Article
Mathematics, Applied
Qingda Wei, Xian Chen
Summary: This paper investigates nonzero-sum games for continuous-time jump processes with Borel state spaces, considering the expected average payoff criterion. The study introduces auxiliary static games and proves the existence of a stationary Nash equilibrium in the class of all randomized history-dependent strategy profiles using a discounted approximation technique. An example is provided to illustrate the optimality conditions.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Operations Research & Management Science
Qingda Wei
Summary: This paper discusses nonzero-sum discrete-time constrained stochastic games under the expected average payoff criteria. By constructing auxiliary models and proving ergodicity, the existence of constrained Nash equilibria is established. Additionally, the existence of a stationary constrained Nash equilibrium for the original game model is proven using an approximation technique.
Article
Mathematics, Interdisciplinary Applications
Qingda Wei, Xian Chen
Summary: This paper studies discrete-time nonzero-sum stochastic games under the risk-sensitive average cost criterion. It introduces risk-sensitive first passage payoff functions and their properties, establishes the existence of a solution to the risk-sensitive average cost optimality equation for the case of unbounded cost functions, and demonstrates the existence of a randomized stationary Nash equilibrium in the class of randomized history-dependent strategies. The main results are illustrated using a controlled population system.
DYNAMIC GAMES AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Qingda Wei
Summary: This paper studies nonzero-sum constrained stochastic games for continuous-time jump processes with denumerable states and possibly unbounded transition rates. It allows player payoff functions to be unbounded and demonstrates the existence of stationary constrained Nash equilibria through approximating game models and constructing a suitable multifunction. The results show that any limit point of the stationary constrained Nash equilibria in the approximating sequence is a constrained Nash equilibrium in the original game model.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Operations Research & Management Science
Qingda Wei, Xian Chen
Summary: This paper studies discrete-time nonzero-sum stochastic games under the risk-sensitive first passage discounted cost criterion. The main results include the existence and uniqueness of the risk-sensitive first passage discounted optimal value function for each player, and the existence of a randomized Markov Nash equilibrium.
Article
Acoustics
Mostafa Bagheri, Peiman Naseradinmousavi
Summary: A Nash-based feedback control law is formulated for an Euler-Lagrange system to address efficiency issues and achieve accuracy in trajectory tracking for robot manipulators. The performance of the control law is examined analytically and experimentally, showing almost perfect tracking with closed-loop stability.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Mathematics, Applied
Xin Guo, Jian Chen, Zechao Li
Summary: This paper explores zero-sum risk-sensitive stochastic games with unbounded payoff functions and varying discount factors. A logarithm growth condition is derived to ensure the finiteness of expected risk-sensitive discounted payoffs, extending the existing literature on bounded reward/cost functions. The paper establishes the existence of a solution to the Shapley equation, the value of the game, and a Nash equilibrium using new boundaries and novel arguments. An iteration algorithm is developed for computing the value and Nash equilibria of the games, and an inventory system is used to illustrate the results and highlight the differences from existing articles.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Automation & Control Systems
Randall Martyr, John Moriarty
Summary: In the nonzero-sum setting, a connection is established between Nash equilibria in Dynkin games and generalized Nash equilibrium problems, revealing novel equilibria with complex structures. Non-differentiable reward functions are considered, leading to new results on the existence and uniqueness of threshold-type equilibria and their stability under perturbations.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Mrinal K. Ghosh, Subrata Golui, Chandan Pal, Somnath Pradhan
Summary: We study the nonzero-sum stochastic games for continuous time Markov decision processes on a denumerable state space with risk-sensitive ergodic cost criterion. Under a Lyapunov type stability assumption, we show the existence of a Nash equilibrium in stationary strategies by proving the solvability of the corresponding system of coupled HJB equations. Utilizing principal eigenvalues associated with the HJB equations, we completely characterize the Nash equilibria in the space of stationary Markov strategies.
APPLIED MATHEMATICS AND OPTIMIZATION
(2022)
Article
Automation & Control Systems
Francois Dufour, Tomas Prieto-Rumeau
Summary: This study examines a nonzero-sum Markov game on an abstract measurable state space with compact metric action spaces, focusing on maximizing each player's respective discounted payoff function under certain constraints. The existence of a constrained stationary Markov Nash equilibrium is established under appropriate conditions.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2022)
Article
Management
Utsav Sadana, Puduru Viswanadha Reddy, Georges Zaccour
Summary: This paper introduces a class of deterministic finite-horizon two-player nonzero-sum differential games where one player uses ordinary controls while the other player uses impulse controls, and formulates necessary and sufficient conditions for the existence of an open-loop Nash equilibrium. Specializing these results to linear-quadratic games, it is shown that the equilibrium strategies can be computed by solving a constrained nonlinear optimization problem. Furthermore, analytical characterizations of equilibrium number, timing, and the level of impulse in terms of the problem data are obtained for the special case of linear-state differential games.
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
(2021)
Article
Mathematics, Interdisciplinary Applications
Said Hamadene, Rui Mu
Summary: This article discusses risk-sensitive nonzero-sum stochastic differential games in the Markovian framework, showing the existence of a Nash equilibrium point under certain conditions. The main tool used is the notion of backward stochastic differential equation, which is multidimensional with continuous generator involving both quadratic and stochastic linear growth components.
DYNAMIC GAMES AND APPLICATIONS
(2021)
Article
Operations Research & Management Science
Wenzhao Zhang, Xiaolong Zou
Summary: This paper studies nonzero-sum continuous-time constrained average stochastic games with independent state processes. By introducing average occupation measures, the existence of constrained Nash equilibria is established, and it is shown that each stationary Nash equilibrium corresponds to a global minimizer of a certain mathematical program.
Article
Physics, Multidisciplinary
Hui-Min Cheng, Ming-Xing Luo
Summary: The Nash equilibrium is crucial in classical game theory, but exploring multipartite zero-sum quantum games has led to interesting findings, such as resolving fairness issues in tripartite classical games and introducing dynamic zero-sum quantum games using single quantum states. These quantum games are robust against preparation noise and measurement errors, providing potential advantages in game settings.
Article
Operations Research & Management Science
Zachary Feinstein, Birgit Rudloff, Jianfeng Zhang
Summary: The text discusses the concept of set value in nonzero sum games and explores its properties and the principle of dynamic programming. Closed-loop controls and path-dependent controls are necessary for achieving the dynamic programming principle. The study covers both discrete and continuous time models.
MATHEMATICS OF OPERATIONS RESEARCH
(2022)
