## 2 Multi-Winner Voting  
Collective decision-making is a significant branch of social choice theory and has wide applications in both economics and computer systems. Examples of applications include political elections, committee selections (e.g., journal editorial board selection), selecting items to display in online shops, recommending multiple items to users in recommender systems, company or institute employee recruitment, heuristic algorithms selection in meta-heuristics, selecting data to load into caches in cloud computing systems, etc. Concretely, collective decision-making is mainly concerned with deriving consensus outcomes based on preferences of a number of decision-making participants over possible outcomes. Without a doubt, voting is one of the most popular approaches for collective decision-making. In this setting, we have a set of candidates \(C\) (possible outcomes), a set of voters \(V\) (decision-making participants) each of whom has a preference over candidates in \(C\), and then we either aim to select a subset of exactly \(k\) candidates as winners for some integer, \(k\), or find a ranking of candidates from the best to the worst for the community. It should be pointed out that voters need not necessarily to be human beings, they can also be certain criteria, robots, functions, or even algorithms. A large number of algorithms or multi-winner voting rules have been proposed for the purpose of the former. However, as fairness properties were not comprehensively taken into account when these rules were coined, many of them may result in unfair outcomes. For instance, assume that we have 100 voters who are divided into two groups: the majority and the minority. In particular, the majority consists of 90 voters, all of whom approve their spoiled candidates \(c_{1},c_{2},\ldots,c_{10}\). The remaining 10 voters, who are a minority, approve only the last candidate denoted by \(c^{\prime}\), probably because only this candidate has positive utility to them. If we aim to select 10 winners and apply the prevalent approval voting, then \(\{c_{1},c_{2},\ldots,c_{10}\}\) will be selected, as they are approved by the maximum number of voters. This result is clearly biased against the minority since their opinion is completely ignored. In this section, we shall survey recent progress on the study of fairness properties in multi-winner decision-making. Regarding fairness, an important concern is fair for whom and fair at which level. These two questions are important guidance for us to define different fairness properties. In voting, we have two types of entities, namely, the candidates and the voters, both of whom may need to be fairly considered. For single-winner voting rules (\(k=1\)), which aim to select exactly one winner, the _neutrality_ property ensures that candidates are treated equally, whereas the _anonymity_ property ensures that voters are treated equally [11]. Recall that neutrality says that the winners' identities remain the same after candidates are renamed, and anonymity says that all voters have the equal power and the order of them have no impact on the results (see the work by the authors of [12] for the formal definitions). These two properties have long been studied in the literature. Neutrality and anonymity are of course also desired for multi-winner voting rules, where a fixed number of winners are selected [13; 14]. However, these two properties only provide individual-level fairness by regarding each voter and each candidate as an independent individual, but do not say anything about group-level fairness, which is of particular importance in some real-world applications. Consider the above example and consider what should be a fair result for both the majority and the minority. As the minority accounts to 10% of all voters, should 10% of the winners also come from their approved candidates? If this is the case, then a fair result would be that selecting \(c^{\prime}\) and nine of \(\{c_{1},c_{2},\ldots,c_{10}\}\) as the winners.