On differential games with Markov perfect triggering

This paper is an attempt to throw some light on the issue of whether defining history-dependent state variables in a differential game does allow for a dynamic play of precommitment equilibria. We suggest the application of trigger strategies in a state-feedback context. Based on a punishment mode in Markov perfect strategies and being able to detect even past deviations followed by histories returning to equilibrium, these subgame-perfect strategies lead to the enforcement of certain outcomes by means of dynamic rules of strategic interaction. The last part of our exposition is devoted to specific game structures under which a trigger equilibrium can be used as well as a punishment mode.     

Formulation and analysis of combat problems as zero-sum bicriterion differential games

In this paper, we present a formulation and analysis of a combat game between two players as a zero-sum bicriterion differential game. Each player's twin objectives of terminating the game on his own target set, while simultaneously avoiding his opponent's target set, are quantified in this approach. The solution in open-loop pure strategies is sought from among the Pareto-optimal security strategies of the players. A specific preference ordering on the outcomes is used to classify initial events in the assured win, draw, and mutual kill regions for the players. The method is compared with the event-constrained differential game approach, recently proposed by others. Finally, a simple example of the turret game is solved to illustrate the use of this method.     

A note on clemhout and Wan's dynamic games of common property resources

Recently, Clemhout and Wan (Ref. 1) provided solutions to a class of stochastic differential game models concerning common property resources. However, they only established closed-loop strategies for the infinite-horizon problem using the Kushner test. In this paper, we provide closed-loop feedback solutions to their game in a finite-time horizon with deterministic evolution dynamics.The helpful comments of an anonymous referee are gratefully acknowledged.     

基于随机微分博弈的动态投入产出问题研究

基于W Leontief提出的动态投入产出模型,讨论了考虑随机因素的离散时间下的最优策略设计问题.利用博弈论的思想,把动态投入产出系统抽象为离散时间的博弈模型,运用鞍点均衡策略设计出求解该投入产出问题的新方法,为宏观经济决策提供依据.     

CIR框架下的投资组合效用微分博弈

建立了Cox-Ingersoll—Ross随机利率下的关于两个投资者的投资组合效用微分博弈模型．市场利率具有CIR动力,博弈双方存在唯一的损益函数,损益函数取决于投资者的投资组合财富．一方选择动态投资组合策略以最大化损益函数,而另一方则最小化损益函数．运用随机控制理论,在一般的效用函数下得到了基于效用的博弈双方的最优策略．特别考虑了常数相对风险厌恶情形,获得了显示的最优投资组合策略和博弈值．最后给出了数值例子和仿真结果以说明本文的结论．     

On a problem of stochastic differential games

The process of bargaining between management and union during a strike is modelled by a nonlinear stochastic differential game. It is assumed that the two sides bargain in the mood of a cooperative game. A pair of Pareto-optimal strategies is obtained.     

具有随机工资的养老金最优投资问题

在三种目标函数下, 研究了具有随机工资的养老金最优投资问题. 第一种是均值-方差准则, 第二种基于效用的随机微分博弈, 第三种基于均值-方差准则的随机微分博弈. 随机微分博弈问题中博弈的双方为养老金计划投资者和金融市场, 金融市场是博弈的虚拟手. 应用线性二次控制理论求得了三种目标函数下的最优策略和值函数的显式解.     

Approximate solution of singularly perturbed nonlinear pursuit-evasion games

A methodology to obtain an approximate solution of a singularly perturbed nonlinear differential game is presented. The outcome of the game with approximate strategies, defined as extended value, is related to the saddle-point value of the game. In an example of a simple pursuit-evasion game, it is shown that the proposed methodology leads to an easily implementable feedback form solution with fairly accurate results. This approach seems to be attractive for analyzing realistic air-combat models without solving a two-point boundary-value problem.This research was partially supported by AFSC Contract No. F-49620-79-6-0135. The authors are grateful to Prof. J. V. Breakwell for encouraging the approach taken in this research. Thanks are also due to Dr. S. Gutman and Dr. J. Lewin for their useful comments.     

Stochastic pursuit-evasion differential games in 3D

A stochastic version of Isaacs's homicidal chauffeur game in the (x, y, z)-space is considered. This is used to solve a pursuit-evasion problem in the (x, y, z)-space in which the pursuer has incomplete information on the evader motion. Optimal feedback strategies for the game, and optimal feedback guidance laws for the pursuer, which uses only the measurements available to the pursuer, are computed. A simple suboptimal guidance law for the pursuer is suggested.     

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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     

Feedback Competitive Advertising Strategies with a General Objective Function

We introduce a general objective function, which incorporates competitive situations, such as conservative, punitive, and predatory advertising. Linking together the particular situations into a two-parameter family of max–min problems, and using the Lanchester model to describe the dynamics of the market, a bilinear-quadratic differential game is obtained. For this game, we find saddle-point feedback time-invariant advertising strategies and show when these strategies are Nash equilibrium strategies. In an empirical application involving duopolistic competition in the cola market, we find evidence of a punitive motivation for the advertising strategies.     

On the minimax principle and zero-sum stochastic differential games

The problem of prior and delayed commitment in zero-sum stochastic differential games is discussed. A new formulation and solution based on the delayed-commitment model is derived and its significant implications to stochastic games and control are considered.     

考虑相互依赖性的信息系统安全投资及协调机制

考虑信息系统安全相互依赖情形下最优化信息系统连续时间安全投资水平是一个值得研究的问题。首先讨论了非合作博弈下信息系统安全投资的最优策略选择,在此基础上讨论了安全投资效率参数、黑客学习能力、传染风险对信息系统脆弱性及信息系统安全投资率的影响。其次,在推导出两企业在合作博弈情形下最优策略选择的基础上,对比两种情形下的博弈均衡结果,得出合作博弈下的投资水平高于非合作博弈下的投资水平。原因是两个企业的相互依赖关系隐含着企业投资的负外部性,从而导致企业投资不足。最后,构建一种双边支付激励机制消除企业投资不足问题,从而使企业达到合作博弈下的最优投资水平,提高两个企业的收益。     

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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.     

考虑声誉效应的网络众包参与者行为博弈模型及仿真分析

为了控制网络众包交易的风险,以网络众包平台中的发包方和接包方为研究对象,基于微分博弈方法建立了网络众包交易中参与者行为的博弈模型,研究了发包方和接包方在合作和非合作两种情形下的策略选择。本文研究发现在声誉效应满足特定阈值的条件时,合作博弈情形下众包系统的总收益更大;而且,发包方和接包方违约控制的收益及努力程度更大。在网络众包平台的违约罚金超过特定阈值时,随时间推移发包方和接包方违约控制的努力程度逐渐提高,实现帕累托改进。本文结合猪八戒网、InnoCentive等网络众包平台的众包交易机制,采用数值分析对博弈模型的求解结果进行模拟。对于网络众包平台来说,可以通过建立有效的信用评价机制,将信用评价或声誉纳入众包交易定价,引导参与者约束自身行为。     

A differential game of surveillance evasion of two identical cars

A partial solution of a differential game of surveillance evasion is presented. The dynamics is that of two identical cars, with constant speeds and bounded turn rates. The pursuer's surveillance zone is circular. The game of kind and the game of degree are solved for ratios of the surveillance radius to the minimal turn radius equal to or greater than +2. A barrier is constructed, and regions of strategies are separated through the use of dispersal and universal surfaces. A synthesis of a composite game of two identical cars is outlined, covering the space outside the surveillance zone as well as that inside it.Part of this research (the game of kind, Ref. 1) was carried out in the Faculty of Mechanical Engineering in the Technion, Haifa under the supervision of J. Lewin. The author is indebted to J. Lewin for his guidance.     

Stochastic Differential Games With a Small Parameter

Much of stochastic game theory is concerned with diffusion models. Such models are often only idealizations of the actual physical process, which might be driven by a wide bandwidth process or be a discrete parameter system with correlated driving noises. For a two person zero-sum game, under quite general conditions, the optimal or nearly optimal strategies derived for the diffusion model are shown to be “nearly optimal” for the physical (say wideband noise driven) process. An approach based on occupation measures are used. We treat the problem of discounted cost, as well as the average cost per unit time problem. Weak convergence methods are utilized in the analysis     

基于微分博弈的数据开放策略及合作收益分配机制研究

数据作为数字经济时代一种重要的生产要素,在国民经济运行中的作用越来越重要。然而,现阶段我国社会数据开放仍处于起步阶段,急需加快推进数据开放共享。为准确分析数据开放主体政府和企业的数据开放策略,运用微分博弈分别构建Nash非合作博弈、Stackelberg主从博弈和协同合作博弈模型,得到三种情形下政府和企业最优开放数据努力程度、社会数据开放水平最优轨迹以及最优收益函数;其次,在协同合作博弈情景下,将时间因素引入Nash谈判模型,进一步分析政府和企业合作时具有时间一致性的动态收益分配机制;最后,结合数值仿真对影响社会数据开放水平的相关参数进行灵敏性分析。研究结果发现,当政府获得收益分配比例大于1/3时,Stackelberg主从博弈下的政府和企业双方收益以及社会数据开放水平均高于Nash非合作情形;协同合作情形下,双方总收益和社会数据开放水平均能达到帕累托最优状态。     

Differential games with noise-corrupted measurements

In this paper, readily computable strategies for zero-sum, linear-quadratic differential games with noise-corrupted measurements are developed. Of particular significance is the fact that the governing differential equations no longer require the solution of an often difficult nonlinear, two-point boundary-value problem, but again satisfy the separation principle of linear-quadratic optimal control. The implications of the payoff relationships are considered.In a subsequent paper, we will apply the theory developed in this paper to a detailed example of a pursuit-evasion game. We discuss a missile and an airplane system where the missile supported by its launch platform has perfect state measurements and the airplane has noise-corrupted measurements.     

Combat games

We propose a mathematical formulation of a combat game between two opponents with offensive capabilities and offensive objectives. Resolution of the combat involves solving two differential games with state constraints. Depending on the game dynamics and parameters, the combat can terminate in one of four ways: (i) the first player wins, (ii) the second player wins, (iii) a draw (neither wins), or (iv) joint capture. In the first two cases, the optimal strategies of the two players are determined from suitable zero-sum games, whereas in the latter two the relevant games are nonzero-sum. Further, to avoid certain technical difficulties, the concept of a -combat game is introduced.Dedicated to G. LeitmannThe first author wishes to acknowledge the friendship and guidance of George Leitmann, beginning in the author's student days at Berkeley and continuing to the present time. All the authors thank George Leitmann for many recent fruitful discussions on differential games.on sabbatical leave from Technion, Israel Institute of Technology, Haifa, Israel.