Misalignedrewards can lead to suboptimal behaviors [Ngo et al., 2022], undermining the potential benefits of RL in practical scenarios.

Estimating the unknown reward functions driving agents' behaviors is of central interest in inverse reinforcement learning and game theory. To tackle this problem, we develop a unified framework for reward function recovery in two-player zero-sum matrix games and Markov games with entropy regularization, where we aim to reconstruct the underlying reward functions given observed players' strategies and actions. This task is challenging due to the inherent ambiguity of inverse problems, the non-uniqueness of feasible rewards, and limited observational data coverage. To address these challenges, we establish the reward function's identifiability using the quantal response equilibrium (QRE) under linear assumptions. Building upon this theoretical foundation, we propose a novel algorithm to learn reward functions from observed actions. Our algorithm works in both static and dynamic settings and is adaptable to incorporate different methods, such as Maximum Likelihood Estimation (MLE). We provide strong theoretical guarantees for the reliability and sample efficiency of our algorithm. Further, we conduct extensive numerical studies to demonstrate the practical effectiveness of the proposed framework, offering new insights into decision-making in competitive environments.

Multivariate continuous latent variables arise in numerous empirical settings in economics, psychology, marketing, epidemiology, and the social sciences. Examples include multidimensional skills, cognitive factors, latent preferences, health indices, productivity components, and risk attitudes. In practice, such latent constructs are rarely observed directly; instead, researchers rely on multiple imperfect measurements that contain potentially correlated forms of noise. The resulting measurement-error problem is especially severe when the latent variable is multidimensional: each measurement typically captures only a low-dimensional projection of the latent vector, and the noise may exhibit dependence or heterogeneity across measurement channels. As a result, classical approaches to continuous measurement error, which often rely on injectivity, offer limited guidance. This paper develops a identification strategy tailored specifically to multivariate continuous latent variables measured with noise. The key innovation is to combine tools from multi-linear algebra--specifically the uniqueness properties of so-called CP tensor decompositions-- with the multivariate extension of Kotlarski's identity, a powerful deconvolution result based on characteristic functions. The starting point is the observation that third-order cross-moments of three separate measurements form a three-way moment tensor whose CP decomposition is governed by the latent factor loading matrices. By invoking Kruskal's theorem (Kruskal, 1977a), I show that these loadings are generically identifiable even when each measurement matrix is rank-deficient and--even more surprisingly--when the stack of all measurement matrices is non-injective.

This paper studies the linear quadratic regulation (LQR) problem of unknown discrete-time systems via dynamic output feedback learning control. In contrast to the state feedback, the optimality of the dynamic output feedback control for solving the LQR problem requires an implicit condition on the convergence of the state observer. Moreover, due to unknown system matrices and the existence of observer error, it is difficult to analyze the convergence and stability of most existing output feedback learning-based control methods. To tackle these issues, we propose a generalized dynamic output feedback learning control approach with guaranteed convergence, stability, and optimality performance for solving the LQR problem of unknown discrete-time linear systems. In particular, a dynamic output feedback controller is designed to be equivalent to a state feedback controller. This equivalence relationship is an inherent property without requiring convergence of the estimated state by the state observer, which plays a key role in establishing the off-policy learning control approaches. By value iteration and policy iteration schemes, the adaptive dynamic programming based learning control approaches are developed to estimate the optimal feedback control gain. In addition, a model-free stability criterion is provided by finding a nonsingular parameterization matrix, which contributes to establishing a switched iteration scheme. Furthermore, the convergence, stability, and optimality analyses of the proposed output feedback learning control approaches are given. Finally, the theoretical results are validated by two numerical examples.