This paper bridges two perspectives: it studies the multi-secretary problem through the fairness lens of social choice, and examines multi-winner elections from the viewpoint of online decision making. After identifying the limitations of the prominent proportionality notion of Extended Justified Representation (EJR) in the online domain, the work proposes a set of mechanisms that merge techniques from online algorithms with rules from social choice---such as the Method of Equal Shares and the Nash Rule---and supports them through both theoretical analysis and extensive experimental evaluation.

We examine an approval-based model of Liquid Democracy with a budget constraint on voting and delegating costs, aiming to centrally select casting voters ensuring complete representation of the electorate. From a computational complexity perspective, we focus on minimizing overall costs, maintaining short delegation paths, and preventing excessive concentration of voting power. Furthermore, we explore computational aspects of strategic control, specifically, whether external agents can change election components to influence the voting power of certain voters.

Participatory budgeting allows citizens to decide how to allocate public funds among projects. Motivated by recent real-world applications in both municipal and blockchain environments, we propose and study a framework where voters can donate additional private funds to enhance their own satisfaction, using cumulative ballots to express preferences. We introduce the first mechanisms for this setting and evaluate them primarily based on the satisfaction of axioms, while also exploring their algorithmic and strategic aspects.

Pure proportional voting rules can sometimes lead to highly suboptimal outcomes. We introduce the Method of Equal Shares with Bounded Overspending (BOS Equal Shares), a robust variant of the Method of Equal Shares that balances proportionality and efficiency. BOS Equal Shares addresses inefficiencies implied by strict proportionality axioms, yet still provides fairness guarantees, similar to the original Equal Shares. Our extensive empirical analysis shows excellent performance of BOS Equal Shares across several metrics. In the course of the analysis, we also study a fractional variant of the Method of Equal Shares.

We focus on a generalization of the classic Minisum approval voting rule, introduced by Barrot and Lang (2016), and referred to as Conditional Minisum ( cms ), for multi-issue elections with preferential dependencies. Under this rule, voters are allowed to declare dependencies between different issues, but the price we have to pay for this higher level of expressiveness is that we end up with a computationally hard rule. Motivated by this, we first focus on finding special cases that admit efficient algorithms for cms . Our main result in this direction is that we identify the condition of bounded treewidth (of an appropriate graph, emerging from the provided ballots) as the necessary and sufficient condition for exact polynomial algorithms , under common complexity assumptions. We then move to the design of approximation algorithms . For the (still hard) case of binary issues, we identify natural restrictions on the voters' ballots, under which we provide the first multiplicative approximation algorithms for the problem. The restrictions involve upper bounds on the number of dependencies an issue can have on the others and on the number of alternatives per issue that a voter can approve. Finally, we also investigate the complexity of problems related to the strategic control of conditional approval elections by adding or deleting either voters or alternatives and we show that in most variants of these problems, cms is computationally resistant against control. Overall, we conclude that cms can be viewed as a solution that achieves a satisfactory tradeoff between expressiveness and computational efficiency, when we have a limited number of dependencies among issues, while at the same time exhibiting sufficient resistance to control.

We consider a multi-issue election setting over a set of possibly interdependent issues with the goal of achieving proportional representation of the views of the electorate. To this end, we employ a proportionality criterion suggested recently in the literature, that guarantees fair representation for all groups of voters of sufficient size. For this criterion, there exist rules that perform well in the case where all the issues have a binary domain and are independent of each other. In particular, this has been shown for Proportional Approval Voting (PAV) and for the Method of Equal Shares (MES). In this paper, we go two steps further: we generalize these guarantees for issues with a non-binary domain, and, most importantly, we consider extensions to elections with dependencies among issues, where we identify restrictions that lead to analogous results. To achieve this, we define appropriate generalizations of PAV and MES to handle conditional ballots. In addition to proportionality considerations, we also examine the computational properties of the conditional version of MES. Our findings indicate that the conditional case poses additional challenges and differs significantly from the unconditional one, both in terms of proportionality guarantees and computational complexity.

Our work focuses on a generalization of the classic Minisum approval voting rule, introduced by Barrot and Lang (2016), and referred to as Conditional Minisum (CMS), for multi-issue elections. Although the CMS rule provides much higher levels of expressiveness, this comes at the expense of increased computational complexity. In this work, we study further the issue of efficient algorithms for CMS, and we identify the condition of bounded treewidth (of an appropriate graph that emerges from the provided ballots), as the necessary and sufficient condition for polynomial algorithms, under common complexity assumptions. Additionally we investigate the complexity of problems related to the strategic control of such elections by the possibility of adding or deleting either voters or alternatives. We exhibit that in most variants of these problems, CMS is resistant against control.

Approval voting provides a simple, practical framework for multi-issue elections, and the most representative example among such election rules is the classic Minisum approval voting rule. We consider a generalization of Minisum, introduced by the work of Barrot and Lang [2016], referred to as Conditional Minisum, where voters are also allowed to express dependencies between issues. The price we have to pay when we move to this higher level of expressiveness is that we end up with a computationally hard rule. Motivated by this, we focus on the computational aspects of Conditional Minisum, where progress has been rather scarce so far. We identify restrictions to every voter's dependencies, under which we provide the first multiplicative approximation algorithms for the problem. The restrictions involve upper bounds on the number of dependencies an issue can have on the others. At the same time, by additionally requiring certain structural properties for the union of dependencies cast by the whole electorate, we obtain optimal efficient algorithms for well-motivated special cases. Overall, our work provides a better understanding on the complexity implications introduced by conditional voting.