a b s t r a c tThe Mathieu operator L(y) = −y ′′ + 2a cos(2x)y, a ∈ C, a ̸ = 0, considered with periodic or anti-periodic boundary conditions has, close to n 2 for large enough n, two periodic (if n is even) or anti-periodic (if n is odd) eigenvalues λ − n , λ + n . For fixed a, we show thatThis result extends the asymptotic formula of Harrell-Avron-Simon by providing more asymptotic terms.

In this paper, we introduce a regularized mean-field game and study learning of this game under an infinite-horizon discounted reward function. The game is defined by adding a regularization function to the one-stage reward function in the classical mean-field game model. We establish a value iteration based learning algorithm to this regularized mean-field game using fitted Q-learning. This regularization term in general makes reinforcement learning algorithm more robust with improved exploration. Moreover, it enables us to establish error analysis of the learning algorithm without imposing restrictive convexity assumptions on the system components, which are needed in the absence of a regularization term.

In the literature, existence of mean-field equilibria has been established for discretetime mean field games under both the discounted cost and the average cost optimality criteria. However, there is no algorithm with convergence guarantee for computing mean-field equilibria for a general class of models. In this paper, we provide a value iteration algorithm to compute mean-field equilibrium for both the discounted cost and the average cost criteria. We establish that the value iteration algorithm converges to the fixed point of a mean-field equilibrium operator. Then, using this fixed point, we construct a mean-field equilibrium. In our value iteration algorithm, we use Q-functions instead of value functions for possible extension of this work to the model-free setting.

No abstract

In this paper, we introduce a regularized mean-field game and study learning of this game under an infinite-horizon discounted reward function. Regularization is introduced by adding a strongly concave regularization function to the one-stage reward function in the classical mean-field game model. We establish a value iteration based learning algorithm to this regularized mean-field game using fitted Q-learning. The regularization term in general makes reinforcement learning algorithm more robust to the system components. Moreover, it enables us to establish error analysis of the learning algorithm without imposing restrictive convexity assumptions on the system components, which are needed in the absence of a regularization term. Keywords Mean-field gamesThis article is part of the topical collection "Multi-agent Dynamic Decision Making and Learning" edited by