Nash Equilibrium

Regularized self-play -- the family behind DeepNash's Stratego play -- drives a two-player zero-sum policy to a Nash equilibrium by best-responding to a slowly moving, entropy-regularized reference policy $ρ$. When the game has a polytope of value-equivalent equilibria, the regularizer silently breaks the tie: with a uniform reference it selects the maximum-entropy member, the I-projection of $ρ$ onto the Nash set. Can the reference be used to choose the equilibrium on purpose? On five exactly solvable games plus a 2-D polytope, with exact best responses and equivalence tests over independent seeds, anchoring the reference at a target member and refining steers self-play to that member with mean coordinate error 0.007 at median exploitability $5\times10^{-5}$, TOST-equivalent to the request within $\pm0.05$; the anchoring persists through refinement and follows the reference, not the initialization. Selection follows the reach-weighted I-projection (slope 0.969 [0.950, 0.987]). We report with equal emphasis where the story breaks: fixed off-manifold references cost 0.08-0.25 exploitability; stiff or flat families require a smaller mirror step, set by a pre-registered rule; boundary targets undershoot; curvature predicts where boundary saturation bites (rank correlation 0.90, p=0.037) while interior precision is curvature-independent. Table and MLP steering maps are equivalent within $\pm0.03$ at every target (30 seeds); matched control arms show attention's robust signature is excess seed variance, any systematic shift bounded at 0.018 and not significant. Against a best response the selection-robustness trade-off is degenerate: steering matters only against fixed, non-equilibrium opponents. The recipe -- anchor the reference at the desired member and refine -- reinterprets the KL anchor of RLHF-style RL as a selection knob, not only a stability leash.

Game-theoretic analyses of cyber defence often compute equilibria of games whose payoffs exist only as the output of a simulator. Iterative equilibrium-finding methods grow a set of attacker and defender policies and need the payoff of every attacker--defender pair, so they are bottlenecked by payoff estimation: each payoff costs many simulator runs. We introduce Regret-Weighted Payoff Sampling (RWPS), which spends a fixed simulation budget on the payoffs the equilibrium actually depends on and predicts the rest with a model trained on every payoff measured so far. Standard error bounds for estimated games are driven by the worst-estimated payoff, so they cannot credit an estimator that is inaccurate only where accuracy does not matter. We prove a bound that weights payoff errors by the opponent's equilibrium strategy, a certificate that can be computed from simulated payoffs alone, and a condition under which errors in the predicted payoffs cannot change either player's regret. On three synthetic general-sum games, one of them a Colonel Blotto game of military resource allocation, the new bounds are four to six times tighter than the standard one, and RWPS finds less exploitable equilibria than minimum-regret-first search, information-gain search and progressive sampling at the same budget. On two cyber-defence simulators, CyGym and a new game whose hosts are LLM agents exposed to prompt injection, it gives the least exploitable equilibria at the smallest budgets.

This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from the simplex domain to the product-of-simplex domain, which is directly applicable to solving Nash equilibria. The fixed-point bundle for VIs on the product-of-simplex domain reveals a composite fiber bundle structure. The key innovation is to construct an equivalent VI on the simplex domain and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, the predictor-corrector path-following algorithm for the VI on the product-of-simplex domain is shown to inherit the convergence guarantee of the simplex-domain framework, namely, global convergence with linear gap reduction near solutions. Numerical experiments on 5600 randomly generated instances with dimensions ranging from 2-player 128-action to 128-player 2-action demonstrate robust performance. The algorithm converges in every tested instance.

We study decentralized learning of Nash equilibria (NE) in infinite-horizon discounted Markov games under bandit feedback, focusing on Markov $α$-potential games. We develop KL-projected natural policy gradient (NPG) algorithms in two settings: an episodic setting with frozen policies during sampling and a fully online setting in which players receive a single realized cost sample per time step and update their policies asynchronously along a continuing trajectory. We establish finite-time high-probability NE regret bounds of order $\widetilde O(T^{-1/4})$ and $\widetilde O(T^{-2/15})$ for the episodic and fully online settings, respectively, up to fixed approximation terms. Crucially, our bounds eliminate the distribution-mismatch coefficient, which can scale prohibitively with the size of the state space, while accommodating potential approximation, estimation-oracle bias, and transition sensitivity. We further identify a state-wise potential structure that yields sharper guarantees with additive dependence on the potential approximation error $α$. We specialize the framework to independent-resource Markov congestion games (IMCGs), establish their approximate-potential and transition-sensitivity properties, and construct decentralized estimation oracles from realized costs. As an application, we introduce strategic online job scheduling on stochastic machines and obtain a scalable decentralized algorithm for learning stable dispatching policies. Overall, our results provide the first finite-time high-probability NE regret guarantees for fully online asynchronous decentralized learning in Markov $α$-potential games, remove distribution-mismatch coefficients from the regret bounds, accommodate fixed estimation-oracle bias, and provide scalable decentralized learning with finite-time guarantees for IMCGs.

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

We study risk-sensitive evolutionary learning dynamics and their long-run equilibrium selection behaviors in coordination games. Agents' risk attitudes enter through the classical entropic risk measure, which evaluates opponent-induced payoff uncertainty and feeds into noisy best responses under two standard revision protocols: best response with mutations and logit choice. We first analyze $2\times 2$ coordination games in both single-population symmetric and two-population asymmetric settings. In the single-population setting, unlike the risk-neutral case where the dynamics are known to favor the risk-dominant equilibrium, we show that risk sensitivity can change the stochastically stable outcome: a greater risk-seeking attitude favors the payoff-dominant equilibrium, while a greater risk-averse attitude favors the maximin equilibrium. Thus, the population's risk attitude may act as a control knob for long-run equilibrium selection. In both population settings, we also identify a robust regime: any super-dominant equilibrium is stochastically stable for all risk attitudes, under both protocols, and across populations. We further extend the single-population analysis to symmetric $k$-action games, which include symmetric $k$-action coordination games as a special case, under risk-sensitive best response with mutations. In this setting, we show that, for sufficiently large populations, sufficiently risk-seeking agents uniquely select the strongly payoff-dominant equilibrium when it exists, whereas sufficiently risk-averse agents uniquely select the strongly maximin equilibrium when it exists. These results show that entropic risk sensitivity may serve as a systematic mechanism for steering equilibrium selection in evolutionary games, beyond the classical risk-neutral benchmark.

We introduce the first Probably Approximately Correct (PAC) learning framework for general-sum concurrent stochastic games (CSGs) with transition uncertainty, while addressing the challenge of Nash equilibrium (NE) existence. Our algorithm maintains data-driven $L^1$ confidence sets over transition kernels and solves a robust CSG to compute a social-welfare optimal $\varepsilon$-NE, using a robust MDP-based exploration mechanism to drive joint state-action coverage. Crucially, we introduce a Nash margin characterisation that enables principled reasoning about equilibrium existence: the framework either returns an $\varepsilon$-approximate NE whose social-welfare value is $\varepsilon$-close to optimal, or provides a sound certificate that no exact NE exists. Under a minimum reachability condition $p_{\mathrm{reach}}>0$ over relevant state-action pairs, the algorithm terminates after a polynomial number of trajectory samples, with sample complexity $\widetilde{O}\left( {R_{\max}^2 H^4 |S|^2 |A| / (p_{\mathrm{reach}} \varepsilon^2)} \right)$. Empirical results on benchmark CSGs demonstrate near-optimal performance, correct handling of equilibrium (non-)existence, and sample complexity consistent with theory.

We study the allocation of indivisible goods among agents with identical additive valuations, focusing on envy-freeness up to one good (EF1) and Nash social welfare (NSW). Since every maximum-NSW allocation is EF1 under additive valuations, the associated threshold problem inherits the known strong NP-hardness of NSW maximization under identical additive valuations and is strongly NP-complete. We therefore focus on welfare guarantees satisfied by arbitrary EF1 allocations. Although every such allocation is known to achieve an $e^{-1/e}$-approximation to the unrestricted optimal NSW, we identify conditions yielding stronger guarantees. Under uniform valuations, every EF1 allocation is NSW-optimal. Under an $\varepsilon$-small-item condition, every EF1 allocation achieves an explicit approximation ratio $ρ_n(\varepsilon)$ satisfying $ρ_n(\varepsilon) = 1-O(\varepsilon^2)$ as $\varepsilon\to 0$ for fixed $n$. We further consider the stronger sequential requirement that $\operatorname{EF1}$ be maintained after every item assignment. For this setting, we introduce \emph{PriorityNet}, a deep reinforcement learning framework trained with Proximal Policy Optimization (PPO) and equipped with prospective $\operatorname{EF1}$ action masking, which guarantees prefix-wise $\operatorname{EF1}$ by construction. Across 3,000 test instances in each of the offline full-information and random-order online regimes ($n\in[2,20]$, $m\in[5,100]$), PriorityNet achieves mean normalized $\operatorname{NSW}$ values of $0.9911$ and $0.9701$, respectively. Relative to the offline Longest Processing Time (LPT) heuristic and the online least-valued-bundle rule, it attains instance-wise win-minus-loss rates of $+27.10\%$ and $+17.87\%$. Its aggregate welfare matches the offline LPT baseline to four decimal places and modestly improves upon the online baseline, from $0.9694$ to $0.9701$.

Inspired by possible future markets of autonomous routing and driving (ARAD), we introduce competitive mediator games and their equilibria which generalize the (coarse) correlated equilibria, which have become a popular research area recently as they not only can be more socially efficient than Nash equilibria but also are limits of algorithmic no-regret multi-agent learning dynamics. We discuss the basic properties of competitive mediator games and prove that in the generic setting of anonymous congestion(routing) games with market-share maximizing mediators all competitive mediator equilibria are monopolies whenever one of the mediators is weakly preferred to other mediators by all users. We apply and interpret these results in the context of new markets of competing ARAD service providers. We also provide a comprehensive overview of these markets and discuss the future mechanism design thereof.

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $ε$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

Attention-only dynamical theories model Transformer residual directions as particles aggregating on a sphere. We extend this framework by incorporating the feed-forward network (FFN) term as a local steering field acting on each token state. The resulting theory predicts that the tangential component of the FFN field is necessary for motion in residual-direction space, that critical residual directions correspond to nonlinear projective equilibria, and that a commutator defect determines when a finite attention--FFN block can be accurately approximated by a parallel, additive flow. Across GPT-2, Pythia, Mistral, and Llama models, the extended theory improves one-step angular prediction relative to an attention-only baseline, with the contribution of the FFN increasing from GPT-2 to Llama-3-8B. Intervention experiments show that retaining only the tangential FFN component preserves most model quality, whereas retaining only the radial component causes performance to collapse. The tangential component also preserves output diversity under aggregation pressure. As a practical application, layers with small commutator defects can be approximately parallelized with only a modest increase in loss, whereas layers with large defects degrade rapidly. These findings support the interpretation of FFN layers as directional steering fields that shape Transformer residual geometry and govern the feasibility of block-level interventions.

Clustered federated learning benefits from organizing heterogeneous participants into coalitions that train coalition-specific models, but such clustering is sustainable only if participants prefer their assigned coalition and the required transfers are affordable. We develop a transferable-surplus model separating learning benefit, system cost, participant cost, and monetary transfers; an allocation rule converts coalition surplus into hedonic preferences, and weak budget feasibility guarantees nonnegative retained coordinator surplus. For symmetric pairwise allocations the induced game is an exact potential game: a Nash-stable partition exists, every strict better-response process converges, and with destination consent accepted better responses reach an individually stable partition. We characterize feasibility of bounded pair incentives and verify the exponentially many budget constraints in polynomial oracle time when retained slack is submodular. Decomposing welfare into participant potential and retained slack yields additive and multiplicative price-of-stability guarantees, the latter asymptotically tight; exact balance gives welfare-optimal stability only on the pairwise-representable class, and budget feasibility alone permits unbounded welfare loss. Global potential maximization equals weighted maximum-agreement correlation clustering, and approximation followed by stabilization satisfies an end-to-end welfare bound governed by retained slack and negative-edge mass, attained by an explicit construction. In a preregistered five-seed CIFAR-10 study the mechanism reaches the certified estimated-table welfare optimum on every primary instance, equal-surplus sharing has no Nash-stable outcome on three, and pairwise validation gain gives far more reliable pair signs than gradient alignment.

Concurrent games are a standard model for multi-agent systems, with Nash equilibrium as their central solution concept. The associated \emph{constrained existence problem}---does a game admit a Nash equilibrium whose expected payoff lies within a prescribed interval for every player?---is undecidable, and remains so even for 10-player \emph{stopping} games, in which a terminal state is reached almost surely under every strategy profile. We give two routes to tractability. We first relax exactness and consider the problem of approximate constrained existence problem, parametrised by $\varepsilon$-NE, which decides whether an $\varepsilon$-Nash equilibrium with the prescribed payoffs exists. The algorithm runs in exponential time, and only polynomially in the bit-size of $\varepsilon$. We complement it with a \PSPACE-hardness lower bound that holds already for turn-based games, and for pure equilibria as well. We then relax the solution concept, turning to \emph{extreme risk-sensitive equilibria} (XRSE), recently introduced for turn-based stochastic games. Here the players are partitioned into optimists and pessimists, who evaluate a strategy profile by the best, respectively the worst, payoff attainable with positive probability, instead of the expected payoff. We prove that the constrained existence problem for XRSE is \NP-complete on concurrent games, as for turn-based games.

Safe and efficient navigation in mixed-traffic environments remains a critical challenge for Autonomous Vehicles (AVs), primarily due to the complex interdependence between the AV's decisions and the unpredictable reactions of human drivers. This paper introduces a comprehensive decision-making framework that formulates the driving interaction as a Generalized Nash Equilibrium Problem (GNEP). Unlike decoupled optimization approaches, this framework explicitly models shared safety and geometric constraints, ensuring that the feasibility of the AV's strategy is dynamically linked to the opponent's actions. To solve this non-convex problem in real-time, we propose a dedicated solver based on Particle Swarm Optimization (PSO). The complete architecture was validated on a test track using a real autonomous Renault Zoé interacting with a human driver. Experimental results demonstrate the system's ability to handle critical scenarios by generating comfortable, human-like trajectories. Benchmarks confirm the solver's operational feasibility, achieving convergence in under 50 ms.

Organizations often pool dispersed information into one ranking and then allow many agents to act on that shared view. In a discovery problem, this can improve beliefs while reducing coverage. We develop an exactly solvable benchmark with sixteen boxes, one target, eight searchers, and noisy private clues. Pooling raises the accuracy of the best single recommendation from 0.20 to 0.3835, but repeating that recommendation lowers group discovery from 0.8322 under decentralized clue-following to 0.3835. A coordinated eight-action portfolio using the same pooled reports reaches 0.8594, and seven coordinated actions recover the decentralized benchmark. The paradox is a protocol failure, not an information failure: a one-answer rule compresses a portfolio of available actions into one repeated choice. We then replace the planner with self-interested searchers who split a prize. The equal-split game is a potential game. Its anonymous symmetric equilibrium obeys a water-filling rule. In the canonical instance it achieves 0.5991: strictly above consensus, but below both private search and the planner. The exact mixed price of anarchy is 2 - 1/N. A sole-rescue reward, which pays only an agent who covers the target alone, makes every pure Nash equilibrium first-best. Finally, a latent common-cue model shows how correlated reports collapse effective discovery channels. The centralized planner gain rises strictly with copying, and in the canonical environment the symmetric market overtakes decentralized report-following at copying probability c = 0.788462. In a proportional large-market limit the five-protocol ordering survives exactly: consensus discovery vanishes while blind, market, private, and portfolio search converge to 0.500, 0.547, 0.847, and 0.874. The contribution is a compact benchmark that separates information, allocation, incentives, and dependence into exact, reusable quantities.

In two-player zero-sum games whose Nash equilibria form a convex set, regularized solvers such as Regularized Nash Dynamics (R-NaD) empirically select the maximum-entropy member: the information projection (I-projection) of a uniform reference onto the Nash set. On a panel of small games this match is exact, with one apparent exception: in Kuhn poker R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member sits at 0.201, a coordinate gap of about 0.021, even though R-NaD attains 99.7 percent of the maximum entropy. We ask whether this gap is a genuine selection bias or an artifact, and answer it quantitatively. We show that for selection on a one-dimensional Nash manifold the coordinate gap factorizes as $\mathrm{gap} \approx \sqrt{2δ/κ}$, where $δ$ is the entropy shortfall of the solver and $κ$ is the curvature of the entropy landscape at its peak. Across five games this relation holds to within $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $δ\approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore no gap regardless of curvature; only the sequential game (Kuhn) has $δ> 0$. A causal sweep of the magnet strength drives $δ\to 0$ and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, $R^2 > 0.999999$, against the exact prediction of 1/2), until the dynamics destabilize at a stability floor: behavior consistent with a removable shortfall and inconsistent with a fixed bias. We quantify the curvature half of the law from measured curvatures and flag a moving-target pitfall in the natural Tsallis-entropy experiment. The Kuhn gap is thus the curvature shadow of a small, removable entropy shortfall on an unusually flat peak; the I-projection account is upheld up to a flatness-limited residual.

For decades, static solution concepts (Nash, Correlated, and Coarse Correlated Equilibria) and the Price of Anarchy (PoA) have formed the bedrock of algorithmic game theory, with no-regret learning proving fast convergence to such game-theoretic equilibria. We show that reducing multi-agent learning to static equilibrium and black-box regret analysis obscures underlying dynamic disequilibrium and game theoretic bounds. First, interior Nash equilibria lack $C^1$ vector field information, meaning agents cannot distinguish aligned from strictly opposing incentives. Inheriting this geometry, the worst-case pure Nash equilibria dictating robust PoA bounds manifest as topologically unstable strict saddles, and in canonical congestion games, as global repellers supported on almost everywhere strictly dominated strategies. Anchoring efficiency guarantees to these unstable states causes algebraic sensitivity; we prove that accommodating all strictly positive affine costs renders the PoA unbounded. Furthermore, projecting learning trajectories onto the discrete simplex of correlated play systematically accommodates non-rationalizable behavior. Evaluating dynamics via Coarse Correlated Equilibria or proximal refinements fails to preclude strictly dominated strategies. Moreover, optimal $O(1/T)$ swap-regret minimization does not preclude macroscopic turbulence, manifesting as chaotic limit sets even in minimal games. Finally, we examine the non-atomic limit of congestion games. Though considered highly stable with tight sub-linear $Θ(p/\ln p)$ PoA bounds (where $p$ is the polynomial degree), we prove that under discrete-time learning, the unique equilibrium destabilizes into Li-Yorke chaos and global attractors whose time-averaged inefficiency degrades exponentially as $2^p$. These results necessitate re-evaluating worst-case equilibrium frameworks for dynamically grounded metrics.

Adversarial team games (ATGs) with asymmetric information, such as adversarial path-finding, goal search, and reachability games on graphs, require strategies that are robust to hidden opponent types, such as a hidden goal flag, and to deception. Under asymmetric information, deception is seen as strategic shifts in the type distribution such that the omniscient opponent can collude with Nature and condition its play on the observed type. Existing risk-neutral solution concepts, such as Bayesian Nash equilibrium (BNE), are sensitive to distribution shifts, while distributionally robust approaches provide guarantees only within a prescribed ambiguity set. To address these limitations, we introduce Probabilistically Robust Minimax-Regret Equilibrium (PR-MRE), a novel equilibrium concept that combines the distribution-free robustness of minimax-regret reasoning with probabilistic information from a nominal type distribution. PR-MRE minimizes worst-case regret over a high-confidence subset of the type space, providing protection against strategic redistribution of probability mass while avoiding the conservatism of fully distribution-free approaches. We show that, for normal-form Bayesian games, PR-MRE can be formulated as a robust bilinear program and derive a tractable semidefinite relaxation. We then adapt this relaxation into a novel meta-solver within a robust double-oracle framework, PRMRE-PSRO, enabling population-based learning of approximate PR-MRE strategies via deep reinforcement learning best responses. Experiments on graph-structured adversarial team games demonstrate that PR-MRE discovers strategies with substantially improved worst-case performance across hidden types compared to risk-neutral equilibrium solutions, resulting in more robust behavior under strategic distribution shifts.

Computing Nash equilibrium policies in multi-agent Pursuit-Evasion games (PEG) is challenging due to the exponential growth of the joint state and action spaces with the number of agents. Existing approaches either rely on offline equilibrium approximations, which may lack adaptability during execution, or online planning methods, which suffer from large branching factors. In this work, we propose Primitive-Guided Tree Search (PGTS), a hybrid framework that integrates offline exact Nash equilibrium computation with online tree search: PGTS first solves a collection of smaller, tractable sub-games offline; at deployment, PGTS performs online tree search at each time step, using the optimal sub-game policies and value functions to guide tree expansion and estimate leaf-node values. Extensive experiments on varied graph topologies, including real-world networks, demonstrate that PGTS significantly outperforms state-of-the-art learning and heuristic baselines, while maintaining robust performance against adversaries.

Many important games have more than two players and imperfect information. Existing approaches for computing Nash equilibrium, the central game-theoretic solution concept, in such games either lack scalability or obtain poor performance. In this paper we introduce a new algorithm called projected exploitability descent (PED) for approximating Nash equilibria in multiplayer games of imperfect information. The algorithm works by running projected subgradient descent minimizing a proxy for the multiplayer generalized exploitability function. The objective is nonconvex and nonsmooth, but can be represented as the sum of the maxima of linear functions, for which a subgradient can easily be computed and projected to the polytope of feasible sequence-form strategies. We explore performance of PED on a generalized version of the well-studied benchmark game three-player Kuhn poker. No prior exact algorithms scale to the version of the game with deck size larger than 4, and we compare performance to the popular algorithms of fictitious play (FP) and counterfactual regret minimization (CFR). We find that PED obtains a consistent near-monotonic improvement throughout all runs, though both FP and CFR perform significantly better in the initial iterations. This inspires a hybrid algorithm FP-PED that runs FP for an initial burn-in period before switching to PED for stable long-run refinement. We can alternatively view this as a multi-step algorithm that runs FP as a pre-processing step to obtain a strong initialization for PED.

The mean mechanism is known to be non-incentive-compatible, namely, rational players are incentivized to misreport their values. Despite this game-theoretic issue, the mean mechanism is prevalent in practice due to its other desirable properties. We give a full characterization of pure Nash equilibria--how the players will misreport--for the affine mechanism, of which the mean is a special case. Furthermore, we characterize both complete-information and Bayesian games under the affine mechanism. Our results highlight the inevitability of extreme exaggeration in such games.

Solving multi-agent optimal control problems in aerospace such as pursuit-evasion and contested space operations can be modeled as non-convex differential games for which, there are limited algorithms. In this work, a relaxation of generalized Nash Equilibrium problems (GNEPs) to exclude inter-agent control coupling in dynamics, which is representative of many multi-agent systems is introduced. The main contribution is an algorithm for solving a broad class of differential games named FALCON: Fast Augmented Lagrangian Convexification for Open-loop Nash equilibria is presented. Methodologically, sequential convex programming (SCP) is utilized to create tractable convex sub-games which can then be solved via standard convex programming methods involving a potential game reformulation. FALCON is demonstrated to have global convergence guarantees to an open-loop Nash equilibrium for non-convex differential games under mild assumptions. This is numerically shown through both cooperative and competitive differential games.

Many two-player zero-sum games admit not a unique Nash equilibrium but a convex set of them: a polytope of profiles that all share the minimax value V* yet prescribe different behaviour. Standard solvers each converge to some equilibrium and are treated as interchangeable. We ask whether they instead select different members of the Nash set, systematically as a function of the algorithm rather than the seed. Using a tabular, exactly solvable testbed of six games with analytically known Nash sets -- including a two-dimensional Nash polytope and Kuhn poker -- we find that (i) selection is determined by the algorithm, not the seed, but families differ only on asymmetric Nash sets; (ii) regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member, the information projection of their uniform reference onto the Nash set -- exactly on the 2-D polytope and at 99.7% of maximum entropy in Kuhn -- while regret-averaging methods (CFR, CFR+, fictitious play) drift to a lower-entropy face; we confirm this on a randomized 180-game ensemble, where R-NaD attains the maximum-entropy member in 100% of converged games while CFR+ sits strictly below it in 94% (paired Wilcoxon p < 10^-27); (iii) the selected member has downstream consequences against sub-optimal opponents that scale with sequential/hidden-information structure but stay bounded -- in Kuhn the max-entropy member is a strictly better hedge, whereas on the matrix games the members differ without either dominating. We also report two negative results correcting common intuitions: removing CFR's positive-orthant (max(R,0)) projection does not eliminate boundary drift; and R-NaD's selection is anchor-following, not initialization-independent. We state the maximum-entropy / I-projection characterization as a strongly data-supported conjecture, checked throughout against analytic ground truth.

Open-source game theory studies agents whose behavior may depend on one another's decision procedures, but most existing models use discrete or symbolic programs. We introduce parametric open-source games, a continuous analogue of program equilibria in which players choose parameter vectors and semantics maps convert the full parameter profile into mixed actions in an underlying finite game. We establish equilibrium existence results, derive an exact coupling threshold at which selfish gradient ascent in symmetric $2\times2$ games switches from defection toward cooperation, and give a one-dimensional boundary test for parametric program Nash equilibria. We further extend the framework to a neural semantics class whose first-order cooperation condition is governed by the ratio of cross-player to self-player sensitivity. Across canonical games, the framework shows how access to internal parameterizations can qualitatively reshape learning dynamics and equilibrium structure, and how sufficiently strong open-source coupling can steer selfish optimization toward cooperative outcomes.

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. Recently, an approach has been presented for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically constrained program based on a nonlinear complementarity problem formulation derived from the sequence-form game representation. This formulation was solved using Gurobi's nonconvex quadratic solver, which employs spatial branch-and-bound to iteratively refine variable bounds by solving convex relaxations of bilinear terms via McCormick envelopes. During presolve, Gurobi introduces auxiliary variables and, in some cases, binary variables, leading to an internal MIQCP reformulation. This approach was demonstrated to outperform prior algorithms from the Gambit software suite and quickly solve three-player Kuhn poker after removal of dominated actions; however, the algorithm was not able to solve the full version of the game within 24 hours. In this paper, we derive finite bounds on slack and multiplier variables in the nonlinear complementarity formulation. These bounds strengthen the convex relaxations used within spatial branch-and-bound and lead to substantial computational improvements. We demonstrate the impact of the proposed bounds on exact Nash equilibrium computation in three-player Kuhn poker.

This paper develops a noncooperative model of directed network formation in which agents create links to access valuable information while sharing the costs generated along the paths through which information is obtained. Each agent is endowed with a positive amount of information and chooses, simultaneously, which other agents to contact. A directed link initiated by one agent allows her to access the information of the contacted agent and of the latter's reachable network, but each link in the resulting information path entails a unit cost. Payoffs therefore depend on the total value of accessible information net of the accumulated connection costs required to obtain it. The paper characterizes the relationship between strategy profiles and directed graphs, defines accessibility, paths, components, and minimal connectedness, and studies the Nash architectures induced by individual best responses. The central result is that strict Nash equilibria must take the form of circular directed networks. Moreover, circular networks are exactly the Nash networks that use the minimum number of links while allowing every agent to access all available information. Although noncircular weak Nash networks may exist, they are structurally redundant and do not satisfy the same minimality property. The model also shows that strict Nash networks are both Pareto optimal and efficient in terms of aggregate welfare. Finally, the paper compares this framework with Bala and Goyal's model, emphasizing that shared path costs and heterogeneous information values generate different equilibrium implications. The analysis supports the equivalence between strict stability and minimal connectivity in directed information networks.

Nash equilibrium (NE) arises from selfish utility maximization, yet its social welfare can be arbitrarily far from optimal. Moreover, computing an NE is intractable in general. We study augmented game models in which players use budget-balanced internal transfers to improve incentives before play. We first introduce \emph{Self-Enforcing Transfer Equilibrium} (SETE), where players commit to nonnegative peer-to-peer transfers that are paid only if the recipient does not deviate from a prescribed strategy. For polymatrix games, we show that every stationary point of the social welfare function, in particular any socially optimal strategy profile, can be sustained as a SETE. This induces a Nash equilibrium in the agent normal form of the corresponding augmented game. We further propose a polynomial-time algorithm and a decentralized learning dynamic to compute such product-form equilibria. We then introduce \emph{Mediated Self-Enforcing Transfer Equilibrium} (M-SETE), where a mediator makes both the payment schedule and the prescribed strategies binding offers. This additional enforcement resolves the agent-normal-form limitation: an M-SETE is a Nash equilibrium of the augmented game itself, not merely of its agent normal form, and any socially optimal strategy profile can be supported as an M-SETE in any finite game while preserving budget balance. Thus, internal transfers improve welfare and computation while preserving independent play on the equilibrium path. When full sequential-game stability is required, binding mediation provides the corresponding implementation.

Safe coordination in networked cyber-physical systems forces learning algorithms to simultaneously handle hybrid discrete-continuous actions, hard training-time safety constraints, and physics-governed dynamics. We show that these three features form a directed cycle of biases that defeats any naive composition of off-the-shelf modules, and formalize this as a three-way coupling lemma. We then introduce TRIDENT, the first MARL framework whose three components are co-designed to cancel each leak: a Richardson-Romberg gradient correction reducing Gumbel-Softmax bias from O(tau) to O(tau^2), a Lyapunov-constrained sequential trust-region update enforcing per-iterate feasibility, and a physics-informed residual critic that decomposes value rather than reward. We prove an O~(1/sqrt(K)) convergence rate to a constrained Nash equilibrium and an O(sqrt(K)) cumulative-violation bound. On multi-UAV mobile-edge computing, autonomous intersection management, and a hybrid SMAC variant, TRIDENT cuts training-time violations by 95.5% over MADDPG and 76.3% over MACPO, while improving reward by 13.5% over the strongest unconstrained baseline.

We study strategic space- and time-constrained cooperation between two self-interested agents through the Intermittent Strategic Cooperation-Based Two-Agent Path Planning (IC2PP) problem, a shortest-path game on graphs in which agents navigate toward individual targets while optionally cooperating at specific nodes to reduce their own travel times. Although such cooperation can strictly benefit both agents, it is strategically fragile: agents may deviate at any point along their paths. Modeled as a 2-player game, we characterize the structure of Pure Nash Equilibrium (PNE) joint strategies in IC2PP, and show that stable cooperation must follow a highly constrained form. We further prove that at least one PNE exists in every instance of IC2PP, and present a polynomial-time algorithm for enumerating all relevant PNEs. When multiple equilibria arise, we study coordination mechanisms based on bargaining-theoretic selection concepts and empirically compare equilibrium outcomes in terms of individual travel times and social welfare.

We study repeated bilateral trade from a fairness perspective. At each round, a fresh seller-buyer pair arrives, and the platform posts a price before observing the traders' valuations. Trade occurs only if both agents accept the price. Rather than maximizing only the gain from trade, we consider platforms that seek balanced divisions of the generated surplus. We show that natural fairness desiderata lead to a one-parameter Rawls-to-Nash family of fair-gain objectives, obtained by aggregating the seller's and buyer's net gains through nonpositive Hölder means. Unlike the standard gain-from-trade objective and the Rawlsian fair-gain objective studied in prior work, our proposed objectives induce a new statistical structure in which expected rewards are recovered from threshold feedback through a two-dimensional singular-kernel integral identity. This leads to a nonstandard pure-exploration problem whose natural estimators are rectangular double sums with row-column dependence and singular weights. Assuming independent i.i.d. seller and buyer valuation sequences with arbitrary unknown marginals, we characterize the optimal learning rates for the whole Rawls-to-Nash family of fair-gain objectives, giving matching fixed-confidence sample-complexity and regret bounds up to polylogarithmic factors.

Cooperation in multi-agent systems is typically optimized through utilitarian objectives that maximize overall efficiency but fail to account for reward distribution, often resulting in inequitable "leader-follower" dynamics. While fairness-based approaches encourage pro-social behaviors where every agent benefits from cooperation, many current algorithms - including those utilizing reward shaping - break the stationarity of Markov Games or lack rigorous theoretical guarantees. This creates a critical gap between fair objective methods and theoretically safe learning frameworks. We propose a novel framework that bridges $α$-fairness with Heterogeneous-Agent Trust Region Learning (HATRL), ensuring monotonic improvement and convergence toward Nash Equilibria. Our approach leverages a fair advantage function that dynamically weights agent utilities based on their expected returns, allowing the global objective to transition from purely utilitarian efficiency to $α$-fairness welfare based on the parameter $α$. We introduce two practical algorithms, $α$-fair HATRPO and $α$-fair HAPPO, and demonstrate through experiments in sequential social dilemmas like CleanUp and CommonHarvest that they perform better than HATRL's algorithms from a utilitarian point of view while achieving socially higher outcomes.

Mean-based algorithms are a class of online learning algorithms that assign low probability to actions with low average rewards. Recent work indicates these algorithms converge favorably to serially undominated actions, which approximate Nash equilibria in economic games. However, empirical studies also show slower convergence compared to established algorithms in bandit-feedback scenarios. We study mean-based algorithms when the time horizon is unknown and only bandit feedback is available. In this setting, we provide the first lower bound on the algorithm-defining sequence $γ_t$ that formally establishes a limit on how fast these algorithms can learn. Additionally, we propose two mean-based algorithms: one generalizes $ε$-greedy, and the other extends the mean-based Exp3 to unknown horizons. Our experiments show that mean-based algorithms, although slightly slower, can perform competitively with other bandit-feedback algorithms. We further analyze the relationship to no-regret algorithms. Depending on the choice of $γ_t$, the intersection with no-regret algorithms is non-trivial, and we show that algorithms exist that are both mean-based and no-regret. This adds context to the "exploitability" of this class of algorithms that previous contributions suggest.

The coordination of heterogeneous autonomous agents in dynamic, adversarial environments requires simultaneous satisfaction of geometric constraints, logical consistency, temporal reasoning, and strategic optimization. Existing sheaf- and topos-theoretic frameworks provide powerful tools for geometric consensus, knowledge alignment, and causal planning, but lack explicit models for value, reward, and strategic choice. This report presents a unified categorical framework that integrates event calculus, SCEL-like ensemble formation, and game-theoretic reward structures into a single Grothendieck topos of time-space histories. We introduce the notion of a \emph{game sheaf} whose stalks contain utility functions and policy distributions, and restriction maps encode both parallel transport and best-response dynamics. We prove that Nash equilibria correspond to global sections of a derived best-response correspondence sheaf, while cohomological obstructions classify failures of strategic consistency. A detailed case study of an immunological ``bastion defense'' scenario -- heterogeneous agents forming attack/defense ensembles under resource constraints -- demonstrates the framework's expressiveness. This synthesis provides a rigorous foundation for verifiable, autonomic, and economically rational multi-agent systems.

Preference alignment is central to improving large language models, but standard reward-based formulations can be restrictive when human preferences are cyclic, non-transitive, or otherwise not representable by a scalar reward. Nash Learning from Human Feedback (NLHF) addresses this limitation by modeling alignment as a preference game and targeting a Nash equilibrium rather than a reward maximizer. However, the learning-theoretic foundations of scalable NLHF remain limited. Existing regret guarantees rely on oracle-based methods that estimate a general preference model and solve KL-regularized minimax problems, while iterative NLHF methods directly optimize policy-level preference losses and are easier to implement but lack regret guarantees. We study online iterative NLHF under general preference models and identify exploration as the key obstacle. First, we show that standard iterative NLHF can suffer an exponential dependence on the KL-regularization parameter, revealing that implicit exploration through policy updates is insufficient for controlling regret. Second, we propose an explicitly exploratory iterative NLHF algorithm that combines SFT-based regularization with adversarial policy exploration. The resulting method retains the direct policy optimization structure of iterative NLHF, avoids explicit preference model estimation, and achieves an $O(\sqrt{T})$ regret bound without an exponential dependence on the KL-regularization parameter. We show that the regret can be improved to $O(\log(T))$ with access to a minimax oracle, clarifying the computational-statistical tradeoff in learning general preference games. Finally, we instantiate our method for LLM fine-tuning and evaluate it on \texttt{Llama-3-8B-Instruct} across multiple benchmarks, where explicit exploration yields consistent improvements over existing NLHF baselines.

Recent advances in artificial intelligence have expanded the focus from classical optimization to include equilibrium analysis in noncooperative games. Many such games involve shared constraints, leading to Generalized Nash Equilibrium Problems (GNEPs). Existing distributed algorithms typically require agents to exchange Lagrange multipliers to enforce consensus and compute variational-GNEs (v-GNEs). This work introduces fully distributed continuous-time algorithms and establishes convergence without requiring multiplier exchange, thereby reducing information exchange per iteration while improving privacy preservation. The analysis focuses on strongly monotone games with convex individual constraints and linear shared constraints. I also propose several discretization schemes for the continuous-time algorithms. The proposed approach converges to general GNEs, rather than being restricted to v-GNEs, with the attained equilibrium depending on the initialization. The effectiveness of the proposed method is demonstrated through applications in multi-robot coordination and placement. In the second part, this work includes research conducted in collaboration with Amazon scientists. One of the most challenging problems in real-world machine learning is labeled data collection, which typically requires substantial human effort and cost. Active learning aims to reduce this labeling requirement. Existing handcrafted active learning strategies, however, generally perform well only on specific types of datasets, which are often unknown in advance. In this work, I propose using contextual bandits to adaptively select the most suitable active learning strategy. The effectiveness of the proposed approach is demonstrated on publicly available external datasets.

We argue that governance must transition from a normative discipline to an engineering discipline, and we develop a formal framework, inspired by the physics of metamaterials, to make this transition quantitative and testable. Artificial General Intelligence affects civilization primarily by increasing decision velocity while human verification capacity remains bounded. When the cost of validating AI-generated outputs exceeds the expected utility of acting on them, rational agents default to inaction: a stable but catastrophic Nash equilibrium we term the Freezing Equilibrium. Drawing on metamaterials, where emergent macro-properties arise from designed microstructure, we develop a phenomenological constitutive law for institutional coordination: $R_{\mathrm{eff}} = β\cdot (1-ρ) \cdot (1-τ) \cdot (1-γρτ)$, where $β$ is the decision branching factor, $ρ$ is provenance fidelity, $τ$ is the verification rate, and $γ\in [0,1]$ captures correlated-detection synergy between provenance and verification failures. The model predicts a sharp phase transition between self-healing ($R_{\mathrm{eff}} < 1$) and self-destabilizing ($R_{\mathrm{eff}} > 1$) regimes. We introduce a three-class provenance taxonomy: cryptographic, institutional, and context binding, and derive four falsifiable hypotheses with a proposed 12-week stepped-wedge cluster-randomized trial in government grant review panels. The framework bridges AI alignment theory and institutional design.

Trauma resuscitation is a clinical process for treating life-threatening physiological disorders in safety-critical environments, driven by the experience of healthcare workers (HCWs). Designing and optimizing quantifiable metrics that accurately capture HCW decisions may augment current resuscitation procedures with the potential to improve patient outcomes. This motivates our socio-technical formulation of trauma resuscitation as a distributed generalized Nash equilibrium (GNE)-seeking game with coupled inequality constraints. This method is optimized over a time-varying communication graph. We introduce novel insights from clinical experience to model HCWs behavior. This work facilitates the best possible resuscitation outcome given HCWs workloads, schedules, competencies, and limited resources.

Vehicle-to-vehicle (V2V) energy trading enables decentralized peer-to-peer energy exchange among electric vehicles (EVs), reducing grid dependency while monetizing surplus capacity. However, coordinating self-interested EV agents with diverse charging needs and uncertain arrival-departure schedules remains challenging. Existing approaches either require centralized optimization with computational limitations or lack fairness guarantees. This paper integrates Nash Bargaining Solution into Multi-Agent Deep Deterministic Policy Gradient, namely Nash-MADDPG, for incentive-aligned V2V energy trading. Nash bargaining determines efficient bilateral pricing, while Nash-guided price proximity rewards align agent learning toward bargaining-optimal strategies. Evaluation over 30-day continuous operation demonstrates an improvement of 61.6% in social welfare and 62.9% improvement in trading volume over Double Auction, while achieving superior fairness, such as 40.1% improvement in Jain's index. Testing across 6-100 agents over a 30-day horizon with continuous vehicle turnover confirms scalability across population size and empirically stable pricing near the Nash Bargaining benchmark.

Multimodal large language models often generate reasoning chains containing subtle errors that lead to incorrect answers. Current verification approaches have notable limitations. Learned critics need extensive labeled data and show inconsistent performance across different tasks. Meanwhile, existing training-free methods simply average scores from different sources, missing a key insight: when these scores disagree, that disagreement itself carries important information about whether a reasoning step is truly valid or not. We propose a training-free verification approach that treats step-wise verification as a coordination problem among specialized judges. We formalize these judges' interaction as a Nash equilibrium game where agreement signals valid steps while disagreement reveals instability. Our method computes equilibrium scores through a closed-form solution, enabling both disagreement-aware filtering and stability-conscious ranking of reasoning steps. Evaluated across six benchmarks, our approach achieves consistent improvements of 2.4% to 5.2% over baseline models and shows competitive performance against learned critics, demonstrating that cross-modal agreement (not just average confidence) provides robust verification signals without task-specific adaptation.

To verify the robustness of a program or protocol, it is common in the computer science community to rely on the theoretical framework of game theory. In particular, if one seeks to enforce a desired property, or specification, despite an unpredictable environment, a useful abstraction is to model the situation as a two-player zero-sum game. The goal is then to find a strategy for the system that guarantees the specification against any strategy of the environment. However, to model more complex situations, such as multiple systems with different objectives or an environment composed of various agents, the richer framework of multiplayer games must be considered. In this setting, a natural question is to identify equilibria, i.e., strategy profiles that are robust in the sense that no player has an incentive to deviate. The most well-known equilibrium concept is the Nash equilibrium, but several alternatives exist. We study five such notions and, for each of them, we provide complexity results for the constrained existence problem, which consists of deciding whether a given game contains an equilibrium that ensures each player a payoff within a specified interval.

Additively separable hedonic games (ASHGs) are a prominent model of coalition formation where agents' preferences are derived from their individual valuations of peers. While social welfare maximization in ASHGs has traditionally focused mostly on utilitarian welfare, Nash welfare -- a well-established metric in economics which balances fairness with efficiency and offers scale invariance -- has been entirely overlooked. In this paper, we initiate the study of Nash welfare in ASHGs. We point out desirable properties fulfilled by partitions with high Nash welfare. This includes guaranteed contractual Nash stability in symmetric games, even for any approximation of Nash welfare. This is particularly appealing since, as for other welfare notions, Nash welfare turns out to be NP-hard to maximize, even for the ASHG subclass of symmetric aversion to enemies games (AEGs). A main focus of our study is on approximation algorithms for the Nash welfare objective. We present packing-based algorithms with approximation ratios for well-established subclasses of ASHGs: $n-1$ for AEGs and $2n$ for appreciation of friends games. This is complemented by a strict inapproximability result showing it is NP-hard to approximate Nash welfare within a factor of $1.0000759$ in general ASHGs. Further, we investigate the restricted settings with an upper bound on the coalition size or number of coalitions, and draw the boundary between the cases admitting efficient algorithms and those yielding NP-hardness: bounding the allowed size or number of coalitions by $2$ admits polynomial-time solvability, whereas bounds of $3$ or more yield NP-hardness or unbounded inapproximability.

Multi-agent policy-gradient methods have been shown to converge locally near stable Nash equilibria. Local convergence, however, does not determine which equilibrium is reached. We study this question through basin-entry probability with respect to a target set of equilibria selected by an external criterion, such as payoff dominance. For finite-unroll Meta-MAPG, we show that the update decomposes into ordinary policy gradient plus own-learning and peer-learning corrections, with controlled sampling noise and finite-unroll bias. We identify the peer-learning correction as the main equilibrium-selection mechanism: under a local alignment condition, the probability of entering the certified attraction region of the target stable-Nash set increases, relative to ordinary policy gradient. Because persistent correction may shift zero-update points of the original game, annealing the correction after entering the basin recovers ordinary policy-gradient dynamics and inherits local stable-Nash convergence guarantees. Experiments in Stag Hunt, iterated Prisoner's Dilemma, and preliminary neural-policy coordination environments support this basin-entry view, showing increased entry into cooperative basins under peer-aware updates.

Responsibility allocation -- determining the extent to which agents are accountable for outcomes -- is a fundamental challenge in the design and analysis of multi-agent systems. In this work, we model such systems as concurrent stochastic multi-player games and introduce a notion of retrospective (backward) counterfactual responsibility, which quantifies an agent's accountability for outcomes resulting from a given strategy profile. To allocate responsibility among agents, we utilise the Shapley value and formally show that this method satisfies key desirable properties, including fairness and consistency. Building on this foundation, we propose a formal framework that supports both verification and strategic reasoning in responsibility-aware multi-agent systems. Furthermore, by adopting Nash equilibrium as the solution concept, we demonstrate how to compute stable strategy profiles in which agents trade off responsibility against expected reward.

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $\widetilde{\mathcal{O}}(1/n)$ convergence rate under \textit{unilateral concentrability}, improving over the standard $\widetilde{\mathcal{O}}(1/\sqrt{n})$ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $\widetilde{\mathcal{O}}(1/n)$ statistical rate up to a vanishing optimization error of order $\widetilde{\mathcal{O}}(1/\sqrt{T})$ in the number of self-play iterations $T$.

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.

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.

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.

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.

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.

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.

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 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.

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.

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 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.

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.

We propose and study distributionally robust Markov games (DR-MGs) with the average-reward criterion as a crucial framework for multi-agent decision-making under model mismatches and over extended horizons. Under a standard irreducible assumption, we first derive a correspondence between the optimal policies and the solutions of the robust Bellman equation, based on which we further show the existence of a stationary Nash Equilibrium (NE) of the game. We further study DR-MGs under a more general weakly communicating setting. We construct a set-valued map based on the constant-gain optimal robust Bellman operator and show that its value is a subset of the best-response policies. We further prove that this map admits a fixed point, which implies the existence of NE. We then design two algorithms, Robust Nash-Iteration and robust TD Descent, with provably convergent guarantees. Finally, we show that the NE under average-reward can be approximated by the ones for the discounted DR-MGs as the discount factor approaches one. Our studies provide a comprehensive theoretical and algorithmic foundation for decision-making in complex, uncertain, and long-running multi-player environments.

This paper develops a simulation-based framework for evaluating the energy implications of quantum and classical computing firms competing in a market with limited energy resources. We model providers as differentiated Cournot competitors whose feasible service capacity is induced by technology-specific energy scaling laws: polylogarithmic for quantum algorithms that achieve an equivalent computational target and polynomial for classical emulation. For symmetric groups of quantum and classical firms, the equilibrium reduces to a tractable two-equation system that supports large scenario sweeps over market size, technology mix, and hardware coefficients. We characterize the capacity-constrained Nash equilibrium, prove the existence of a demand scale beyond which quantum service becomes more energy efficient, and report numerical experiments calibrated to trapped-ion and Rydberg platforms. The results identify when quantum energy advantage is only asymptotic and when it becomes operationally relevant.

While Nash equilibrium has emerged as the central game-theoretic solution concept, many important games contain several Nash equilibria and we must determine how to select between them in order to create real strategic agents. Several Nash equilibrium refinement concepts have been proposed and studied for sequential imperfect-information games, the most prominent being trembling-hand perfect equilibrium, quasi-perfect equilibrium, and recently one-sided quasi-perfect equilibrium. These concepts are robust to certain arbitrarily small mistakes, and are guaranteed to always exist; however, we argue that neither of these is the correct concept for developing strong agents in sequential games of imperfect information. We define a new equilibrium refinement concept for extensive-form games called observable perfect equilibrium in which the solution is robust over trembles in publicly-observable action probabilities (not necessarily over all action probabilities that may not be observable by opposing players). Observable perfect equilibrium correctly captures the assumption that the opponent is playing as rationally as possible given mistakes that have been observed (while previous solution concepts do not). We prove that observable perfect equilibrium is always guaranteed to exist, and demonstrate that it leads to a different solution than the prior extensive-form refinements in no-limit poker. We expect observable perfect equilibrium to be a useful equilibrium refinement concept for modeling many important imperfect-information games of interest in artificial intelligence.