Equilibrium

We propose a game-theoretic framework for adaptive multi-agent intelligent systems. Unlike classical game theory, which often treats strategies as primitive objects chosen by perfectly rational agents, the proposed framework provides a mathematical foundation for studying equilibrium in NeuroAI and can be viewed as an extension of game theory under relaxed assumptions, including partial observability, bounded computation, and uncertainty. At its core, Multilevel Interactive Equilibrium (MIE) generalizes the classical Nash equilibrium to intelligent systems with internal computation. Rather than being defined solely at the level of observable behavior, equilibrium emerges when neural learning dynamics, cognitive representations, and behavioral strategies mutually stabilize between interacting agents. This framework applies uniformly to interactions between two biological brains, two artificial agents, or hybrid human-AI systems. We discuss applications of multilevel game theory to human-autonomous vehicle driving, human-machine interaction, human-large language model (LLM) interaction, and computational psychiatry. We also outline experimental strategies and computational methods for estimating MIE and discuss challenges and prospects for future research.

In this paper we study team-symmetric games with $m\ge 2$ teams. Players within a team have symmetric identity and have a common payoff function. We show that team-symmetric games always have a team-symmetric Nash equilibrium. We develop and solve a linear complementarity problem of team-symmetric Nash equilibria. We propose an actor-critic based multi-agent reinforcement learning algorithm for team-symmetric games. Through simulations, we show that this multi-agent reinforcement learning algorithm performs much better than many existing algorithms.

Large language models can score well on named game-theory benchmarks while failing on the same strategic computation once semantic cues are removed. We show this gap with procedurally generated zero-sum matrix games: a model that recognizes familiar games drops to 34%, 18%, and 2% success on anonymous $2{\times}2$, $3{\times}3$, and $5{\times}5$ payoff matrices. The benchmark separates semantic recall, learned approximate Nash computation, and an output-interface bottleneck that limits scale. Training only on $2{\times}2$ and $3{\times}3$ games, supervised fine-tuning raises unseen $5{\times}5$--$7{\times}7$ success from 2% to 61%, while exploitability-reward training averages 37% with high seed variance. We prove that the exploitability residual is $2$-Lipschitz in payoff perturbations, unlike discontinuous vertex-returning LP equilibrium selectors, explaining why residual training can transfer under payoff shifts even when formatting instability limits mean performance. A dominated-action padding experiment provides causal evidence: trained models solve $3{\times}3$ games embedded in much larger matrices, while random-padded controls fail and dense $12{\times}12$ games remain near failure. Procedural evaluation is therefore necessary for measuring strategic reasoning, and residual rewards expose a real but format-limited route to approximate equilibrium computation.

Reachability games are two-player games played on a graph, where the objective of $\texttt{REACH}$ player is to reach the target set whereas the objective of $\texttt{SAFE}$ player is to stay away from the target set. Reachability games have important applications in artificial intelligence and reactive synthesis, and many of these applications give rise to infinite-state reachability games. In this paper, we study turn-based reachability games on infinite-state graphs defined over valuations of a finite set of real variables. We consider the problem of determining the existence of and computing a winning strategy for $\texttt{REACH}$ player. Our contributions are twofold. First, we propose ranking certificates for reachability games, a sound and complete proof rule for proving that $\texttt{REACH}$ player has a winning strategy from the specified initial state. Second, we consider polynomial reachability games, where transitions and objectives are described by polynomial constraints over real variables, and propose a fully automated algorithm for computing a winning strategy for $\texttt{REACH}$ player together with a formal correctness witness in the form of a ranking certificate. The algorithm is sound, semi-complete, and runs in sub-exponential time. Our experiments demonstrate the ability of our method to solve challenging examples from the literature that were out of the reach of existing methods. Specifically, for the classical Cinderella-Stepmother game, we are able to compute an optimal winning strategy for an arbitrary precision parameter for the first time.

Accurate stock price forecasting has consistently remained a pivotal yet challenging FinTech task that underpins quantitative trading and investment decision making. Recent efforts have been dedicated to modeling various complex relationships among stocks in the stock market toward more reliable stock price forecasting.These methods depend heavily on strong static prior assumptions by modeling either temporal dependencies within individual stocks or spatial dependencies across different stocks based on predefined structures, while the complex market dynamics that drive stock price movements remain unexplored. To alleviate this issue, we propose a novel game-theoretic modeling method that captures heterogeneous investor interactions for stock price forecasting. The core idea is to embed game-theoretic mechanisms into the heterogeneous graph structure to finely model the dynamic strategic interactions among heterogeneous investors with respect to target stocks. Additionally, temporal positional encoding is adopted to reflect the differentiated influences of each game event at different time steps within the time window on future stock price movements. Leveraging heterogeneous graph networks, we proxy the intricate dynamics of the stock market through investor games and enable real-time information propagation and node updates among all nodes. Extensive experiments conducted on two real-world benchmark dataset demonstrate that our method effectively outperforms state-of-the-art stock price forecasting methods.

Last-iterate convergence of learning dynamics in games has attracted significant recent attention. In two-player zero-sum games with bandit feedback, where only the loss of the selected action pair is observed, Fiegel et al. (2025) show a separation between average-iterate and last-iterate convergence in duality gap: while the optimal t^(-1/2) rate after t rounds is achievable for the former via standard no-regret algorithms, the latter cannot converge faster than t^(-1/3) in expectation or t^(-1/4) with high probability. However, in many practical settings, such as preference learning, the players observe not only their loss but also the opponent's action. This raises a natural question: can such additional information enable faster last-iterate convergence? We answer this question affirmatively, showing that t^(-1/2) last-iterate convergence is achievable with high probability in this setting, via an efficient algorithm that updates its strategy infrequently by solving an estimated log-barrier-regularized game. We identify fundamental obstacles preventing standard analysis for multi-armed bandits, the single-player case, from generalizing to games, and develop a novel analysis to overcome them. Experiments confirm that our algorithm indeed converges faster than naive baselines and prior methods that do not exploit opponent-action feedback. Finally, we note that our results also improve those for dueling bandits, a special case with skew-symmetric game matrices.

Poker is an imperfect information game that has served as a long-standing benchmark for decision-making under uncertainty. To maximize utility beyond the Nash equilibrium, an agent can deviate from Nash-equilibrium policies to exploit suboptimal play. We introduce AlphaExploitem, which extends the competitive RL poker agent AlphaHoldem by using a hierarchical transformer encoder that enables reasoning over previously played hands and modifying the training procedure with the inclusion of a diverse pool of exploitable opponents to facilitate learning to exploit. We train and evaluate AlphaExploitem on two standard benchmarks for imperfect-information games. Empirically, AlphaExploitem successfully exploits weak play by both in- and out-of-distribution opponents, without losing performance against NE opponents.

Agents built on large language models (LLMs) rely on a range of reliability techniques, including retry, majority voting, and self-consistency, that have been developed in parallel rather than within a common analytical framework. We observe that an LLM sampled at temperature $T$ is a discrete stochastic channel $p(y \mid x)$ in the sense of Shannon's coding theory, and use this identity as the entry point for such a framework grounded in communication theory. Each of these techniques is a special case of one of six classical reliability operators: diversity combining, hybrid retransmission, iterative generator-critic decoding, rateless sampling, structured redundant verification, and difficulty-adaptive routing. Within the framework we give two closed-form results: a noise-variance threshold above which uniform averaging beats quality-weighted averaging, and a contractivity criterion for generator-critic refinement, consistent with a contractive-to-divergent transition we observe between 3B- and 14B-parameter models. We further introduce a cost-aware semantic-nearest-neighbor router whose single Lagrangian knob traverses the quality-cost frontier without retraining. Across six channel configurations spanning local and cloud models on 69 hard tasks, no fixed model-technique-budget choice dominates, motivating per-task allocation. On a 300-item hard split of MMLU, GSM8K, and HumanEval, our router occupies the full empirical Pareto frontier: at matched quality, its normalized cost is ${\approx}56$% lower than the strongest fixed technique; at matched normalized cost, it improves quality by ${\approx}7$% ($26$% over single-shot decoding). These results argue for consolidating these reliability techniques into a single tunable layer informed by channel coding.

Activated PI3K8 Syndrome (APDS) is a rare genetic immune disorder caused by variants in PIK3CD or PIK3R1, with highly heterogeneous symptoms that often delay diagnosis. Early recognition is hampered by overlapping clinical presentations and limited clinician awareness, motivating systematic, data-driven approaches to detect APDS-associated phenotypic patterns in routine electronic health records. Traditional linear scoring systems cannot capture complex symptom interactions, while deep learning models, though expressive, often lack interpretability. To bridge this gap, we propose Shapley regression, a novel game-theoretic model replacing the linear predictor with a k-additive cooperative game, explicitly modeling co-occurrence of symptoms while maintaining the transparency and convexity of logistic regression. We carry out an empirical study of our lightweight method on eight public biomedical datasets, showing that a 2-additive model with $l_{2}$ regularization achieves an optimal trade-off between predictive power and noise robustness. We also apply it to a real-world cohort of 222 patients, on which Shapley regression accurately distinguished APDS cases from matched controls, confirming and validating phenotypes known to be associated with APDS, and facilitating the exploration of pairwise interactions between symptoms, validated by clinical experts.

Self-play red team is an established approach to improving AI safety in which different instances of the same model play attacker and defender roles in a zero-sum game, i.e., where the attacker tries to jailbreak the defender; if self-play converges to a Nash equilibrium, the model is guaranteed to respond safely within the settings of the game. Although the parameter sharing enforced by the use of the same model for the two roles improves stability and performance, it introduces fundamental theoretical and architectural limitations. We show that the set of Nash equilibria that can be reached corresponds to a broad class of behaviours that includes trivial always refuse strategies and oracle-like defenders, thus limiting practical applicability. We then show that when attacker and defender share and update the same base model, the dynamics collapse to self-consistency, so that attacks do not enforce adversarial pressure on the defender. In response, we propose Anchored Bipolicy Self-Play, which trains distinct role-specific LoRA adapters on top of a frozen base model, thereby maintaining stable optimisation while preserving adversarial pressure through explicit role separation. In relation to standard self-play, we show up to 100x greater parameter efficiency than finetuning and consistent improvements in safety compared to self-play fine-tuned models. We evaluate on Qwen2.5-{3B, 7B,14B}-IT models across widely used safety benchmarks, showing improved robustness without loss of reasoning ability. Cross-play experiments further show that our attacker and defender models are superior to self-play in terms of adversarial defence and safety.

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.

Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary, discontinuous, and harder to learn than the underlying solution set. We study bifurcation models, a weight-tied dynamical view in which different initializations can converge to different stable equilibria, so the model represents an attractor landscape rather than one chosen branch. We prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics and that the induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular. Experiments on frustrated Ising models show that such dynamics can discover multiple valid equilibria without branch labels and outperform single-branch supervision. Allen--Cahn experiments further show that diversity is not automatic: it can be encouraged explicitly, but with an accuracy--diversity tradeoff.

Current research applying N-level Stackelberg Game to multi-agent systems often uses the default decision order of agents provided by the environment. However, this raises the question: does the order of agents necessarily affect the final equilibrium point of the game? To address this, we formally analyze the N-level Stackelberg Game, where changing the order in which agents make decisions typically leads to an overdetermined system. As a result, the equilibrium point shifts unless special structural conditions are satisfied. Based on this analysis, we propose the Hierarchical Priority Adjustment (HPA) method, which adjusts and selects the agents' decision order. At the upper level, an upper policy dynamically selects the optimal decision order of agents based on the current game state. At the lower level, agents execute strategies in the Spatio-Temporal Sequential Markov Game (STMG) according to the selected order. To coordinate learning across time scales, we employ a slow-fast update scheme with shared intrinsic rewards derived from the advantage function of the upper policy. Experimental results on high-precision control tasks, including multi-agent MuJoCo, show that HPA outperforms benchmark algorithms and robustly adapts to changing environments. These results highlight the crucial role of optimizing the agents' decision order in N-level Stackelberg Game.

We study multi-objective multi-agent multi-armed bandits (MO-MA-MAB) under stochastic rewards, where agents observe heterogeneous reward vectors and communicate over time-varying graphs. We formulate this emerging problem setting to address \emph{efficient learning}, measured by Pareto regret, and incorporate \emph{fair learning} as an additional goal, captured via social welfare. To measure efficiency, we formulate Pareto regret and develop \textsc{Pareto UCB1 Gossip}, whose novel exploration radius explicitly separates statistical uncertainty in Pareto-based inference from consensus error. To express the fairness constraint, we formulate a Nash Social Welfare objective over preference-scalarized rewards and propose \textsc{Simulated NSW UCB Gossip}, which integrates preference-based reward simulation, gossip-based utility estimation, and UCB-style exploration. We prove that \textsc{Pareto UCB1 Gossip} achieves $\mathcal{O}(\log T)$ regret and an instance-independent rate of $\mathcal{O}(\sqrt{T})$, while \textsc{Simulated NSW UCB Gossip} achieves an instance-independent regret bound of $\mathcal{O}(T^{3/4})$. This separation reveals the cost of imposing the fairness constraint to our efficiency objective: fairness limits information aggregation and slows convergence. Experiments show that our methods consistently outperform baselines, improving performance by approximately $100\%$ and $50\%$ in the efficiency and fairness settings, respectively.

Large language models (LLMs) are increasingly used to provide instructions to many agents who interact with one another. Such shared reliance couples agents who appear to act independently: they may in fact be guided by a common model. This coupling can change the prospects for cooperation among agents with misaligned incentives. We study settings in which multiple LLMs each advise a population of clients who participate in instances of an underlying game, creating strategic interaction at the level of the LLMs themselves. This induces a meta-game among the LLMs, mediated through clients. We first analyze the one-shot setting, where shared instructions can change equilibrium behavior only when an LLM may influence more than one role in the same interaction; in such cases, cooperation may emerge, and the effect of client share can be beneficial, harmful, or non-monotone, depending on the base game. Our main result concerns the repeated setting. We prove a folk theorem for LLMs: despite indirect observation and the clients' inability to identify which LLM advised their opponents, all feasible and individually rational outcomes can be sustained as $\varepsilon$-equilibria. The result does not follow from the standard folk theorem and requires new proof techniques. Together, these results show that shared LLM guidance can sustain cooperation among populations of agents even when the underlying incentives are misaligned.

We introduce the metagame, a conceptual framework for quantifying second-order interaction effects of model explanations. For any first-order attribution $φ(f)$ explaining a model $f$, we measure the directional influence of feature $j$ on the attribution of feature $i$, denoted as meta-attribution $\varphi_{j \to i}(f)$, by treating the attribution method itself as a cooperative game and computing its Shapley value. Theoretically, we prove that attributions hierarchically decompose into meta-attributions, and establish these as directional extensions of existing interaction indices. Empirically, we demonstrate that the metagame delivers insights across diverse interpretability applications: (i) quantifying token interactions in instruction-tuned language models, (ii) explaining cross-modal similarity in vision-language encoders, and (iii) interpreting text-to-image concepts in multimodal diffusion transformers.

A natural intuition about the economics of AI agents is that, because agents can be replicated at very low marginal cost, agent labor may be supplied highly elastically, placing downward pressure on cognitive-labor wages when it closely substitutes for human labor. We argue this framing is wrong in mechanism but partially correct in conclusion, and that the correction matters for both theory and policy. \textbf{Agents are not labor; they are a production technology that converts compute capital $K_c$ into effective units of cognitive labor $L_A$.} Once this is recognized, the elastic-supply margin that anchors the equilibrium wage migrates from the labor market to the compute capital market. Building on the classic factor-pricing framework \citep{mankiw2020}, we derive a \emph{Compute-Anchored Wage} (CAW) bound stating that, on tasks where human and agent-produced cognitive labor are substitutes, the competitive human wage is bounded above by $λ\cdot k \cdot r_c$, where $r_c$ is the rental rate of compute capital, $k$ is the compute intensity of one effective agent-produced cognitive labor unit, and $λ$ is the relative human-to-agent productivity. We generalize the result through constant elasticity of substitution (CES) aggregation, separate substitutable from complementary tasks, and discuss factor-share consequences. The conclusion is concise: \emph{the price-setter for cognitive labor is no longer the labor market.}

In the modern age of large-scale AI, federated learning has become an increasingly important tool for training large populations of AI agents; however, its computational and communication costs can rapidly fail to scale with the number of agents. This is precisely where decentralized agentic strategies shine: each agent acts autonomously, using only its own state together with a minimal summary of the ensemble, namely the mean-field. We derive the unique optimal decentralized policy in closed form. Optimality is characterized through a worst-client/minimax criterion: minimizing the under-performer regret, namely the maximal online cost incurred by the weakest agent in the ensemble. We further prove that the resulting decentralized policy asymptotically converges, in the large-population limit, to the Nash-optimal centralized policy, whose direct computation is not scalable. We use an online weighting mechanism to optimize the server-computed mixture of client predictions, thereby improving the mean prediction in addition to the previously optimized weakest-client prediction. Numerical experiments verify our theoretical guarantees and demonstrate that our decentralized policy typically outperforms natural greedy decentralized baselines.

As Large Language Models (LLMs) evolve into interactive agents, understanding their behavioral alignment within human social dynamics becomes essential. While behavioral game theory offers a framework to study these interactions, previous work has predominantly relied on outcome-based metrics such as average scores. This focus overlooks the mechanisms that facilitate sustainable cooperation, as identical scores can be derived from vastly different strategies. To bridge this gap, we introduce SODE (Social Dynamics Evaluation), a framework that evaluates LLM agents across three evolutionary dimensions: Direct Reciprocity for strategy adaptation, Indirect Reciprocity for reputation sensitivity, and Group Dynamics for cooperative resilience. Applying SODE reveals systematic divergences: instruction-tuned models often exhibit "passive compliance" that renders them vulnerable to exploitation, while reasoning models prioritize short-horizon optimization, destabilizing long-term cooperation. Notably, we demonstrate that a "long-horizon framing" can unlock reciprocal capabilities in reasoning models. Thus, SODE offers a systematic, mechanism-grounded benchmark for aligning AI agents with complex human social dynamics.

We consider the extensive-form bandit problem, where on each trial the learner (a user coordinated by a server) plays an extensive-form game against an oblivious adversary, observing the information sets it finds itself in as well as the resulting payoff/loss. We give an algorithm for this problem that satisfies $ε$-local differential privacy and attains a regret of $\tilde{O}(\sqrt{A\ln(S)T}/ε)$, where $A$ is the total number of actions that the learner can possibly take, $S$ is the number of the learner's possible reduced strategies, and $T$ is the number of trials. On each trial, the time complexity of our algorithm is, up to a factor logarithmic in the maximum number of actions at an infoset, equal to the time required for the server to transmit the reduced strategy to the user. We note that local differential privacy is the strongest version of differential privacy and, to the best of our knowledge, this is the first work to study differential privacy of any form in the extensive-form bandit problem.

This article introduces S-AI-Recursive, a bio-inspired Sparse Artificial Intelligence architecture in which reasoning is operationalized as a hormonal closed-loop iteration rather than a single feed-forward pass. Building upon the S-AI foundational framework [1], the hormonal-probabilistic unification doctrine [2], and the formal mathematical methodology established in S-AI-IoT [3], the present work formalizes the Recursive Reasoning Cycle (RRC) as a dynamical system governed by two novel hormones: Clarifine, a convergence signal, and Confusionin, an uncertainty detector, whose antagonistic regulation drives iterative state refinement toward a stable cognitive equilibrium. The complete mathematical framework is developed, including recursive state dynamics, Lyapunov stability proof, entropic contraction theorem, hormonal stopping criterion with finite-time termination guarantee, Euler-Maruyama discretization with projection, primal-dual agent selection under iteration budget, and recursive engram memory with warm-start acceleration. Experimental validation on the SAI-UT+ testbench demonstrates that S-AI-Recursive achieves competitive reasoning performance on abstract and symbolic benchmarks with fewer than ten million parameters, confirming the central principle of temporal parsimony: iterative cognitive depth substitutes for architectural width.

Strategic classification studies the problem where self-interested individuals or agents manipulate their response to obtain favorable decision outcomes made by classifiers, typically turning to dishonest actions when they are less costly than genuine efforts. Prior works have demonstrated a fundamental inability to get out of this conundrum by only focusing on the design of a classifier. We note that prior work also heavily focuses on either one-shot settings or repeated interaction with the same classifier. Real-world decision making is often multi-stage, involving a sequence of potentially different classifiers as an agent progresses. This paper introduces a sequential, stochastic, multi-stage model of strategic classification, by capturing how agents adapt their behavior, through improvement actions (enhancing both observable features and true attributes) and gaming actions (enhancing only observable features), over multiple levels of classification with increasing difficulty as well as reward. For each level, we adopt a selective classifier that can abstain from making a prediction at low confidence. Consequently, a positive (resp. negative) outcome leads to promotion (resp. demotion) of the agent to the next higher (resp. lower) level, while abstention keeps the agent at the same level. We fully characterize the agent's optimal instantaneous action under selective classifiers and compare the long-term properties and utility of the agent repeatedly following an optimal myopic policy of either no-improvement (never choose the improvement action) or no-gaming (never choose the gaming action). We further examine design principles over the sequence of classifiers that yield higher long-term utility for the latter policy, thereby effectively incentivizing genuine effort in the long run.

This paper asks whether large language models (LLMs) can be used to study the strategic foundations of conflict and cooperation. I introduce LLMs as experimental subjects in a repeated security dilemma and evaluate whether they reproduce canonical mechanisms from international relations theory. The baseline game is extended along three theoretically central dimensions: multipolarity, finite time horizons, and the availability of communication. Across multiple models, the results exhibit systematic and consistent patterns: multipolarity increases the likelihood of conflict, finite horizons induce universal unraveling consistent with backward-induction logic, and communication reduces conflict by enabling signaling and reciprocity. Beyond observed behavior, the design provides access to agents' private reasoning and public messages, allowing choices to be linked to underlying strategic logics such as preemption, cooperation under uncertainty, and trust-building. The contribution is primarily methodological. LLM-based experiments offer a scalable, transparent, and replicable approach to probing theoretical mechanisms.

Many physical AI tasks require sequential implicit computation: at each step, boundary controls are applied, and the resulting configuration is obtained by solving an equilibrium problem. This setting arises naturally in deformable object manipulation, where even bending a deformable linear object (DLO) to a target shape can be nonlinear and multistable: identical boundary conditions may produce different configurations depending on actuation history. Unlike explicit transition models, the control-to-configuration relation is implicit and history-dependent, making long-horizon learning and control brittle; backpropagating through iterative solves is also memory- and compute-intensive. We propose Neural Control, a boundary-control framework that propagates gradients through branch-dependent sequences of equilibrium solves rather than a single fixed point. Neural Control computes trajectory-dependent proxy gradients by differentiating equilibrium conditions with an adjoint formulation, avoiding solver unrolling while keeping forward rollouts on converged equilibria. Combined with receding-horizon continuation, Neural Control re-anchors optimization to realized equilibria and mitigates basin switching. We validate Neural Control on simulated and real DLO manipulation, compare against SPSA and iCEM, and demonstrate applicability to a learned DEQ-style implicit equilibrium model.

This paper uses game theory to argue that, contrary to the prevailing view, a moratorium on Artificial Superintelligence (ASI) can be in a state's self-interest. By formalizing trategic interactions between geopolitical superpowers, we model the trade-off between the benefits of technological supremacy and the catastrophic risks of uncontrolled ASI. The analysis reveals that as the perceived cost of loss of control increases sufficiently relative to other parameters, it becomes in each state's self-interest to impose a moratorium. We further provide empirical evidence suggesting that the global perception of ASI risk is rising, making a stable, rational moratorium increasingly plausible in the current geopolitical landscape.

As large language models grow increasingly capable, concerns about their safe deployment have intensified. While numerous alignment strategies aim to restrict harmful behavior, these defenses can still be circumvented through carefully designed adversarial prompts. In this work, we introduce a theoretical framework that formalizes a game between an attacker and a defender. Within this framework, we design a theoretical best-response attack strategy and show that it is closely related to many existing adversarial prompting methods. We further analyze the resulting game, characterize its equilibria, and reveal inherent advantages for the attacker. Drawing on our theoretical analysis, we also derive a provably optimal defense strategy. Empirically, we evaluate a practical instantiation of the theoretically optimal attack and observe stronger performance relative to existing adversarial prompting approaches in diverse settings encompassing different LLMs and benchmarks.

The advancement of generalized deepfake disruption is constrained by the interruption imbalance, a fundamental bottleneck inherent to the generation of universal perturbations. We reveal that conventional static gradient normalization fundamentally struggles to resolve architectural conflicts, causing the optimization to bias towards susceptible models while neglecting resistant ones. We argue that achieving high and uniform effectiveness requires resolving this imbalance by reaching an adaptive equilibrium. We propose the Adaptive Equilibrium Framework (AEF), which employs a dynamic weighting mechanism that utilizes real-time loss feedback to adaptively assign greater interruption weights to the most resistant models. This approach shifts the optimization from an average-case problem to finding a dynamic balance, driving the perturbation to a uniformly effective equilibrium state. Comprehensive experiments validate that AEF achieves a more balanced interruption performance, maintaining a consistent interruption success rate across the evaluated diverse architectures.

Offline multi-agent reinforcement learning in general-sum settings is challenged by the distribution shift between logged datasets and target equilibrium policies. While standard methods rely on manual pessimistic penalties, we demonstrate that KL regularization suffices to stabilize learning and achieve equilibrium recovery. We propose General-sum Anchored Nash Equilibrium (GANE), which recovers regularized Nash equilibria at an accelerated statistical rate of $\widetilde{O}(1/n)$. For computational tractability, we develop General-sum Anchored Mirror Descent (GAMD), an iterative algorithm converging to a Coarse Correlated Equilibrium at the standard rate of $\widetilde{O}(1/\sqrt{n}+1/T)$. These results establish KL regularization as a standalone mechanism for pessimism-free offline learning that achieves equivalent or accelerated rates in multi-player general-sum games.

Most familiar equilibrium concepts, such as Nash and correlated equilibrium, guarantee only that no single player can improve their utility by deviating unilaterally. They offer no guarantees against profitable coordinated deviations by coalitions. Although the literature proposes solution concepts that provide stability against multilateral deviations (\emph{e.g.}, strong Nash and coalition-proof equilibrium), these generally fail to exist. In this paper, we study an alternative solution concept that minimizes coalitional deviation incentives, rather than requiring them to vanish, and is therefore guaranteed to exist. Specifically, we focus on minimizing the average gain of a deviating coalition, and extend the framework to weighted-average and maximum-within-coalition gains. In contrast, the minimum-gain analogue is shown to be computationally intractable. For the average-gain and maximum-gain objectives, we prove a lower bound on the complexity of computing such an equilibrium and present an algorithm that matches this bound. Finally, we use our framework to solve the \emph{Exploitability Welfare Frontier} (EWF), the maximum attainable social welfare subject to a given exploitability (the maximum gain over all unilateral deviations).

Collective intelligence emerges across biological, physical, and artificial systems without central coordination, yet a unifying principle governing such behaviour remains elusive. The Free Energy Principle explains how individual agents adapt through variational inference, while game theory formalises strategic interactions. Here we introduce the Game-Theoretic Free Energy Principle, a unified framework showing that multi-agent systems performing local free-energy minimisation implicitly implement a stochastic game. We prove that, under bounded rationality and local information constraints, stationary points of collective free energy correspond to approximate Nash equilibria of an induced game. Conversely, a broad class of cooperative games admits a variational representation in which equilibria arise as Gibbs distributions over coalitions, establishing a bridge between Bayesian inference and strategic interaction. To characterise higher-order effects, we introduce a free-energy formulation of the Harsanyi dividend, isolating irreducible multi-agent synergy. This yields a predictive theory of cooperation, including a falsifiable non-monotonic relationship between sensory precision and agent influence. We validate this prediction across neural, biological, and artificial multi-agent systems. These results identify a common variational principle underlying inference, thermodynamics, and game-theoretic equilibrium.

The volume of scientific manuscripts is growing faster than the capacity to evaluate them, yet the institutions that govern peer review have remained largely unchanged. The result is a widening mismatch: reviewer scarcity, noisier assessments, and declining confidence in editorial decisions. Every scientist wants better reviews, but review quality depends on the total burden, which no single author can shift. To isolate this tension, we provide a game-theoretic thought experiment: a voluntary lottery in which authors accept a chance of random pre-review rejection, reducing reviewer burden and improving the quality of surviving evaluations. We show that a Nash equilibrium emerges in which authors voluntarily enter the lottery. Scientists who care about the literature they read, not just the papers they publish, will opt in, raising the quality of published science for all.

LLM agents are known to deviate from Nash equilibria in strategic interactions, but nobody has looked inside the model to understand why, or asked whether the deviation can be reversed. We do both. Working with four open-source models (Llama-3 and Qwen2.5, 8B to 72B parameters) playing four canonical two-player games, we establish the behavioral picture through self-play and cross-play experiments, then open up the 32-layer Llama-3-8B model and examine what actually happens during a strategic decision. The mechanistic findings are clear. Opponent history is encoded with near-perfect fidelity at the first layer (96% probe accuracy) and consumed progressively, while Nash action encoding is weak throughout, never exceeding 56%. There is no dedicated Nash module. Instead, the model privately favors the Nash action through most of its forward pass, but a prosocial override rooted in pretraining on human text concentrated in the final layers reverses this, reaching 84% probability of cooperation at layer 30. Injecting a learned Nash direction into the residual stream shifts behavior bidirectionally and causally, confirmed through concept clamping. The behavioral experiments surface six scale- and architecture-dependent findings, the most notable being that chain-of-thought reasoning worsens Nash play in small models but achieves near-perfect Nash play above 70B parameters. The cross-play experiments reveal three phenomena invisible in self-play: a small model can unravel any partner's cooperation by defecting early; two large models reinforce each other's cooperative instincts indefinitely; and who moves first determines which Nash equilibrium the system reaches. LLMs do not lack Nash-playing competence. They compute it, then suppress it.

We present StratFormer, a transformer-based meta-agent that learns to simultaneously model and exploit opponents in imperfect-information games through a two-phase curriculum. The first phase trains an opponent modeling head to identify behavioral patterns from action histories while the agent plays a game-theoretic optimal (GTO) policy. The second phase progressively shifts the policy toward best-response (BR) exploitation, guided by a per-opponent regularization schedule tied to exploitability. Our architecture introduces dual-turn tokens -- feature vectors constructed at both agent and opponent decision points -- coupled with bucket-rate features that encode opponent tendencies across five strategic contexts. On Leduc Hold'em, a small poker variant with six cards and two betting rounds, we test against six opponent archetypes at two strength levels each, with exploitability ranging from 0.15 to 1.26 Big Blinds (BB) per hand. StratFormer achieves an average exploitation gain of +0.106 BB per hand over GTO, with peak gains of +0.821 against highly exploitable opponents, while maintaining near-equilibrium safety.

In this work, we introduce the Deceptive Resource Allocation Game (DRAG), which studies purposeful deception within a Bayesian game framework. In DRAG, a Defender allocates resources across the true asset and several decoys to influence an Attacker's beliefs and actions, with the goal of diverting the Attacker away from the true asset. We seek to characterize purposeful deception, whereby the Defender deceives only when doing so improves its performance. To this end, we solve for the Perfect Bayesian Nash Equilibrium (PBNE) of the corresponding game. We show that, despite the coupled belief-policy interdependence, the problem admits an efficient, non-iterative linear programming formulation. Numerical results demonstrate that the resulting policies naturally balance effective allocation and belief manipulation, giving rise to purposeful and emergent deceptive behaviors.

Vision-Language-Action (VLA) models are a promising paradigm for generalist robotic manipulation by grounding high-level semantic instructions into executable physical actions. However, prevailing approaches typically adopt a monolithic generation paradigm, directly mapping visual-linguistic features to high-frequency motor commands in a flat, non-hierarchical fashion. This strategy overlooks the inherent hierarchy of robotic manipulation, where complex actions can be naturally modeled in a Hybrid Action Space, decomposing into discrete macro-directional reaching and continuous micro-pose alignment, severely widening the semantic-actuation gap and imposing a heavy representational burden on grounding high-level semantics to continuous actions. To address this, we introduce Libra-VLA, a novel Coarse-to-Fine Dual-System VLA architecture. We explicitly decouple the learning complexity into a coarse-to-fine hierarchy to strike a training equilibrium, while simultaneously leveraging this structural modularity to implement an asynchronous execution strategy. The Semantic Planner predicts discrete action tokens capturing macro-directional intent, while the Action Refiner conditions on coarse intent to generate high-frequency continuous actions for precise alignment. Crucially, our empirical analysis reveals that performance follows an inverted-U curve relative to action decomposition granularity, peaking exactly when the learning difficulty is balanced between the two sub-systems. With the asynchronous design, our approach offers a scalable, robust, and responsive solution for open-world manipulation.

Cloud-hosted Large Language Models (LLMs) offer unmatched reasoning capabilities and dynamic knowledge, yet submitting raw queries to these external services risks exposing sensitive user intent. Conversely, relying exclusively on trusted local models preserves privacy but often compromises answer quality due to limited parameter scale and knowledge. To resolve this dilemma, we propose Game-theoretic Trustworthy Knowledge Acquisition (GTKA), a framework that formulates the trade-off between knowledge utility and privacy as a strategic game. GTKA consists of three components: (i) a privacy-aware sub-query generator that decomposes sensitive intent into generalized, low-risk fragments; (ii) an adversarial reconstruction attacker that attempts to infer the original query from these fragments, providing adaptive leakage signals; and (iii) a trusted local integrator that synthesizes external responses within a secure boundary. By training the generator and attacker in an alternating adversarial manner, GTKA optimizes the sub-query generation policy to maximize knowledge acquisition accuracy while minimizing the reconstructability of the original sensitive intent. To validate our approach, we construct two sensitive-domain benchmarks in the biomedical and legal fields. Extensive experiments demonstrate that GTKA significantly reduces intent leakage compared to state-of-the-art baselines while maintaining high-fidelity answer quality.

Benchmark hacking refers to tuning a machine learning model to score highly on certain evaluation criteria without improving true generalization or faithfully solving the intended problem. We study this phenomenon in a generic machine learning contest, where each contestant chooses two types of effort: creative effort that improves model capability as desired by the contest host, and mechanistic effort that only improves the model's fitness to the particular task in contest without contributing to true generalization. We establish the existence of a symmetric monotone pure strategy equilibrium in this competition game. It also provides a natural definition of benchmark hacking in this strategic context by comparing a player's equilibrium effort allocation to that of a single-agent baseline scenario. Under our definition, contestants with types below certain threshold (low types) always engage in benchmark hacking, whereas those above the threshold do not. Furthermore, we show that more skewed reward structures (favoring top-ranked contestants) can elicit more desirable contest outcomes. We also provide empirical evidence to support our theoretical predictions.

We study learned memory tokens as a computational scratchpad for a single-block Universal Transformer with Adaptive Computation Time (ACT) on Sudoku-Extreme, a combinatorial reasoning benchmark. Memory tokens are empirically necessary: no configuration without them reaches non-trivial performance. The optimal count has a sharp lower threshold (T=0 always fails, T=8 reliably succeeds) followed by a stable plateau (T=8-32, 57.4% +/- 0.7% exact-match) and a dilution boundary at T=64. Under halt-side pressure (lambda warmup), mean halt drops monotonically with memory size across the plateau (from 11.6 at T=8 to 8.3 at T=64), showing that memory tokens and ponder depth substitute as resources at fixed accuracy. We also identify a router initialization trap that causes the majority of training runs to fail: both default zero-bias and Graves' recommended positive bias settle into a shallow halt equilibrium the model cannot escape. Inverting the bias to -3 ("deep start") eliminates the failure mode, and ablation shows the trap is inherent to ACT initialization rather than an artifact of our architecture. With reliable training, ACT yields an order of magnitude lower seed variance than fixed-depth processing (+/-0.7 vs +/-9.3 pp); lambda warmup recovers 34% of compute at matched accuracy; and attention heads specialize into memory readers, constraint propagators, and integrators across recursive depth. Code: https://github.com/che-shr-cat/utm-jax.

We cast the design of parameterized quantum circuits as a four-player potential game whose state is a circuit directed acyclic graph (DAG) and whose players encode trainability, non-stabilizerness, task performance, and hardware cost. Per-player restricted action sets factorize the move space into append, remove, retype, and rewire operations; a block-coordinate $\varepsilon$-Nash residual $δ_\text{Nash}$ certifies that no single player can improve unilaterally. A single weight sweep on MaxCut $K_4$ traces a Pareto frontier from a Clifford endpoint $(M_2/n,\langle H\rangle)=(0,4.00)$ to a non-Clifford endpoint $(0.48,3.30)$. On three four-qubit hardware topologies (heavy-hex, $2\times 2$ grid, Rydberg all-to-all), Nash search achieves the highest mean potential; on the $2\times 2$ grid Nash reaches the theoretical ceiling $Φ_\text{max}=4.10$ on two of five seeds while the simulated-annealing baseline does so on one; paired Wilcoxon tests over five seeds cannot reject the null on any single topology ($p\ge 0.22$). On LiH/STO-3G, seeding Nash from a 58-gate Givens-doubles ansatz produces a 48-operation, depth-25 circuit retaining $97.7\%$ of the correlation energy while simultaneously reducing gate count, increasing non-stabilizerness, and controlling trainability. The framework is complementary to energy-only searches such as ADAPT-VQE and k-UpCCGSD, which reach chemical accuracy with fewer operations but do not optimize the other three axes.

When training neural networks with full-batch gradient descent (GD) and step size $η$, the largest eigenvalue of the Hessian -- the sharpness $S(\boldsymbolθ)$ -- rises to $2/η$ and hovers there, a phenomenon termed the Edge of Stability (EoS). \citet{damian2023selfstab} showed that this behavior is explained by a self-stabilization mechanism driven by third-order structure of the loss, and that GD implicitly follows projected gradient descent (PGD) on the constraint $S(\boldsymbolθ)\leq 2/η$. For mini-batch stochastic gradient descent (SGD), the sharpness stabilizes below $2/η$, with the gap widening as the batch size decreases; yet no theoretical explanation exists for this suppression. We introduce stochastic self-stabilization, extending the self-stabilization framework to SGD. Our key insight is that gradient noise injects variance into the oscillatory dynamics along the top Hessian eigenvector, strengthening the cubic sharpness-reducing force and shifting the equilibrium below $2/η$. Following the approach of \citet{damian2023selfstab}, we define stochastic predicted dynamics relative to a moving projected gradient descent trajectory and prove a stochastic coupling theorem that bounds the deviation of SGD from these predictions. We derive a closed-form equilibrium sharpness gap: $ΔS = ηβσ_{\boldsymbol{u}}^{2}/(4α)$, where $α$ is the progressive sharpening rate, $β$ is the self-stabilization strength, and $σ_{ \boldsymbol{u}}^{2}$ is the gradient noise variance projected onto the top eigenvector. This formula predicts that smaller batch sizes yield flatter solutions and recovers GD when the batch equals the full dataset.

Consider betting against a sequence of data in $[0,1]$, where one is allowed to make any bet that is fair if the data have a conditional mean $m_0 \in (0,1)$. Cover's universal portfolio algorithm delivers a worst-case regret of $O(\ln n)$ compared to the best constant bet in hindsight, and this bound is unimprovable against adversarially generated data. In this work, we present a novel mixture betting strategy that combines insights from Robbins and Cover, and exhibits a different behavior: it eventually produces a regret of $O(\ln \ln n)$ on almost all paths (a measure-one set of paths if each conditional mean equals $m_0$ and intrinsic variance increases to $\infty$), but has an $O(\log n)$ regret on the complement (a measure zero set of paths). Our paper appears to be the first to point out the value in hedging two very different strategies to achieve a best-of-both-worlds adaptivity to stochastic data and protection against adversarial data. We contrast our results to those in Agrawal and Ramdas [2026] for a sub-Gaussian mixture on unbounded data: their worst-case regret has to be unbounded, but a similar hedging delivers both an optimal betting growth-rate and an almost sure $\ln\ln n$ regret on stochastic data. Finally, our strategy witnesses a sharp game-theoretic upper law of the iterated logarithm, analogous to Shafer and Vovk [2005].

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order O~(1/epsilon^4) for a desired accuracy epsilon, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

Games offer a compelling paradigm for developing general reasoning capabilities in language models, as they naturally demand strategic planning, probabilistic inference, and adaptive decision-making. However, existing self-play approaches rely solely on terminal game outcomes, providing no mechanism to distinguish transferable reasoning patterns from game-specific heuristics. We present STRATAGEM, which addresses two fundamental barriers to reasoning transfer: domain specificity, where learned patterns remain anchored in game semantics, and contextual stasis, where static game contexts fail to cultivate progressive reasoning. STRATAGEM selectively reinforces trajectories exhibiting abstract, domain-agnostic reasoning through a Reasoning Transferability Coefficient, while incentivizing adaptive reasoning development via a Reasoning Evolution Reward. Experiments across mathematical reasoning, general reasoning, and code generation benchmarks demonstrate substantial improvements, with particularly strong gains on competition-level mathematics where multi-step reasoning is critical. Ablation studies and human evaluation confirm that both components contribute to transferable reasoning.

In this article, we generalize Unbounded Minimax, the state-of-the-art search algorithm for zero sums two-player games with perfect information to the framework of multiplayer games with perfect information. We experimentally show that this generalized algorithm also achieves better performance than the main multiplayer search algorithms.

A key challenge in distributed coalition formation within characteristic function games is determining how to allocate the calculation of coalition values across a set of agents. The number of possible coalitions grows exponentially with the number of agents, and existing distributed approaches may produce uneven or redundant allocations, or assign coalitions to agents that are not themselves members. In this article, we present the \emph{Necklace-based Distributed Coalition Algorithm} (N-DCA), a communication-free algorithm in which each agent independently determines its own coalition value calculation allocation using only its identifier and the total number of agents. The approach builds on the notion of Increment Arrays (IAs), for which we develop a complete mathematical framework: equivalence classes under circular shifts, periodic IAs, and a rotated designation scheme with formal load-balance guarantees (tight bounds). We establish a bijection between canonical representative IAs and two-colour combinatorial necklaces, enabling the use of efficient necklace generation algorithms to enumerate allocations in constant amortised time. N-DCA is, to the best of our knowledge, the only distributed coalition value calculation algorithm for unrestricted characteristic function games to provably satisfy five desirable properties: no inter-agent communication, equitable allocation, no redundancy, balanced load, and self-interest. An empirical evaluation against DCVC (Rahwan and Jennings 2007) demonstrates that, although DCVC is faster by a constant factor, this difference becomes negligible under realistic characteristic-function evaluation costs, while N-DCA offers advantages in working memory, scalability, and the self-interest guarantee.

We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, the convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performance, with the best attainable rate being $Ω(T^{-1/4})$ in contrast to the usual $Ω(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate up to constant and logarithmic factors. The first algorithm leverages a straightforward trade-off between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.

We introduce the first version of KWBench (Knowledge Work Bench), a benchmark for unprompted problem recognition in large language models: can an LLM identify a professional scenario before attempting to solve it. Existing frontier benchmarks have saturated, and most knowledge-work evaluations to date reduce to extraction or task completion against a specification. KWBench targets the step before that: recognizing the governing structure of the situation from raw inputs alone. The benchmark contains 223 tasks sourced from practitioners across acquisitions, contract negotiations, clinical pharmacy, organizational politics, fraud analysis, and incentive design. Each task encodes a formal game-theoretic pattern (principal-agent conflict, signaling, mechanism design failure, strategic omission, coalitional dynamics, strategic interdependence) and carries structured ground truth recording the expert reading of the situation and the anticipated failure modes. Models receive raw data and a task prompt with no indication of problem type. Scoring is a three-tier rubric gated by a mandatory conjunctive check. Mandatory criteria encode the predicted wrong paths. We evaluate 16 models. The best model passes on 27.9% of tasks. The top two models agree on only 31.7% of their passes. Among the top 8, 44 tasks are solved by exactly one model; routing across the top 8 covers 50.7% of the benchmark, nearly double the best single model. Conditional on passing, quality scores converge (approx 83% across models); unconditional scores do not. Same models articulate the relevant game-theoretic concept correctly when asked, then fail to apply it unprompted. We release KWBench to shift how frontier models are evaluated on knowledge work, scoring them on whether they recognize the right problem from the situation alone, not only on how well they execute once the problem has been framed for them.

Automated negotiation in complex, multi-party and multi-issue settings critically depends on accurate opponent modeling. However, conventional numerical-only approaches fail to capture the qualitative information embedded in natural language interactions, resulting in unstable and incomplete preference estimation. Although Large Language Models (LLMs) enable rich semantic understanding of utterances, it remains challenging to quantitatively incorporate such information into a consistent opponent modeling. To tackle this issue, we propose a novel preference estimation method integrating natural language information into a structured Bayesian opponent modeling framework. Our approach leverages LLMs to extract qualitative cues from utterances and converts them into probabilistic formats for dynamic belief tracking. Experimental results on a multi-party benchmark demonstrate that our framework improves the full agreement rate and preference estimation accuracy by integrating probabilistic reasoning with natural language understanding.

We study the problem of learning minimax policies in zero-sum matrix games. Fiegel et al. (2025) recently showed that achieving last-iterate convergence in this setting is harder when the players are uncoupled, by proving a lower bound on the exploitability gap of Omega(t^{-1/4}). Some online mirror descent algorithms were proposed in the literature for this problem, but none have truly attained this rate yet. We show that the use of a log-barrier regularization, along with a dual-focused analysis, allows this O-tilde(t^{-1/4}) convergence with high-probability. We additionally extend our idea to the setting of extensive-form games, proving a bound with the same rate.

We present a new solution concept called evolutionarily stable Stackelberg equilibrium (SESS). We study the Stackelberg evolutionary game setting in which there is a single leading player and a symmetric population of followers. The leader selects an optimal mixed strategy, anticipating that the follower population plays an evolutionarily stable strategy (ESS) in the induced subgame and may satisfy additional ecological conditions. We consider both leader-optimal and leader-pessimal selection among ESSs, which arise as special cases of our framework. Prior approaches to Stackelberg evolutionary games either define the follower response via evolutionary dynamics or assume rational best-response behavior, without explicitly enforcing stability against invasion by mutations. We present algorithms for computing SESS in discrete and continuous games, and validate the latter empirically. Our model applies naturally to biological settings; for example, in cancer treatment the leader represents the physician and the followers correspond to competing cancer cell phenotypes.

Offline learning of strategies takes data efficiency to its extreme by restricting algorithms to a fixed dataset of state-action trajectories. We consider the problem in a mixed-motive multiagent setting, where the goal is to solve a game under the offline learning constraint. We first frame this problem in terms of selecting among candidate equilibria. Since datasets may inform only a small fraction of game dynamics, it is generally infeasible in offline game-solving to even verify a proposed solution is a true equilibrium. Therefore, we consider the relative probability of low regret (i.e., closeness to equilibrium) across candidates based on the information available. Specifically, we extend Policy Space Response Oracles (PSRO), an online game-solving approach, by quantifying game dynamics uncertainty and modifying the RL objective to skew towards solutions more likely to have low regret in the true game. We further propose a novel meta-strategy solver, tailored for the offline setting, to guide strategy exploration in PSRO. Our incorporation of Conservatism principles from Offline reinforcement learning approaches for strategy Exploration gives our approach its name: COffeE-PSRO. Experiments demonstrate COffeE-PSRO's ability to extract lower-regret solutions than state-of-the-art offline approaches and reveal relationships between algorithmic components empirical game fidelity, and overall performance.

We study the mean-field Langevin descent-ascent (MFL-DA), a coupled optimization dynamics on the space of probability measures for entropically regularized two-player zero-sum games, together with its associated interacting particle system. For general nonconvex-nonconcave payoffs, Wang and Chizat (COLT 2024) asked whether the original single-timescale MFL-DA converges to the mixed Nash equilibrium and, if so, at what rate. We prove a local affirmative answer in Wasserstein space: if the initial datum is sufficiently close to the mixed Nash equilibrium, then the mean-field dynamics converges to it exponentially fast at a quantitative rate. We further show that the finite-$N$ particle system inherits this stability up to times exponential in $N$, with an $N$-independent exponential rate modulo a finite-particle error floor. Combined with the recent counterexample of Mourrat and Pillaud-Vivien for MFL-DA, which shows that global convergence cannot hold in general, our theorem completes the positive local counterpart of the Wang-Chizat question: the mixed Nash equilibrium has a robust basin of attraction, stable under both the mean-field flow and its finite-particle approximation.

We present an algorithm for computing evolutionarily stable strategies (ESSs) in symmetric perfect-recall extensive-form games of imperfect information. Our main algorithm is for two-player games, and we describe how it can be extended to multiplayer games. The algorithm is sound and computes all ESSs in nondegenerate games and a subset of them in degenerate games which contain an infinite continuum of symmetric Nash equilibria. The algorithm is anytime and can be stopped early to find one or more ESSs. We experiment on an imperfect-information cancer signaling game as well as random games to demonstrate scalability.

We present an algorithm for computing all evolutionarily stable strategies in nondegenerate normal-form games with three or more players.

Decentralized air traffic management systems offer a scalable alternative to centralized control, but often assume high levels of cooperation. In practice, such assumptions frequently break down since airspace sectors operate independently and prioritize local objectives. We address the problem of sector overload in decentralized air traffic management by proposing a regulated decentralized protocol that models self-interested behaviors based on best response dynamics. Each sector adjusts the departure times of flights under its control to reduce its own congestion, without requiring centralized joint optimization. A tunable cooperativeness factor models the degree to which each sector accounts for overload in other sectors, while a minimal admissibility rule prevents local updates from creating new overloads. We prove that the proposed protocol satisfies a potential game structure, ensuring that best response dynamics converge to a pure Nash equilibrium under this restriction. In addition, we identify a sufficient condition under which an overload-free solution corresponds to a global minimizer of the potential function. Numerical experiments using 24 hours of European flight data demonstrate that the proposed algorithm substantially reduces overload even with only minimal cooperation between sectors, while maintaining scalability and achieving solution quality comparable to the centralized benchmark.

We analyze an algorithm for assigning weights prior to scalarization in discrete multi-objective problems arising from data analysis. The algorithm evolves weights (interpreted as the relevance of features) by a replicator-type dynamic on the standard simplex, with update indices computed from a normalized data matrix. We prove that the resulting sequence converges globally to a unique interior equilibrium, yielding non-degenerate limiting weights.

Large language models (LLMs) are increasingly deployed as decision-making agents in high-stakes domains and as imitators of human behavior in the social and behavioral sciences. Yet how closely LLMs mirror human decision-making remains poorly understood. This gap is critical: misalignment could produce harmful outcomes in practice, while failure to replicate human behavior renders LLMs ineffective as social simulators. Here, we address this gap by replicating large-scale game-theoretic experiments and by introducing a systematic prompting and probing framework for machine-behavioral evaluation. We test three open models typically used to power agents (Llama, Mistral, and Qwen). Across 121 dyadic games spanning four classical game types, Llama reproduces human cooperation patterns with high fidelity, while Qwen aligns closely with Nash equilibrium predictions. Characterizing models through behavioral phenotyping, we find that humans and Llama share an envious decision profile, while Qwen and Mistral exhibit different profiles. An attention-based analysis of payoff salience reveals Llama processes payoff information in a structured, layer-dependent manner absent in Qwen and Mistral, suggesting a mechanistic basis for its closer alignment with human behavior. Population-level behavioral replication is achieved without persona-based prompting, simplifying the simulation process. Extending the experimental parameter space beyond the original human-tested games, we generate and preregister testable hypotheses for novel game configurations. Our findings demonstrate appropriately configured LLMs can replicate aggregate human behavioral patterns, exhibit human-like decision phenotypes, and enable systematic exploration of unexplored experimental spaces, offering a complementary approach to traditional behavioral research that generates new empirical predictions about human social decision-making.

We extend the study of learning in games to dynamics that exhibit non-asymptotic stability. We do so through the notion of uniform stability, which is concerned with equilibria of individually utility-seeking dynamics. Perhaps surprisingly, it turns out to be closely connected to economic properties of collective rationality. Up to strategic equivalence, if a mixed equilibrium is uniformly stable, then it is weakly Pareto optimal; there is no way for all players to improve by jointly deviating from the equilibrium. This is a form of collective rationality that rules out the types of behaviors in the prisoner's dilemma or the tragedy of the commons. Moreover, we show that uniform stability determines the last-iterate convergence behavior for the family of incremental smoothed best-response dynamics, used to model individual and corporate behaviors in the markets. Unlike dynamics around strict equilibria, which can stabilize to socially-inefficient solutions, individually utility-seeking behaviors near mixed Nash equilibria lead to collective rationality.

Games have long been a microcosm for studying planning and reasoning in both natural and artificial intelligence (AI), often focusing on expert-level or even super-human play. But real life also pushes human intelligence along a different frontier, requiring people to flexibly navigate decision-making problems that they have never thought about before. Here, we use novice gameplay to study how people reason about new problem settings. Through a series of large-scale behavioral studies with over 1000 participants and 121 two-player strategic board games (almost all novel to our participants), we show that people are systematic and adaptively rational in how they play a game for the first time, or evaluate a game (e.g., how fair or how fun it is likely to be) before they have played it even once. We explain these capacities via a computational cognitive model that we call the 'Intuitive Gamer', a model based on mechanisms of fast and flat (depth-limited) goal-directed probabilistic simulation. Our work offers new insights into how people rapidly evaluate, act, and make suggestions when encountering novel problems, and could inform the design of more flexible and human-like AI systems that can determine not just how to solve new tasks, but whether a task is worth thinking about at all.

There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer strategic-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. We present an approach for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically-constrained program based on a nonlinear complementarity problem formulation from the sequence-form game representation. This approach capitalizes on recent advances for solving nonconvex quadratic programs. Our algorithm is able to quickly solve three-player Kuhn poker after removal of dominated actions. Of the available algorithms in the Gambit software suite, only the logit quantal response approach is successfully able to solve the game; however, the approach takes longer than our algorithm and also involves a degree of approximation. Our formulation also leads to a new approach for computing Nash equilibrium in multiplayer strategic-form games which we demonstrate to outperform a previous quadratically-constrained program formulation.