Infinite-Horizon Stochastic Differential Games with Branching Payoffs

In this paper, we consider infinite-horizon stochastic differential games with an autonomous structure and steady branching payoffs. While the introduction of additional stochastic elements via branching payoffs offers a fruitful alternative to modeling game situations under uncertainty, the solution to such a problem is not known. A theorem on the characterization of a Nash equilibrium solution for this kind of games is presented. An application in renewable resource extraction is provided to illustrate the solution mechanism.     

A Type of General Forward-Backward Stochastic Differential Equations and Applications

The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with It?o’s stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations. The existence and uniqueness results of the general FBSDEs are obtained. In the framework of the general FBSDEs in this paper, the explicit form of the optimal control for linearquadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.     

Two Different Approaches to Nonzero-Sum Stochastic Differential Games

We make the link between two approaches to Nash equilibria for nonzero-sum stochastic differential games: the first one using backward stochastic differential equations and the second one using strategies with delay. We prove that, when both exist, the two notions of Nash equilibria coincide.     

A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

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 nonzero-sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-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 and to find optimal investment of an insurance firm under model uncertainty.     

Separable and low-rank continuous games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game. This research was funded in part by National Science Foundation grants DMI-0545910 and ECCS-0621922 and AFOSR MURI subaward 2003-07688-1.     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.     

Randomized stopping games and Markov market games

We study nonzero-sum stopping games with randomized stopping strategies. The existence of Nash equilibrium and ɛ-equilibrium strategies are discussed under various assumptions on players random payoffs and utility functions dependent on the observed discrete time Markov process. Then we will present a model of a market game in which randomized stopping times are involved. The model is a mixture of a stochastic game and stopping game. Research supported by grant PBZ-KBN-016/P03/99.     

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

In this paper we deal with the problem of existence of a smooth solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.     

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.     

Maximum Principle for Stochastic Differential Games with Partial Information

In this paper, we first deal with the problem of optimal control for zero-sum stochastic differential games. We give a necessary and sufficient maximum principle for that problem with partial information. Then, we use the result to solve a problem in finance. Finally, we extend our approach to general stochastic games (nonzero-sum), and obtain an equilibrium point of such game.     

Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t _f. In this paper, we study the behavior of feedback Nash equilibria for t _f.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.     

On a class of nonzero-sum,linear-quadratic differential games

A class of two-player, nonzero-sum, linear-quadratic differential games is investigated for Nash equilibrium solutions when both players use closed-loop control and when one or both of the players are required to use open-loop control. For three formulations of the game, necessary and sufficient conditions are obtained for a particular strategy set to be a Nash equilibrium strategy set. For a fourth formulation of the game, where both players use open-loop control, necessary and sufficient conditions for the existence of a Nash equilibrium strategy set are developed. Several examples are presented in order to illustrate the differences between this class of differential games and its zero-sum analog.This research was supported by the National Science Foundation under Grant No. GK-3341.     

A finite step algorithm via a bimatrix game to a single controller non-zero sum stochastic game

Given a non-zero sum discounted stochastic game with finitely many states and actions one can form a bimatrix game whose pure strategies are the pure stationary strategies of the players and whose penalty payoffs consist of the total discounted costs over all states at any pure stationary pair. It is shown that any Nash equilibrium point of this bimatrix game can be used to find a Nash equilibrium point of the stochastic game whenever the law of motion is controlled by one player. The theorem is extended to undiscounted stochastic games with irreducible transitions when the law of motion is controlled by one player. Examples are worked out to illustrate the algorithm proposed.The work of this author was supported in part by the NSF grants DMS-9024408 and DMS 8802260.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDEs

In this Note, we deal with one kind of stochastic nonzero-sum differential game problem for N players. Using the theory of backward stochastic differential equations and Malliavin calculus, we give the explicit form of a Nash equilibrium point. To cite this article: J.-P. Lepeltier et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On linear stochastic differential games with average cost criterions

In this paper, we consider scalar linear stochastic differential games with average cost criterions. We solve the dynamic programming equations for these games and give the synthesis of saddle-point and Nash equilibrium solutions.The authors wish to thank A. Ichikawa for providing the initial impetus and helpful advice.     

Sensitive equilibria for ergodic stochastic games with countable state spaces

We consider stochastic games with countable state spaces and unbounded immediate payoff functions. Our assumptions on the transition structure of the game are based on a recent work by Meyn and Tweedie [19] on computable bounds for geometric convergence rates of Markov chains. The main results in this paper concern the existence of sensitive optimal strategies in some classes of zero-sum stochastic games. By sensitive optimality we mean overtaking or 1-optimality. We also provide a new Nash equilibrium theorem for a class of ergodic nonzero-sum stochastic games with denumerable state spaces.     

On stochastic games in economics

In this paper we show that many results on equilibria in stochastic games arising from economic theory can be deduced from the theorem on the existence of a correlated equilibrium due to Nowak and Raghavan. Some new classes of nonzero-sum Borel state space discounted stochastic games having stationary Nash equilibria are also presented. Three nontrivial examples of dynamic stochastic games arising from economic theory are given closed form solutions. Research partially supported by MNSW grant 1 P03A 01030.     

On Nash equilibrium points and games of imperfect information

This paper considers a class of two-player, nonzero-sum games in which the players have only local, as opposed to global, information about the payoff functions. We study various modes of behavior and their relationship to different stability properties of the Nash equilibrium points.     

带随机跳跃的线性二次非零和微分对策问题

对于一类以布朗运动和泊松过程为噪声源的正倒向随机微分方程,在单调性假设下,给出了解的存在性和唯一性的结果．然后将这些结果应用于带随机跳跃的线性二次非零和微分对策问题之中,由上述正倒向随机微分方程的解得到了开环Nash均衡点的显式形式．