Question: | Is there a k-committee $w\subseteq C$ which provides the $\tau $ property at $(C,V)$? |

$\tau $-Testing | |

Input: | An election $(C,V)$ and a committee $w\subseteq C$. |

Question: | Does w satisfy $\tau $ at $(C,V)$? |

**Committee scoring rules.**Under a committee scoring rule, each voter provides a score to each committee based on the positions of the committee-members in the preference of this voter, and winning committees are those with the maximum total score. Committee scoring rules were first studied by Elkind et al. [17] as a general framework to encapsulate many concrete multi-winner voting rules, including, e.g., Bloc, k-Borda, Chamberlin–Courant, etc.**k-Borda.**Each voter gives $m-i$ points to each candidate ranked in the i-th position, where m denotes the number of candidates. The score of a committee from a voter is the sum of the scores of all its members from the voter.**Bloc.**Every voter gives 1 point to all of their top k ranked candidates. The score of a committee from a voter is the sum of the scores of all its members from the voter.**Single nontransferable vote (SNTV).**Every voter gives 1 point to her top ranked candidate. The score of a committee from a voter is the sum of the scores of all its members from the voter.**Chamberlin–Courant (CC).**Different from the above three rules where all members of the winning committee are counted to accumulate the satisfaction of a voter, in CC, for each voter, only the best candidate in the winning committee contributes to the satisfaction of this voter. In other words, each voter is assumed to be only represented by her best candidate in the winning committee. Precisely, each voter has a nonincreasing mapping $\alpha :\mathbb{N}\to \mathbb{N}$, such that $\alpha \left(i\right)$ is a voter’s satisfaction of a candidate ranked in the i-th position. For a voter v with preference ${\succ}_{v}$ and a nonempty committee $w\subseteq C$, let ${\mathsf{top}}^{w}\left({\succ}_{v}\right)$ be the top-ranked candidate of v among w, i.e., ${\mathsf{top}}^{w}\left({\succ}_{v}\right)$ is the candidate $c\in w$, such that $c{\succ}_{v}{c}^{\prime}$ for all ${c}^{\prime}\in w\backslash \left\{c\right\}$. The CC score of a committee $w\subseteq C$ from a voter with mapping $\alpha $ is then $\alpha \left({\mathsf{pos}}_{{\succ}_{v}}\left({\mathsf{top}}^{w}\left({\succ}_{v}\right)\right)\right)$. In this section, we consider only the Borda satisfaction function $\alpha :\mathbb{N}\to \mathbb{N}$, which, for m candidates, holds that $\alpha \left(i\right)=m-i$.

**Monroe’s rule.**This rule is similar to the CC rule but with a further restriction that every candidate can represent at most $\lceil \frac{n}{k}\rceil $ voters. Let $g:V\to C$ be an assignment function and ${g}^{-}\left(c\right)$, $c\in C$ be the set of voters, $\succ \in V$, such that $g(\succ )=c$. Moreover, let $\mathcal{G}$ be the set of all assignment functions from V to C. The Monroe score of a k-committee $w\subseteq C$ is then defined as$$\underset{\begin{array}{c}g\in \mathcal{G}\phantom{\rule{3.33333pt}{0ex}}s.t.\\ |{g}^{-}\left(c\right)|\le n/k\phantom{\rule{3.33333pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{3.33333pt}{0ex}}c\in C\end{array}}{max}\left\{\sum _{\succ \in V}{\alpha}_{\succ}\left(g\left(v\right)\right)\right\},$$**Single-transferable voting (STV).**STV rules are a large class of voting rules each of which is featured by a rational number q and some vote-reweighting approach. A common principle of these rules is to guarantee certain groups of voters are proportionally represented. Fixing a rational quota q and a vote-reweighting approach, the STV rule selects winning committees iteratively as shown below. For a candidate c, let ${V}^{\mathsf{top}}\left(c\right)$ be the set of voters ranking c in the top.- Initially, we associate to each voter $v\in V$ a weight denoted by $\mathsf{weight}\left(v\right)$. (Usually, all voters have weight 1 initially, but this is not necessarily the case.)
- If there is a candidate, $c\in C$, that is ranked in the top by at least q voters, that candidate is added to the winning committee. Then, we apply the vote-reweighting approach so that the total weight of all votes ranking c in the top are reduced by $min\{q,p\}$, where $p={\sum}_{v\in {V}^{\mathsf{top}}\left(c\right)}\mathsf{weight}\left(v\right)$ is the sum of the weights of all voters ranking c in the top before the reweighting. Moreover, the candidate c is deleted from C and from all votes.
- If there is no such a candidate c as discussed above, then a candidate that is ranked in the top by the least number of voters is eliminated.
- The procedure terminates until k candidates are selected.

- Many of concrete STV rules have been considered in the literature (see the works by the authors of [18,19] for a history and a summary of many important STV rules). However, for simplicity, in this survey, we discuss only STV rules where initially all voters have weight 1, and the uniform reweighting approach is used in Step 2. Particularly, according to this reweighting approach, in Step 2, the weight of a voter v which ranks c in the top is reduced to $min\{0,\mathsf{weight}\left(v\right)\xb7(1-\frac{q}{p})\}$. Two important STV rules are those when q is equal to the Hare quota or the Droop quota, i.e., $q=\frac{n}{k}$ and $q=\u230a\frac{n}{k+1}\u230b+1$. We denote these two special STV rules as D-STV and H-STV, respectively.

#### 2.2. Voter Fairness in Approval-Based Voting

**Justified representation (JR).**A k-committee, $w\subseteq C$, provides JR, if, for every subset $U\subseteq V$ of at least $\frac{n}{k}$ votes such that ${\bigcap}_{u\in U}u\ne \xd8$, at least one of the candidates approved by some vote in U is included in w, i.e.,$$w\cap \left(\bigcup _{u\in U}u\right)\ne \xd8.$$**Proportional justified representation (PJR).**A k-committee, $w\subseteq C$, provides PJR if for every positive integer $\ell \le k$, and for every subset $U\subseteq V$ of at least $\ell \xb7\frac{n}{k}$ votes such that $|{\bigcap}_{u\in U}u|\ge \ell $, the committee w contains at least ℓ candidates from ${\bigcup}_{u\in U}u$, i.e., $|w\cap \left({\bigcup}_{u\in U}u\right)|\ge \ell $. This property was proposed in the work by the authors of [24].**Extended justified representation (EJR).**A k-committee $w\subseteq C$ provides EJR if for every positive integer $\ell \le k$ and for every subset $U\subseteq V$ of at least $t\ge \ell \xb7\frac{n}{k}$ votes such that $|{\bigcap}_{u\in U}u|\ge \ell $, the committee w contains at least ℓ candidates from every vote $u\in U$, i.e., $|w\cap u|\ge \ell $ for all $u\in U$. This property was proposed by Aziz et al. [23].**Perfect representation (PR).**PR is defined for special elections. Particularly, let $(C,V)$ be an election such that $\left|V\right|=t\xb7k$ for some integer t. A k-committee $w=\{{c}_{1},{c}_{2},\dots ,{c}_{k}\}$ provides PR if there is a partition $({V}_{1},{V}_{2},\dots ,{V}_{k})$ of V such that $|{V}_{i}|=t$ for all $1\le i\le k$ and ${c}_{i}$ is approved by all votes in ${V}_{i}$. This property was studied in the work by the authors of [24].

**Approval voting (AV).**The AV score of a candidate is the number of votes approving this candidate, and a winning k-committee consists of k candidates with the highest AV scores.**Satisfaction approval voting (SAV)**. The SAV score of a candidate c is defined as$$\sum _{c\in v\in V}\frac{1}{\left|v\right|}-\sum _{\begin{array}{c}v\in V,\\ c\notin v\end{array}}\frac{1}{m-\left|v\right|}$$**Minimax approval voting (MAV).**This rule aims to find a committee that is most close to every voter’s opinion. More precisely, the Hamming distance between a committee w and a vote v is ${d}_{\mathrm{H}}(v,w)=|w\backslash v|+|v\backslash w|$, and this rule selects a k-committee w minimizing ${max}_{v\in V}{d}_{\mathrm{H}}(v,w)$.**Proportional approval voting (PAV).**The PAV score of a committee w is defined as$$\sum _{\begin{array}{c}v\in V,\\ v\cap w\ne \xd8\end{array}}\left(1+\frac{1}{2}+\dots +\frac{1}{|v\cap w|}\right).$$A winning k-committee is an one with the maximum score.**Sequential proportional approval voting (seq-PAV).**This rule provides an approximation solution to PAV rule. It selects k winners in k rounds, one in each round. Precisely, initially we let $w=\xd8$. Assume that we have an i-committee w after round $i<k$. Then, in the next round, we find a candidate c which offers the maximum PAV score of $w\cup \left\{c\right\}$, and we extend w by resetting $w:=w\cup \left\{c\right\}$. After k rounds, w contains exactly k candidates.**Chamberlin–Courant approval voting (CCAV).**This rule is a variant of CC rule for approval-based voting. In particular, a voter satisfies with a committee if and only if this committee contains at least one of her approved candidates. This rule selects a k-committee that satisfies the maximum number of voters.**Monroe’s approval voting (MonAV).**This is a variant of Monroe’s rule for approval-based voting and is similar to CCAV. In CCAV, a candidate can satisfy all voters who approve this candidate. However, in MonAV, we require that each candidate is assigned to at most $\u2308\frac{n}{k}\u2309$ voters approving this candidate and, moreover, each voter can be assigned to at most one candidate. The MonAV score of a committee is the maximum number of voters who are satisfied by this committee and fulfill the above conditions.

- For each $v\in V$ and $c\in C$ it holds that$$0\le {x}_{v,c}\le 1.$$
- For every $c\in C$ and $v\in V$, if $c\notin v$, then$${x}_{v,c}=0.$$This corresponds to winner that is only distributed over voters approving that winner.
- It holds that$$\sum _{v\in V,c\in C}{x}_{v,c}=k.$$That is, there are in total k pointes to be distributed.
- For every $c\in C$, it holds that$$\sum _{v\in V}{x}_{v,c}\in \{0,1\}.$$This together with the previous restriction ensure that exactly k candidates have points to distribute.

**max-Phragmén.**This rule first calculates a load distribution $\mathbf{x}$ such that ${max}_{v\in V}{x}_{v}$ is minimized. Then, $f\left(\mathbf{x}\right)$ is the winning committee.**var-Phragmén**. This rule first calculates a load distribution $\mathbf{x}$ such that ${\sum}_{v\in V}{x}_{v}^{2}={\sum}_{v\in V}{\left({\sum}_{c\in C}{x}_{v,c}\right)}^{2}$ is minimized. Then, $f\left(\mathbf{x}\right)$ is the winning committee.**seq-Phragmén.**This rule takes k rounds to select the winners, one for each round. For a candidate c, let ${V}_{c}=\{v\in V:c\in v\}$ be the set of voters approving c. Initially, let $w=\xd8$. Let ${x}_{v}^{\left(j\right)}$ denote the voter loads after round j. At first, all voters have a load of 0, i.e., ${x}_{v}^{\left(0\right)}=0$ for all $v\in V$. As a first candidate, we select one $c\in C$ that receives the most approvals and add c into w. Then, the voter load of each voter approving this selected candidate is increased to $\frac{1}{|{V}_{c}|}$. In the next round, we choose a candidate that induces a (new) maximal voter load that is as small as possible, but now we have to take into account that some voters already have a non-zero load. The new maximal load if some candidate $c\in C$ is chosen as the $(j+1)$-st committee member is measured as$${s}_{c}^{(j+1)}=\frac{1+{\sum}_{c\in v\in V}{x}_{v}^{\left(j\right)}}{|{V}_{c}|}.$$

Is there a natural rule (or an algorithm) whose outcome always provide JR, EJR, PJR, and PR simultaneously?

#### 2.3. Fairness for Candidates with Sensitive Attributes

#### 2.4. Stable Fairness

## 3. Machine Learning Algorithms

#### 3.1. Fairness Notions

**Disparate Treatment.**Given dataset $D=(A,X,Y)$, with a set of sensitive attributes A (such as race, gender, etc.), remaining attributes X, and binary class to be predicted Y, predicted binary class $\widehat{Y}$, disparate treatment is said to exist in data D if$$Pr\left(\widehat{Y}\right|X)\ne Pr\left(\widehat{Y}\right|X,A).$$**Disparate Impact.**Given dataset $D=(A,X,Y)$, with a set of sensitive attributes A (such as race, gender, etc.), remaining attributes X, and binary class to be predicted Y, disparate impact is said to exist in data D if$$\frac{Pr(Y=1|A=0)}{Pr(Y=1|A=1)}\le \tau =0.8$$

**disparate mistreatment**. A classifier is said to suffer from disparate mistreatment if the misclassification rates for different groups of individuals having different values of the sensitive attribute A are different. Zafar et al. [52] proposed that disparate mistreatment in binary classification task can be specified with respect to various misclassification measures such as overall misclassification rate, false positive rate, false negative rate, false omission rate, and false discovery rate. We have also witnessed recent works drawing on fairness concepts from economics and social welfare such as equality, Gini distribution, etc., in its conceptualization of fairness.

- Anticlassification, also known as unawareness, seeks to achieve fairness in ML outcomes by excluding the use of protected features such as race, gender, or ethnicity from the statistical model. This notion is consistent with disparate treatment. Despite being intuitive, easy-to-use and having legal support, a crucial difficulty of this approach is that a protected feature might be correlated with many other unprotected features, and it is practically infeasible to identify all such covariate “proxies” and remove them from the statistical model. For example, protected class race might be correlated with various other features, such as education level, salary, life-expectancy, etc., and removing all these proxies from the statistical model could have detrimental effects in predictive performance. Consider we have a vector ${x}_{i}\in {\mathbb{R}}^{t}$ that represents the visible attributes of individual i such as race, gender, education level, age, etc. An algorithmic decision can be represented as a function $d:{\mathbb{R}}^{t}\mapsto \{0,1\}$, where $d\left(x\right)=k$, $k\in \{0,1\}$, means that action ${a}_{k}$ is taken. Suppose that x can be partitioned into protected and unprotected features: $x=({x}_{p},{x}_{u})$. Let ${X}_{p}$ denote the set of all protected features. Then, anticlassification requires that decisions do not consider protected attributes, more formally,$$d\left(x\right)=d\left({x}^{\prime}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}x,{x}^{\prime}\phantom{\rule{4.pt}{0ex}}\mathrm{such}\phantom{\rule{4.pt}{0ex}}\mathrm{that}\phantom{\rule{4.pt}{0ex}}{x}_{u}={x}^{\prime}.$$
- Statistical parity (also known by the names of demographic parity, independence, statistical parity, and classification parity) requires that common measures of predictive accuracy and performance errors remain uniform across various groups segmented by the protected features. This includes notions such as statistical parity, equality of accuracy, equality of false positive/false negative rates, and equality of positive/negative predictive values [55,56,57]. The main idea of this notion is to quantify and equate benefit and harm of the impact of the ML prediction to groups segmented by protected attributes equally and to distribute the errors among different stakeholders equally [55]. This notion of fairness has recently found application in criminal justice [58] and is consistent with disparate impact.The measure of classification parity based on false positive rate and the proportion of decisions that are positive have received considerable attention in machine learning domain [55,59,60]. For formal definition, please refer to Table 5.Recent research by Hu and Chen [61] suggests that the enforcement of statistical parity criteria in the short-term benefits building up the reputation of the disadvantageous minority in labor market in the long run. Note that, a critical flaw of notion of statistical parity is that it is easy to satisfy it by some arbitrary configuration, for example selecting best and qualified candidates from one group and random alternatives from the other group can still satisfy statistical parity. Moreover, the definition also ignores any possible correlation between positive outcome and protected attributes.
- Calibration requires that ML outcomes remain independent of protected features after controlling for estimated risk. Calibration relates to the fairness of risk scores and requires that for a given risk score, the proportion of individuals re-offending remains uniform across protected groups. Calibration is beneficial as a fairness condition as it does not require much intervention in the existing decision-making process [62]. A major disadvantage of calibration is that it has been shown that risk score can be manipulated to appear calibrated by ignoring information about the favored group [63]. Formally, given risk scores $s\left(x\right)$, calibration is satisfied when$$\mathrm{Pr}(Y=1\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}s\left(x\right),A)=\mathrm{Pr}(Y=1\phantom{\rule{3.33333pt}{0ex}}\left|\phantom{\rule{3.33333pt}{0ex}}s\right(x\left)\right).$$

#### 3.2. Fairness Mechanisms

**A.**- Preprocessing. Preprocessing methods deal with removing the protected features or their covariates before training the model. Similar to anticlassification, this method come with severe disadvantages as the protected feature might be correlated with many other unprotected features, and it is practically infeasible to identify all such covariates and exclude them without losing a lot on predictive accuracy. Kamiran and Calders [71] suggest a set of data processing techniques aimed at ensuring fairness for classification tasks. These include suppression, massaging the dataset, reweighting, and sampling.
**Suppression.**In this process, exactly like anticlassification, all the features that correlate with the protected set of features ${X}_{p}$ are first identified which are then removed from the classification model.**Massaging the dataset.**In this process, labels of some data points are manipulated in order to remove existing discrimination from the training data. In order to find a good set of labels to change, Kamiran and Calders [71] proposed a combination of ranking and learning.**Reweighting.**Instead of changing the labels, in this method the tuples in the training dataset are assigned asymmetric weights in order to overcome the bias.**Sampling.**Kamiran and Calders [71] introduced “uniform sampling” and “preferential sampling”, where the training data is sampled with the help of a ranker as a debiasing method.

Kamiran and Calders [71] found that suppression of the protected attributes does not always result in the removal of bias and massaging and preferential sampling techniques performed best for debiasing with a minimal loss in accuracy.Another idea developed in preprocessing is to learn a new representation of the data such that it removes the information correlated to the sensitive attribute [50,72,73]. The central algorithm such as classification then use the cleaned data. An advantage of this method is that the analyst can avoid the need to modify the classifier or access sensitive attributes during test time. **B.**- In-processing. In this method, the optimization procedure is modified to incorporate cost of unfairness. This is typically done by addition of a constraint to the optimizing problem or addition of cost of fairness as a regularizer. For example, Agarwal et al. incorporate cost-sensitive classification into their original objective function [59]. Given a dataset, ${\left\{({x}_{i},{c}_{i}^{0},{c}_{i}^{1})\right\}}_{i=1}^{n}$, where ${c}_{i}^{0}$ is the cost of predicting 0 on ${x}_{i}$ and ${c}_{i}^{1}$ is the cost of predicting 1 on ${x}_{i}$, a cost-sensitive classification algorithm given the dataset outputs$$\widehat{h}=arg\underset{h\in H}{min}h\left({x}_{i}\right){c}_{i}^{1}+(1-h\left({x}_{i}\right)){c}_{i}^{0}.$$More generally, the reduction approach by Agarwal et al. suggests the reduction of training with fairness constraints and solving a series of cost-sensitive classifications using off-the-shelf methods [59].An important advantage of this method is that there is no need to access sensitive attributes at test time. This method also provides higher flexibility in terms of trade-off between accuracy and fairness measures. An important disadvantage is that this method is task specific and requires modification of classifier which can often exponentially increase the computational complexity.The method to optimize counterfactual fairness also falls into this category. Kusner et al. [68] propose “counterfactual fairness” that explicitly specifies the assumptions about the data generating process. This can be done by adding a linear or convex surrogate for the fairness constraint in the learning models. For example, consider a predictive problem with fairness considerations, where A, X, and Y represent the protected attributes, remaining attributes, and the output of interest, respectively.
**C.**- Postprocessing. Postprocessing methods require editing the posteriors in order to satisfy the fairness constraints. The method searches for a proper threshold using the original score function for each group. We refer to Hard et al. [55] for more details on this postprocessing method. This method requires test-time access to the protected attribute and lacks flexibility in terms of trade-off between accuracy and fairness. However, this method benefits from being general and applicable to any classifier without any modification.

## 4. Recommender Systems

#### 4.1. Fairness for Users and Groups of Users

**Proportionality.**Given a package, P, and a parameter, $\u25b5$, we say that a user u likes an item $i\in P$ if i is ranked in the top-$\u25b5\%$ of the preferences of u over all items. Consequently, for a user, u, and a package, P, we say that P is m-proportional for u, for $m\ge 1$, if there exists at least m items in P, which are liked by u.**Envy-freeness.**Given a group G, a package P, and a parameter $\u25b5$, we say that a user $u\in G$ is envy-free for an item $i\in P$, if $r(u,i)$ is in top-$\u25b5\%$ of the preferences in the set $\left\{r\right(u,i):v\in G\}$. Consequently, for a user u, a package P and a group G, we say that the package P is m-envy-free for u, for $m\ge 1$, if u is envy-free for at least m items in P.

#### 4.2. Fairness for Items

#### 4.3. Multiple Stakeholder Fairness

## 5. Conclusions

#### 5.1. Challenges and Future Research Directions

#### 5.2. Limitations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Complexity of computing a committee satisfying a proportional property, or testing whether a given committee satisfies a proportional property. In the table, “P” stands for “polynomial-time solvable”. All results are from the work by the authors of [16].

Complexity of Computing | Complexity of Testing | |
---|---|---|

q-PSC | P | P |

weak q-PSC | P | P |

**Table 2.**A summary of the PSC properties satisfied by several important multi-winner voting rules and the complexity of computing a winning committee with respect to these rules. In the table, “N” means that the rule in the corresponding row does not satisfy the property in the corresponding column, and “Y” means that the rule satisfies the property. Observing that weak q-PSC is a too strong property for many rules to satisfy, Elkind et al. [17] studied three weak versions, namely, solid coalitions, consensus committee, and unanimity. They showed that each of SNTV, Bloc, k-Borda, CC, and Monroe fails at least one of these weak versions, and these results imply the ones for these rules in the table.

q_{H}-PSC | q_{D}-PSC | Weak q_{H}-PSC | Weak q_{D}-PSC | Complexity | |
---|---|---|---|---|---|

k-Borda | N [17] | N [17] | N [17] | N [17] | P (trivial) |

Bloc | N [17] | N [17] | N [17] | N [17] | P (trivial) |

SNTV | N [17] | N [17] | N [17] | N [17] | P (trivial) |

CC | N [17] | N [17] | N [17] | N [17] | NP-complete [20] |

Monroe | N [17] | N [17] | N [17] | N [17] | NP-complete [21] |

H-STV | Y [16] | Y [16] | Y [16] | Y [16] | P (trivial) |

D-STV | Y [16] | Y [16] | Y [16] | Y [16] | P (trivial) |

**Table 3.**Complexity of computing a committee satisfying a proportional property or testing whether a given committee satisfies a proportional property.

Complexity of Computing | Complexity of Testing | |
---|---|---|

justified representation | P [23] | P [23] |

extended Justified representation | P [25] | co-NP-complete [23] |

proportional Justified representation | P [24] | co-NP-complete [25] |

perfect representation | NP-complete [24] | P [24] |

**Table 4.**A summary of proportional properties of important approval-based multi-winner voting rules and the complexity of winner determination for these rules. In the table, “N” means that the rule in the corresponding row does not satisfy the property in the corresponding column, and “Y” means that the rule satisfies the property.

EJR | PJR | JR | PR | Complexity | |
---|---|---|---|---|---|

AV | N [23] | N [23,24] | N [23] | N [24,35] | P (trivial) |

SAV | N [23] | N [23,24] | N [23] | N [24,26,35] | P [26] |

seqPAV | N [23] | N [23,24] | N [23] | N [24,26,35] | P [26] |

MAV | N [23] | N [23,24] | N [23] | N [23] | NP-complete [36] |

CCAV | N [23] | N [24] | Y [23] | Y [35] | NP-complete [37] |

MonAV | N [23] | N [24] | Y [23] | Y [24] | NP-complete [21] |

var-Phragmén | N [38] | N [38] | Y [38] | Y [38] | NP-complete [38] |

seq-Phragmén | N [38] | Y [38] | Y [38] | N [38] | P [38] |

max-Phragmén | N [38] | Y [38] | Y [38] | Y [38] | NP-complete [38] |

PAV | Y [23] | Y [24] | Y [23] | N [24] | NP-complete [26] |

Fairness Definition | Description |
---|---|

Equalized Odds | Predicted outcome $\widehat{Y}$ satisfies equalized odds with respect to protected attribute A and true outcome Y, if $\widehat{Y}$ and A are independent conditional on Y, more specifically $P(\widehat{Y}=1|A=0,Y=y)=P(\widehat{Y}=1|A=1,Y=y)$ [55] |

Equal Opportunity | A binary predictor $\widehat{Y}$ satisfies equal opportunity with respect to A and Y if $P(\widehat{Y}=1|A=0,Y=1)=P(\widehat{Y}=1|A=1,Y=1)$ [55] |

Statistical Parity | A predictor $\widehat{Y}$ satisfies demographic parity if $P(\widehat{Y}|A=0)=P(\widehat{Y}|A=1)$ [64] |

Counterfactual Fairness | For a given causal model $(U,V,F)$ where $V\equiv A\cup X$, predictor $\widehat{Y}$ is said to be “counterfactually fair” if under any context $X=x$ and $A=a$, $P({\widehat{Y}}_{A\u21a4a})(U)=y|X=x,A=a)=P({\widehat{Y}}_{A\u21a4{a}^{\prime}}(U)=y|X=x,A=a),$ for all y and for any value ${a}^{\prime}$ attainable by A [68] |

Fairness through awareness | An algorithm is fair if it gives similar predictions to similar individuals. Any two individuals who are similar with respect to a similarity metric defined for a particular task should be classified similarly [64]. |

Individual fairness | Let $\mathcal{O}$ be a measurable space and $\delta (\mathcal{O})$ be the space of the distribution over $\mathcal{O}$. If $M:\mathcal{X}\mapsto \delta (\mathcal{O})$ denotes a map that maps each individual to a distribution of outcomes, the formulation of individual fairness is then $D(M(\mathcal{X}),M({\mathcal{X}}^{\prime}))\le d(\mathcal{X},{\mathcal{X}}^{\prime})$, where $\mathcal{X},{\mathcal{X}}^{\prime}\in {\mathbb{R}}^{d}$ are two metric functions on the input space and the output space, respectively [64]. |

Type of RecSys Fairness | Focus | References |
---|---|---|

User & Group Fairness | Ensure fairness for individual or a group of individuals, protected group incurs rating prediction errors in parity with the nonprotected group. | Yao and Huang [87] Ning and Karypis [94] |

Item Fairness | Fairness among item categories when recommended to users | Steck [95] Tsintzou et al. [96] |

Multiple Stakeholder Fairness | Fairness for multiple parties involved | Burke [91] Abdollahpouri et al. [97] Mehrotra et al. [98] |

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## Share and Cite

**MDPI and ACS Style**

Shrestha, Y.R.; Yang, Y.
Fairness in Algorithmic Decision-Making: Applications in Multi-Winner Voting, Machine Learning, and Recommender Systems. *Algorithms* **2019**, *12*, 199.
https://doi.org/10.3390/a12090199

**AMA Style**

Shrestha YR, Yang Y.
Fairness in Algorithmic Decision-Making: Applications in Multi-Winner Voting, Machine Learning, and Recommender Systems. *Algorithms*. 2019; 12(9):199.
https://doi.org/10.3390/a12090199

**Chicago/Turabian Style**

Shrestha, Yash Raj, and Yongjie Yang.
2019. "Fairness in Algorithmic Decision-Making: Applications in Multi-Winner Voting, Machine Learning, and Recommender Systems" *Algorithms* 12, no. 9: 199.
https://doi.org/10.3390/a12090199