Stochastic Discrete-Time Nash Games with Constrained State Estimators

In this paper, we consider stochastic linear-quadratic discrete-time Nash games in which two players have access only to noise-corrupted output measurements. We assume that each player is constrained to use a linear Kalman filter-like state estimator to implement his optimal strategies. Two information structures available to the players in their state estimators are investigated. The first has access to one-step delayed output and a one-step delayed control input of the player. The second has access to the current output and a one-step delayed control input of the player. In both cases, statistics of the process and statistics of the measurements of each player are known to both players. A simple example of a two-zone energy trading system is considered to illustrate the developed Nash strategies. In this example, the Nash strategies are calculated for the two cases of unlimited and limited transmission capacity constraints.     

离散时间大种群随机多智能体系统的线性二次分散动态博弈

研究具有耦合二次型随机性能指标的离散时间大种群随机多智能体系统的分散博弈问题.系统所受的噪声干扰为条件二阶矩有界的鞅差序列,比以往研究所考虑的高斯白噪声情形更具有广泛性.采用状态聚集方法构造了对种群状态平均的估计,基于Nash必然等价原理设计了分散控制律,并利用概率极限理论分析了闭环系统的稳定性和最优性.主要结果包括(1)证明了对种群状态的平均的估计在某种范数意义下的强一致性,即种群状态的平均与其估计值之间的误差在该范数意义下将随系统个体数N的增加几乎必然收敛于0;(2)证明了闭环系统的几乎必然一致稳定性,即系统的稳定性与种群个体数N无关;(3)证明了所设计的分散控制律是几乎必然渐近Nash均衡策略.     

多目标的随机结盟对策的ZS-值

本文研究的是多目标随机结盟对策的问题，是将单目标的随机结盟对策的ZS-值拓展到多目标的随机结盟对策上，同时考虑了局中人对不同目标的偏好程度，从而，给出了多目标随机结盟对策的ZS-值的定义，并讨论了该值的性质及定理。     

均值-方差准则下具有负债的随机微分博弈

研究了均值-方差准则下,具有负债的随机微分博弈.研究目标是:在终值财富的均值等于k的限制下,在市场出现最坏的情况下找到最优的投资策略使终值财富的方差最小.即:基于均值-方差随机微分博弈的投资组合选择问题.使用线性-二次控制的理论解决了该问题,获得了最优的投资策略、最优市场策略和有效边界的显示解.并通过对所得结果进行进一步分析,在经济上给出了进一步的解释.通过本文的研究,可以指导金融公司在面临负债和金融市场情况恶劣时,选择恰当的投资策略使自身获得一定的财富而面临的风险最小.     

Polytope Games

Starting from the definition of a bimatrix game, we restrict the pair of strategy sets jointly, not independently. Thus, we have a set , which is the set of all feasible strategy pairs. We pose the question of whether a Nash equilibrium exists, in that no player can obtain a higher payoff by deviating. We answer this question affirmatively for a very general case, imposing a minimum of conditions on the restricted sets and the payoff. Next, we concentrate on a special class of restricted games, the polytope bimatrix game, where the restrictions are linear and the payoff functions are bilinear. Further, we show how the polytope bimatrix game is a generalization of the bimatrix game. We give an algorithm for solving such a polytope bimatrix game; finally, we discuss refinements to the equilibrium point concept where we generalize results from the theory of bimatrix games.     

Approximation of Noncooperative Semi-Markov Games

An approximation of a general V-ergodic semi-Markov game with Borel state space by discrete-state space strongly-ergodic games is studied. The standard expected ratio-average criterion as well as the expected time-average criterion are considered. New theorems on the existence of ∊-equilibria are given.Communicated by D. A. CarlsonThe authors thank an anonymous referee for constructive comments. This work is supported by MEiN Grant 1P03A 01030.     

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.     

Harnack's Inequality for p-Harmonic Functions via Stochastic Games

We give a proof of asymptotic Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Lipschitz continuity and Harnack's inequality for p-harmonic functions in the case p > 2. The proof avoids classical techniques like Moser iteration, but instead relies on suitable choices of strategies for the stochastic tug-of-war game.     

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??Under inflation influence, this paper investigate a stochastic differential game with reinsurance and investment. Insurance company chose a strategy to minimizing the variance of the final wealth, and the financial markets as a game ``virtual hand' chosen a probability measure represents the economic ``environment' to maximize the variance of the final wealth. Through this double game between the insurance companies and the financial markets, get optimal portfolio strategies. When investing, we consider inflation, the method of dealing with inflation is: Firstly, the inflation is converted to the risky assets, and then constructs the wealth process. Through change the original based on the mean-variance criteria stochastic differential game into unrestricted cases, then application linear-quadratic control theory obtain optimal reinsurance strategy and investment strategy and optimal market strategy as well as the closed form expression of efficient frontier are obtained; finally get reinsurance strategy and optimal investment strategy and optimal market strategy as well as the closed form expression of efficient frontier for the original stochastic differential game.     

Zero-Sum Stochastic Games with Partial Information

We study a zero-sum stochastic game on a Borel state space where the state of the game is not known to the players. Both players take their decisions based on an observation process. We transform this into an equivalent problem with complete information. Then, we establish the existence of a value and optimal strategies for both players.     

Static and Dynamic Equilibria in Games With Continuum of Players

This paper is a study of a general class of deterministic dynamic games with an atomless measure space of players and an arbitrary time space. The payoffs of the players depend on their own strategy, a trajectory of the system and a function with values being finite dimensional statistics of static profiles. The players' available decisions depend on trajectories of the system.The paper deals with relations between static and dynamic open-loop equilibria as well as their existence. An equivalence theorem is proven and theorems on the existence of a dynamic equilibrium are shown as consequences.     

Hierarchical Production Control in Dynamic Stochastic Jobshops with Long-Run Average Cost

We consider a production planning problem for a dynamic jobshop producing a number of products and subject to breakdown and repair of machines. The machine capacities are assumed to be finite-state Markov chains. As the rates of change of the machine states approach infinity, an asymptotic analysis of this stochastic manufacturing systems is given. The analysis results in a limiting problem in which the stochastic machine availability is replaced by its equilibrium mean availability. The long-run average cost for the original problem is shown to converge to the long-run average cost of the limiting problem. The convergence rate of the long-run average cost for the original problem to that of the limiting problem together with an error estimate for the constructed asymptotic optimal control is established.     

On Applied Nonlinear and Bilevel Programming or Pursuit-Evasion Games

Motivated by the benefits of discretization in optimal control problems, we consider the possibility of discretizing pursuit-evasion games. Two approaches are introduced. In the first approach, the solution of the necessary conditions of the continuous-time game is decomposed into ordinary optimal control problems that can be solved using discretization and nonlinear programming techniques. In the second approach, the game is discretized and transformed into a bilevel programming problem, which is solved using a first-order feasible direction method. Although the starting points of the approaches are different, they lead in practice to the same solution algorithm. We demonstrate the usability of the discretization by solving some open-loop representations of feedback solutions for a complex pursuit-evasion game between a realistically modeled aircraft and a missile, with terminal time as the payoff. The solutions are compared with those obtained via an indirect method.     

Optimal Production Planning in a Stochastic Manufacturing System with Long-Run Average Cost

This paper is concerned with the optimal production planning in a dynamic stochastic manufacturing system consisting of a single machine that is failure prone and facing a constant demand. The objective is to choose the rate of production over time in order to minimize the long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach that the Hamilton–Jacobi–Bellman equation for the average cost problem has a solution giving rise to the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem. Finally, the optimal control policy is specified in terms of the potential function.     

An Approach to Fuzzy Noncooperative Nash Games

Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility function over a set of the player decision space. Each player attempts to maximize/minimize his/her own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends.In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and operation of future military operations.     

随机利率下DC型养老金的随机微分博弈

本文研究了Vasicek随机利率下DC型养老金的随机微分博弈.金融市场是博弈的"虚拟"手,博弈中养老金计划投资者占主导.研究目标是:通过养老金计划投资者和金融市场之间的博弈,寻找最优的策略使得终止时刻财富的期望效用达到最大.在幂效用函数下,运用随机控制理论求得了最优策略和值函数的显式解.最后,解释了所研究的结果在经济上的意义,并通过数值计算分析了一些参数对最优策略的影响.     

Stochastic systems in Riemannian manifolds

A formulation of stochastic systems in a Riemannian manifold is given by stochastic differential equations in the tangent bundle of the manifold. Brownian motion is constructed in a compact Riemannian manifold as well as the horizontal lift of this process to the bundle of orthonormal frames. The solution of some stochastic differential equations in the tangent bundle of the manifold is defined by the transformation of the measure for the manifold-valued Brownian motion by a suitable Radon-Nikodym derivative. Real-valued stochastic integrals are defined for this Brownian motion using parallelism along the Brownian paths. A stochastic control problem is formulated and solved for these stochastic systems where a suitable convexity condition is assumed.This research was supported by NSF Grants Nos. GK-32136, ENG-75-06562, and MCS-76-01695.The author wishes to thank D. Gromoll, J. Simons, and J. Thorpe for some helpful conversations on differential geometry.     

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This paper investigate a stochastic differential games for DC (defined contribution plans) pension under Vasicek stochastic interest rate. The finance market as the hypothetical counterpart, the investor as pension the leader of game. Our goal is through the game between pension plan investor and financial market, obtain optimal strategies to maximizes the expected utility of the terminal wealth. Under power utility function, by using stochastic control theory, we obtain closed-form solutions for the value function as well as the strategies. Finally, explain the research results in the economic sense, and though numerical calculation given the influence of some parameters on the optimal strategies     

On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games

The purpose of this paper is to investigate whether theorems known to guarantee the existence and uniqueness of Nash equilibria, provide also sufficient conditions for the Tikhonov well-posedness (T-wp). We consider several hypotheses that ensure the existence and uniqueness of a Nash equilibrium (NE), such as strong positivity of the Jacobian of the utility function derivatives (Ref. 1), pseudoconcavity, and strict diagonal dominance of the Jacobian of the best reply functions in implicit form (Ref. 2). The aforesaid assumptions imply the existence and uniqueness of NE. We show that the hypotheses in Ref. 2 guarantee also the T-wp property of the Nash equilibrium.As far as the hypotheses in Ref. 1 are concerned, the result is true for quadratic games and zero-sum games. A standard way to prove the T-wp property is to show that the sets of -equilibria are compact. This last approach is used to demonstrate directly the T-wp property for the Cournot oligopoly model given in Ref. 3. The compactness of -equilibria is related also to the condition that the best reply surfaces do not approach each other near infinity.