# Jérôme Lang's contributions at the interface of economic theory with artificial intelligence

Ulle Endriss, Nicolas Maudet

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Ulle Endriss, Nicolas Maudet. Jérôme Lang's contributions at the interface of economic theory with artificial intelligence. Theory and Decision, 2026, 10.1007/s11238-025-10098-0. http-05519686

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# Jérôme Lang's Contributions at the Interface of Economic Theory with Artificial Intelligence

Ulle Endriss

ILLC, University of Amsterdam

ulle.endriss@uva.nl

Nicolas Maudet

LIP6, Sorbonne Université, CNRS, Paris

nicolas.maudet@lip6.fr

Article in the Special Issue in Honour of Jérôme Lang published in *Theory and Decision* in 2026.

## Abstract

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.

## 1 Introduction

Jérôme Lang is one of the leading figures working at the interface of Economic Theory with Artificial Intelligence (AI). Especially in the domain of collective decision making, he is known for numerous innovative ideas that have shaped the interdisciplinary field of Computational Social Choice (Brandt et al., 2016) over the past two decades. Just as importantly, throughout this time, he has been a mentor to countless individual researchers, both in his own group and around the world, enabling many of them to do influential work themselves. Both of us have had the great fortune to benefit from this mentorship for many years. It therefore is a source of great joy for us—on the occasion of his 60th birthday—to have the opportunity to present this overview of Jérôme's scientific contributions at the interface of Economic Theory with AI.

To set the stage, let us briefly recall where he came from. Jérôme was born in 1965 in Neufchâteau, France, and studied Computer Science and Applied Mathematics at the engineering school ENSEEIHT in Toulouse. He received his PhD from Paul Sabatier University in Toulouse in 1991 for work conducted under the supervision of Henri Prade on the topic of possibilistic logic—a logic for reasoning about uncertain, incomplete, and potentially inconsistent information (Lang, 1991). He belonged to the research group of Didier Dubois and Henri Prade, which from 1990 was part of the newly formed Institut de Recherche en Informatique de Toulouse (IRIT). The group was one of the leading centres of excellence in decision-theoretic AI at the time, which meant that Jérôme had more exposure to mathematical economics than most computer scientists, even if collective decision making as such was not part of his portfolio in those early days. Right after completing his PhD, he managed to secure a tenured position as a full-time researcher at the CNRS, the French National Centre for Scientific Research. In this role, he stayed
on at IRIT until 2008, when he was promoted to CNRS Research Director and moved to LAMSADE, a research centre with a long history in Operations Research, Decision Support, and related fields based at Paris-Dauphine University. Since 2024 he has been serving as the director of LAMSADE. He is a Fellow of the European Association for Artificial Intelligence (EurAI) and received the CNRS Silver Medal in 2017.

During the years following his PhD and throughout the 1990s, Jérôme continued to focus his research on decision-theoretic AI, the modelling of uncertainty, and other forms of knowledge representation. But around the turn of the century he got more and more interested in questions of collective decision making. It turned out that his background in Computer Science granted him a fresh perspective on a topic that, until then, had largely been the prerogative of economists. In particular, he found that many of the tools and techniques developed in the area of Knowledge Representation and Reasoning in AI naturally lent themselves to answering crucial questions about the design of algorithms to support collective decision making—questions that were not at all obvious to even ask for researchers trained in Economics.

For instance, many real-world decision-making problems have a *combinatorial structure*, where we need to settle on a value for several variables, meaning that the overall number of alternatives is exponential in the number of variables. Modelling such a problem as if each decision maker were in a position to simply rank all of these exponentially many alternatives might be possible in the abstract and perfectly suitable for a purely mathematical analysis, but it not only is cognitively unrealistic but also algorithmically naïve. Jérôme pioneered the use of carefully designed logic-based languages for the compact representation of preferences in the area of collective decision making and started to systematically explore the impact of using such languages for a wide range of decision-making scenarios. His initial focus was on voting, soon followed by fair division.

We are going to review a selection of his contributions of this kind in the body of this paper. Of course, what we can summarise here merely amounts to a selection of a few highlights—albeit highlights we feel go some way towards illustrating his manner of thinking about a problem. Indeed, Jérôme has been and continues to be a very productive researcher who has worked on a great variety of different topics.<fnref n="1" />

Despite his training in and great talent for theory, in recent years Jérôme increasingly felt the need to see theoretical results more broadly applied, and he got involved in several initiatives aiming for more direct an impact on policy making. For instance, together with colleagues from Political Science, Economics, and Computer Science, he contributed to several field and online experiments on alternative voting systems (“Voter Autrement”), notably for the French presidential elections in 2017 (Bouveret et al., 2019; Laslier, 2019), and then again in 2022. Responding to a request by former parliamentarian Cédric Villani, he also was involved in designing several redistricting models used to simulate different voting systems for the French national assembly, aiming for different levels of proportionality (Blanch and Bouveret, 2018). On the same topic, he contributed to the publication of two reports discussing scenarios regarding the introduction of proportionality for the election of members of parliament (Cohendet et al., 2018; Laslier et al., 2023). On these occasions, he wrote several opinion pieces published in newspapers, in which he advocated

<footnote n="1">¹ Jérôme himself often ascribes his own productivity to his frequent and enthusiastic use of the French—and indeed the European—train network, with regular trips between Paris and Toulouse providing ample opportunity for undisturbed thought and writing. He is known to at least attempt to reach even fairly far-away conference locations and the like by train—and did so long before this became a more widely adopted practice. He is equally well known to willingly offer recommendations for how to find the best connections (usually in favour of the online information portal of the German railway company).</footnote>
using voting systems with provably better properties than what is currently in use.<fnref n="2" />

Beyond his individual scientific contributions, Jérôme has been instrumental in creating the field of Computational Social Choice as a whole. Back in 2006, he was one of the two programme chairs of the first edition of COMSOC, the International Workshop on Computational Social Choice (Endriss and Lang, 2006), he was the lead author of the first survey article attempting to characterise the field shortly afterwards (Chevaleyre et al., 2007), and he was one of the editors of the main reference work in the field, the *Handbook of Computational Social Choice*, published a decade later (Brandt et al., 2016). At the same time, at the national level, he played a key role, through setting up a series of collaborative projects, in stimulating collaboration between the Computer Science, Economics, and Political Science research communities.

Throughout all this time, he has been an important mentor to many individual members of the research community he helped create, starting of course with his own PhD students but reaching far beyond this. This includes the authors of this paper. To give but one of many examples, while both of us were postdocs in London shortly after the turn of the century working in the area of Multiagent Systems, when Jérôme heard that we were getting interested in what at the time we still called ‘formal models of social interaction’ and struggled to orient ourselves, he decided to photocopy the first couple of chapters of Hervé Moulin’s excellent *Axioms of Cooperative Decision Making* (Moulin, 1988) and snail-mailed them to us, thereby making sure we got a proper grounding in social choice and fair division before attempting to contribute to these fields ourselves.

He has helped countless others in similar ways. Jérôme graduated more than 15 PhD students, and at the time of writing, the DBLP Computer Science Bibliography (dblp.org) lists a total of 293 coauthors for Jérôme, many of them younger colleagues for whom he acted as an informal advisor. At any given time, he is involved in a large number of big and small paper projects with individuals from his extensive network. This includes local colleagues, people he met at conferences,<fnref n="3" /> and the many fellow researchers he graciously invited for a research visit to Paris.

Mentorship can take many forms. Jérôme also is a dedicated reviewer who, over the years, must have helped hundreds of individuals with their research through providing detailed and helpful feedback on their work submitted for peer review.

His arrival at LAMSADE, a lab with a long tradition of interdisciplinary research at the interface of Economics, Operations Research, and Computer Science, was a natural move. It coincided with international efforts to shape a community on these topics. Together with Alexis Tsoukiàs, Jérôme played a key role in these years in establishing LAMSADE as a leading lab in Algorithmic Decision Theory and Computational Social Choice. One key ingredient that Jérôme brought to the mix was his habit of inviting many people to visit the lab.<fnref n="4" />

In the remainder of this paper, we are going to outline some of the many research directions pursued by Jérôme over the years and shall try to highlight themes and ideas that have shaped how the research community has come to think about questions of

<footnote n="2">²High-stakes politics is not the only area of application Jérôme has contributed to. He also has a long-standing interest in improving the peer review process, including in particular the process of assigning papers to reviewers based on their preferences (Benferhat and Lang, 2001; Meir et al., 2021).</footnote>

<footnote n="3">³Conferences provide a crucial opportunity for scientific exchange for all of us. For Jérôme, in addition, they also provide an excellent opportunity to practise speaking some of the impressive number of languages in which he managed to attain a certain degree of linguistic competence. (According to Jérôme, your L-index is the greatest integer $\ell$ such that you can speak at least $\ell$ words in each of $\ell$ different languages.)</footnote>

<footnote n="4">⁴One colleague described Jérôme’s office to us as a “continuously ongoing workshop”: If you visited LAMSADE in the past decade or so, chances are that you would have encountered at least one visitor sitting at the spare desk in Jérôme’s office—the birth place of so many ideas and joint paper projects.</footnote>
collective decision making ‘through a computational lens’.

## 2 Compact Representation of Preferences

The notion of preference is central to the challenge of modelling, analysing, and supporting the process of decision making. But how best to model the preferences of a given individual is far from obvious. As an economist you might debate whether it would be appropriate to use a cardinal utility function, mapping any of the available alternatives to a real number, or whether an ordinal preference relation defined on the same set of alternatives might be more appropriate. On the one hand, having access to cardinal utilities offers a lot of room for fine-grained modelling. For instance, we can now talk about the ‘social welfare’ of a given alternative, by summing up the utilities for that alternative for all the individuals affected by the decision in question. On the other hand, performing calculations of this kind presupposes that comparisons of utility levels across individuals are in fact meaningful ("you like apples exactly as much as I like pears"). And this is a conclusion we might not always be willing to accept.

What sort of fundamental questions we ask about preferences, more often than not, will be strongly influenced by the academic discipline we have been trained in. A philosopher, a political scientist, a psychologist likely would all focus on different questions. This is also true for the computer scientist, which might explain why Jérôme approached the question of preference modelling somewhat differently from the economists who had worked on such problems before him. For a computer scientist arguably the most fundamental question to ask is this:

> When trying to encode someone's preferences on a computer, how should we represent those preferences? In other words, what is the right data structure when working with preferences?

For instance, should we encode a utility function as a large table, associating each alternative with its utility, or might there be a better representation? And in case we want to work with ordinal preferences—say, strict rankings of the available alternatives—should we represent such a ranking as a list from best to worst alternative, should we use a matrix of 1's and 0's to indicate for each pair of alternatives whether the first is strictly preferred to the second, or should we look for yet some other data structure? Of course, the choice of data structure will greatly impact the computational difficulty of answering all sorts of questions about an individual's preferences (this is the so-called *query complexity*) and thus the range of algorithms we might be able to develop to support the taking of decisions on the basis of the preferences reported.

Such questions become most pressing when the number of alternatives to consider is very large. This is the case for many natural decision-making problems, as these often involve choosing a value for each of several different variables, leading to a combinatorial explosion in the overall number of alternatives to consider. For instance, if a decision maker might in principle receive any subset of some set of 10 resources, then describing her utility function will involve specifying up to $2^{10} = 1024$ real numbers. Another example is one of Jérôme's favourites, featuring in many of his papers and talks:

> "[Suppose we] have to agree on a common menu to be composed of a first course dish, a main course dish, a dessert and a wine, with a choice of 6 items for each. This makes $6^4$ [possible menus]." (Lang, 2004)
Your preferences over the choice of wine most likely are not independent from the main course being served: you might prefer a *Riesling* when having *quiche lorraine* but a *Pinot Noir* when eating a *vegetable tagine*. Therefore, the problem of representing your preferences over alternative menus cannot simply be reduced to four separate preference relations (one over alternative main courses, one over alternative wines, etc.). In the early 2000s, Jérôme was one of a small number of researchers with roots in the field of Knowledge Representation and Reasoning in AI who recognised the importance of devising adequate preference representation formalisms for situations of this kind.<fnref n="5" /> An early example is a joint paper with his second PhD student, Céline Lafage, on the logic-based representation of utility functions (Lafage and Lang, 2000), and maybe the clearest articulation of the challenges raised by this perspective can be found in his 2004 paper on logic-based preference representation and its applications to voting (Lang, 2004).

An example for such a logic-based language for preference representation is the idea of using prioritised goals to represent preferences. Suppose we need to decide on a value for *k* different variables (*X*₁, *X*₂, ..., *X*ₖ), and for the sake of simplicity, suppose these are binary variables. So, in effect, there are *k* questions we need to answer with either a ‘yes’ or a ‘no’. Suppose we want to model ordinal preferences. Rather than asking individuals to each rank the 2*k* possible combinations of answers to our *k* questions, we could ask them to specify goals they care about. For instance, the goal *X*₁ ∧ *X*₇ expresses that you would want both *X*₁ and *X*₇ to be set to true. *X*₂ ∨ *X*₄ says that you would want one of *X*₂ and *X*₄ to be true, but you do not mind which, and (*X*₂ ∧ ¬*X*₆) → *X*₃ says that you want *X*₃ to be true whenever *X*₂ happens to be true while *X*₆ is false. If you have multiple goals, we can ask you to prioritise, i.e., to order your goals in terms of their importance to you (with the possibility of several goals having the same priority level). We can induce an ordinal preference relation over alternatives from such a database of prioritised goals in a number of ways. One of them is the lexicographic approach: when comparing two alternatives, we go through all goals in order of their priority levels (from most to least important); once we reach a priority level for which the number of goals satisfied by the two alternatives differ from one another we say that we prefer the alternative that satisfies more goals at this level. We will encounter a concrete example in the next section.

How difficult, in computational terms, is it to reason about someone’s preferences when those preferences are communicated to us using such a language? Several of the technical results in Jérôme’s programmatic paper on logic-based preference representation languages answer different variations of this question (Lang, 2004). For example, for the language of prioritised goals with lexicographic ordering sketched earlier, deciding whether one given alternative is preferred to another given alternative can be done in polynomial time, while deciding whether a given alternative is optimal (in the sense of not being dispreferred to some other alternative) is coNP-complete.<fnref n="6" />

The extent to which a chosen language of preference representation enables efficient computation is important. But an even more fundamental question is this: What range of preference structures can we model using the language of prioritised goals? That is, what is the *expressivity* of this language? Can we actually describe every preference order of interest using this approach? Also here there are multiple ways of making this question fully precise, as there are multiple ‘dialects’ to the language of prioritised goals. Together with his coauthors, Jérôme studied such questions systematically (Coste-Marquis et al.,

<footnote n="5">⁵Others include Bacchus and Grove (1995) as well as Boutilier et al. (2004).</footnote>

<footnote n="6">⁶The complexity class coNP is closely related to the better known class NP. The coNP-complete problems are the hardest problems in this class. If a problem is proven to be coNP-complete, then this means that it is highly unlikely that a polynomial algorithm to solve the problem will ever be found. For a full definition we refer to the reader to standard textbooks, such as the one by Arora and Barak (2009).</footnote>
2004). For example, if agents are only permitted to specify a single goal each, then we clearly cannot describe the full range of ordinal preference structures. However, we still can describe every conceivable *dichotomous* preference relation. So when working in an application domain where agents can be assumed to always have dichotomous preferences, this might be exactly the right language to encode those preferences. If we do allow for an arbitrary number of prioritised goals and use the aforementioned lexicographic ordering, then one of the results of Coste-Marquis et al. (2004) is that we can express every conceivable weak ordering on the set of alternatives. Interestingly, this remains true if we use a more simplistic approach to ordering and now associate the desirability of an alternative with the priority level of the most important goal it fails to satisfy (so-called *best-out ordering*): we still can express every conceivable preference!

This is surprising, given that the lexicographic approach intuitively seems much more powerful. This intuition is not misguided, as Coste-Marquis et al. (2004) managed to make clear by asking a further question: For two languages that can express the same range of preference structures, can one of them do so more efficiently? This is about *compactness* (or *representational efficiency*). Is it the case that the first language can express anything the second language can with at most a polynomial increase in the amount of space (number of bits) required, while there are also some preferences where the first language requires exponentially less space than the second? One of the results of Coste-Marquis et al. (2004) is that the lexicographic variant of the prioritised-goal language is strictly more compact than the best-out variant in this sense. The central step in the proof involves showing that the former language can express certain *exponentially long chains* (relative to the number of variables) using only a polynomial number of goals, while the latter cannot. It in fact is not too difficult to grasp the essence of the argument:

Think of alternatives as numbers in binary representation. For instance, 110 represents the alternative where $X_1$ and $X_2$ are true while $X_3$ is false. Now consider the following chain of alternatives, the length of which is exponential in the number of variables:

$$111 \succ 110 \succ 101 \succ 100 \succ 011 \succ 010 \succ 001 \succ 000$$

Under the *lexicographic ordering*, we can compactly represent this ordering using just three prioritised goals: $X_1$ with priority level 4, $X_2$ with priority level 2, and $X_3$ with priority level 1. But under the *best-out ordering*, as overall desirability of an alternative by definition always equals the priority level of just one single goal, we must have as many goals as there are alternatives (or at most one less, given that we can arrange it so that the least preferred alternative does not satisfy any goals at all).

This example easily generalises from $k = 3$ variables to arbitrary $k$. We will need just $k$ goals (polynomial) under the lexicographic ordering but $2^k - 1$ goals (exponential) under the best-out ordering.

In later work, Jérôme and his coauthors carried out a similar research agenda for logic-based languages for the representation of cardinal preferences, i.e., for the representation of utility functions (Uckelman et al., 2009). One of the many areas of application for such languages is in the domain of combinatorial auctions, where we need to be able to represent bids in a compact manner and then design algorithms to clear the auction that can operate on the bids thus represented (Nisan, 2006).

Concrete technical results aside, these early papers promoted a shift in perspective: preferences are mathematical objects we can formally represent and reason about using
computers, and there are better and worse ways of doing so. This fresh perspective directly influenced important work on both voting and fair division, as we shall see next.

# 3 Voting in Combinatorial Domains

It is well-known that designing a good voting rule—one that would adequately operationalise people's intuitions about fairness and other normative requirements—is a major challenge. It is the challenge at the very heart of Social Choice Theory (Arrow et al., 2002). But when the alternatives to choose between have a combinatorial structure, requiring us to pick a value for several different variables, then there are additional difficulties that introduce an algorithmic dimension into an otherwise purely axiomatic problem. Let us illustrate the issue with a variant of a well-known example that makes an appearance in several of Jérôme's papers (see, e.g., Chevaleyre et al., 2007; Lang and Xia, 2009, 2016) and that has been a regular feature in his talks, usually inviting active audience participation along the way:

> Imagine 101 voters need to decide whether to invest in building a swimming pool ($S$ or $\bar{S}$) and whether to invest in building a tennis court ($T$ or $\bar{T}$). How should we organise this election? We do not know in advance what preferences each voter will report, but the voting rule we pick should perform adequately, for instance, in the following situation. Suppose almost all voters would want to realise one of the two projects but think that investing into both of them would be disastrous: 50 individuals have the preference relation $ST \succ \bar{S}T \succ \bar{S}\bar{T} \succ ST$ and a further 50 have $\bar{S}T \succ ST \succ \bar{S}\bar{T} \succ \bar{S}\bar{T}$. Only a single individual wants to realise as many projects as possible: $ST \succ \bar{S}T \succ \bar{S}T \succ \bar{S}\bar{T}$.

One approach would be to ignore the combinatorial structure of the problem and to treat this as an election with four alternatives. Most common voting rules would return $\bar{S}\bar{T}$ as the winner. For instance, $\bar{S}\bar{T}$ wins under the Borda rule, and it also is a Condorcet winner.<fnref n="7" /> The Plurality rule would return a tie between $\bar{S}\bar{T}$ and $\bar{S}T$, which also is not entirely unreasonable.

But this approach does not scale. If $k$ is the number of (binary) issues to decide on, the number of alternatives generated is $2^k$. So the number of alternatives increases exponentially with the number of issues under consideration. Using a sophisticated voting rule such as Borda or Copeland would require each voter to rank all alternatives, which is hardly possible for a scenario with even just a handful of issues. For the Plurality rule, on the other hand, where voters only need to indicate their most preferred alternative, this is not a problem. But when the number of alternatives is huge, chances are that each alternative that receives any voter support at all will receive just a single point under Plurality, meaning that the final decision will depend heavily on the tie-breaking rule. This ‘information-theoretic dilemma’ cannot be avoided when voting directly on combinatorial alternatives: we must either ask each voter for an unreasonably large amount of information or we must take a decision on the basis of an amount of information that is unreasonably small relative to the size of the space of potential outcomes.

An altogether different approach would be to hold separate elections for each issue. For our example, this means holding one election on the question of whether to build a swimming pool and another one on the question of whether to build a tennis court.

<footnote n="7">⁷We shall assume that the reader is familiar with common voting rules such as Plurality, Borda, and Copeland, as well as with the concept of Condorcet winner. Definitions can be found in any textbook on voting theory as well as the introductory chapter on voting theory by Zwicker (2016).</footnote>
Suppose we query voters on their preferences regarding both of these issues in parallel. It is not obvious what preference a voter belonging to the first group, with preference relation $ST \succ \bar{S}T \succ \bar{S}T \succ ST$, would report when asked about her preferences regarding $S$ vs. $\bar{S}$. If such a voter is optimistic, they might choose $S \succ \bar{S}$. If all voters are optimistic in this sense (with respect to both issues), then we end up with a slight majority in favour of both $S$ and $T$, and thus with combinatorial alternative $ST$ as the overall winner—the arguably worst possible outcome. This is an instance of the well-known paradox of multiple elections (Brams et al., 1998). Whatever assumptions we might make on how an individual voter tries to account for the preferential dependencies between issues when queried about her preferences regarding one of them, we cannot hope to always circumvent this paradox.

So neither directly voting on combinatorial alternatives nor directly voting on individual issues is a particularly good idea, at least not when approached in a naïve manner. Jérôme has done much to put order into the space of potential solutions to this problem of voting in combinatorial domains. Let us briefly sketch two approaches he worked on extensively: (i) voting with compactly represented preferences and (ii) sequential voting.

If we feel we cannot ask voters to rank an exponential number of alternatives, maybe we can instead ask them to report their preferences using one of the languages for compact preference representation discussed in the previous section. Suppose we use the language of prioritised goals with lexicographic ordering and suppose there are just three voters.

Voter 1 has three goals: $\neg(S \wedge T)$ (high priority), $S$ (medium priority), and $T$ (low priority). This corresponds to $ST \succ \bar{S}T \succ \bar{S}T \succ ST$.

Voter 2 has two goals of equal priority: $\neg S$ and $T$. This does not correspond to a strict ranking but rather a weak preference relation: $\bar{S}T \succ \bar{S}T \sim ST \succ \bar{S}T$.

Voter 3 only has one goal, namely $S \vee T$, inducing $ST \sim \bar{S}T \sim \bar{S}T \succ \bar{S}T$.

Suppose we want to use the Borda rule. Of course, the standard Borda rule is well-defined only for profiles of preferences each strictly ranking all the alternatives, but generalisations to weak orders are possible, e.g., by averaging the Borda scores a given alternative would receive over all linear refinements of a given weak order. Under this rule, the four combinatorial alternatives receive the following scores:

$ST$ receives a score of $0 + 1.5 + 2 = 3.5$.

$\bar{S}T$ receives a score of $3 + 0 + 2 = 5$.

$\bar{S}T$ receives a score of $2 + 3 + 2 = 7$.

$\bar{S}T$ receives a score of $1 + 1.5 + 0 = 2.5$.

So $\bar{S}T$ wins this election, which indeed is a reasonable outcome in view of the goals reported by the voters. This is an example for what Jérôme calls ‘combinatorial vote’ (Lang, 2004). Any combinatorial vote procedure is defined in terms of (i) a preference representation language (in our example this was the language of prioritised goals) and (ii) a voting rule that can operate on the class of preference structures definable using the chosen language (in our example this was the adaption of the Borda rule to weak preferences). While in our example we first ‘unfolded’ the compactly represented voter preferences into explicit preference structures, the ultimate goal would be to have the voting rule operate directly on the compactly represented preferences. Most of the technical results in Jérôme’s 2004 paper on combinatorial vote were complexity results illustrating the computational intractability of this approach for many natural choices of a language and a voting rule. For example, even if we work with a very simple language where each voter can only
specify a single goal and even if we use one of the simplest voting rules available, such as a natural adaption of the Plurality rule to weak orders, the problem of deciding whether a given combinatorial alternative will win is coNP-complete (Lang, 2004).

The other major approach Jérôme contributed to significantly is sequential voting. Here the idea is to let voters vote on one variable at a time, but to do so sequentially, meaning that they learn about the outcome for one variable before they get asked to report their preferences on the next one. Intuitively speaking, this certainly helps and should allow us to circumvent at least some instances of the paradox of multiple elections. For instance, Lacy and Niou (2000) point out that using this regime (when all issues are binary), we will never elect an alternative that is a Condorcet loser, i.e., an alternative that is dispreferred to every other alternative by a strict majority of the voters.<fnref n="8" />

So we can avoid the very worst of outcomes in this manner. But can we do better than that? Together with Lirong Xia, Jérôme investigated this question in depth (see, e.g., Lang and Xia, 2009). Let us briefly sketch one of their results.<fnref n="9" /> A powerful language for representing preferences in combinatorial domains are *CP-nets* (Boutilier et al., 2004). Such a CP-net consists of two components. First, there is a directed graph between the variables we need to instantiate. This graph represents the preferential dependencies between variables: an edge from *X* to *Y* indicates that we can comfortably report our preferences on *Y* only once the instantiation of *X* has been fixed.<fnref n="10" /> Second, each variable *X* (i.e., each node in the graph) is associated with a table that lists for every possible instantiations of the variables that *X* depends on a preference ranking over the possible values we could instantiate *X* with.<fnref n="11" /> Jérôme and Lirong studied sequential voting in situations where all voters have preferences that are represented in the form of CP-nets and they looked for conditions on these CP-nets that would guarantee favourable properties of the overall voting procedure. Amongst other things, they were interested in conditions that would ensure that the sequential voting procedure is *Condorcet-consistent*, meaning that it would elect the Condorcet winner amongst the combinatorial alternatives whenever such a Condorcet winner exists at all. They found that, if there exists a linear order on the set of variables that is a refinement of every single one of the preferential dependency graphs of the individual voters, then using sequential voting (following this linear order) will guarantee the Condorcet consistency of the sequential voting procedure whenever the voting rules used at each individual stage are Condorcet-consistent as well. This is a simple but beautiful result that is exemplary in how it brings together axiomatic concerns typically investigated by economists and representational concerns with an algorithmic flavour naturally considered by computer scientists.

The state of the art on voting in combinatorial domains as it stood about a decade ago, reflecting the contributions of a large group of authors, is expertly summarised in the relevant chapter of the *Handbook of Computational Social Choice* co-authored by Jérôme (Lang and Xia, 2016). This chapter also includes an exposition of some of the

<footnote n="8">⁸The proof is immediate: In the very final round, when asking voters to vote on the final variable, we in fact are asking them about their preferences over the two combinatorial alternatives we can still construct at this point, given the variable instantiations already decided upon. If we then pick the variable instantiation preferred by a majority we cannot possibly end up with the Condorcet loser—at the very least the alternative selected will beat the other alternative still on the table at the start of the final round.</footnote>

<footnote n="9">⁹In fact, the specific result we sketch here already makes an appearance in Jérôme’s earlier work (Lang, 2007).</footnote>

<footnote n="10">¹⁰For instance, in our menu choice example, your preference over wines might depend on the choice made regarding the main course to be served.</footnote>

<footnote n="11">¹¹CP stands for ceteris paribus (‘all else being equal’): each row in such a preference table lists the decision maker’s preferences regarding different values for *X* under the assumption that all else remains the same.</footnote>
research challenges in the area of *multiwinner voting*, i.e., voting to elect not a single winning alternative but rather a set of alternatives (such as when voting for the members of a committee or a parliament). Since the publication of the *Handbook*, multiwinner voting has become a major research trend in Computational Social Choice (Faliszewski et al., 2017; Lackner and Skowron, 2023). We can think of multiwinner voting as an instance of voting in combinatorial domains (with the variables representing the seats on the committee to be filled), and the combinatorial perspective is natural when we want to enrich multiwinner voting with additional constraints on the composition of the set of alternatives to be elected. Fully understanding and exploiting these connections continues to be an exciting research challenge. Recent contributions along these lines include those by Lang and Skowron (2018), Brill et al. (2023), Mavrov et al. (2023), Masařík et al. (2024), and Chingoma et al. (2024).

# 4 Preference Elicitation and Communication in Voting

In classical voting theory (Arrow et al., 2002; Zwicker, 2016), the focus of attention is on the design of good voting rules and we tend to assume that the voters' preferences are 'given': every voter is able to strictly rank all of the available alternatives and our voting rule can freely query any part of the full preference profile to determine an adequate collective decision. In practice, of course, things are not quite so clean and simple. We need to elicit everyone's preferences step by step. This process is costly, both for the authority executing the voting rule and for the individual voters. We might not want to burden voters with eliciting information we do in fact not need to determine the best collective decision. We also might not want to clutter the communication network used to transmit preferences reported by the voters with unnecessary data.

In a seminal paper from 2005, Jérôme and his coauthor Kathrin Konczak recognised this situation and proposed a highly influential solution concept, the *possible winner* of an election, designed to address at least some of these concerns (Konczak and Lang, 2005).

Imagine we want to run an election to select one from a given set of alternatives, based on the preferences of a given set of voters. We assume that each voter, in principle, is able to strictly rank all alternatives, but that becoming aware of those preferences (and then communicating them) is costly. We want to use a classical voting rule, such as Borda or Copeland. Now suppose we elicit everyone's preferences step by step: at any given time, we ask one more voter about their relative ranking of one more pair of alternatives. So at any point in time, we have access to a profile of partial preferences, one for each voter. Even before having elicited all preferences in their entirety, we might be able to say something about the eventual election winner. Let's consider a simple example, with three voters and three alternatives.

Imagine that for Voter 1 we have already elicited a full ranking: $A \succ_1 B \succ_1 C$. For Voter 2 we only know $A \succ_2 C$, and for Voter 3 we only know that $B$ is their top alternative (so $B \succ_3 A$ and $B \succ_3 C$).

Suppose we want to use the Borda rule. We know that $A$ will get 2 points from Voter 1, between 1 and 2 points from Voter 2, and between 0 and 1 points from Voter 3. So $A$ will get between 3 and 5 points overall. The same kind of analysis reveals that $B$ will also get between 3 and 5 points, while $C$ will get between 0 and 2 points once all preferences have been elicited.

Now, while we cannot yet predict the eventual winner, we can say with certainty that $C$ will not win. So the *possible winners* at this stage are $A$ and $B$.
Suppose that in the next step we find out that $C \succsim_2 B$. Then we can update our counts: A will get between 4 and 5 points, while B will definitely get just 3 points. So now we know that the only remaining possible winner is A and we can stop eliciting further preferences.

The possible winner problem (for a particular voting rule) is the problem of deciding, given a profile of partially defined preferences and the name of one of the alternatives, whether there exists a way of completing those preferences so that the given alternative wins the election. The closely related necessary winner problem asks whether the given alternative will win the election under all possible completions of the partial preference profile.

Observe that the set of necessary winners is always a (not necessarily proper) subset of the set of possible winners. In fact, the former is typically the empty set. As elicitation progresses, the set of possible winners becomes smaller and the set of necessary winners becomes larger. Once the two sets coincide, we know that we have elicited all the information we require to decide the outcome of the election. In particular, whenever we have reached a stage where there is only a single possible winner, that one alternative must also be the necessary winner. But in view of the fact that ties are possible, we might also find a necessary winner before the possible and necessary winner sets coincide.

The need to solve the possible winner problem (and to a lesser extent also the necessary winner problem) arises in a wide range of scenarios. The step-by-step elicitation of preference information, designed to avoid burdening either voters or communication networks, is one example. Several special cases are of particular interest. For instance, when for some voters we already have access to their full preferences and for others we do not yet have any information at all, then solving the possible winner problem for a given alternative is equivalent to determining whether those 'missing voters' have a strategy to make that alternative win. This is known as the constructive manipulation problem. Similarly, they have a strategy to bar the alternative of interest from winning precisely when that alternative is not a necessary winner. This is the destructive manipulation problem. The special case where we have elicited preferences that are complete with respect to a subset of the alternatives but where we do not have any information on the relative standing of the remaining alternatives is also of interest. This case of 'missing alternatives' can be used to model scenarios where some alternatives become available only at a later point in time. For instance, when trying to agree on a time to meet, we might add additional time slots to the poll some time after participants have voted on the original options already.

The publication of the 2005 paper on the possible winner problem created a great deal of interest in the research community, and a large number of authors contributed to improving our understanding of the algorithmic challenges of solving the problem for specific voting rules. Not entirely unsurprisingly, solving the problem is computationally intractable for many voting rules. For instance, for the Borda rule the possible winner problem is NP-complete (Xia and Conitzer, 2011). This result extends to most other voting rules from the family of positional scoring rules (Betzler and Dorn, 2010; Baumeister and Rothe, 2012), but not to all of them. In particular, by relating the underlying algorithmic problem to a network flow problem, Betzler and Dorn (2010) showed that for the Plurality rule the problem can be solved in polynomial time.

Jérôme, together with several coauthors, also contributed to the complexity-theoretic analysis of the possible winner problem himself, focusing in particular on the aforementioned special case of 'missing alternatives' (Chevaleyre et al., 2012). As this variant of the problem is less general, there in principle is room for more favourable results than the rather damning complexity results for the general possible winner problem. For instance, they were able to show that for the Borda rule the possible winner problem for missing
alternatives is solvable in polynomial time. But there are other, seemingly simple, positional scoring rules where the problem remains intractable. For instance, for k-approval rules<fnref n="12" /> with $k \ge 3$ the problem is NP-complete (as long as the number of initially unranked alternatives is at least 3).

A related range of questions of a fundamental nature concern the amount of information (measured as number of bits) required to be able to compute the winner for a given preference profile under different voting rules. This kind of question was first studied by Conitzer and Sandholm (2005) who asked how much information we must elicit from the individual voters and who used the technical tool of communication complexity to analyse this question. Take the Plurality rule, with $n$ voters and $m$ alternatives: certainly, winners can always be computed by having each voter report the name of a single alternative (requiring $\lceil \log m \rceil$ bits), so transmitting $n[\log m]$ bits overall. But is this minimal? To prove this to be a lower bound (which indeed it is), one has to show that transmitting less information could lead to a mistake in determining the winner.

Jérôme and his coauthors initiated the study of a related problem regarding the compila-
lation complexity of voting rules (Chevaleyre et al., 2009), which was then further explored
by Xia and Conitzer (2010). Imagine we have elicited the preferences of part of the pop-
ulation already. How much information do we need to retain to be able to compute the
eventual election winner, whatever preferences the remainder of the population might end
up reporting later on? Given that an individual voter can choose between $m!$ rankings of
the alternatives, one voter's preference can be stored using $\lceil \log(m!) \rceil$ bits. So we never re-
quire more than $n[\log(m!)]$ bits to record the entire profile for the first $n$ votes. But often
we can do better. For example, for any anonymous voting rule (i.e., any rule treating all
voters the same), we can instead store how many times each possible ballot got reported,
which requires $m!\lceil \log(n+1) \rceil$ bits: for each of the $m!$ possible ballots there are between
0 and $n$ voters who might have reported that ranking. When $n$ (the number of voters) is
much larger than $m$ (the number of alternatives), then the latter is a much better bound:
$n\log(m!) \gg m!\lceil \log(n+1) \rceil$. For specific voting rules, we can get further improvements.
For example, for the Borda rule it is sufficient to store the Borda score accumulated by
each alternative, which can be done using (in the order of) $m \log(nm)$ bits, given that any
one alternative's Borda score must be an integer between 0 and $n(m-1)$. Chevaleyre
et al. (2009) showed that this bound in fact is tight, so under Borda we cannot do better
than compiling intermediate election results by simply storing Borda scores.

For a broader discussion of the theme of accounting for incomplete preferences and
modelling issues related to the elicitation and communication of preferences in voting we
refer to the in-depth chapter by Boutilier and Rosenschein (2016) in the *Handbook of
Computational Social Choice*. Matters related most immediately to the possible winner
problem are also reviewed by Jérôme himself in his recent survey on the topic (Lang,
2020), where he emphasises the fact that, while these questions were first raised in the
context of voting, they now are widely studied in other areas of collective decision making
as well. We are going to see one such example in the next section.

5 Fair Allocation of Indivisible Goods

Voting is a form of collective decision making where, based on the reported preferences of
everyone involved, we need to pick a single alternative for the collective of participants as
a whole. That ‘single alternative’ might be a complex object, such as the composition of a

<footnote n="12">¹²A k-approval rule awards 1 point to each of the alternatives ranked in the top-k positions of a voter's ballot, and no points to any of the other alternatives (Zwicker, 2016).</footnote>
national parliament, but it still is the case that the very same outcome is experienced by each individual member of the collective. Resource allocation, in contrast to voting, is a form of collective decision making where the outcome experienced by each participant is unique to them. Here we need to divide up a common set of resources and decide which items to allocate to which individuals based on the preferences reported by those individuals. Just as for voting, we should care both about normative matters, such as fairness, and about algorithmic matters, such as the computational complexity of computing outcomes.

Jérôme worked on a wide range of questions regarding the fair allocation of indivisible goods. (Here indivisibility means that individual goods cannot be broken up into smaller pieces or shared amongst several individuals.) His early work on the topic was directly inspired by a concrete application scenario, namely choosing a fair and efficient schedule for the images to be taken by an earth observation satellite on a given day on the basis of the requests of the organisations funding the satellite (Fargier et al., 2004).

The basic setting of fair allocation with indivisible goods is that of a finite number of agents and a finite number of goods, with each agent reporting their preferences over all possible bundles (i.e., subsets of the set of goods). Much of the literature assumes that preferences are cardinal, so each agent has a utility function mapping each possible bundle of goods to a real number. The task then is to decide on an allocation, specifying for each good which agent should receive it. We are especially interested in allocations that are, in some sense, fair. There are many possible definitions of ‘fairness’ one might use (Moulin, 1988). For this exposition, let us focus on just two of them, both of which played an important role in Jérôme’s work, namely *envy-freeness* and *egalitarian social welfare*. We say that an allocation is envy-free if no agent envies one of the other agents in the sense of strictly preferring that other agent’s bundle of goods over their own. And we say that an allocation maximises egalitarian social welfare if no other allocation would give the worst-off agent a strictly higher level of utility. Observe that envy-freenes can be defined also if we assume that agents merely report ordinal preferences, but egalitarian social welfare cannot.

Following seminal work by Lipton et al. (2004) who studied algorithmic and complexity issues of envy-freeness (with a focus on approximation), Jérôme and his PhD student Sylvain Bouveret were the first to study the computational complexity of finding envy-free allocations of goods when preferences are represented in a compact way, using the kinds of languages we discussed in Section 2 (Bouveret and Lang, 2008). They specifically focussed on scenarios where agents only distinguish good and bad bundles (*dichotomous preferences*). Such preferences can be modelled by assuming that each agent has a single goal, expressible in propositional logic, they would like to see met by the bundle they receive. How hard is it to decide whether there exists an envy-free allocation once we have learned about the goals (and thus the preferences) of the agents? In fact, there is a trivial answer: the allocation where we discard of all items and do not give any of them to any of the agents is always envy-free by definition. So a more interesting question is this: How hard is it to decide whether there exists an allocation that is both envy-free and Pareto-efficient? Sylvain and Jérôme showed that the general problem is highly intractable (it is complete for $\Sigma_2^p$, a complexity class located ‘above’ NP). But they also showed that when individual preferences are subject to certain restrictions, the complexity of the problem can reduce significantly. For instance, if every agent has a goal that can be expressed as a disjunction of literals—meaning that they are satisfied as soon as they receive one from a given list of ‘desirable’ items (or as soon as they definitely do not receive one from a given list of ‘undesirable’ items)—the problem can be reduced to the problem of finding a maximal matching in a bipartite graph and thus is solvable in polynomial time.
Recall that, while much of the literature on fair division assumes that agents are endowed with cardinal preferences (utility functions), when we are interested in finding envy-free (or Pareto-efficient) allocations, we only really need to access the ordinal content of people's preferences, as these solution concepts are defined in terms of comparisons between different bundles of goods. Recognising this mismatch, Jérôme and his coauthors suggested to investigate fair allocation problems with tailor-made preference representation languages for modelling ordinal preferences (Bouveret et al., 2010). If we are looking for languages to describe preferences using only a limited amount of information, those preferences might be incomplete. In the same paper, mirroring the possible winner problem for voting we discussed in Section 4, they also introduced the idea of a possibly envy-free allocation, i.e., an allocation that would be envy-free for at least one way of completing the incompletely specified preferences elicited at a given point in time. In another paper, Jérôme and his coauthors studied the related idea of epistemic variants of envy-freeness, where agents are unaware of the allocation of some of the goods in circulation and merely have to consider it possible that they would not envy anyone if they were to learn about the full allocation (Aziz et al., 2018).

Besides questions of representational and algorithmic efficiency, another recurrent concern of Jérôme's is to propose mechanisms that have an intuitive appeal and that can be used in practice. *Picking sequences* (Bouveret and Lang, 2011) meet these desiderata:<fnref n="13" /> fix a sequence of agents (with repetition), and let them pick one item at a time from those still available when it is their turn. Let us illustrate the idea with an example.

Suppose there are three agents and eight goods (A, B, ...), and suppose the agents have the following preferences regarding individual items:

$$ 
\begin{array}{l}
A \succ_1 B \succ_1 C \succ_1 D \succ_1 E \succ_1 F \succ_1 G \succ_1 H \\
A \succ_2 B \succ_2 C \succ_2 D \succ_2 E \succ_2 F \succ_2 G \succ_2 H \\
A \succ_3 D \succ_3 G \succ_3 E \succ_3 B \succ_3 H \succ_3 F \succ_3 C
\end{array}
$$

Now let us compare the picking sequences $\pi_1 = [12312312]$, $\pi_2 = [12332112]$, and $\pi_3 = [11223323]$. For the first one, $\pi_1$, agent 1 first picks A, then agent 2 picks B (since A, their most preferred item, is no longer available), then agent 3 picks D, and so on. In the end, agent 1 gets $\{A, C, F\}$, agent 2 gets $\{B, E, H\}$, and agent 3 gets $\{D, G\}$. Similarly, for the second sequence, $\pi_2$, the final allocation is 1: $\{A, E, F\}$, 2: $\{B, C, H\}$, 3: $\{D, G\}$. Finally, the third sequence, $\pi_3$, yields the allocation 1: $\{A, B\}$, 2: $\{C, D, F\}$, 3: $\{G, E, H\}$.

How do these allocations compare? Assuming preferences to be Borda-like<fnref n="14" /> and additive, it can be observed that $\pi_1$ yields a utility vector of $\langle 17, 12, 13 \rangle$, indicating the utilities enjoyed by agents 1, 2, and 3, respectively, while $\pi_2$ yields $\langle 15, 14, 13 \rangle$ and $\pi_3$ yields $\langle 15, 14, 14 \rangle$.

The first sequence is a *round robin* sequence, while the second one is a *balanced alternation* sequence. Such sequences are quite natural and often used in practice. Observe also that the protocol is very frugal in terms of communication: it suffices to elicit in each round the item selected by the agent whose turn it is, thus requiring $m[\log m]$ bits overall. However, it is difficult to be sure such a protocol offers specific guarantees, for instance in terms of

<footnote n="13">¹³While Bouveret and Lang (2011) appear to be the first to have studied picking sequences as a general approach to designing resource allocation mechanisms, the basic idea is very intuitive and has been discussed repeatedly in earlier work, such as that of Kohler and Chandrasekaran (1971) and Brams and Taylor (2000).</footnote>

<footnote n="14">¹⁴Agents assign 8 points to their most preferred item, then 7 points to the next one, and so on.</footnote>
expected egalitarian social welfare (that is, in terms of maximising, the expected utility of the worst-off agent, when assuming utilities to be drawn from specific ‘cultures’). Bouveret and Lang (2011) initiated the formal study of this family of protocols for the allocation of indivisible items, showing in particular that computing optimal sequences can be difficult for some social welfare objectives, while also identifying optimal sequences for small numbers of agents and items (it turns out that $\pi_3$ is the optimal sequence for egalitarian social welfare, under the assumption that agents’ preferences are fully independent).

They also opened up a fertile area of research on the strategic manipulation of such sequences. Indeed, for almost any picking sequence, it is possible to conceive of a scenario where an agent would rather pick an item different from their most preferred item. The only exception are sequences based on *serial dictatorships*, where agents are never interleaved in the sequence (that is, each of them picks in batch all their items). Since agents never appear again in the sequence they have no better strategy than to pick their preferred items when they have the opportunity to do so. In a recent paper (Bouveret et al., 2025), Jérôme and colleagues study how these strategyproof sequences can be designed to still ensure a reasonably fair allocation.

When utility functions are additive, allocation protocols based on picking sequences have been shown to have remarkable properties: even though picking sequences do not necessarily produce Pareto-optimal allocations, any Pareto-optimal allocation can result from a picking sequence. Furthermore, a simple argument suffices to show that round robin sequences guarantee envy-freeness *up to one good* (Lipton et al., 2004; Caragiannis et al., 2019): indeed, by considering the round robin subsequence starting with the first occurrence of any agent *i*, it can be observed that this agent picks all of their items before any other agent. If envy should occur, it can only be so because of the single item that was picked before the first occurrence of *i*.

Picking sequences have now become a standard tool in the study of fair division. They have been adapted to several contexts, e.g., settings with both goods and chores (Aziz et al., 2019) and resource allocation subject to cardinality constraints (Biswas and Barman, 2018). Interestingly, the same principle can also be applied in voting in contexts where there are few voters and many candidates, by letting voters proceed to sequential elimination until only a single candidate remains (Bouveret et al., 2017).

Readers interested in finding out more—about the issues briefly discussed in this section or some of the many other exciting topics in the area of fair division—might want to consult the survey recently authored by Jérôme and Jörg Rothe (Lang and Rothe, 2024).

# 6 Other Topics

We have tried to illustrate some of the main themes in the œuvre of Jérôme Lang regarding the analysis of methods for collective decision making, and more specifically the field of Computational Social Choice at the intersection of Economic Theory and AI.<sup>15</sup> Key notions that come back again and again include the need to think about preferences in terms of the *languages* we use to represent them; the need to think of the processes occurring in the context of decision making as being fuelled by *algorithms* that can and should be analysed in those terms; and the need to analyse all instances of *communication*.

<footnote n="15"><sup>15</sup>But we have taken the discussion of any of his scientific contributions *strictly within AI* as being beyond the scope of this paper. This includes topics such as the aforementioned possibilistic logic as a tool to model incomplete and uncertain knowledge (Dubois et al., 1994), nonmonotonic reasoning (Benferhat et al., 1993; Dupin de Saint-Cyr et al., 1994), the algorithmic difficulty of identifying redundant information in a knowledge base (Lang et al., 2003), and foundational issues in automated planning (Herzig et al., 2003) and constraint satisfaction (Fargier and Lang, 1993).</footnote>
between decision makers in precise information-theoretic terms. This comes paired with
regular appeals to relax common assumptions often found in the classical literature (e.g.,
regarding the format of individual preferences) so as to better account for features of
decision-making scenarios we encounter in practice, and the realisation that enormous
benefits can be reaped by combining the axiomatic perspective common in Economics and
the algorithmic perspective common in Computer Science and AI.

While we have focused on topics where Jérôme has published extensively over multiple
years, he also has greatly enriched the literature on Computational Social Choice on many
other topics, sometimes just with a one-off paper or a small set of papers on a specific
issue. He did so in collaboration with a large and diverse group of coauthors. We conclude
by briefly mentioning some of these further directions.

**Preference learning.** In the late 2000s, motivated by applications such as recommender
systems, the problem of learning preferences—that is, of automatically acquiring prefer-
ences from observations (Fürnkranz and Hüllermeier, 2011)—started to attract a lot of
interest. Note that this topic is broader than the preference elicitation problem discussed
earlier (which focuses on an active process typically based on queries addressed to the
user), since it includes passive learning from existing examples.

When preferences are expressed over a multi-attribute combinatorial domain, making assumptions about the structure of preferences to be learned is a way to reduce the dimension of the hypothesis space. Taking this model-based perspective, Jérôme and his colleagues investigated this learning process when preferences are conditionally lexicographic (Booth et al., 2010) or represented as CP-nets (Chevaleyre et al., 2011), in both a passive and an active setting. Examples may also naturally exhibit specific constraints, for instance swaps (a comparison of two alternatives differing on a single attribute) in the context of CP-nets. While learning utility functions had already received some attention, these questions at the time were largely unexplored in ordinal settings, and those papers stimulated this new direction of research.

**Recognising single-peaked preference profiles.** Many of the limitative results in Social Choice Theory, such as the impossibility of ruling out strategic manipulation in voting, can be circumvented by giving up on the idea that voters can report arbitrary preferences and to instead focus on situations where those preferences are drawn from some *restricted domain*. The best-known example is the domain of *single-peaked preferences*. A profile of preferences, each consisting of a strict linear order on the set of available alternatives, is single-peaked if it is possible to arrange the alternatives along some axis such that, for a voter with most preferred alternative $\hat{X}$, between two other alternatives that are either both to the left or both to the right of $\hat{X}$ with respect to that axis, our voter will prefer the one that is closer to $\hat{X}$. When preferences are single-peaked in this sense, it is possible to design attractive voting rules that cannot be manipulated, we can rule out the existence of Condorcet cycles, and more generally the analysis of the will of the people will become a lot easier than in the general case. Single-peakedness and other domain restrictions therefore have received significant attention in both classical Social Choice Theory (Gaertner, 2001) and Computational Social Choice (Elkind et al., 2017).

Jérôme and his coauthors were amongst the first to contribute to the latter strand of literature by designing an efficient algorithm to recognise whether a given profile of preferences is single-peaked, i.e., whether we can find an axis with respect to which the graph of each individual voter's preference order has a single peak (Escoffier et al., 2008).<fnref n="16" />

<footnote n="16">¹⁶It later emerged that a very similar idea for designing an algorithm to recognise single-peaked profiles</footnote>
Given that the number of possible axes is *m*! when the number of alternatives is *m*, the existence of such an algorithm is not immediately obvious. The key insight underlying the design of the algorithm is that any alternative ranked in the bottom position by one of the voters must be either the leftmost or the rightmost alternative on the axis to be found.

**Complexity of coalitional manipulation.** The most hotly debated topic in the early days of Computational Social Choice was the computational complexity of the strategic manipulation problem for different voting rules, following seminal work by Bartholdi et al. (1989). As is well known, by virtue of the Gibbard-Satterthwaite Theorem, every reasonable voting rule is manipulable, meaning that it must allow for the possibility that a voter might benefit from strategically misrepresenting their true preferences. But what if we can prove that identifying a successful manipulation is computationally intractable? Maybe this would constitute some kind of protection against such strategic behaviour? Initially, many researchers felt that the answer to this question might be a cautious ‘yes’, although nowadays the scientific consensus is that this was too optimistic: too many of the theoretical intractability results obtained only really have an effect in practice for certain edge cases and only once the number of alternatives is very large.

Jérôme and his coauthors contributed to this stream of literature with an analysis of the computational complexity of the *coalitional manipulation problem*, where a coalition of voters is trying to coordinate their votes, so as to obtain an unfair advantage (Conitzer et al., 2007). They focused on *weighted voting rules*, where different voters might have different weights (like, e.g., in the shareholder meeting of a large company). For example, they found that the problem a coalition would have to solve to find out whether they can force a given alternative to win under the Borda rule is NP-complete (while finding out whether they can *prevent* that alternative from winning can be done in polynomial time). A distinguishing feature of their results is that they apply even in case the number of alternatives is very small (in this case, the intractability result applies as soon as there are at least three alternatives), thereby addressing some of the criticism often levied against the approach of using complexity barriers as a protection against strategic manipulation.

**Strategic candidacy.** The decision to actually run for an election can be highly strategic. Indeed, candidates themselves have preferences (well, they usually prefer to win the election, but if that's out of the question, they might still have preferences on the other candidates). And it can happen that by attracting to your name some voters who otherwise would have voted for a candidate you like, the outcome might be worse when standing as a candidate yourself. You might thus have an incentive to withdraw from the election, which might lead to other candidates reacting strategically themselves. This type of question has been modelled in the setting of *strategic candidacy* (Dutta et al., 2001), leading to a game-theoretical analysis of the games induced by different voting rules. Is it the case that the situation where all candidates run is a Nash equilibrium? Is there a guarantee to have a Nash equilibrium? Is it the case that the candidate elected in an equilibrium is the same as the one who would be elected if all candidates were to run?

Lang et al. (2013, 2025) showed that different voting rules exhibit very different properties, with the Copeland rule standing out as the only rule amongst those studied guaranteeing an equilibrium faithful to the situation where all candidates run.

<footnote>was already present in earlier work by Doignon and Falmagne (1994). The paper by Escoffier et al. (2008) helped not only only to draw the research community's attention to this fundamental problem but, indirectly, also contributed to getting Doignon and Falmagne the recognition they deserve for their hitherto largely unknown contribution.</footnote>
**Condorcet winning sets.** When a profile of preferences has a Condorcet winner, then simply selecting that Condorcet winner represents a convincing solution concept. But what when there is no Condorcet winner? Jérôme and his coauthors proposed a natural relaxation of the notion of Condorcet winner for such cases (Elkind et al., 2015). A *Condorcet winning set* is a set of alternatives such that, for any alternative not in the set, a strict majority of voters will prefer their most preferred alternative from within the set to that other alternative. The *Condorcet dimension* of a profile is the size of its smallest Condorcet winning set. Thus, a profile has a Condorcet dimension of 1 if and only if it has a Condorcet winner. As a solution concept, (minimal) Condorcet winning sets seem attractive if we can offer a guarantee that Condorcet dimensions tend to be small (even if it is clear that guaranteeing a dimension of 1 is out of the question in the general case).

Elkind et al. (2015) proved several results providing some initial understanding of this issue. First, they showed that, if the number of voters is large and preferences are drawn uniformly at random from the set of all possible preferences, then the probability of ending up with a profile with a Condorcet dimension of at most 2 is close to 1. Second, they showed that profiles with a Condorcet dimension of (at least) 3 exist. Third, they established a first upper bound on the Condorcet dimension of any profile, showing that it can be at most logarithmic in the number of alternatives. These initial results created a flurry of activity in the research community, with many researchers attempting to obtain sharper bounds, using both analytical and computer-supported methods. To date, no profile with a Condorcet dimension strictly above 3 has been found, and in very recent work, Charikar et al. (2025) managed to show that the Condorcet dimension of any profile is at most 6. Remarkably, this bound applies independently of the number of voters and the number of alternatives. Whether it can be lowered further is the subject of ongoing research.

**Boolean games.** In noncooperative games in normal form, the most widely studied class of games in Game Theory, the number of outcomes is exponential in the number of players. Furthermore, when choosing an action for a given player amounts to instantiating several variables with some values, the number of outcomes, and thus the number of alternatives each player needs to express their preferences over, also is exponential in the number of these variables. *Boolean games* provide a formalism to represent such games in a compact manner. They are a class of noncooperative games where the preferences of each player over the possible outcomes are encoded in terms of a goal expressed in propositional logic (Harenstein, 2004). Together with his PhD student Elise Bonzon and further coauthors, Jérôme worked on generalising the basic model of Boolean games, for instance by allowing for the modelling of a wider range of possible preferences (Bonzon et al., 2009).

**Judgment aggregation.** While much of Social Choice Theory is concerned with the aggregation of preferences, similar questions arise in the context of the aggregation of judgments regarding the truth or falsehood of a number of (possibly inter-related) statements (List, 2012). Jérôme has made several contributions to the study of *judgment aggregation*.<fnref n="17" /> One line of work has culminated in the publication of a *taxonomy* of aggregation rules (Lang et al., 2017), at a time when the literature in judgment aggregation was shifting from its initial focus on impossibility results to a more optimistic focus on devising practically useful aggregation rules. Many of the rules considered are inspired by simi-

<footnote n="17">¹⁷He also contributed to the study of *belief merging*, e.g., in joint work with Sébastien Konieczny and Pierre Marquis (Konieczny et al., 2004), a framework similar to judgment aggregation—but with roots in Logic and AI rather than Economics and Philosophy.</footnote>
lar rules familiar from the theory of voting (Lang and Slavkovik, 2013).<fnref n="18" /> For instance, Jérôme and his coauthors appear to have been the first to have adapted Young's rule and Tideman's Ranked-Pairs rule to judgment aggregation.

Jérôme and his coauthors also contributed to the complexity-theoretic analysis of the *outcome determination problem* for a range of rules for judgment aggregation (Lang and Slavkovik, 2014; Endriss et al., 2020).

**Group activity selection.** Imagine you and some 40 fellow researchers find yourselves on a week-long research retreat at Schloss Dagstuhl and need to decide what to do on the Wednesday afternoon, which traditionally is left free for one or more social activities. Will it be a guided tour through the city of Trier, a boat trip along the river Saar, or a visit to one of the many local vineyards? Your preferences might depend not only on the activity you end up participating in yourself but also on the number of people choosing said activity together with you. For example, splitting the cost of the bus to Trier is much more attractive when you are a group of 20 rather than a group of 10. During one of the many retreats on Computational Social Choice at Schloss Dagstuhl, Jérôme initiated the idea of taking a principled approach to this problem of coalition formation, leading to him and his coauthors developing a formal model of *group activity selection problems* (Darmann et al., 2018), which since has spawned a flourishing literature considering variants and extensions of the basic model as well as strategic, algorithmic, and complexity-theoretic issues.

**Portioning.** Portioning is the problem of dividing a fixed budget over a number of alternatives in view of the preferences of a number of voters. For instance, we may want to divide a fixed amount of money over several projects or we may want to determine the amount of time to be allocated to each of several programme items at an event. When preferences are uninominal, i.e., when each voter simply reports their most preferred alternative, it is clear what to do: the budget allocated to a given alternative should be proportional to the number of voters who voted for that alternative. Jérôme and his coauthors instead considered the problem of portioning when voters report ordinal preferences (Airiau et al., 2023). For example, one of the allocation rules they propose and study, the *Borda-Nash solution*, works as follows. We first define each voter's disutility for any given alternative as the number of other alternatives they rank higher (that is, disutilities are closely linked to Borda scores). We then determine each voter's disutility for any given budget division as a weighted average by, for each alternative, multiplying its disutility with the budget it receives. Finally, we aggregate across voters by defining the collective disutility of a given budget division as the (Nash) product of its individual disutilities, and we aim to select a budget division that minimises this value.

**Empirical analysis and policy recommendation.** As alluded to already in the introductory section, in recent years Jérôme has become ever more interested in the empirical side of his research and in its potential to affect public policy for the betterment of the people. Regarding empirical work, an example is the recent study of real-world voter behaviour in situations that might be subject to the paradox of multiple elections discussed earlier (Grandi et al., 2024).<fnref n="19" /> Regarding policy recommendations, besides the initiatives

<footnote n="18">¹⁸As Jérôme’s coauthor Marija Slavkovik informs us, the idea that judgment aggregation rules can be constructed by taking inspiration from voting rules, quite characteristically, occurred to Jérôme on a train trip from Luxembourg to France.</footnote>

<footnote n="19">¹⁹A footnote in that paper helpfully informs the uninitiated reader that “[o]ne of the authors gave many talks for over a decade about multiple referenda in which the audience was presented with a two-issue</footnote>
mentioned earlier, in recent years Jérôme and his coauthors have proposed and studied
several voting mechanisms that are simple enough to conceivably replace mechanisms cur-
rently in common use yet known to have serious normative defects. Examples include the
works of Ayadi et al. (2019) and Delemazure et al. (2025).

More later.

**Acknowledgements.** We are grateful to Remzi Sanver and Ali Ozkes for their suggestion to write this overview of some of Jérôme Lang’s scientific contributions, and to Elise Bonzon, Sylvain Bouveret, Yann Chevaleyre, Edith Elkind, Umberto Grandi, Pierre Marquis, Dominik Peters, Marija Slavkovik, and Lirong Xia for their most helpful and encouraging feedback on an earlier version of this paper.

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