Mixed Strategies for Deterministic Differential Games

Journal de Mathématiques Pures et Appliquées, 2018

We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean-Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for openloop controls. We also show that the upper and lower value functions are viscosity solutions to a corresponding upper and lower Master Bellman-Isaacs equation. Our results extend the seminal work of Fleming and Souganidis [15] to the McKean-Vlasov setting.

1995

Abstract We consider in this paper a continuous time stochastic hybrid system with a finite time horizon, controlled by two players with opposite objectives (zero-sum game). Player one wishes to maximize some linear function of the expected state trajectory, and player two wishes to minimize it. The state evolves according to a linear dynamic. The parameters of the state evolution equation may change at discrete times according to a MDP, ie, a Markov chain that is directly controlled by both players, and has a countable state space.

Stochastic Analysis and Applications, 2020

We study nonzero-sum stochastic differential games with risk-sensitive discounted cost criteria. Under fairly general conditions on drift term and diffusion coefficients, we establish a Nash equilibrium in Markov strategies for the discounted cost criterion. We achieve our results by studying relevant systems of coupled HJB equations.

2016

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward-backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for non zero-sum stochastic differential game problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for non zero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty. We also apply the results to find optimal investment of an insurance firm under model uncertainty.

Nonlinear Differential Equations and Applications NoDEA, 2009

We study a zero sum differential game of mixed type where each player uses both control and stopping times. Under certain conditions we show that the value function for this problem exists and is the unique viscosity solution of the corresponding variational inequalities. We also show the existence of saddle point equilibrium for a special case of differential game.

We present a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers. We describe its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this property in terms of Hamilton-Jacobi-Isaacs equa-tions. We also give several sufficient conditions for ergodicity and describe some extensions of the theory to stochastic differential games.

Dynamic Games and Applications, 2017

It is generally admitted that a correct forecasting of uncertain variables needs Markov decision rules. In a dynamic game environment, this belief is reinforced if one focuses on credible actions of the players. Usually, subgame perfectness requires equilibrium strategies to be constructed on Markov rules. It comes as a surprise that there are interesting classes of stochastic differential games where the equilibrium based on open loop strategies is subgame perfect. This fact is well known for deterministic games. We explore here the stochastic case, not dealt with up to now, identifying different game structures leading to the subgame perfectness of the open-loop equilibrium.

Nonlinear Analysis-theory Methods & Applications, 1993

IEEE Transactions on Automatic Control, 1968

Abstracf-The solution for a class of stochastic pursuit-evasion dfierential games between two linear dynamic systems is given. This class includes the classical interception game in Euclidean space. The performance index which is optimized is quadratic, and one of the two players has imperfect (noisy) knowledge of the states of the two systems. The "certainty-equivalence principle" or, equivalently, the technique of separating the estimator and the controller which characterizes the standard stochastic control problem is shown to be applicable to this class of differential games.

Journal of Optimization Theory and Applications, 1982

A differential game of prescribed duration with generaltype phase constraints is investigated, The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.