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Mean Field Games (MFGs) have been introduced to efficiently approximate games with very large populations of strategic agents. Recently, the question of learning equilibria in MFGs has gained momentum, particularly using model-free…
In the context of simple finite-state discrete time systems, we introduce a generalization of mean field game solution, called correlated solution, which can be seen as the mean field game analogue of a correlated equilibrium. Our notion of…
We develop a splitting method to prove the well-posedness, in short time, of solutions for two master equations in mean field game (MFG) theory: the second order master equation, describing MFGs with a common noise, and the system of master…
We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for…
We present the concept of a Generalized Feedback Nash Equilibrium (GFNE) in dynamic games, extending the Feedback Nash Equilibrium concept to games in which players are subject to state and input constraints. We formalize necessary and…
We analyze two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm. While classical FW algorithms require solving a linear minimization problem over the domain at each iteration, the proposed method only…
In this work, we consider a first order mean field games system with non-local couplings. A Lagrange-Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton-Jacobi-Bellman equation, is proposed to…
The designs of many large-scale systems today, from traffic routing environments to smart grids, rely on game-theoretic equilibrium concepts. However, as the size of an $N$-player game typically grows exponentially with $N$, standard game…
We use the Markov chain approximation method to construct approximations for the solution of the mean field game (MFG) with reflecting barriers studied in Bayraktar, Budhiraja, and Cohen (2017). The MFG is formulated in terms of a…
Many economic games and machine learning approaches can be cast as competitive optimization problems where multiple agents are minimizing their respective objective function, which depends on all agents' actions. While gradient descent is a…
We study the Frank-Wolfe algorithm for constrained optimization problems with relatively smooth objectives. Building upon our previous work, we propose a fully adaptive variant of the Frank-Wolfe method that dynamically adjusts the step…
We obtain global, non-asymptotic convergence guarantees for independent learning algorithms in competitive reinforcement learning settings with two agents (i.e., zero-sum stochastic games). We consider an episodic setting where in each…
We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean…
The convexification numerical method with the rigorously established global convergence property is constructed for a problem for the Mean Field Games System of the second order. This is the problem of the retrospective analysis of a game…
A variety of practical problems can be modeled by the decision-making process in multi-player games where a group of self-interested players aim at optimizing their own local objectives, while the objectives depend on the actions taken by…
We develop a generalized gradient expansion of the inhomogeneous dynamical mean-field theory method for determining properties of ultracold atoms in a trap. This approach goes beyond the well-known local density approximation and at higher…
We introduce a new algorithm for the numerical computation of Nash equilibria of competitive two-player games. Our method is a natural generalization of gradient descent to the two-player setting where the update is given by the Nash…
We consider seeking generalized Nash equilibria (GNE) for noncooperative games with coupled nonlinear constraints over networks. We first revisit a well-known gradientplay dynamics from a passivity-based perspective, and address that the…
In this paper, we characterize the asymptotic behavior of a first-order stationary mean-field game (MFG) with a logarithm coupling, a quadratic Hamiltonian, and a periodically oscillating potential. This study falls into the realm of the…
Generative adversarial networks (GANs) are powerful tools for learning generative models. In practice, the training may suffer from lack of convergence. GANs are commonly viewed as a two-player zero-sum game between two neural networks.…