Social Choice Theory

Suppose a committee, expert panel, or other group is making judgments on some issues, where these may be not just yes/no-questions, such as whether a defendant is guilty, but also variables with many possible values, such as macroeconomic or meteorological variables or travel directions. Furthermore, there may be interconnections between different issues, as in the case of economic or climate variables. How can the group arrive at "intelligent" collective judgments, based on the group members' individual judgments? We investigate three challenges raised by this judgment-aggregation problem. First, reasonable methods of aggregation (such as defining the collective judgment for each issue as the average or median judgment) can produce inconsistent collective judgments. Secondly, many methods of aggregation are manipulable by strategic voting. Finally, not all methods of aggregation are conducive to tracking the truth on the issues in question. We prove new impossibility or possibility theorems on all three challenges, identifying what it takes to produce collective judgments in a consistent, non-manipulable, and truth-tracking manner and thereby to achieve collective intelligence through aggregation. Overall, the median method, though imperfect, performs reasonably well. We also note the relevance of our analysis for non-human group decisions.

Preference-based alignment methods such as reinforcement learning from human feedback (RLHF) and Nash learning from human feedback (NLHF) aggregate pairwise preferences to learn an LLM policy, but a natural goal is maximizing social welfare (average cardinal utility), which comparisons alone need not identify. Gölz, Haghtalab, and Yang (GHY) measure the gap by distortion: the worst-case ratio between the welfare of the best fixed lottery (distribution over responses) and of the learned lottery. They show NLHF is optimal when every user receives the same lottery. Account-based LLMs, however, have information about their users and can serve different lotteries to different people. We give an efficient algorithm, GLHF, that learns a single group-conditioned policy from one comparison per user. Under individual Bradley--Terry comparisons, GLHF asymptotically matches GHY's optimal population distortion bound simultaneously on every group in a prespecified, possibly overlapping collection, with sample complexity growing logarithmically in the number of groups and inversely with the smallest group mass. A sharper guarantee for groups with similar preferences approaches distortion of one when members share a feasible favorite response. In experiments using human coffee ratings and synthetic LLM-generated ratings, GLHF lowers distortion in every evaluated group and substantially reduces worst-group distortion relative to NLHF and other group-agnostic baselines.

We study the problem of computing an allocation of indivisible items that maximizes egalitarian welfare, i.e., the utility of the worst-off agent, when agents' item values or marginal values belong to a small set. For additive valuations with values in $\{p,q\}$, where $q>p>0$ and $\gcd(p,q)=1$, we give a polynomial-time algorithm when $p=2$ and prove constant-gap hardness when $p\geq3$, already with exactly three high-valued goods per agent. We also give an $\sqrt{3/2}$-approximation for common positive bi-valued additive valuations. For mixed additive valuations in $\{-p,0,c\}$, where $p\in\{1,2\}$ and $c$ is a positive integer, a reduction to maximum-weight perfect matching resolves the conjectured tractability of $\{-2,0,c\}$-valuations. For submodular valuations with marginals in $\{-2,0,c\}$, where $c$ is odd, we establish an exact unit-gap hardness result and exponential value-query lower bounds, even when all but one agent are additive. Finally, for $\{-1,0,1\}$-submodular valuations, we prove that no finite multiplicative approximation exists unless $\p=\np$. Together, our results resolve open questions and provide a complete picture of the computational complexity of max-min allocation with ternary valuations.

A Condorcet domain is a set of rankings over a given candidate set, such that every election that consists only of (an odd number of) votes from the domain has a transitive majority relation. We study outer diversity of Condorcet domains, i.e., a measure that quantifies expected swap distance from a random vote to a closest one in the domain. We numerically analyze outer diversity for maximal Condorcet domains with few candidates, and then we establish its asymptotic behavior for several special domains, mostly obtaining theoretical results.

In a district-based election, N voters are partitioned into k districts, and each voter votes for one of m candidates. Each district elects a winner using the plurality rule (i.e. the candidate getting the largest number of votes is declared the winner, breaking ties as per some fixed rule), and the overall winner is determined by applying plurality to the district winners; we assume that there is a unique winner amongst the district winners. The margin of victory of such an election is the minimum number of votes that must be altered so that the current winner ceases to be the unique district winner. We study the problem of predicting the winner of a district-based election in the query complexity model, where one has query access to individual votes. The objective is to minimise the number of queries. This setting captures exit polling, where queries correspond to interviewing voters, and is closely related to problems in query complexity and property testing. Assuming that the margin of victory of the election is at least eps N, Dey, Kar and Sanyal (AAMAS 2023) gave algorithms for the case of two candidates with error probability del and query complexity tilde{O}(1/eps^6 log^2 1/del), which improves to tilde{O}(1/eps^4 log^2 1/del) under the additional assumption that district populations are balanced. Our main result is an adaptive randomised algorithm that, for an arbitrary district-based election and any error parameter del, with probability at least 1-del, predicts the winner correctly using tilde{O}(1/eps^2 log m/del log 1/del) queries. In particular, we improve the bounds of Dey et al. for arbitrary district populations and extend their results to any number of candidates. Furthermore, for constantly many candidates, our algorithm nearly matches a lower bound of Omega(1/eps^2 log 1/del) on the query complexity that holds even for two candidates and a single district.

We study the Reverse Sequential Proportional Approval Voting Rule (RevSeqPAV) in approval-based committee elections. Despite its historical prominence and practical use, its properties and guarantees are much less understood than those of Sequential PAV. We analyze it along two dimensions: proportional representation (measured by Extended Justified Representation, its approximations, and proportionality degree) and approximation of the maximum PAV score of instances. We first establish strong negative results for general, unrestricted election instances and then identify settings in which the rule provides meaningful fairness and optimization guarantees.

In school choice, a lottery number is often used by the matching mechanism to break ties when there are more students who prefer the same school than the number of seats available. There has been growing theoretical and empirical interest in understanding the impact of revealing the lottery number to students. In practice, in recent years, the NYC Public Schools started revealing the lottery number to students to improve transparency. Theoretical findings from prior literature also suggest that revealing the lottery number strictly improves the number of matches under the deferred acceptance algorithm. However, these theoretical results are based on the over-simplifying assumption that all students share the same preference ranking for schools. In this work, we relax this assumption and allow students to have heterogeneous preference rankings. Under a game-theoretic model where student strategies form a Bayesian Nash equilibrium, we characterize scenarios where revealing the lottery number can either improve or worsen the matching outcome, measured by two metrics of match rate and social welfare. We further consider revealing partial information about the lottery, and demonstrate non-monotonic effects in the amount of information available to students. These results together illustrate complex tradeoffs induced by the lottery revealing policy.

We study the problem of computing how close a given election is to being group-separable, measuring proximity by swaps of adjacent candidates in the votes. We also consider several other domains, including caterpillar group-separable, balanced group-separable, single-peaked, and single-crossing ones. Our problem is generally intractable, but we find practical FPT algorithms parameterized by the number of candidates or swaps. For the latter case, our algorithm applies to all domains characterized by finite forbidden subelections, resolving a well-established open problem. We supplement our theoretical findings with experimental analysis.

We study a partial-participation variation of the Hotelling-Downs model. Voters each have a cost to vote, and only vote when the comparative gain from their preferred candidate exceeds the cost. Under this model the median voter theorem breaks, and we study the extent of polarization under equilibria in different voters and cost distributions. We find that the main predictor of polarization is the reverse-hazard-rate of the cost distribution, indicating that the driver of polarization under our model is the willingness of voters to respond to changes in positions of candidates. We then extend the model by adding parameters governing alienation and candidate competitiveness, showing that our results are robust even when taking into account other realistic factors.

Sequential deliberation is a mechanism for collective decision making: at each round, a uniformly randomly selected pair is asked to revise a collective outcome, which then becomes the input for the next round. Existing theory by Fain et al.~\cite{fain2017sequential} treats the current outcome solely as the disagreement alternative in bargaining. Yet the existing outcome might also carry social influence and anchor participants' positions toward the status quo. In this paper, we introduce \emph{anchored sequential deliberation}. At each round, two participants with bliss points $U$ and $V$ shift their positions toward the previous outcome $O_{t-1}$ with anchoring strength $0\leq λ<1$. They then bargain using $O_{t-1}$ as the disagreement alternative. For the sake of analysis, we assume that the decision space is one-dimensional, the anchoring effect is linear, and participants use Nash bargaining. The update simplifies to $O_t=(1-λ)Med\{U,V,O_{t-1}\}+λO_{t-1}$. Our analysis reveals a trade-off. Through a coupling of two outcomes, we find that the process contracts in $1$-Wasserstein distance with a factor of at most $\frac{1+λ}{2}$, which implies that stronger anchoring slows mixing. On the other hand, stationary distortion weakly decreases with $λ$, although the worst-case distortion remains $\frac{1+\sqrt{2}}{2}$ for every feasible $λ$. We also identify a unique \emph{deliberative fixed point}, at which the expected movement is zero, and prove that the stationary distribution concentrates around it as $λ$ increases. For symmetric populations, we further provide a tighter bound on the stationary variance around this fixed point. For the uniform distribution, we establish upper and lower bounds on stationary distortion, both of which approach $1$ as $λ$ increases.

When the same question is asked of multiple AI advisers, as in self-consistency and LLM-as-a-judge panels, Condorcet's jury theorem predicts that adding independent, competent advisers makes the majority more reliable. The theorem, however, has a latent dimension when viewed from the user's vantage: adding advisers also makes disagreement more visible. A binomial model reveals that this ``visible dissent'' becomes nearly inevitable as the number of advisers grows, and that reliability and disagreement approach certainty at rates that cross at an adviser accuracy of 4/5 (0.8); below it, visible dissent eventually becomes more likely than a correct majority. Even ideal panels of independent and competent advisers can be correct in aggregate but appear divided; such disagreement does not by itself indicate aggregation failure. The way advisers split also provides a common basis for predictive multiplicity, reconciliation load, and reliance miscalibration. These results separate aggregation from disclosure and turn the latter into testable questions about how disagreement should be presented and interpreted.

Recent work in fair division has focused on either simultaneously satisfying closely related fairness notions or achieving a single notion across the ex-ante and ex-post worlds. We study the compatibility of two fundamentally different fairness notions: envy-freeness and equitability. For indivisible goods-only and chores-only settings, we study the existence and complexity of simultaneously satisfying their relaxations, revealing sharp contrasts between the two settings. For normalized binary goods, we give a polynomial-time algorithm for computing an EF1+EQ1 allocation with at most seven agents, but also construct a normalized instance with a larger number of agents for which no such allocation exists. In sharp contrast, binary chores admit the stronger EFX+EQX guarantee for any number of agents, even without normalization. We further initiate the study of cross-notion ex-ante and ex-post guarantees, asking whether randomized allocations can provide ex-ante guarantees for one notion while preserving ex-post guarantees for another.

We study the problem of formally explaining why a candidate was not selected by a given tournament rule, by identifying sub-tournaments in which the candidate loses independently of how the rest of the tournament is completed. We define destructive minimal supports as any minimal sub-tournament satisfying this property, which in formal explainable artificial intelligence corresponds to abductive explanations for the question "Why does the loser lose the tournament?". For six common tournament solutions (maximin, uncovered set and its weighted variant, top cycle, Copeland, and Borda) we provide characterizations of when a candidate is either a necessary loser or a possible winner, we determine the size of the smallest destructive minimal supports, complemented by polynomial-time algorithms for their computation except for the case of Borda and Copeland rules which we conjecture to also be polynomial.

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

A common problem in social choice is to determine whether there is a social choice procedure, such as a voting method, satisfying some desired criteria. Computer-aided methods such as SAT solving can sometimes answer these questions. However, under typical encodings, a SAT solver may only synthesize a voting method on a finite domain, while we may want one on an infinite domain, such as the domain of all preference profiles for a fixed number of candidates but any finite number of voters. In this paper, we use an approach based on reasoning with constrained Horn clauses and computation with polyhedra to synthesize a voting method on an infinite domain. We then use SMT and Lean to verify its properties. Our main result is a possibility theorem about four well-known criteria from voting theory: the Condorcet winner and loser criteria, positive involvement, and resolvability. Previous work has shown that for five or more candidates, there is no voting method satisfying these axioms, and that for four candidates, there is no method satisfying these core axioms plus one more invariance axiom. Here we show that for four candidates, there does exist a method satisfying the core axioms and more.

We study the computational complexity of winner determination problems in approval-based committee elections under Thiele voting rules. These form a class of rules parameterized by a fixed weight vector that specifies how a voter's satisfaction depends on the number of approved candidates elected. We first analyze the structure of optimal solutions based on the sets of voters who approve each candidate---that is, how voters' approval ballots induce dependencies between candidates---revealing constraints on a winning committee under any fixed Thiele voting rule. Using this, we design FPT algorithms for Proportional Approval Voting (PAV) and other Thiele rules on a natural restricted domain known as the Voter Interval (VI) domain---that is, after a suitable ordering of voters, each candidate is approved by a consecutive interval of voters. In particular, we show that every Thiele rule on VI is FPT with respect to a parameter for which the problem is NP-hard on general instances, even when the parameter takes constant values. Our results advance the understanding of the computational complexity of PAV on Voter Interval instances, which remains one of the central open questions in this area. We further resolve two open questions from the literature on PAV (and other Thiele voting rules) by providing a polynomial-time algorithm for instances where each candidate is approved by at most two voters, and an FPT algorithm parameterized by the total score of a winning committee.

We study whether strictly positive marginal values restore the compatibility of envy-freeness up to one good (EF1) and Pareto optimality (PO) for indivisible goods. For two agents, we identify the exact threshold in the number of goods. Every instance with at most seven goods and strictly increasing valuations admits an allocation that is both EF1 and PO, without any submodularity assumption. In contrast, we construct an eight-good instance with normalized, integer-valued, strictly increasing, submodular valuations in which every EF1 allocation is strictly Pareto dominated. Thus, eight goods are necessary and sufficient for a two-agent counterexample. Finally, we strengthen the three-agent NP-hardness result of Chandramouleeswaran and Nimbhorkar (2026): deciding whether an EF1 and PO allocation exists remains NP-hard for normalized, integer-valued, monotone submodular valuations even when zero marginals are confined to eight fixed agent-good pairs, all involving a single agent.

From housing allocation for households experiencing homelessness to triage in emergency departments, LLMs are increasingly being considered as judges of consequential decisions that require ranking people for scarce resources. Ranking large groups simultaneously is cognitively demanding and error-prone. A natural solution, drawing on decades of social choice theory, elicits pairwise comparisons and aggregates them into a total order. However, a fundamental question remains when LLMs serve as the pairwise judge: how can a practitioner tell, before committing to a ranking, whether the LLM's judgments are sufficiently consistent to trust the result? We discuss two different ways of identifying consistency. A classical diagnostic, the coefficient of consistency $ζ$, originally developed to measure judge reliability by counting circular triads in tournament graphs, provides a cheap, model-free measure of intra-run consistency. Various standard measures of distance between rankings, for example Kendall's $τ$, can measure inter-run variability. We show, in both theory and practice, that these measures are independently valuable, and advocate for using both to assess reliability of rankings. We demonstrate the practical importance of our results across two high-stakes prioritization tasks: homelessness service allocation and emergency department triage. Three different leading LLMs have considerably different performance profiles across these two axes of consistency. We provide guidelines for how practitioners could think about measuring and assessing consistency before committing to a model for ranking or prioritization.

This theoretical note studies the finite axiomatizability of strict majority reasoning in finite social decision frames. Moss and Pedersen (2026) <doi: 10.48550/arXiv.2606.23853> introduce a coherence criterion that characterizes exactly when qualitative majority judgments are representable by a finitely additive measure. The question addressed here is whether that coherence criterion can be replaced, in the finite setting, by any bounded finite fragment. We prove that it cannot. For every $k\ge 1$, we construct a maximal standard frame whose shortest coherence violation has length exactly $2k+2$. Hence there is no uniform finite bound on the incoherence index of social decision frames, resolving Conjecture 5.7 stated by Moss and Pedersen (2026). The construction is geometric, in the sense that it proceeds via orthogonality and dimension in rational vector spaces, and self-contained: it isolates a symmetric family of half-sized voting blocs and extends it to a maximal frame in which every shorter balanced obstruction is excluded. Along the explicit infinite sequence of universe sizes obtained in the construction, this also establishes the middle-layer family predicted by Conjecture B.25 by Moss and Pedersen (2026). Together with the soundness and completeness theorem for the Moss-Pedersen minimal logic for strict majorities, this establishes that measurable social decision frames are not finitely axiomatizable in that language.

This paper studies strict majority reasoning in finite electorates using so-called $\textit{social decision frames}$: finite sets of voters equipped with distinguished families of coalitions interpreted as those voting blocs evaluated to form a strict majority. A coherence criterion for qualitative majority judgments is identified and shown to give an exact characterization for representability of strict majorities by finitely additive measures. In addition, a minimal natural logic for reasoning about strict majorities is shown to be sound and complete. These developments motivate examination of associated combinatorial questions concerning incoherence in finite families of sets; partial results and a conjecture are given. Finally, the results of this paper are applied to correct a classical representation theorem for weak qualitative probability structures due to Patrick Suppes and to establish a May-type characterization for ordinary strict majority rule for social decision frames.

Laws and institutions shape individual outcomes through complex interactions with citizens' diverse circumstances, yet how different voting methods navigate this coupled landscape remains poorly understood. We model collective governance as optimization on NK fitness landscapes, where shared bits (laws) are updated by voting while individual bits (personal traits) remain fixed. A cross-dependency parameter $α$ controls how legislation's effects depend on individual circumstances. We compare eight standard voting methods and a generalized scoring family across landscape ruggedness $K \in \{1,\ldots,20\}$ and $α\in [0,1]$ with 1000 runs per configuration. Under direct democracy, the optimal voting method undergoes sharp phase transitions as a function of landscape complexity: cardinal score voting dominates on smooth landscapes, ordinal scoring with $p=0.35$ at low-to-moderate ruggedness, Borda count across a wide middle range, and STAR voting at the highest complexity. A two-parameter empirical formula reduces the $(K, α)$ plane to a single complexity axis for visualization. Borda count achieves the highest mean fitness and lowest variance across most of the parameter space. We further introduce a representative democracy model parameterized by identity weight $β$ and candidate self-interest $p_{\mathrm{self}}$. Representation reshapes the complexity-dependent structure even under favorable conditions: cardinal score voting dominates across most regimes, with plurality emerging as the top method at high $β$ and low-to-moderate $p_{\mathrm{self}}$.

Current approaches to aligning large language models (LLMs) aggregate diverse human preferences into a single reward signal, effectively optimizing for a hypothetical ``average user'' who represents no real person particularly well. This position paper argues that LLMs should learn personalized, individual preferences rather than aggregated ones. We show that aggregation masks critical information about preference diversity, individual values, and contextual dependencies, which is a limitation both theoretically grounded in social choice theory and empirically evident across demographic groups. We analyze the rich structure that human preferences encode, survey technical approaches to personalization, and systematically address counterarguments on scalability, shared standards, and manipulation risk. While personalization introduces genuine safety challenges including filter bubbles, value lock-in, and psychological manipulation, we argue these are manageable through bounded personalization frameworks that preserve universal safety constraints while accommodating legitimate individual variation. We conclude with a concrete research and policy agenda for developing preference-aware models that respect both individual autonomy and collective safety.

In RLHF, each training example contains a prompt $x$ and two candidate responses $y,y'$, and annotators provide pairwise preferences between these responses. The learning problem is to convert these heterogeneous pairwise judgments into a single scalar reward $r(x,y)$ that measures response quality for each prompt. Classical social choice implies an impossibility because heterogeneous annotator samples can induce pooled preferences with Condorcet cycles, so no scalar reward can evaluate all compared response pairs consistently. A growing literature analyzes RLHF as a social-choice problem, but usually assumes a fixed finite set of alternatives, i.e., a pre-enumerated finite set of candidate responses for each prompt. Modern pipelines instead score responses through a learned representation $φ(x,y)$ before a scalar head, so $φ$ determines which responses are treated as distinguishable alternatives and which comparisons are visible to the reward model. Once this embedding is part of the problem, the impossibility results from social choice theory become a tradeoff. We show that the excess cross-entropy loss of any reward built on $φ$ decomposes exactly into a representational term, which a richer $φ$ shrinks, and an aggregation term, which a richer $φ$ enlarges by exposing more comparisons that no scalar can rank consistently. The same results extend to direct preference optimization (DPO), and jointly training the embedding and the reward cannot guarantee to recover the sweet spot of this tradeoff. Experiments on synthetic data and real preference datasets corroborate our results.

The communication complexity of a voting rule is the worst-case number of bits that n voters must transmit to a central authority under the most efficient elicitation protocol in an election with m candidates. We study the communication complexity of Instant-Runoff Voting (IRV). Conitzer and Sandholm [2005] established an upper bound of O(n (log m)${}^2$), but did not provide a matching lower bound beyond $Ω$(n log m). We resolve this open problem by raising the lower bound to $Ω$(n (log m)${}^2$) using the fooling set technique, thereby showing that the communication complexity of IRV is $Θ$(n (log m)${}^2$). We further show that this complexity drops to $Θ$(n log m) under the single-peakedness restriction, and that both the IRV-Average variant and Single Transferable Vote (STV), the multiwinner extension of IRV, have the same asymptotic communication complexity as IRV.

Judgment aggregation studies how to combine individual judgments on logically related propositions into a collective judgment. Classical impossibility results show that sufficiently strong logical interconnections force dictatorship under natural aggregation axioms. In this paper, we ask whether such impossibility can still arise when the objects of aggregation are required to be genuinely modal judgments rather than plain factual propositions. Since modal logic contains propositional logic, this question is meaningful only if one excludes fact-based aggregation in disguise. We show that Arrow-type impossibility already re-emerges in a strikingly sparse modal setting. We prove an impossibility theorem on a simple cyclic frame for an agenda generated from a single propositional variable by repeated applications of a single modal operator, and we further demonstrate this phenomenon for an alternative family of frames satisfying a natural symmetry condition. Thus, even under a modal-operator requirement, semantic structure alone can generate the logical interconnections needed for dictatorship. Technically, our analysis has two layers. First, we prove a semantic reduction theorem showing that certain iterated modal patterns can be collapsed by shifting the evaluation point. Second, building on this reduction, we identify a local-to-global frame mechanism by which frame geometry yields minimally inconsistent modal judgment sets and the strong path-connectivity required for impossibility. The same reduction also turns consistency checking into a small combinatorial covering problem, which yields efficient implementations of non-dictatorial aggregation procedures.

We analyze the structure of the disagreement among a population of voters over a set of alternatives. Surveys typically ask either for pairwise comparisons, simple and intuitive for participants, or full rankings over alternatives, eliciting the entire voters' preferences. Building on the observation that pairwise comparisons cannot distinguish structural disagreement from noise, we propose a stratified framework to identify the minimal aggregated preference information needed to compute a number of disagreement measures from the literature. Specifically, we introduce the plurality matrix, a generalization of pairwise comparisons that records, for every subset $S$ of alternatives, the probability that each $a \in S$ ranks first in $S$. We define the level of a disagreement measure as the smallest subset size needed to express it, showing that many existing notions, including rank-variance and divisiveness, sit at level $3$, proving that pairwise comparisons are not enough. In addition, we demonstrate the interest of going beyond level $3$ both theoretically and experimentally. To make these results actionable, we design two elicitation protocols to estimate the plurality matrix, exploring the trade-off between the number of required participants and the cognitive load requested to each of them.

An index is a function that given an election outputs a value between 0 and 1, indicating the extent to which this election has a particular feature. We seek indices that capture agreement, diversity, and polarization among voters in approval elections, and that are normalized with respect to saturation. By the latter we mean that if two elections differ by the fraction of candidates approved by an average voter, but otherwise are of similar nature, then they should have similar index values. We propose several indices, analyze their properties, and use them to (a) derive a new map of approval elections, and (b) show similarities and differences between various real-life elections from Pabulib, Preflib and other sources.

We study proportional representation in the temporal voting model, where collective decisions are made repeatedly over time over a fixed horizon. Prior work has extensively investigated how proportional representation axioms from multiwinner voting (e.g., justified representation (JR) and its variants) can be adapted, satisfied, and verified in this setting. However, much less is understood about their interaction with social welfare. In this work, we quantify the efficiency cost of enforcing proportionality. We formalize the welfare-proportionality tension via the worst-case ratio between the maximum achievable utilitarian welfare and the maximum welfare attainable subject to a proportionality axiom. We show that imposing proportional representation in the temporal setting can incur a growing, yet sublinear, welfare loss as the number of voters or rounds increases. We further identify a clean separation among axioms: for JR, the welfare loss diminishes as the time horizon grows and vanishes asymptotically, whereas for stronger axioms this conflict persists even with many rounds. Moreover, we prove that welfare maximization under each axiom is NP-complete and APX-hard, even under static preferences and bounded-degree approvals, and provide fixed-parameter algorithms under several natural structural parameters.

Nash equilibrium serves as a fundamental mathematical tool in economics and game theory. However, it classically assumes knowledge of player utilities, whereas economics generally regards preferences as more fundamental. To leverage equilibrium analysis in strategic scenarios, one must first elicit numerical utilities consistent with player preferences, a delicate and time-consuming process. In this work, we forgo precise utilities and generalize the Nash equilibrium to a setting where we only assume a player is capable of providing an ordinal ranking of their actions within the context of other players' joint actions. The key technical challenge is to rethink the definition of a best-response. While the classical definition identifies actions maximizing expected payoff, we naturally look towards social choice theory for how to aggregate preferences to identify the most preferred actions. We define this generalized notion of a context-ordinal Nash equilibrium, establish its existence under mild conditions on aggregation methods, introduce notions of regularization, approximation, and regret, explore complexity for simple settings, and develop learning rules for computing such equilibria. In doing so, we provide a generalization of Nash equilibrium and demonstrate its direct applicability to elicited preferences in human experiments.

Approval-based committee voting has received significant attention in the social choice community. Among the studied rules, Thiele rules, and especially Proportional Approval Voting (PAV), stand out for desirable properties such as proportional representation, Pareto optimality, and support monotonicity. Their main drawback is that computing a Thiele outcome is NP-hard in general. A glimpse of hope comes from the fact that Thiele rules are better behaved under structured preferences. On the candidate interval (CI) domain, they are computable in polynomial time via a linear program (LP) that has a totally unimodular constraint matrix. Surprisingly, this approach fails for the related voter interval (VI) domain, and the complexity of the problem has repeatedly been posed as an open question. Our main result resolves this question: although the relevant matrix is not totally unimodular, the ``standard'' LP still admits at least one optimal integral solution, and we provide a fast algorithm for finding it. Our technique naturally extends to the voter-candidate interval (VCI) domain, also known as the 1-dimensional voter-candidate range (1D-VCR) domain, and to the linearly consistent (LC) domain, both of which generalize the candidate and voter interval domains. Although both the VCI and LC domains have been studied in social choice, their relationship was unknown. We show, through connections to graph theory, that LC strictly contains VCI. We also provide an alternative definition of LC that is closer in spirit to VCI and has a natural interpretation in approval elections; this equivalence may be of independent interest. Finally, we study an alternative tree-based generalization of VCI and show that Thiele rules become NP-hard to compute on this domain.