best-response approximation

This work addresses the challenge of deploying nonlinear classifiers in strategic classification settings, where computing agents’ optimal responses is typically intractable. The authors model an agent’s strategic response as a constrained optimization problem and leverage Lagrangian duality to reformulate it into a first-order differentiable form. By applying the implicit function theorem, they explicitly establish gradient-based relationships between classifier parameters and strategic behaviors. This approach enables, for the first time, scalable end-to-end training of nonlinear strategic classifiers and provides a unified framework for approximating optimal responses in both linear and nonlinear settings. Experimental results on multiple benchmark datasets demonstrate significant improvements in strategic accuracy, confirming the method’s effectiveness and practical utility.

This paper addresses the problem of computing optimal responses in strategic classification, where classifier deployment induces strategic user behavior that shifts the data distribution. Existing methods are largely restricted to linear models and fail to yield tractable solutions for nonlinear classifiers. To overcome this limitation, we propose a novel framework based on Lagrangian dual optimization of a surrogate objective, unifying optimal-response computation with classifier training. Our approach exactly recovers known optimal solutions in the linear case and exposes theoretical shortcomings of several prior methods. For the first time, it enables scalable, differentiable estimation of optimal responses for nonlinear models—including kernel SVMs and neural networks. Experiments demonstrate that our method significantly improves model robustness and generalization under strategic manipulation, establishing the first general-purpose, practical paradigm for nonlinear strategic classification.

This work addresses the computation of Nash equilibria in two-player constrained games under asymmetric information, where one player lacks full knowledge of the other’s objective and constraints and can only interact via a best-response mapping. The authors propose an iterative algorithm that combines projected gradient descent with best-response updates, requiring no complete disclosure of either player’s optimization problem and applicable to settings with decoupled feasible sets. Under standard regularity conditions, they establish—for the first time—the global linear convergence of the algorithm in such an asymmetric regime relying solely on best-response access. Moreover, when the best-response is subject to uniformly bounded errors of magnitude ε, the iterates converge to an O(ε)-neighborhood of the equilibrium, with an explicit error bound provided. Numerical experiments corroborate the theoretical convergence rates and sensitivity to response inaccuracies.

This work studies Markov games with time-varying cost constraints, modeling safety-critical cooperative multi-agent reinforcement learning scenarios—e.g., autonomous teams operating under energy- or time-limited budgets. Conventional methods struggle with joint-action-dependent and state-coupled constraints. To address this, we introduce Lagrangian relaxation to constrained Markov games for the first time, proposing a Lagrangian game framework and an iterative primal-dual algorithm. Theoretically, under mild assumptions, the generated policy sequence converges to a non-stationary Nash equilibrium of the original constrained problem; moreover, online updates of the Lagrange multipliers ensure dynamic satisfaction of time-varying constraints. Empirical evaluations demonstrate the framework’s effectiveness in safety-aware decision-making under resource constraints. This work establishes the first verifiable, convergence-guaranteed optimization framework for safety-critical multi-agent systems.

This study addresses the challenge of designing an optimal recommendation mechanism in a finite-horizon discrete-time dynamic system where a system designer cannot directly control the actions of two strategic agents. The designer aims to maximize their own objective by recommending actions based on shared historical information, while ensuring that the agents find it sequentially rational to follow these recommendations, thereby forming a sequential rationality equilibrium. To this end, the paper proposes a novel recommendation mechanism that explicitly satisfies sequential rationality constraints and develops a computationally tractable solution framework combining backward induction with linear programming. This approach achieves, for the first time, the optimization of the designer’s objective under strict sequential rationality conditions, demonstrating both the effectiveness and computational feasibility of the proposed mechanism.

Computing Nash (or generalized Nash) equilibria in dynamic games is highly challenging due to coupled optimality conditions, nested optimization structures, and numerical ill-conditioning. This work proposes a data-driven, structured decomposition approach that circumvents these difficulties by offline construction of each agent’s best-response mapping, which is then embedded as a feasibility constraint. This formulation eliminates nested optimization and derivative coupling while preserving equilibrium consistency—without requiring explicit modeling of all objectives and constraints or approximation via policy prediction. By integrating best-response embedding, structured optimization reformulation, and large-scale Monte Carlo validation, the method significantly improves computational efficiency and constraint satisfaction in a two-player open-loop autonomous racing game, yielding high-quality approximate equilibrium solutions.

This study investigates the statistical properties of Lagrange multipliers in constrained maximum likelihood estimation and least squares problems, along with their implications for numerical optimization. Leveraging large-sample theory, it establishes that under correctly specified models, Lagrange multipliers converge in probability to zero as the sample size grows, a result extended to high-dimensional settings such as deep learning. Building on this asymptotic behavior, the work provides the first statistical justification for initializing Lagrange multipliers at zero and integrates this insight into constrained optimization algorithms, including augmented Lagrangian methods and sequential quadratic programming. Numerical experiments demonstrate that this initialization strategy substantially enhances algorithmic stability and convergence efficiency in applications such as constrained regression and dynamic discrete choice models.

This study investigates the computational complexity of computing optimal linear contracts in combinatorial contract design, particularly when the reward function exhibits both substitutable and complementary relationships among outcomes. By establishing a geometric connection between combinatorial contracts and demand types in consumer theory, the authors reduce the problem of bounding the number of critical values to counting how many regions of best responses a contract ray intersects. They introduce a novel class of reward functions—termed ASC (All Substitutes and Complements)—which strictly generalizes and unifies all previously known classes admitting polynomially many critical values, and conjecture it to be maximal with this property. Leveraging this geometric framework and efficiently simulating demand queries via value queries, the paper presents a general algorithm for “succinct” demand types and achieves, for the first time, polynomial-time computation of optimal contracts for a new class of reward functions incorporating both substitutes and complements.

This work investigates the impact of utility perturbation on the convergence rate of optimal-response-based Nash equilibrium algorithms—specifically Double Oracle and Fictitious Play—in zero-sum games. To address the common issue where iteration counts grow linearly or superlinearly with strategy space size, we introduce controlled utility perturbations during best-response computation. We theoretically prove that this mechanism reduces the expected number of iterations to logarithmic in the strategy space size. Furthermore, we design efficient, structure-aware perturbation schemes tailored to pure-strategy spaces exhibiting intrinsic structure—such as decomposability, sparsity, or low-rankness. Experiments across multiple classes of structured zero-sum games demonstrate that perturbation significantly accelerates convergence without compromising equilibrium accuracy. Our core contribution is the first systematic, quantitative characterization of the relationship between utility perturbation and convergence rates for best-response algorithms, coupled with a structure-adaptive perturbation optimization framework.