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arXiv2026

The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections

Core stability gives approval-based committee elections a strong form of coalitional proportionality, but deterministic existence is not known for an arbitrary number of seats. We prove that every finite election with nine seats has a deterministic Hare-quota core committee. Starting from an eight-seat committee that is both a global Proportional Approval Voting (PAV) maximizer and core stable, we suppose that all one-candidate extensions are blocked and extract an inclusion-minimal closed response system. Singleton responses are eliminated by exact equality rigidity. For the remaining responses, rational Farkas certificates bound every nonpositive PAV add-marginal drift below by $-n/189$ and force every successor drift above $n/126$. The uniform response chain is irreducible, so stationarity makes these bounds incompatible. The computer-assisted component comprises an exact pointwise check over 7,356 voter types, 36 one-blocker cells, and a symmetry-complete family of 19 two-blocker motifs. Exact certificate archives and standard-library verifiers support these checks. The theorem settles the nine-seat case but does not decide deterministic core existence for arbitrary committee size.

cs.GT↗

arXiv2026

Sparse Disapproval Guarantees a Nonempty Hare Core

An approval committee is Hare-core stable if no coalition meeting the Hare quota can strictly improve by moving to another candidate set. Whether every approval election has such a committee remains open. We prove nonemptiness when each voter disapproves at most two candidates, with no bounds on the numbers of candidates, seats, or voter types. The result also permits arbitrary positive rational voter weights. Our deterministic rule represents a committee by its missing set. It first maximizes weighted coverage of two-candidate disapproval sets and then maximizes total disapproval incidence. An exact coverage inequality excludes targets one seat below the committee. The incidence objective excludes unanimous equal-size targets, while targets of size at most $k-2$ cannot improve any voter. Two implementation-level independent verifiers audit overlapping finite grids. The symbolic proof, not this bounded enumeration, establishes the theorem's unbounded quantifiers. The argument identifies complement-side coverage as a tractable mechanism for a broad parameter range within a sharply defined preference domain.

cs.GT↗

arXiv2026

Data Market Design through Deep Learning

The data market design problem is a problem in economic theory to find a set of signaling schemes (statistical experiments) to maximize expected revenue to the information seller, where each experiment reveals some of the information known to a seller and has a corresponding price [Bergemann et al., 2018]. Each buyer has their own decision to make in a world environment, and their subjective expected value for the information associated with a particular experiment comes from the improvement in this decision and depends on their prior and value for different outcomes. In a setting with multiple buyers, a buyer's expected value for an experiment may also depend on the information sold to others [Bonatti et al., 2022]. We introduce the application of deep learning for the design of revenue-optimal data markets, looking to expand the frontiers of what can be understood and achieved. Relative to earlier work on deep learning for auction design [Dütting et al., 2023], we must learn signaling schemes rather than allocation rules and handle obedience constraints $-$ these arising from modeling the downstream actions of buyers $-$ in addition to incentive constraints on bids. Our experiments demonstrate that this new deep learning framework can almost precisely replicate all known solutions from theory, expand to more complex settings, and be used to establish the optimality of new designs for data markets and make conjectures in regard to the structure of optimal designs.

cs.GT↗

arXiv2026

A Minimax Perspective on Almost-Stable Matchings

Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either unachievable or comes at the cost of leaving agents unmatched. In these cases, algorithmicists and market designers face a critical question: how should instability be measured and distributed among participants? Existing approaches to "almost-stable" matchings focus on aggregate measures, minimising the number of blocking pairs or the count of agents involved in blocking pairs. However, these objectives can result in concentrated instability on a few agents, raising concerns about fairness and incentives to deviate. We introduce a fairness-oriented approach to approximate stability based on the minimax principle: we seek matchings that minimise the maximum number of blocking pairs any agent is in. Equivalently, we minimise the maximum number of agents that anyone has justified envy towards. This distributional objective protects the worst-off agents from bearing a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees with respect to the distribution of instability prove computationally intractable: we show that it is NP-complete to decide whether a matching exists in which no agent is in more than one blocking pair, even when preference lists are bounded. This result applies to both Stable Roommates and maximum cardinality Stable Marriage. On the positive side, we provide polynomial-time algorithms when agents rank at most two others, and present approximation algorithms and integer programs for general settings. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees on justified envy and computational feasibility in matching market design.

cs.GT↗

arXiv2026

How Wasteful is Signaling?

Signaling is wasteful. But how wasteful? We study the fraction of surplus dissipated in a separating equilibrium. For isoelastic environments, this waste ratio has a simple formula: $β/(β+σ)$, where $β$ is the benefit elasticity (reward to higher perception) and $σ$ is the elasticity of higher types' relative cost advantage. The ratio is constant across types and is independent of other parameters, including convexity of cost in the signal. We show that the directional effects of $β$ and $σ$ on waste extend to non-isoelastic environments. In an application to signaling tournaments, more competitors or fewer prizes increase waste, with full dissipation in large tournaments.

econ.GN↗

arXiv2026

Near-Feasible Stable Matchings: Incentives and Optimality

Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works leave open deviation incentive issues that arise when the centralised authority modifies agents' capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.

cs.GT↗

arXiv2026

Towards Affordable Energy: A Gymnasium Environment for Electric Utility Demand-Response Programs

Extreme weather and volatile wholesale electricity markets expose residential consumers to catastrophic financial risks, yet demand response at the distribution level remains an underutilized tool for grid flexibility and energy affordability. While a demand-response program can shield consumers by issuing financial credits during high-price periods, optimizing this sequential decision-making process presents a unique challenge for reinforcement learning despite the plentiful offline historical smart meter and wholesale pricing data available publicly. Offline historical data fails to capture the dynamic, interactive feedback loop between an electric utility's pricing signals and customer acceptance and adaptation to a demand-response program. To address this, we introduce DR-Gym, an open-source, online Gymnasium-compatible environment designed to train and evaluate demand-response from the electric utility's perspective. Unlike existing device-level energy simulators, our environment focuses on the market-level electric utility setting and provides a rich observational space relevant to the electric utility. The simulator additionally features a regime-switching wholesale price model calibrated to real-world extreme events, alongside physics-based building demand profiles. For our learning signal, we use a configurable, multi-objective reward function for specifying diverse learning objectives. We demonstrate through baseline strategies and data snapshots the capability of our simulator to create realistic and learnable environments.

cs.AI↗

arXiv2026

Hospitals/Residents with Inseparable Couples: Finding a Coalition-Stable Assignment Is NP-Hard

In recent work on course allocation, Rodríguez and Manlove consider the complexity of finding a stable assignment under four notions of stability, including two coalitional notions. In one case, which they call pair-size stability, they show that a stable assignment always exists and they provide a polynomial-time algorithm to find one. In a second case, called pair stability, they observe that an earlier NP-hardness result of McDermid and Manlove holds for a special case of course allocation called Hospitals/Residents with Sizes ($\mbox{HRS}$). In a third case, called first-coalition stability, they use a reduction from $\mbox{HRS}$ to show it is NP-hard to find a stable assignment. They leave open the complexity of finding a stable assignment under so-called coalition stability. Building on ideas from McDermid and Manlove, we resolve the open problem of Rodríguez and Manlove by showing that it is NP-hard to find a coalition-stable assignment for $\mbox{HRS}$. Indeed, our proof shows that the problem remains NP-hard when the hospital capacities and resident sizes are at most two. Accordingly, our NP-hardness result applies to the special case of $\mbox{HRS}$ known as Hospitals/Residents with Inseparable Couples ($\mbox{HRIC}$). Finally, we introduce a novel and natural notion of coalitional stability for both $\mbox{HRS}$ and course allocation, and we show that our NP-hardness result extends to this notion, which we call unitwise-coalition stability.

cs.GT↗

arXiv2026

The Complexity of Justified Representation with Additive Utilities

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.

cs.GT↗

arXiv2026

Fair Stable Matching: A Nash Social Welfare Approach

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 \emph{Nash social welfare} (NSW) as a fairness objective in the classic \emph{stable marriage problem}. We develop \texttt{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 \texttt{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 \texttt{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.

cs.GT↗

arXiv2026

Almost Envy-Freeness for Additive Mixed Manna with Entitlements: Deterministic and Randomized Guarantees

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.

cs.GT↗

arXiv2026

Approximately Efficient Multidimensional Bilateral Trade

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.

cs.GT↗

arXiv2026

A Bayesian Correlated Equilibrium for Early Insider-Threat Detection

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.

cs.CR↗

arXiv2026

Turn-Based Combat Arena: A New Framework for Multiagent Training and Game Balancing

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.

cs.GT↗

arXiv2026

Structural and Algorithmic Results for Stable Cycles and Partitions in the Roommates Problem

In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable partition always does and provides a succinct certificate for the unsolvability of an instance. Furthermore, apart from being a useful structural tool to study the problem, every stable partition corresponds to a stable half-matching, which has applications, for example, in sports scheduling and time-sharing. We establish new structural results for stable partitions and show how to enumerate all stable partitions and the cycles included in such structures efficiently. We also adapt optimality criteria from stable matchings to stable partitions and give complexity and approximability results for the problems of computing such "fair" and "optimal" stable partitions. Through this research, we contribute to a deeper understanding of stable partitions from a combinatorial point of view, as well as the computational complexity of computing "fair" or "optimal" stable half-matchings in practice, closing the gap between integral and fractional stable matchings and paving the way for further applications of stable partitions to unsolvable instances and computationally hard stable matching problems.

cs.DS↗

arXiv2026

Unsolvability and Beyond in Many-To-Many Non-Bipartite Stable Matching

We study the Stable Fixtures problem, a many-to-many generalisation of the classical non-bipartite Stable Roommates matching problem. Building on the foundational work of Tan on stable partitions, we extend his results to this significantly more general setting and develop a rich framework for understanding stable structures. Our main contribution, the notion of a generalised stable partition (GSP), not only characterises the solution space but also serves as a versatile tool for ordinal preference systems with capacity constraints. We show that a GSP can be computed efficiently and can provide an elegant representation of key aspects of a preference system. Leveraging a connection to stable half-matchings, we also establish an analogous Rural Hospitals Theorem for stable half-matchings and GSPs, and connect our results to recent work on near-feasible matchings, providing a simpler algorithm and tighter analysis. Our work also addresses the computational challenges of finding optimal stable half-matchings and GSPs, presenting a flexible integer linear programming model for various objectives. Beyond theoretical insights, we conduct the first empirical analysis of random Stable Fixtures instances. Our work unifies and extends classical and recent perspectives on stability in non-bipartite stable matching and establishes new tools and techniques for stable matchings and their applications.

cs.DS↗

arXiv2026

Bridging the Gap Between Stable Marriage and Stable Roommates: A Parameterized Algorithm for Optimal Stable Matchings

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable: one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an optimal stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding an optimal stable matching in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the minimum crossing distance. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find an optimal stable matching efficiently. More generally, we show that when an SR instance has minimum crossing distance $k$, an optimal stable matching can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed-parameter tractable (FPT) with respect to minimum crossing distance.

cs.DS↗

arXiv2026

Security Games on Series-Parallel Attack Graphs with Adaptive Attackers

We study security games on attack graphs, where an adaptive attacker seeks to reach a target by sequentially attempting stochastic controls along the current attack frontier, while a defender allocates limited resources across controls to delay compromise. The attacker may choose among exponentially many attack routes and freely pivot between them as successes and failures are observed, yielding an exponentially large space of contingent attack policies. For any fixed defender allocation, we show that an optimal attacker policy on a two-terminal series-parallel attack graph is an index policy: at each step, the attacker selects an available control with the largest value of an extension of the classical Gittins index. The indices and the resulting attacker best response can be computed in polynomial time, without explicitly enumerating attack paths or contingent policies. To the best of our knowledge, this is the first optimal index characterization for adaptive attackers in security games on general series-parallel attack graphs. We further develop efficient algorithms for computing the attacker's exact utility and an exact defender subgradient, enabling deterministic first-order optimization of defensive resource allocations without sampling attack trajectories. Our framework strictly generalizes prior approaches restricted to parallel chains and out-trees, while exploiting the compositional structure of series-parallel graphs to support interpretable attacker policies and parallel computation across independent subgraphs. Experiments demonstrate that the resulting methods scale substantially better than naive explicit-state approaches while producing effective defensive allocations.

cs.GT↗