Papers

This paper is the first in a series on Turn-Based Combat Arena, a configurable framework for turn-based strategy games designed to support the efficient training and evaluation of machine learning agents. The proposed framework enables flexible modification of game rules and parameters, allowing rapid experimentation across diverse scenarios. Its architecture is optimized for high-throughput simulation, supporting tens of thousands of games per second and enabling the storage and processing of billions of gameplay records on a single machine. The problem of balancing the game, and particularly the parameters of game units, is investigated in detail. We evaluate several optimization approaches and show that multiple methods converge to comparable solutions, suggesting robustness in identifying balanced game configurations. These results indicate that the framework can serve as a practical platform for both game design analysis and agent training.

We model insider threat detection as a dynamic Bayesian game in which a platform coordinates a committee of strategic certifiers to sustain equilibrium among honest users and detect malicious deviations before exfiltration. Certifiers and users operate under a Bayesian Temporal Correlated Equilibrium (BTCE), where a sealed-envelope correlating device issues private recommendations over time and obedience is verified at every on-path information state. Unlike Stackelberg formulations, BTCE coordinates heterogeneous certifiers without requiring commitment power. We incorporate present bias and loss aversion to capture impulsive escalation dynamics, enabling 1.7–4.5 days earlier detection than rational baselines. We prove three guarantees: (1) calibrated intervention losses make recommended behavior a current-self best response despite behavioral biases, (2) controlled evidence accumulation guarantees intervention in bounded expected time before exfiltration, and (3) median aggregation confines implemented actions to the honest recommendation range when fewer than half of certifiers are Byzantine. On CERT r6.2 our mechanism achieves up to 28.3% pre-exfiltration detection with false positives below 1.6%, while both a transformer baseline and a streaming provenance approximation (HOLMESLite) achieve near-zero pre-exfiltration detection under comparable constraints.

A central challenge in mechanism design is to develop truthful trade mechanisms that maximize the expected gains-from-trade (GFT) in two-sided markets. Because achieving the full GFT is generally impossible, the literature has focused on constant-factor approximations—a notoriously difficult problem even in simple settings. It was only recently that a breakthrough result by [DMSW22] achieved a constant-factor approximation for single-item bilateral trade. The same guarantee was later extended to single-dimensional matching markets with general downward-closed constraints [BRTW26]. Most existing results, however, are limited to single-dimensional agents. A notable multi-dimensional exception is [CGMZ21]. They considered a market with one constrained-additive buyer and $n$ single-dimensional sellers and provided a mechanism that achieves a $\log^2(n)$ approximation to the second-best GFT, i.e., the maximum expected GFT theoretically achievable by any mechanism satisfying Bayesian Incentive Compatibility (BIC), Interim Individual Rationality (IIR), and ex-ante Weak Budget Balance (WBB). In this paper, we study multi-dimensional bilateral trade problem where both sides of the market are multi-dimensional. We start with one buyer with XOS valuation and one seller with an additive cost function. We then generalize to a market with $n$ XOS buyers and one additive seller. Assuming independent items' values and costs, in both settings we propose simple mechanisms that are BIC, IIR, and ex-ante WBB, while achieving a constant fraction of the optimal (first-best) expected GFT.

We investigate the fair allocation of indivisible items among agents with asymmetric entitlements in mixed manna settings, where the items consist of both goods and chores. For additive valuations, we establish that weighted envy-free up to one item (WEF1) allocations always exist and can be computed in polynomial time. We also study fair and efficient allocation and show that weighted envy-freeness up to one transfer (WEF1T) is compatible with fractional Pareto optimality (fPO) for every mixed-manna instance. This relaxation from WEF1 to WEF1T is tight, as demonstrated by our impossibility result. We further show a best-of-both-worlds result via a finite lottery that guarantees weighted envy-freeness (WEF) in expectation, with every realized allocation satisfying WEF1T and achieving the tight characterization complemented by the existing impossibility result.

While traditional stable matching algorithms, such as the Gale-Shapley algorithm, prioritize stability, they may fall short of achieving equitable outcomes among participants. We study the role of Nash social welfare (NSW) as a fairness objective in the classic stable marriage problem. We develop SNSW-Alg that finds a stable matching that maximizes Nash social welfare under rank-induced utilities in $\tilde{\mathcal{O}}(n^4)$ time, where $n$ is the number of men or women. We demonstrate that SNSW-Alg balances equity while preserving stability. We empirically evaluate our methods across diverse preference distributions, demonstrating significant gains in fairness without substantial losses in other key measures such as regret, egalitarian criterion, and sex equality. Our findings suggest that the stable matching produced by SNSW-Alg is statistically Pareto-undominated by stable matchings based on other fairness measures - regret, egalitarian, and sex equality. This study offers compelling insights for designing fair-stable matching.

We study the computational complexity of satisfying proportional representation – in particular proportional, extended, and fully justified representation (PJR, EJR, and FJR) – in participatory budgeting and committee elections with additive utilities. First, we give a complete picture of the complexity of the axioms for a constant number of voters or voter types. Second, we show that even for committee elections with integer utilities bounded above by a small constant, satisfying FJR is intractable, giving the first strong NP-hardness result for a justified representation axiom. Third, we extend the Expanding Approvals Rule to committee elections with additive utilities and show that it satisfies PJR. Lastly, we show that no sequential voting rule can improve on the known positive result, thus proving that novel, substantially different voting rules are needed to surpass these boundaries. Beyond their theoretical merit, our results carry practical importance, as multi-winner voting with additive utilities has recently been gaining prominence in online deliberation and real-world participatory budgeting.

Information sharing can improve a pooled estimate while eliminating independent rescue actions. This paper separates those effects in exact finite discovery models. A centralized action-budget profile shows that equal one-person accuracy can coexist with different portfolio values. Under a registered incremental-sharing protocol, a sharing step improves discovery exactly when pooled residual error contracts faster than an independent rescue attempt. Exact bounded registries exhibit compression, aggregation, neutral curves, and a bounded zero mixed class. In a two-agent Bayesian game with a hidden mixture of common and independent signal sources, the registered selected equilibrium yields a strict positive sharing interval at signal accuracy 3/5, while alternative equilibria show that the result is selection-dependent rather than universal. The models are synthetic and finite; no human or organizational data are used.

We develop a framework for mechanism design with AI agents whose alignment (preferences) and capabilities (feasible actions and information) are unknown. We want such agents to act on our behalf so mechanisms must incentivize both honesty and obedience. A one-sided imitation structure—capabilities can be concealed but not counterfeited—yields a revelation principle, a characterization of implementable policies via nested cyclical monotonicity, and conditions under which eliciting higher-order beliefs can discipline multiple agents. We apply our framework to stylized examples of (i) sandbagging in which a more capable agent pretends to be less capable; (ii) an alignment–interpretability trade-off, where the two are substitutes in the instrument but complements in value; (iii) discipline via peer scoring; (iv) coupling rewards to induce competition among multiple agents; and (v) scalable oversight and reward shaping.

We study weighted fair division of indivisible mixed manna under additive valuations. First, we resolve the general existence open question for weighted envy-freeness up to one item (WEF1), and show that every instance with arbitrary positive entitlements admits a complete WEF1 allocation computable in polynomial time. We then show that existence does not imply any welfare guarantee, i.e., the utilitarian price of WEF1 is infinite, even for two unweighted agents with normalized valuations, common item signs, and singleton values in a fixed four-value set; a welfare-maximizing WEF1 allocation in the construction is fractionally Pareto optimal. Second, suppose each agent $i$ has a number $a_i>0$ such that their valuation for any item is $-a_i$, $0$, or $a_i$. Then, for arbitrary entitlements, a weighted maximin share (WMMS) allocation always exists, is computable in polynomial time, and can be chosen to be fractionally Pareto optimal. An exact formula for each WMMS value leads to a polynomial-time flow algorithm. In this class, every WEF1 allocation satisfies a best possible additive WMMS guarantee whose loss depends on the agent's entitlement relative to the largest entitlement. Thus maximum entitlement agents receive exact WMMS and, under equal entitlements, every WEF1 allocation is also MMS-fair. Allowing a second positive magnitude can violate exact WMMS, while unrestricted entitlement ratios rule out any fixed multiplicative WMMS guarantee compatible with WEF1 for chores.

Inoue et al. have introduced the successful derivation game (SDG) on context-free grammars (CFGs), which is a generalization of classic heap-based games including subtraction games and Keyles, and shown that the least upper bound of the Grundy numbers in the SDG on a given CFG G is undecidable in general even when we restrict G to be a linear CFG. This paper shows that for the SDG on a right-linear grammar (RLG), we can construct a DFA for computing the Grundy number of a given position. In other words, for the SDG on an RLG, the set of positions with a given Grundy number c is regular. As a corollary, the least upper bound of the Grundy numbers in the SDG on a given RLG is decidable. We also investigate the complexity of computing the least upper bound of the Grundy numbers in the SDG on a given RLG, and it is shown to be PSPACE-complete.

We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $\chi_\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $\chi_\mathrm{stable}(G)$. This shows that $\chi_\mathrm{stable}(G)$ is well-defined. We also show that $O(t \log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.

In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming “$\mathsf{ETH}$ for $\mathsf{PPAD}$", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first radically uncoupled algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a layered variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.

Fair allocation of scarce, indivisible resources is an important challenge in many societal problems. While there are several formal theories of fairness, no single definition can always be satisfied. As large language models (LLMs) are increasingly used to support decisions and act as agents, they raise new concerns about distributional justice: their judgments are not directly tied to any specific fairness framework and may violate key normative principles. In this work, we introduce a general method for evaluating fairness reasoning in LLMs. We study first-person fairness judgments across a broad set of models and compare them directly with human responses on matched scenarios and elicitation conditions. We find that LLMs tend to prefer stricter fairness constraints than humans, show more self-interested behavior, are sensitive to how information is framed, and are difficult to align with human judgments using fine-tuning with current datasets.

House allocation is a fundamental problem in which each agent is assigned exactly one house. While the classical model assumes that all houses are available before the allocation is computed, many practical settings require decisions to be made as houses become available over time. We introduce the online house allocation problem, where houses arrive sequentially and the algorithm must maintain an allocation without knowledge of future arrivals. Unlike online fair division, the one-house-per-agent constraint makes recourse an inherent part of the problem, as accepting a newly arrived house may require reassigning previously allocated houses. We study online house allocation under subsidy-based fairness, where monetary subsidies eliminate envy among agents. We show that envy-freeability can always be maintained online using bounded recourse and that reassignment chains of length linear in the number of agents are unavoidable in the worst case. In contrast, minimizing the total subsidy is fundamentally harder: no deterministic online algorithm against an adaptive adversary, and no randomized online algorithm against a non-adaptive adversary, admits a bounded competitive ratio, even for two agents and four houses. We complement these impossibilities by showing that exact online subsidy minimization is possible whenever there is at most one extra house beyond the number of agents, and that this guarantee is best possible with respect to the number of extra houses. Finally, we develop learning-augmented algorithms that recover the offline optimum under accurate predictions while providing explicit robustness guarantees when predictions are inaccurate.

In the approval-based committee selection problem, a committee is said to be in the core if no subset of voters has an incentive to deviate by selecting a blocking committee of proportional size, such that every voter in the deviating group strictly prefers the blocking committee. We consider the setting where candidates can be selected fractionally. Under a mild regularity assumption, we show that there always exists a weighting of candidates such that the fractional committee maximizing the candidate-weighted Nash Social Welfare is in the core. We refer to such a solution as being in the Nash core. Additionally, we show that a Nash core solution admits a payment assignment between voters and candidates, where each voter pays a candidate they approve in proportion to the weight. For the discrete setting, where each candidate is either included or excluded from the committee, we prove that every approval-based committee election with at most eight equally weighted voters has a core committee by rounding the fractional Nash core solution. Although the non-emptiness of the core in this setting remains an open question and checking core membership is coNP-hard, we extend the notion of the Nash core to the discrete case, yielding a formulation that is efficiently verifiable and offers a promising path toward establishing core existence in discrete settings. Finally, we test our approach on real voting data using a payment-guided heuristic. We empirically show that the Nash core solution can be efficiently computed through an iterative algorithm in both the fractional and discrete settings.

We seek to understand the effect of adding disruptive highly-capable new technologies to competitions by assessing the addition of Dynamite to Rock-Paper-Scissors. We find that providing a versatile Dynamite move to only one player provides limited value (win probability increases from 50% to 55.5%) and is played rarely. That value decreases further if the game is expanded beyond just the original three moves. We also observe several mechanisms by which prior moves can become strategically unplayable, or obsolete. We hope that this model illustrates some non-intuitive aspects of developing new versatile technologies. We also hope that it illustrates some pitfalls for developers and integrators to avoid in order to create value rather than merely capability.

Uncoupled no-regret dynamics provide a decentralized route to equilibrium, but prior guarantees for individual regret retain a polylogarithmic dependence on the horizon. We remove this dependence for every finite $N$-player normal-form game under full-information feedback. We introduce ECHO-OFTRL: optimistic follow-the-regularized-leader (OFTRL) equipped with an EMA cascade for high-order optimism (ECHO), where EMA denotes exponential moving average. The algorithm is deterministic and fully uncoupled. If $m_{\max}$ denotes the largest action-set size, then, simultaneously for every horizon $T\geq1$, it guarantees that each of the $N$ players in the game incurs regret upper bounded by $O(\textrm{poly}(N, \log m_{\max}))$. Our algorithm leverages a new form of optimism inspired by modern filter design.

We study fair allocation of indivisible goods under additive valuations and matroid constraints. A challenging open question is whether a complete and feasible envy-free up to one good (EF1) allocation exists under every matroid that admits a complete and feasible allocation. The state-of-the-art result by Biswas and Barman [2018] positively resolves this question for partition matroids. Our first result positively resolves it for laminar matroids, which generalize partition matroids, when there are three agents. Our technique reduces this general existence question to finding an EF1 allocation satisfying a mild additional condition under a single finite-sized key laminar matroid, and we establish the required allocation by case analysis. We show that our technique somewhat extends to four agents, reducing the analogous problem to finding EF1 allocations under two finite-sized laminar matroids, although we are unable to establish their existence. We also use recent matroid decomposition results to establish EF1 existence under broader classes of matroids. Specifically, we show that EF1 allocations always exist under transversal matroids whenever a complete allocation is feasible, and obtain existence results for graphic matroids and gammoids under stronger assumptions.

Bidding games are graph games in which a token is placed on a vertex, each player starts with an initial budget, and a simultaneous auction determines which player moves the token; the players' budgets are then updated accordingly. Motivated by scenarios such as resource-allocation systems in which agents receive periodic rewards (e.g., credits, energy) while competing for control, we introduce and study bidding games with rewards, in which, at each vertex, players may receive additional budget, incentivizing desired behaviors. We focus on reachability discrete poorman bidding games with rewards (DPBGr). The main challenge when compared to discrete bidding games without rewards is that the configuration graph is infinite. To this end we introduce a novel technique to eliminate plays with suboptimal infixes. This enables focusing on a finite part of the infinite configuration graph in order to solve the game via approximation to continuous bidding games with overall complexity in EXP. Finally, we discuss a new type of strategy, usable on a subclass of DPBGr, which guarantee a winning strategy for the reachability player. Membership in this subclass is shown to be in NP.

Fair allocation of indivisible goods has largely been studied under the assumption that no prior allocation exists. Motivated by practical settings with pre-existing (and possibly unfair) allocations, we study how to achieve fairness through limited reallocations. Building on recent work on reformability/reallocations, we consider three fairness notions—envy-freeness (EF), envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX)—and optimize the number of goods reallocated. We analyze both the classical and parameterized complexity of these problems, providing a comprehensive analysis across multiple fairness notions.

Test-time reasoning has significantly improved performance in domains ranging from games to language models. However, test-time policy changes with formal guarantees on the performance of the resulting strategy remain a challenge in two-player zero-sum imperfect-information games. Existing solutions are limited to tabular methods or single gradient step updates. In this work, we investigate policy-gradient algorithms as a method for scalable test-time reasoning. We extend the concept of gadget game, tabular technique for test-time search, to the reinforcement learning setting. Unlike prior approaches, we represent the gadget game implicitly by modified sampling and neural policy rather then explicitly by constructing it, thereby removing constraints on subgame size. Furthermore, we formally prove that, unlike prior tabular algorithms, regularized policy-gradient algorithms limit possible strategy degradation caused by test-time reasoning, even without the gadget games. Our evaluation across small- and large-scale games confirms that additional test-time training often substantially improves performance relative to the blueprint strategy.

Auto-bidding is a long-horizon sequential decision problem for maximizing conversion value under budget and key performance indicator (KPI) constraints. Recent work extends this task from bidding alone to joint bidding and pricing, where a policy controls bidding decisions and pricing corrections. Existing methods mainly rely on numerical trajectory modeling, which offers limited support for interpreting campaign context and expressing high-level strategies. Large language models (LLMs) can complement this paradigm with their reasoning capabilities. However, existing language-guided methods have two limitations. First, they condition actions on language strategies without modeling the corresponding state changes, making it difficult to distinguish errors in strategy understanding from errors in action generation. Second, different instructions can produce similar execution effects, leading to imbalanced policy updates across effects. We propose LangBP, a hierarchical framework for language-guided joint bidding and pricing. LangBP's Semantic Decision Transformer (S-DT) predicts target states from the instruction and the trajectory history, then recovers the joint action via inverse dynamics. We further propose Execution-Grouped Policy Optimization (EGPO), which scores candidate effects with a Context–Effect Verifier (CEV) and balances policy updates across effect groups. Experiments on AuctionNet show that LangBP outperforms strong baselines, and online A/B tests further demonstrate business gains in real-world deployment on a large-scale e-commerce platform.

We study the problem of locating a new homogeneous facility under a prelocated facility. Here, a set of $n$ agents is located on a real line or a circle, each of whom has her location as private information, and her cost is the (expected) distance from her location to the nearest facility. Our goal is to design mechanisms which can approximately minimize the maximum cost or the social cost while eliciting agents' private information truthfully (i.e., strategy-proof). Based on real-life scenarios, we consider the problem in two settings: the general setting where each agent can be located at both sides of the prelocated facility, and the special setting where all the agents are located at the same side of the prelocated facility. For agents on a line, in the general setting, we design the best possible deterministic strategy-proof mechanism with $2$-approximation and provide a lower bound of $1.5-\epsilon\textbf{ }(\epsilon>0)$ for any randomized strategy-proof mechanism under the maximum cost objective. For the social cost, we obtain an upper bound of $n$ for deterministic strategy-proof mechanisms and lower bounds of $1.5$ and $1.0425$ for any deterministic strategy-proof mechanism and any randomized strategy-proof mechanism, respectively. In the special setting, we further provide a randomized strategy-proof $5/3$-approximation mechanism for the maximum cost and a deterministic strategy-proof $(n-1)$-approximation mechanism for the social cost. For agents on a circle, we provide a deterministic strategy-proof 2-approximation mechanism under the maximum cost objective.

Envy-freeness up to any good (EFX) and pairwise maximin share (PMMS) are standard local fairness criteria for indivisible goods, whereas maximin share (MMS) is a global benchmark. We determine the exact quantitative relationship between these local fairness notions and the global MMS guarantee under nonnegative additive valuations. We show that the optimal universal factor for both notions is $\rho^{\mathrm{EFX}\to\mathrm{MMS}}=\rho^{\mathrm{PMMS}\to\mathrm{MMS}}=\frac{10}{17}$. We prove the lower bound by a combinatorial charging argument. After an initial reduction, both EFX and PMMS imply the same local condition on every foreign bundle from the perspective of a focal agent: deleting its least valuable good leaves value at most the focal bundle. A three-piece concave weight function translates this local condition into the global $10/17$ guarantee. We then construct an explicit family of complete allocations that are simultaneously PMMS and EFX$_0$, whose MMS ratios converge to $10/17$, showing that both constants are tight even in the presence of zero-valued goods. The argument also gives $\alpha$-EFX $\Rightarrow (10\alpha/17)$-MMS. Finally, we establish an exact correspondence between these fair-division guarantees and scheduling equilibria. For every fixed number of agents $n$, the EFX-to-MMS extremal ratio equals the reciprocal of the pure price of anarchy for selfish identical-machine covering. Similarly, the PMMS-to-MMS ratio equals the reciprocal of a locality gap based on exact pairwise machine repartition. These correspondences explain why the constant $10/17$ governs both problems.

Residual maximin share (RMMS) is the largest share threshold that remains guaranteeable throughout dynamic allocation processes, even after previously allocated, lower-valued bundles are removed from the item pool. For additive valuations, recent density-balance analyses established finite-agent lower bounds comparing RMMS with the classical maximin share (MMS). In this paper, we prove that these finite-agent lower bounds are exact. Specifically, if $d_n$ denotes the largest odd integer at most $n$, the worst-case ratio satisfies $\inf_{M,v:\operatorname{MMS}>0}\frac{\operatorname{RMMS}(M,v,n)}{\operatorname{MMS}(M,v,n)}=\frac{2d_n}{3d_n-1}$. Consequently, the exact additive frontier forms consecutive odd-even plateaus and converges monotonically to $2/3$. We then investigate the combinatorial structure of extremal instances. While naive witnesses require $\Theta(n^2)$ items, we construct an explicit three-valued family achieving the exact boundary with only linear support: $(5n-3)/2$ items for odd $n$ and $(5n-4)/2$ items for even $n$. Its low-valued block supports two exact partitions that simultaneously certify the MMS benchmark and the residual obstruction. By modeling these dual partitions as a bipartite transportation graph, we prove that this block attains the absolute minimum support $q+d-1=3q$. At minimum support, any two-valued filler is uniquely rigid up to relabeling. Finally, we establish structural characterizations of RMMS. A general min–max representation applies to all finite monotone valuations. For integer additive valuations, we prove that a threshold $T$ is residual self-feasible if and only if every subset cut satisfies a packing-covering condition. Because RMMS is pointwise maximal among residual self-feasible shares, these exact constants establish a tight limitation on the fairness guarantees achievable by share-based lone-divider algorithms.

We study complete allocations under nonnegative additive valuations through the positive supports of goods, focusing on the all-good form of envy-freeness up to any good (EFX$_0$). We define rival-injective (RI) endpoint ownership: every good with nonempty positive support is assigned to an agent who values it positively, and every ordered observer–owner pair is used by at most one good. RI ownership is exactly proper list coloring of the graph joining goods whose positive supports overlap in at least two agents. For pair-supported goods with arbitrary exceptional goods, fixing the exceptional owners yields a necessary-and-sufficient pair-capacity criterion and an exact finite-domain owner constraint satisfaction problem (CSP). Every RI allocation in which each agent's own bundle is worth at least her maximum-valued singleton is all-good EFX$_0$. A specified injective choice of maximum-singleton anchors, together with residual support-list degeneracy, constructs such an allocation by reverse greedy coloring in $O(nm^2)$ time. We give two explicit witness families: a nonempty relatively open, 22-dimensional cone on a fixed $4\times10$ support face, and a family for every $n\ge4$ with two universal-support goods and $m=(n-1)(n-2)+2$ goods. Both fail unanchored list degeneracy and are not implied by the explicit pure-multigraph or published high-girth/controlled-multiplicity hypotheses compared here. We also study recognition of the maximum-anchor certificate class, leaving its general complexity unresolved. The exact list-coloring and pair-capacity results concern RI ownership, not general EFX$_0$ existence; unrestricted four-agent, ten-good all-good EFX$_0$ remains unresolved.

We study violation-feedback learning of proportionally representative approval-based committees. In each round, a learner proposes a committee of size $k$. An oracle either accepts the proposal or adversarially selects a representation violation with respect to a single fixed hidden approval profile. We compare full-witness feedback, which reveals the violation level, an omitted candidate, and the affected voter group, with candidate-only feedback, which reveals only that candidate. The target notions are proportional justified representation plus (PJR+) and extended justified representation plus (EJR+). In every setting we study, the number of rejected proposals can be bounded solely in terms of $k$, with no dependence on the numbers of voters and candidates. For PJR+, the optimal deterministic and randomized rejection complexities equal $k$ under both feedback models. For EJR+, the picture is more nuanced. Under full-witness feedback, we prove an $\Omega(k^{3/2})$ deterministic lower bound and give a deterministic polynomial-time algorithm using $O(k^2\log k)$ rejections. Under candidate-only feedback, randomization achieves $O(k^2\log k)$ expected rejections via uniform random deletion, while deterministic exhaustive branching gives a $2^{O(k^2(\log k)^2)}$ rejection bound. Even with full-witness feedback, randomized learners may require $k$ rejections.

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle – a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

We study fair division of indivisible items when agents have arbitrary two-level preferences: the value of each agent for any set of items is Boolean, which need not be monotone or additive. Notably, we do not impose the standard assumption of normalization, i.e., different agents may value the empty set at different Boolean levels. Since the preferences are nonmonotone, envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) each admit several variants, depending on which items are tested for removal and whether they are removed from the envious agent's bundle or the envied agent's bundle. This paper investigates the existence of these variants of EF1 and EFX, on their own and together with economic efficiency, incentive compatibility, feasibility constraints, and lottery-based randomization. Our results highlight that the existence landscape depends crucially on the number of normalized agents, who value the empty bundle at the lower Boolean level. The authors used significant assistance from GPT-5.6-Sol for deriving theoretical results, verified any AI-generated proofs for correctness, and expanded on the exposition and simplified arguments, with the aid of GPT-5.6-Sol and Claude Opus 5.

Recent machine learning research has increasingly focused on equilibrium analysis in non-cooperative games rather than solely on optimal solutions. Many such problems involve shared constraints and can be formulated as Generalized Nash Equilibrium Problems (GNEPs). For strongly monotone games, existing methods compute consensus-based variational GNEs (v-GNEs) by exchanging Lagrange multipliers. We propose a fully distributed continuous-time algorithm for shared linear equality constraints that converges without multiplier exchange and reaches any GNE, reducing communication overhead and improving privacy. Discrete-time schemes are also provided, and the method is validated on a multi-robot placement task.