The fair division of indivisible goods is not only a subject of theoretical research, but also an important problem in practice, with solutions being offered on several online platforms. Little is known, however, about the characteristics of real-world allocation instances and how they compare to synthetic instances. Using dimensionality reduction, we compute a map of allocation instances: a 2-dimensional embedding such that an instance's location on the map is predictive of the instance's origin and other key instance features. Because the axes of this map closely align with the utility matrix's two largest singular values, we define a second, explicit map, which we theoretically characterize.

We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that an EF1 outcome is guaranteed to exist even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are n agents and the number of goods m is divisible by the number of groups k, then an envy-free outcome exists with high probability if m = ω(log n), and this bound is tight. On the other hand, if m is not divisible by k, then an envy-free outcome is unlikely to exist as long as m = o(√n).

In citizens' assemblies, a group of constituents is randomly selected to weigh in on policy issues. We study a two-stage sampling problem faced by practitioners in countries such as Germany, in which constituents' contact information is stored at a municipal level. As a result, practitioners can only select constituents from a bounded number of cities ex post, while ensuring equal selection probability for constituents ex ante. We develop several algorithms for this problem. Although minimizing the number of contacted cities is NP-hard, we provide a pseudo-polynomial time algorithm and an additive 1-approximation, both based on separation oracles for a linear programming formulation. Recognizing that practical objectives go beyond minimizing city count, we further introduce a simple and more interpretable greedy algorithm, which additionally satisfies an ex-post monotonicity property and achieves an additive 2-approximation. Finally, we explore a notion of ex-post proportionality, for which we propose two practical algorithms: an optimal algorithm based on column generation and integer linear programming and a simple heuristic creating particularly transparent distributions. We evaluate these algorithms on data from Germany, and plan to deploy them in cooperation with a leading nonprofit organization in this space.

We study the fair allocation of indivisible goods across groups of agents, where each agent fully enjoys all goods allocated to their group. We focus on groups of two (couples) and other groups of small size. For two couples, an EF1 allocation — one in which all agents find their group's bundle no worse than the other group's, up to one good — always exists and can be found efficiently. For three or more couples, EF1 allocations need not exist. Turning to proportionality, we show that, whenever groups have size at most k, a PROPk allocation exists and can be found efficiently. In fact, our algorithm additionally guarantees (fractional) Pareto optimality and PROP1 to the first agent in each group, PROP2 to the second, and so on, for an arbitrary agent ordering. In special cases, we show that there are PROP1 allocations for any number of couples.

Obtaining meaningful and informed consent from users is essential for ensuring autonomy and control over one’s data. Notice and consent, the standard for collecting consent, has been criticized. While other individualized solutions have been proposed, this paper argues that a collective approach to consent is worth exploring. First, individual consent is not always feasible to collect for all data collection scenarios. Second, harms resulting from data processing are often communal in nature, given the interconnected nature of some data. Finally, ensuring truly informed consent for every individual has proven impractical.

The mathematical study of voting, social choice theory , has traditionally only been applicable to choices among predetermined alternatives, but not to open-ended decisions such as collectively selecting a textual statement. We introduce generative social choice , a design methodology for open-ended democratic processes that combines the rigor of social choice theory with the capability of large language models to generate text and extrapolate preferences. Our framework divides the design of AI-augmented democratic processes into two components: first, proving that the process satisfies representation guarantees when given access to oracle queries; second, empirically validating that these queries can be approximately implemented using a large language model. We apply this framework to the problem of summarizing free-form opinions into a proportionally representative set of opinion statements; specifically, we develop a democratic process with representation guarantees and use this process to portray the opinions of participants in a survey about abortion policy. In a trial with 100 representative US residents, we find that 84 out of 100 participants feel “excellently” or “exceptionally” represented by the set of five statements we extracted.

After pre-training, large language models are aligned with human preferences based on pairwise comparisons. State-of-the-art alignment methods (such as PPO-based RLHF and DPO) are built on the assumption of aligning with a single preference model, despite being deployed in settings where users have diverse preferences. As a result, it is not even clear that these alignment methods produce models that satisfy users \emph{on average} --- a minimal requirement for pluralistic alignment. Drawing on social choice theory and modeling users' comparisons through individual Bradley-Terry (BT) models, we introduce an alignment method's \emph{distortion}: the worst-case ratio between the optimal achievable average utility, and the average utility of the learned policy. The notion of distortion helps draw sharp distinctions between alignment methods: \emph{Nash Learning from Human Feedback} achieves the minimax optimal distortion of $(\frac{1}{2} + o(1)) \cdot \beta$ (for the BT temperature $\beta$), robustly across utility distributions, distributions of comparison pairs, and permissible KL divergences from the reference policy. RLHF and DPO, by contrast, suffer $\geq (1 - o(1)) \cdot \beta$ distortion already without a KL constraint, and $e^{\Omega(\beta)}$ or even unbounded distortion in the full setting, depending on how comparison pairs are sampled.

Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed to overcome this impossibility by randomizing the apportionment, which can achieve quota as well as perfect proportionality and monotonicity --- at least in terms of the expected number of seats awarded to each party. Still, the correlations between the seats awarded to different parties may exhibit bizarre non-monotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these non-monotonicities can cause paradoxes, including incentives for strategic voting.

The mathematical study of voting, social choice theory, has traditionally only been applicable to choices among a few predetermined alternatives, but not to open-ended decisions such as collectively selecting a textual statement. We introduce generative social choice, a design methodology for open-ended democratic processes that combines the rigor of social choice theory with the capability of large language models to generate text and extrapolate preferences. Our framework divides the design of AI-augmented democratic processes into two components: first, proving that the process satisfies representation guarantees when given access to oracle queries; second, empirically validating that these queries can be approximately implemented using a large language model. We apply this framework to the problem of summarizing free-form opinions into a proportionally representative slate of opinion statements; specifically, we develop a democratic process with representation guarantees and use this process to represent the opinions of participants in a survey about chatbot personalization. In a trial with 100 representative US residents, we find that 93 out of 100 participants feel "mostly" or "perfectly" represented by the slate of five statements we extracted. By providing rigorous guarantees through social choice, our work alleviates concerns about AI-driven democratic innovation and helps unlock its potential.

Citizens’ assemblies are groups of randomly selected constituents who are tasked with providing recommendations on policy questions. Assembly members form their recommendations through a sequence of discussions in small groups (deliberation), in which group members exchange arguments and experiences. We seek to support this process through optimization, by studying how to assign participants to discussion groups over multiple sessions, in a way that maximizes interaction between participants and satisfies diversity constraints within each group. Since repeated meetings between a given pair of participants have diminishing marginal returns, we capture interaction through a submodular function, which is approximately optimized by a greedy algorithm making calls to an ILP solver. This framework supports different submodular objective functions, and we identify sensible options, but we also show it is not necessary to commit to a particular choice: Our main theoretical result is a (practically efficient) algorithm that simultaneously approximates every possible objective function of the form we are interested in. Experiments with data from real citizens' assemblies demonstrate that our approach substantially outperforms the heuristic algorithm currently used by practitioners.

We study the problem of allocating m indivisible items to n agents with additive utilities. It is desirable for the allocation to be both fair and efficient, which we formalize through the notions of envy-freeness and Pareto-optimality. While envy-free and Pareto-optimal allocations may not exist for arbitrary utility profiles, previous work has shown that such allocations exist with high probability assuming that all agents’ values for all items are independently drawn from a common distribution. In this paper, we consider a generalization of this model where each agent’s utilities are drawn independently from a distribution specific to the agent. We show that envy-free and Pareto-optimal allocations are likely to exist in this asymmetric model when m=Ω(n log n), which is tight up to a log log gap that also remains open in the symmetric subsetting. Furthermore, these guarantees can be achieved by a polynomial-time algorithm.

When allocating indivisible items across agents, it is desirable for the allocation to be envy-free, which means that each agent prefers their own bundle over every other bundle. Even though envy-free allocations are not guaranteed to exist for worst-case utilities, they frequently exist in practice. To explain this phenomenon, prior work has shown that, if utilities are drawn from certain probability distributions, then envy-free allocations exist with high probability (as long as the number of items is sufficiently large relative to the number of agents).

Apportionment is the problem of distributing h indivisible seats across states in proportion to the states' populations. In the context of the US House of Representatives, this problem has a rich history and is a prime example of interactions between mathematical analysis and political practice. Grimmett [2004] suggested to apportion seats in a randomized way such that each state receives exactly their proportional share qi of seats in expectation (ex ante proportionality) and receives either ↾qi↿ or ⇂qi⇃ many seats ex post (quota). However, there is a vast space of randomized apportionment methods satisfying these two axioms, and so we additionally consider prominent axioms from the apportionment literature. Our main result is a randomized method satisfying quota, ex ante proportionality and house monotonicity — a property that prevents paradoxes when the number of seats changes and which we require to hold ex post. This result is based on a generalization of dependent rounding on bipartite graphs, which we call cumulative rounding and which might be of independent interest, as we demonstrate via applications beyond apportionment.

Employment outcomes of resettled refugees depend strongly on where they are placed inside the host country. While the United States sets refugee capacities for communities on an annual basis, refugees arrive and must be placed over the course of the year. We introduce a dynamic allocation system based on two-stage stochastic programming to improve employment outcomes. Our algorithm is able to achieve over 98 percent of the hindsight-optimal employment compared to under 90 percent of current greedy-like approaches. This dramatic improvement persists even when we incorporate a vast array of practical features of the refugee resettlement process including indivisible families, batching, and uncertainty with respect to the number of future arrivals. Our algorithm is now part of the Annie™? MOORE optimization software used by a leading American refugee resettlement agency. The full version of this paper is available at https://arxiv.org/pdf/2105.14388.pdf.

Motivated by kidney exchange, we study the following mechanism-design problem: On a directed graph (of transplant compatibilities among patient--donor pairs), the mechanism must select a simple path (a chain of transplantations) starting at a distinguished vertex (an altruistic donor) such that the total length of this path is as large as possible (a maximum number of patients receive a kidney). However, the mechanism does not have direct access to the graph. Instead, the vertices are partitioned over multiple players (hospitals), and each player reports a subset of her vertices to the mechanism. In particular, a player may strategically omit vertices to increase how many of her vertices lie on the path returned by the mechanism. Our objective is to find mechanisms that limit incentives for such manipulation while producing long paths. Unfortunately, in worst-case instances, competing with the overall longest path is impossible while incentivizing (approximate) truthfulness, i.e., requiring that hiding nodes cannot increase a player's utility by more than a factor of 1 + o(1). We therefore adopt a semi-random model where a small ($o(n)$) number of random edges are added to worst-case instances. While it remains impossible for truthful mechanisms to compete with the overall longest path, we give a truthful mechanism that competes with a weaker but non-trivial benchmark: the length of any path whose subpaths within each player have a minimum average length. In fact, our mechanism satisfies even a stronger notion of truthfulness, which we call matching-time incentive compatibility. This notion of truthfulness requires that each player not only reports her nodes truthfully but also does not stop the returned path at any of her nodes in order to divert it to a continuation inside her own subgraph.

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.

Sortition is a political system in which decisions are made by panels of randomly selected citizens. The process for selecting a sortition panel is traditionally thought of as uniform sampling without replacement, which has strong fairness properties. In practice, however, sampling without replacement is not possible since only a fraction of agents is willing to participate in a panel when invited, and different demographic groups participate at different rates. In order to still produce panels whose composition resembles that of the population, we develop a sampling algorithm that restores close-to-equal representation probabilities for all agents while satisfying meaningful demographic quotas. As part of its input, our algorithm requires probabilities indicating how likely each volunteer in the pool was to participate. Since these participation probabilities are not directly observable, we show how to learn them, and demonstrate our approach using data on a real sortition panel combined with information on the general population in the form of publicly available survey data.

Migration presents sweeping societal challenges that have recently attracted significant attention from the scientific community. One of the prominent approaches that have been suggested employs optimization and machine learning to match migrants to localities in a way that maximizes the expected number of migrants who find employment. However, it relies on a strong additivity assumption that, we argue, does not hold in practice, due to competition effects; we propose to enhance the data-driven approach by explicitly optimizing for these effects. Specifically, we cast our problem as the maximization of an approximately submodular function subject to matroid constraints, and prove that the worst-case guarantees given by the classic greedy algorithm extend to this setting. We then present three different models for competition effects, and show that they all give rise to submodular objectives. Finally, we demonstrate via simulations that our approach leads to significant gains across the board.

Equalized odds is a statistical notion of fairness in machine learning that ensures that classification algorithms do not discriminate against protected groups. We extend equalized odds to the setting of cardinality-constrained fair classification, where we have a bounded amount of a resource to distribute. This setting coincides with classic fair division problems, which allows us to apply concepts from that literature in parallel to equalized odds. In particular, we consider the axioms of resource monotonicity, consistency, and population monotonicity, all three of which relate different allocation instances to prevent paradoxes. Using a geometric characterization of equalized odds, we examine the compatibility of equalized odds with these axioms. We empirically evaluate the cost of allocation rules that satisfy both equalized odds and axioms of fair division on a dataset of FICO credit scores.

Most approaches to the synthesis of reactive systems study the problem in terms of a two-player game with complete observation. In many applications, however, the system's environment consists of several distinct entities, and the system must actively communicate with these entities in order to obtain information available in the environment. In this paper, we model such environments as a team of players and keep track of the information known to each individual player. This allows us to synthesize programs that interact with a distributed environment and leverage multiple interacting sources of information. The synthesis problem in distributed environments corresponds to solving a special class of Petri games, i.e., multi-player games played over Petri nets, where the net has a distinguished token representing the system and an arbitrary number of tokens representing the environment. While, in general, even the decidability of Petri games is an open question, we show that the synthesis problem in distributed environments can be solved in polynomial time for nets with up to two environment tokens. For an arbitrary but fixed number of three or more environment tokens, the problem is NP-complete. If the number of environment tokens grows with the size of the net, the problem is EXPTIME-complete.