Rent division is the well-studied problem of fairly assigning rooms and dividing rent among a set of roommates within a single apartment. A shortcoming of existing solutions is that renters are assumed to be considering apartments in isolation, whereas in reality, renters can choose among multiple apartments. In this paper, we generalize the rent division problem to the multi-apartment setting, where the goal is to both fairly choose an apartment among a set of alternatives and fairly assign rooms and rents within the chosen apartment. Our main contribution is a generalization of envy-freeness called *negotiated envy-freeness*. We show that a solution satisfying negotiated envy-freeness is guaranteed to exist and that it is possible to optimize over all negotiated envy-free solutions in polynomial time. We also define an even stronger fairness notion called *universal envy-freeness* and study its existence when values are drawn randomly.

We study online fair division when there are a finite number of item types and the player values for the items are drawn randomly from distributions with unknown means. In this setting, a sequence of indivisible items arrives according to a random online process, and each item must be allocated to a single player. The goal is to maximize expected social welfare while maintaining that the allocation satisfies proportionality in expectation. When player values are normalized, we show that it is possible to with high probability guarantee proportionality constraint satisfaction and achieve O(√T) regret. To achieve this result, we present an upper confidence bound (UCB) algorithm that uses two rounds of linear optimization. This algorithm highlights fundamental aspects of proportionality constraints that allow for a UCB algorithm despite the presence of many (potentially tight) constraints. This result improves upon the previous best regret rate of O(T^(2/3)).

A key challenge in training Large Language Models (LLMs) is properly aligning them with human preferences. Reinforcement Learning with Human Feedback (RLHF) uses pairwise comparisons from human annotators to train reward functions and has emerged as a popular alignment method. However, input datasets in RLHF can be unbalanced due to adversarial manipulation or inadvertent repetition. Therefore, we want RLHF algorithms to perform well even when the set of alternatives is not uniformly distributed. Drawing on insights from social choice theory, we introduce robustness to approximate clones, a desirable property of RLHF algorithms which requires that adding near-duplicate alternatives does not significantly change the learned reward function. We first demonstrate that the standard RLHF algorithm based on regularized maximum likelihood estimation (MLE) fails to satisfy this property. We then propose the weighted MLE, a new RLHF algorithm that modifies the standard regularized MLE by weighting alternatives based on their similarity to other alternatives. This new algorithm guarantees robustness to approximate clones while preserving desirable theoretical properties.

A common problem in LLM evaluation is how to choose a subset of metrics from a full suite of possible metrics. Subset selection is usually done for efficiency or interpretability reasons, and the goal is often to select a "representative" subset of metrics. However, "representative" is rarely clearly defined. In this work, we use ideas from social choice theory to formalize two notions of representation for the selection of a subset of evaluation metrics. We first introduce *positional representation*, which guarantees every alternative is sufficiently represented at every position cutoff. We then introduce *positional proportionality*, which guarantees no alternative is proportionally over- or under-represented by more than a small error at any position. We prove upper and lower bounds on the smallest number of metrics needed to guarantee either of these properties in the worst case. We also study a generalized form of each property that allows for additional input on groups of metrics that must be represented. Finally, we tie theory to practice through real-world case studies on both LLM evaluation and hospital quality evaluation.

In the incentivized exploration model, a principal aims to explore and learn over time by interacting with a sequence of self-interested agents. It has been recently understood that the main challenge in designing incentive-compatible algorithms for this problem is to gather a moderate amount of initial data, after which one can obtain near-optimal regret via posterior sampling. With high-dimensional contexts, however, this \emph{initial exploration} phase requires exponential sample complexity in some cases, which prevents efficient learning unless initial data can be acquired exogenously. We show that these barriers to exploration disappear under mild geometric conditions on the set of available actions, in which case incentive-compatibility does not preclude regret-optimality. Namely, we consider the linear bandit model with actions in the Euclidean unit ball, and give an incentive-compatible exploration algorithm with sample complexity that scales polynomially with the dimension and other parameters.

We consider the problem of online fair division of indivisible goods to players when there are a finite number of types of goods and player values are drawn from distributions with unknown means. Our goal is to maximize social welfare subject to allocating the goods fairly in expectation. When a player's value for an item is unknown at the time of allocation, we show that this problem reduces to a variant of (stochastic) multi-armed bandits, where there exists an arm for each player's value for each type of good. At each time step, we choose a distribution over arms which determines how the next item is allocated. We consider two sets of fairness constraints for this problem: envy-freeness in expectation and proportionality in expectation. Our main result is the design of an explore-then-commit algorithm that achieves $\tilde{O}(T^{2/3})$ regret while maintaining either fairness constraint. This result relies on unique properties fundamental to fair-division constraints that allow faster rates of learning, despite the restricted action space.

In computational social choice, the distortion of a voting rule quantifies the degree to which the rule overcomes limited preference information to select a socially desirable outcome. This concept has been investigated extensively, but only through a worst-case lens. Instead, we study the expected distortion of voting rules with respect to an underlying distribution over voter utilities. Our main contribution is the design and analysis of a novel and intuitive rule, binomial voting, which provides strong distribution-independent guarantees for both expected distortion and expected welfare.

We introduce approximate trace reconstruction, a relaxed version of the trace reconstruction problem. Here, instead of learning a binary string perfectly from noisy samples, as in the original trace reconstruction problem, the goal is to output a string that is close in edit distance to the original string using few traces. We present several algorithms that can approximately reconstruct strings that belong to certain classes, where the estimate is within$n$/ polylog (n) edit distance and where we only use polylog (n) traces (or sometimes just a single trace). These classes contain strings that require a linear number of traces for exact reconstruction and that are quite different from a typical random string. From a technical point of view, our algorithms approximately reconstruct consecutive substrings of the unknown string by aligning dense regions of traces and using a run of a suitable length to approximate each region. A full version of this paper is accessible at: https://arxiv.org/abs/2012.06713.pdf