arXiv subjects

Lirong Xia

Publications and source records attributed to Lirong Xia.

At least 19 recordsLinked to original sources

arXiv2026

Simultaneous Envy and Equitability Guarantees

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. We show that EF1+EQ1 may fail to exist even for normalized binary goods: we construct an instance with 113 agents and 341 goods in which every agent approves exactly 165 goods, but no complete allocation satisfies both notions. Our main algorithmic result computes an EF1+EQ1 allocation for every normalized binary goods instance with at most seven agents. Thus, the smallest number of agents admitting a counterexample lies between 8 and 113, leaving the cases from 8 through 112 unresolved. 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.

cs.GT↗

arXiv2026

Mixture of Complementary Agents for Robust LLM Ensemble

Multi-AI collaboration, such as ensembling or debating large language models (LLMs), is a promising paradigm for aggregating information and boosting performance. A foundational step in these pipelines is to feed the responses of several proposer LLMs into a summarizer LLM, which synthesizes a better answer. However, choosing which proposers to include is non-trivial. Existing approaches primarily focus either on accuracy (picking the strongest models) or diversity (ensuring variety), and often overlook the interactions among proposers and with the summarizer. We reframe proposer selection as a combinatorial selection problem akin to feature selection, where the value of an LLM lies in its complementarity with others. However, directly applying standard feature-selection algorithms is impractical in the LLM setting due to prohibitive time complexity. Motivated by this limitation, we explore an extensive range of computationally feasible, greedy-style selection algorithms that assess complementarity using a small labeled set. Our experiments validate complementarity as a guiding principle for proposer selection and identify methods that achieve the best performance-cost trade-offs in practice.

cs.LG↗

arXiv2026

Winning in the Limit: Average-Case Committee Selection with Many Candidates

We study the committee selection problem in the canonical impartial culture model with a large number of voters and an even larger candidate set. Here, each voter independently reports a uniformly random preference order over the candidates. For a fixed committee size $k$, we ask when a committee can collectively beat every candidate outside the committee by a prescribed majority level $α$. We focus on two natural notions of collective dominance, $α$-winning and $α$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques. We first consider $α$-winning sets. A set $S$ of $k$ candidates is $α$-winning if, for every outside candidate $a \notin S$, at least an $α$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ α_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $α$-winning set of size $k$ exists with high probability when $α< α_{\mathrm{win}}^\star$, and is unlikely to exist when $α> α_{\mathrm{win}}^\star$. We then study the stronger notion of $α$-dominating sets. A set $S$ of $k$ candidates is $α$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $α$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ α_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $α$-dominating sets: for every $k$ and every $α> α_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $α$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.

cs.GT↗

arXiv2026

Shortlisting: a Principled Approach

Shortlisting is the process of selecting a subset of alternatives from a larger pool for further consideration or final decision-making. It is widely applied in social choice and multi-agent system scenarios. The growing demand for participatory decision-making and the continuously expanding space of candidates create an urgent need for efficient and fair shortlisting procedures. However, little principled study has been done on this problem. This blue-sky paper aims to highlight the overlooked significance of shortlisting, distinguish it from related problems, provide initial thoughts, and, more importantly, serve as a call to arms. We envision that principled shortlisting can reduce cognitive burden, enable fair collective decisions, encourage broader participation, and ultimately build trust in democratic systems.

cs.GT↗

arXiv2026

Truth-Revealing Participatory Budgeting

Participatory Budgeting (PB) is commonly studied from an axiomatic perspective, where the aim is to design procedurally fair and economically efficient rules for voters with full information regarding their preferences. In contrast, we take an epistemic perspective and consider a framework where PB projects have different levels of underlying quality, indicating how well the project will take effect, which cannot be directly observed before implementation. Agents with noisy information cast votes to aggregate their information, and aim to elect a high-quality set of projects. We evaluate the performance of common PB rules by measuring the expected utility of their outcomes, compared to the optimal set of projects. We find that the quality of approximation improves as the range of project costs shrinks. When projects have unit cost, these common rules can identify the ``best'' set with probability converging to 1. We also study whether strategic agents have incentives to honestly convey their information in the vote. We find that it happens only under very restrictive conditions. We also run numerical experiments to examine the performance of different rules empirically and support our theoretical findings.

cs.GT↗

arXiv2025

On the Edge of Core (Non-)Emptiness: An Automated Reasoning Approach to Approval-Based Multi-Winner Voting

Core stability is a natural and well-studied notion for group fairness in multi-winner voting, where the task is to select a committee from a pool of candidates. We study the setting where voters either approve or disapprove of each candidate; here, it remains a major open problem whether a core-stable committee always exists. In this work, we develop an approach based on mixed-integer linear programming for deciding whether and when core-stable committees are guaranteed to exist. In contrast to SAT-based approaches popular in computational social choice, our method can produce proofs for a specific number of candidates independent of the number of voters. In addition to these computational gains, our program lends itself to a novel duality-based reformulation of the core stability problem, from which we obtain new existence results in special cases. Further, we use our framework to reveal previously unknown relationships between core stability and other desirable properties, such as notions of priceability.

cs.GT↗

arXiv2025

Likelihood of the Existence of Average Justified Representation

We study the approval-based multi-winner election problem where $n$ voters jointly decide a committee of $k$ winners from $m$ candidates. We focus on the axiom \emph{average justified representation} (AJR) proposed by Fernandez, Elkind, Lackner, Garcia, Arias-Fisteus, Basanta-Val, and Skowron (2017). AJR postulates that every group of voters with a common preference should be sufficiently represented in that their average satisfaction should be no less than their Hare quota. Formally, for every group of $\lceil\ell\cdot\frac{n}{k}\rceil$ voters with $\ell$ common approved candidates, the average number of approved winners for this group should be at least $\ell$. It is well-known that a winning committee satisfying AJR is not guaranteed to exist for all multi-winner election instances. In this paper, we study the likelihood of the existence of AJR under the Erdős--Rényi model. We consider the Erdős--Rényi model parameterized by $p\in[0,1]$ that samples multi-winner election instances from the distribution where each voter approves each candidate with probability $p$ (and the events that voters approve candidates are independent), and we provide a clean and complete characterization of the existence of AJR committees in the case where $m$ is a constant and $n$ tends to infinity. We show that there are two phase transition points $p_1$ and $p_2$ (with $p_1\leq p_2$) for the parameter $p$ such that: 1) when $p p_2$, an AJR committee exists with probability $1-o(1)$, 2) when $p_1<p<p_2$, an AJR committee exists with probability $o(1)$, and 3) when $p=p_1$ or $p=p_2$, the probability that an AJR committee exists is bounded away from both $0$ and $1$.

cs.GT↗

arXiv2025

Epistemic vs. Counterfactual Fairness in Allocation of Resources

Resource allocation is fundamental to a variety of societal decision-making settings, ranging from the distribution of charitable donations to assigning limited public housing among interested families. A central challenge in this context is ensuring fair outcomes, which often requires balancing conflicting preferences of various stakeholders. While extensive research has been conducted on theoretical and algorithmic solutions within the fair division framework, much of this work neglects the subjective perception of fairness by individuals. This study focuses on the fairness notion of envy-freeness (EF), which ensures that no agent prefers the allocation of another agent according to their own preferences. While the existence of exact EF allocations may not always be feasible, various approximate relaxations, such as counterfactual and epistemic EF, have been proposed. Through a series of experiments with human participants, we compare perceptions of fairness between three widely studied counterfactual and epistemic relaxations of EF. Our findings indicate that allocations based on epistemic EF are perceived as fairer than those based on counterfactual relaxations. Additionally, we examine a variety of factors, including scale, balance of outcomes, and cognitive effort involved in evaluating fairness and their role in the complexity of reasoning across treatments.

cs.GT↗

arXiv2025

Average-Case Analysis of Iterative Voting

Iterative voting is a natural model of repeated strategic decision-making in social choice theory when agents have the opportunity to update their votes prior to finalizing the group decision. Prior work has analyzed the efficacy of iterative plurality on the welfare of the chosen outcome at equilibrium, relative to the truthful vote profile, via an adaptation of the price of anarchy. However, prior analyses have only studied the worst- and average-case performances when agents' preferences are distributed by the impartial culture. This work extends average-case analysis comprehensively across three alternatives and distinguishes under which of agents' preference distributions iterative plurality improves or degrades asymptotic welfare.

cs.GT↗

arXiv2025

Bridging Theory and Perception in Fair Division: A Study on Comparative and Fair Share Notions

The allocation of resources among multiple agents is a fundamental problem in both economics and computer science. In these settings, fairness plays a crucial role in ensuring social acceptability and practical implementation of resource allocation algorithms. Traditional fair division solutions have given rise to a variety of approximate fairness notions, often as a response to the challenges posed by non-existence or computational intractability of exact solutions. However, the inherent incompatibility among these notions raises a critical question: which concept of fairness is most suitable for practical applications? In this paper, we examine two broad frameworks -- threshold-based and comparison-based fairness notions -- and evaluate their perceived fairness through a comprehensive human subject study. Our findings uncover novel insights into the interplay between perception of fairness, theoretical guarantees, the role of externalities and subjective valuations, and underlying cognitive processes, shedding light on the theory and practice of fair division.

cs.GT↗

arXiv2025

Distribution of Chores with Information Asymmetry

A well-regarded fairness notion when dividing indivisible chores is envy-freeness up to one item (EF1), which requires that pairwise envy can be eliminated by the removal of a single item. While an EF1 and Pareto optimal (PO) allocation of goods can always be found via well-known algorithms, even the existence of such solutions for chores remains open, to date. We take an epistemic approach utilizing information asymmetry by introducing dubious chores--items that inflict no cost on receiving agents but are perceived costly by others. On a technical level, dubious chores provide a more fine-grained approximation of envy-freeness than EF1. We show that finding allocations with minimal number of dubious chores is computationally hard. Nonetheless, we prove the existence of envy-free and fractional PO allocations for $n$ agents with only $2n-2$ dubious chores and strengthen it to $n-1$ dubious chores in four special classes of valuations. Our experimental analysis demonstrates that often only a few dubious chores are needed to achieve envy-freeness.

cs.GT↗

arXiv2025

The Art of Two-Round Voting

We study the voting problem with two alternatives where voters' preferences depend on a not-directly-observable state variable. While equilibria in the one-round voting mechanisms lead to a good decision, they are usually hard to compute and follow. We consider the two-round voting mechanism where the first round serves as a polling stage and the winning alternative only depends on the outcome of the second round. We show that the two-round voting mechanism is a powerful tool for making collective decisions. Firstly, every (approximated) equilibrium in the two-round voting mechanisms (asymptotically) leads to the decision preferred by the majority as if the state of the world were revealed to the voters. Moreover, there exist natural equilibria in the two-round game following intuitive behaviors such as informative voting, sincere voting [Austen-Smith and Banks, 1996], and the surprisingly popular strategy [Prelec et al., 2017]. This sharply contrasts with the one-round voting mechanisms in the previous literature, where no simple equilibrium is known. Finally, we show that every equilibrium in the standard one-round majority vote mechanism gives an equilibrium in the two-round mechanisms that is not more complicated than the one-round equilibrium. Therefore, the two-round voting mechanism provides a natural equilibrium in every instance, including those where one-round voting fails to have a natural solution, and it can reach an informed majority decision whenever one-round voting can. Our experiments on generative AI voters also imply that two-round voting leads to the correct outcome more often than one-round voting under some circumstances.

cs.GT↗

arXiv2025

Aggregating Information and Preferences with Bounded-Size Deviations

We investigate a voting scenario with two groups of agents whose preferences depend on a ground truth that cannot be directly observed. The majority's preferences align with the ground truth, while the minorities disagree. Focusing on strategic behavior, we analyze situations where agents can form coalitions up to a certain capacity and adopt the concept of ex-ante Bayesian $k$-strong equilibrium, in which no group of at most $k$ agents has an incentive to deviate. Our analysis provides a complete characterization of the region where equilibria exist and yield the majority-preferred outcome when the ground truth is common knowledge. This region is defined by two key parameters: the size of the majority group and the maximum coalition capacity. When agents cannot coordinate beyond a certain threshold determined by these parameters, a stable outcome supporting the informed majority emerges. The boundary of this region exhibits several distinct segments, notably including a surprising non-linear relationship between majority size and deviation capacity. Our results reveal the complexity of the strategic behaviors in this type of voting game, which in turn demonstrate the capability of the ex-ante Bayesian $k$-strong equilibrium to provide a more detailed analysis.

cs.GT↗

arXiv2025

A Linear Theory of Multi-Winner Voting

We introduces a general linear framework that unifies the study of multi-winner voting rules and proportionality axioms, demonstrating that many prominent multi-winner voting rules-including Thiele methods, their sequential variants, and approval-based committee scoring rules-are linear. Similarly, key proportionality axioms such as Justified Representation (JR), Extended JR (EJR), and their strengthened variants (PJR+, EJR+), along with core stability, can fit within this linear structure as well. Leveraging PAC learning theory, we establish general and novel upper bounds on the sample complexity of learning linear mappings. Our approach yields near-optimal guarantees for diverse classes of rules, including Thiele methods and ordered weighted average rules, and can be applied to analyze the sample complexity of learning proportionality axioms such as approximate core stability. Furthermore, the linear structure allows us to leverage prior work to extend our analysis beyond worst-case scenarios to study the likelihood of various properties of linear rules and axioms. We introduce a broad class of distributions that extend Impartial Culture for approval preferences, and show that under these distributions, with high probability, any Thiele method is resolute, CORE is non-empty, and any Thiele method satisfies CORE, among other observations on the likelihood of commonly-studied properties in social choice. We believe that this linear theory offers a new perspective and powerful new tools for designing and analyzing multi-winner rules in modern social choice applications.

cs.GT↗

arXiv2025

Strong Equilibria in Bayesian Games with Bounded Group Size

We study the group strategic behaviors in Bayesian games. Equilibria in previous work do not consider group strategic behaviors with bounded sizes and are too ``strong'' to exist in many scenarios. We propose the ex-ante Bayesian $k$-strong equilibrium and the Bayesian $k$-strong equilibrium, where no group of at most $k$ agents can benefit from deviation. The two solution concepts differ in how agents calculate their utilities when contemplating whether a deviation is beneficial. Intuitively, agents are more conservative in the Bayesian $k$-strong equilibrium than in the ex-ante Bayesian $k$-strong equilibrium. With our solution concepts, we study collusion in the peer prediction mechanisms, as a representative of the Bayesian games with group strategic behaviors. We characterize the thresholds of the group size $k$ so that truthful reporting in the peer prediction mechanism is an equilibrium for each solution concept, respectively. Our solution concepts can serve as criteria to evaluate the robustness of a peer prediction mechanism against collusion. Besides the peer prediction problem, we also discuss two other potential applications of our new solution concepts, voting and Blotto games, where introducing bounded group sizes provides more fine-grained insights into the behavior of strategic agents.

cs.GT↗

arXiv2024

Differentially Private Approval-Based Committee Voting

In this paper, we investigate tradeoffs among differential privacy (DP) and several representative axioms for approval-based committee voting, including justified representation, proportional justified representation, extended justified representation, Pareto efficiency, and Condorcet criterion. Without surprise, we demonstrate that all of these axioms are incompatible with DP, and thus establish both upper and lower bounds for their two-way tradeoffs with DP. Furthermore, we provide upper and lower bounds for three-way tradeoffs among DP and every pairwise combination of such axioms, revealing that although these axioms are compatible without DP, their optimal levels under DP cannot be simultaneously achieved. Our results quantify the effect of DP on the satisfaction and compatibility of the axioms in approval-based committee voting, which can provide insights for designing voting rules that possess both privacy and axiomatic properties.

cs.GT↗

arXiv2024

Computing Most Equitable Voting Rules

How to design fair and (computationally) efficient voting rules is a central challenge in Computational Social Choice. In this paper, we aim at designing efficient algorithms for computing most equitable rules for large classes of preferences and decisions, which optimally satisfy two fundamental fairness/equity axioms: anonymity (every voter being treated equally) and neutrality (every alternative being treated equally). By revealing a natural connection to the graph isomorphism problem and leveraging recent breakthroughs by Babai [2019], we design quasipolynomial-time algorithms that compute most equitable rules with verifications, which also compute verifications about whether anonymity and neutrality are satisfied at the input profile. Further extending this approach, we propose the canonical-labeling tie-breaking, which runs in quasipolynomial-time and optimally breaks ties to preserve anonymity and neutrality. As for the complexity lower bound, we prove that even computing verifications for most equitable rules is GI-complete (i.e., as hard as the graph isomorphism problem), and sometimes GA-complete (i.e., as hard as the graph automorphism problem), for many commonly studied combinations of preferences and decisions. To the best of our knowledge, these are the first problems in computational social choice that are known to be complete in the class GI or GA.

cs.GT↗

arXiv2023

Smoothed Differential Privacy

Differential privacy (DP) is a widely-accepted and widely-applied notion of privacy based on worst-case analysis. Often, DP classifies most mechanisms without additive noise as non-private (Dwork et al., 2014). Thus, additive noises are added to improve privacy (to achieve DP). However, in many real-world applications, adding additive noise is undesirable (Bagdasaryan et al., 2019) and sometimes prohibited (Liu et al., 2020). In this paper, we propose a natural extension of DP following the worst average-case idea behind the celebrated smoothed analysis (Spielman & Teng, May 2004). Our notion, smoothed DP, can effectively measure the privacy leakage of mechanisms without additive noises under realistic settings. We prove that any discrete mechanism with sampling procedures is more private than what DP predicts, while many continuous mechanisms with sampling procedures are still non-private under smoothed DP. In addition, we prove several desirable properties of smoothed DP, including composition, robustness to post-processing, and distribution reduction. Based on those properties, we propose an efficient algorithm to calculate the privacy parameters for smoothed DP. Experimentally, we verify that, according to smoothed DP, the discrete sampling mechanisms are private in real-world elections, and some discrete neural networks can be private without adding any additive noise. We believe that these results contribute to the theoretical foundation of realistic privacy measures beyond worst-case analysis.

cs.CR↗