We give a quantitative analysis of the independence of irrelevant alternatives (IIA) axiom. IIA says that the society's preference between x and y should depend only on individual preferences between x and y: we show that, in several contexts, if the individuals express their preferences about additional (``irrelevant'') alternatives, this information helps to estimate better which of x and y has higher social welfare. Our contribution is threefold: (1) we provide a new tool to measure the impact of IIA on social welfare (pairwise distortion), based on the well-established notion of voting distortion, (2) we study the average impact of IIA in both general and metric settings, with experiments on synthetic and real data and (3) we study the worst-case impact of IIA in the 1D-Euclidean metric space.

We consider a voting model, where a number of candidates need to be selected subject to certain feasibility constraints. The model generalizes committee elections (where there is a single constraint on the number of candidates that need to be selected), various elections with diversity constraints, the model of public decisions (where decisions need to be taken on a number of independent issues), and the model of collective scheduling. A critical property of voting is that it should be fair---not only to individuals but also to groups of voters with similar opinions on the subject of the vote; in other words, the outcome of an election should proportionally reflect the voters' preferences.

We provide a library of participatory budgeting data (Pabulib) and open source tools (Pabutools and Pabustats) for analysing this data. We analyse how the results of participatory budgeting elections would change if a different selection rule was applied. We provide evidence that the outcomes of the Method of Equal Shares would be considerably fairer than those of the Utilitarian Greedy rule that is currently in use. We also show that the division of the projects into districts and/or categories can in many cases be avoided when using proportional rules. We find that this would increase the overall utility of the voters.

We consider approval-based committee elections, in which a size-k subset of available candidates must be selected given approval sets for each voter, indicating the candidates approved by the voter. A number of axioms capturing ideas of fairness and proportionality have been proposed for this framework. We argue that even the strongest of them, such as priceability and the core, only rule out certain undesirable committees, but fail to ensure that the selected committee is fair in all cases. We propose two new solution concepts, stable priceability and balanced stable priceability, and show that they select arguably fair committees. Our solution concepts come with a non-trivial-to-construct but easy-to-understand market-based explanation for why the chosen committee is fair. We show that stable priceability is closely related to the notion of Lindahl equilibrium from economics.

We study voting rules for participatory budgeting, where a group of voters collectively decides which projects should be funded using a common budget. We allow the projects to have arbitrary costs, and the voters to have arbitrary additive valuations over the projects. We formulate two axioms that guarantee proportional representation to groups of voters with common interests. To the best of our knowledge, all known rules for participatory budgeting do not satisfy either of the two axioms; in addition we show that the most prominent proportional rule for committee elections, Proportional Approval Voting, cannot be adapted to arbitrary costs nor to additive valuations so that it would satisfy our axioms of proportionality. We construct a simple and attractive voting rule that satisfies one of our axioms (for arbitrary costs and arbitrary additive valuations), and that can be evaluated in polynomial time. We prove that our other stronger axiom is also satisfiable, though by a computationally more expensive and less natural voting rule.

We consider elections where both voters and candidates can be associated with points in a metric space and voters prefer candidates that are closer to those that are farther away. It is often assumed that the optimal candidate is the one that minimizes the total distance to the voters. Yet, the voting rules often do not have access to the metric space M and only see preference rankings induced by M. Consequently, they often are incapable of selecting the optimal candidate. The distortion of a voting rule measures the worst-case loss of the quality being the result of having access only to preference rankings. We extend the idea of distortion to approval-based preferences. First, we compute the distortion of Approval Voting. Second, we introduce the concept of acceptability-based distortion---the main idea behind is that the optimal candidate is the one that is acceptable to most voters. We determine acceptability-distortion for a number of rules, including Plurality, Borda, k-Approval, Veto, Copeland, Ranked Pairs, the Schulze's method, and STV.