On the occasion of his 60th birthday, we review some of Jérôme Lang’s influential contributions to the study of collective decision making, bringing together ideas from Economic Theory and Artificial Intelligence. Over the past two decades, his work in this domain—often inspired by fundamental questions regarding the most appropriate way of modelling the preferences of individual agents and how answers to those questions might impact the design of algorithms to support decision making—has significantly shaped the interdisciplinary field of Computational Social Choice.

Given the stated preferences of several people over a number of proposals regarding public policy initiatives, some of those proposals might be judged to be more “divisive” than others. When designing online participatory platforms to support digital democracy initiatives enabling citizens to deliberate over such proposals, we might wish to equip those platforms with the functionality to retrieve the most divisive proposals currently under discussion. Such a service would be useful for analysing the progress of deliberation and steering discussion towards issues that still require further debate. Guided by this use case, we explore possibilities for providing a clear definition of what it means to select a set of most divisive proposals on the basis of people's stated preferences over proposals. Then, employing the axiomatic method familiar from social choice theory, we show that the task of selecting the most divisive proposals in a manner that satisfies certain seemingly mild normative requirements faces a number of fundamental difficulties.

We present pabuviz.org, an interactive website that provides in-depth visualisation tools to compare the outcomes that would have been returned by a range of different voting rules for participatory budgeting when applied to historical election data. This information can help policy makers with choosing an appropriate voting rule.

Approval-based scoring rules require participants to submit ballots that correspond to the candidates they approve of, and each candidate then receives some score from each ballot it appears on. One particularly natural class of approval-based scoring rules is the class of size-approval rules. For rules in this class, each ballot is weighted based on the number of candidates it contains. We provide a deep axiomatic analysis of the class of size-approval rules and related classes of approval-based scoring rules. We organise existing work by shedding light on the exact role played by core axioms involved in previous characterisations of size-approval rules, while also providing multiple new characterisations of the same class and its closest relatives. We furthermore provide an original characterisation of even-and-equal cumulative voting, the most paradigmatic representative of the size-approval voting rules, where each ballot's weight is inversely proportional to its size.

Apportionment is the task of assigning resources to entities with different entitlements in a fair manner, and specifically a manner that is as proportional as possible. The best-known application is the assignment of parliamentary seats to political parties based on their share in the popular vote. Here we enrich the standard model of apportionment by associating each seat with a weight representing the (objective) value of that seat. A seat's weight reflects the fact that different seats might come with different roles, such as chair or treasurer. We define several apportionment methods and natural fairness requirements for this new setting, and we study the extent to which our methods satisfy these requirements. Our findings show that full fairness is harder to achieve than in the standard apportionment setting. Yet, for several natural relaxations of those requirements we can achieve stronger results than in the more expressive model of fair division with entitlements, where the values of objects are subjective.

We initiate the study of participatory budgeting using the epistemic approach, where one interprets votes as noisy estimates of some ground truth regarding the objectively best set of projects to fund.

We present game-academy.org, an online platform hosting interactive games that can be used for both teaching and outreach activities to illustrate concepts from game theory and related fields.

In the theory of voting, the Plurality rule for preferences that come in the form of linear orders selects the alternatives most frequently appearing in the first position of those orders, while the Anti-Plurality rule selects the alternatives least often occurring in the final position. We explore extensions of these rules to preferences that are partial orders, offering axiomatic characterisations for them.

We develop an approach for collective decision making from first principles. In this approach, rather than using a---necessarily imperfect---voting rule to map any given scenario where individual agents report their preferences into a collective decision, we identify for every concrete such scenario the most appealing set of normative principles (known as axioms in social choice theory) that would entail a unique decision and then implement that decision. We analyse some of the fundamental properties of this new approach, from both an algorithmic and a normative point of view.

This white paper outlines a long-term scientific vision for the development of digital-democracy technology. We contend that if digital democracy is to meet the ambition of enabling a participatory renewal in our societies, then a comprehensive multi-methods research effort is required that could, over the years, support its development in a democratically principled, empirically and computationally informed way. The paper is co-authored by an international and interdisciplinary team of researchers and arose from the Lorentz Center Workshop on “Algorithmic Technology for Democracy” (Leiden, October 2022).

Voting platforms can offer participants the option to sequentially modify their preferences, whenever they have a reason to do so. But such iterative voting may never converge, meaning that a state where all agents are happy with their submitted preferences may never be reached. This problem has received increasing attention within the area of computational social choice. Yet, the relevant literature hinges on the rather stringent assumption that the agents are able to rank all alternatives they are presented with, i.e., that they hold preferences that are linear orders. We relax this assumption and investigate iterative voting under partial preferences. To that end, we define and study two families of rules that extend the well-known k-approval rules in the standard voting framework. Although we show that for none of these rules convergence is guaranteed in general, we also are able to identify natural conditions under which such guarantees can be given. Finally, we conduct simulation experiments to test the practical implications of our results.

We consider scenarios where a group of agents wish to simplify a given abstract argumentation framework—specifying a set of arguments and the attacks between them—by eliminating cycles in the attack-relation on the basis of their preferences over arguments. They do so by first aggregating their individual preferences into a collective preference order and then removing any attacks involved in a cycle that go against that order. Our analysis integrates insights from formal argumentation and social choice theory. We obtain sweeping impossibility results for essentially all standard methods of preference aggregation, showing that no Condorcet method and no positional scoring rule can uphold the fundamental principle expressing that views held by every single member of the group must be respected. But we also find that so-called representative-agent rules do offer this guarantee.

We introduce a family of normative principles to assess fairness in the context of participatory budgeting. These principles are based on the fundamental idea that budget allocations should be fair in terms of the resources invested into meeting the wishes of individual voters. This is in contrast to earlier proposals that are based on specific assumptions regarding the satisfaction of voters with a given budget allocation. We analyse these new principles in axiomatic, algorithmic, and experimental terms.

We introduce a new approach for designing rules for participatory budgeting, the problem of deciding on the use of public funds based directly on the views expressed by the citizens concerned. The core idea is to embed instances of the participatory budgeting problem into judgment aggregation, a powerful general-purpose framework for modelling collective decision making. Taking advantage of the possibilities offered by judgment aggregation, we enrich the familiar setting of participatory budgeting with additional constraints, namely dependencies between projects and quotas regarding different types of projects. We analyse the rules obtained in both algorithmic and axiomatic terms.

We develop an approach for collective decision making from first principles. In this approach, rather than using a---necessarily imperfect---voting rule to map any given scenario where individual agents report their preferences into a collective decision, we identify for every concrete such scenario the most appealing set of normative principles (known as axioms in social choice theory) that would entail a unique decision and then implement that decision. We analyse some of the fundamental properties of this new approach, from both an algorithmic and a normative point of view.

In the context of aggregating intervals reflecting the views of several agents into a single interval, we investigate the impact of the form of representation chosen for the intervals involved. Specifically, we ask whether there are natural rules we can define both as rules that aggregate separately the left and right endpoints of intervals and as rules that aggregate separately the left endpoints and the interval widths. We show that on discrete scales it is essentially impossible to do so, while on continuous scales we can characterise the rules meeting these requirements as those that compute a weighted average of the endpoints of the individual intervals.

We analyse the problem of finding an allocation of resources in a multiagent system that is as fair as possible in terms of minimising inequality between the utility levels enjoyed by the individual agents. We use the well-known Atkinson index to measure inequality and we focus on the distributed approach to multiagent resource allocation, where new allocations emerge as the result of a sequence of local deals between groups of agents who agree on an exchange of some of the items in their possession. Our results show that it is possible to design systems that provide theoretical guarantees for optimal outcomes that minimise inequality, but also that there are significant computational hurdles to be overcome in the worst case. In particular, finding an optimal allocation is computationally intractable and under the distributed approach a large number of structurally complex deals, possibly involving many agents and items, may be required before convergence to a socially optimal allocation. This remains true even in severely restricted resource allocation scenarios where all agents have the same utility function. From a methodological point of view, while much work in multiagent resource allocation relies on combinatorial arguments, here we instead use insights from basic calculus.

We simulate voting rules for multiwinner elections in a model of judgment aggregation that distinguishes between rationality and feasibility constraints. These constraints restrict the structure of the individual judgments and of the collective outcome computed by the rule, respectively. We extend known results regarding the simulation of single-winner voting rules to the multiwinner setting, both for elections with ordinal preferences and for elections with approval-based preferences. This not only provides us with a new tool to analyse multiwinner elections, but it also suggests the definition of new judgment aggregation rules, by generalising some of the principles at the core of well-known multiwinner voting rules to this richer setting. We explore this opportunity with regards to the principle of proportionality. Finally, in view of the computational difficulty associated with many judgment aggregation rules, we investigate the computational complexity of our embeddings and of the new judgment aggregation rules we put forward.

In the context of social choice theory, we develop a tableau-based calculus for reasoning about voting rules. This calculus can be used to obtain structured explanations for why a given set of axioms justifies a given election outcome for a given profile of voter preferences. We then show how to operationalise this calculus, using a combination of SAT solving and answer set programming, to arrive at a flexible framework for presenting human-readable justifications to users.

We put forward a formal model of participatory budgeting where projects can incur costs with respect to several different resources, such as money, energy, or emission allowances. We generalise several well-known mechanisms from the usual single-resource setting to this multi-resource setting and analyse their algorithmic efficiency, the extent to which they are immune to strategic manipulation, and the degree of proportional representation they can guarantee. We also prove a general impossibility theorem establishing the incompatibility of proportionality and strategyproofness for this model.