On pursuit problems in controlled distributed systems

In the present paper, we study the problem of transferring a system from one state into another state in the presence of a player trying to prevent the occurrence of this transfer in the system. The system dynamics is described by partial differential equations whose right-hand sides contain the player controls in additive form. In a similar setting, the problem was solved in several papers, but it has not been considered for the case in which various constraints are imposed on the controls of the players. Here, in contrast to several other papers, we consider games in the entire scale of spaces H _r , r ≥ 0. We propose a new approach for completing the pursuit under various constraints on the player controls.     

Some problems in the theory of differential pursuit games in systems with distributed parameters

A game with dynamics described by partial differential equations is considered. The equations of the players are additively represented on the right-hand side and are subject to integral or pointwise restrictions. The goal of the first player, who is informed of the instantaneous value of the control of his partner, is to bring the system into an unperturbed state. To solve the problem the decomposition method developed in [1] for a controlled system (with one player) is used. Three combinations of restrictions on the players are considered. In all cases the control of the first player is presented explicitly. The main complication, compared with the problem considered previously [1] is that this control consists of two terms estimated in different norms.     

Research and Development with Stock-Dependent Spillovers and Price Competition in a Duopoly

This paper investigates the research and development accumulation and pricing strategies of two firms competing for consumer demand in a dynamic framework. A firm’s research and development is production-cost-reducing and can benefit from part of the competitor’s research and development stock without payment. We consider decisions in a game characterized by Nash equilibrium. In this dynamic game, a player’s action depends on whether the competitor’s current research and development stock are observable. If the competitor’s current research and development stock are not observable or observable only after a certain time lag, a player’s action can be solely based on the information on the current period t (open-loop strategy). In the converse case, it can also include the information on the competitor’s reaction to a change in the current value of the state vector (closed-loop strategy), which allows for strategic interaction to take place throughout the game. Given the cumulative nature of research and development activities, a primary goal of this paper is to determine whether, regardless of the observability of the competitor’s current research and development stock, free research and development spillovers generate a lower level of scientific knowledge than research and development appropriability. A second objective of the paper is to determine how the observability of the rival’s current research and development stock affects a firm’s research and development and pricing decisions and payoffs under imperfect research and development appropriability.     

Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler’s ruin problem. The generalization is best interpreted as a game of one player against d other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of Möbius monotonicity.     

汇改与外汇市场波动性:大玩家理论的视角

央行在外汇市场的行动对市场波动性有重要影响。本文从主观预期概念出发,基于“大玩家”理论解释了2005年和2015年中国外汇管理体制两次改革前后人民币外汇市场的波动性差异。汇改前,央行积极地承担着“大玩家”的角色,人民币对美元汇率波动性较强;汇改后,央行的“大玩家”角色减弱,汇率波动性程度显著下降,市场效率增强。利用汇率数据进行的R/S分析支持了前述解释。     

Solving the vendor–buyer integrated inventory system with arithmetic–geometric inequality

In the past, economic order quantity (EOQ) and economic production quantity (EPQ) were treated independently from the viewpoints of the buyer or the vendor. In most cases, the optimal solution for one player was non-optimal to the other player. In today’s competitive markets, close cooperation between the vendor and the buyer is necessary to reduce the joint inventory cost and the response time of the vendor–buyer system. The successful experiences of National Semiconductor, Wal-Mart, and Procter and Gamble have demonstrated that integrating the supply chain has significantly influenced the company’s performance and market share (Simchi-Levi et al. (2000) [1]). Recently, Yang et al. (2007) [2] presented an inventory model to determine the economic lot size for both the vendor and buyer, and the number of deliveries in an integrated two stage supply chain. In this paper, we present an alternative approach to determine the global optimal inventory policy for the vendor–buyer integrated system using arithmetic–geometric inequality.     

Explicit Solution for a Network Control Problem in the Large Deviation Regime

We consider optimal control of a stochastic network, where service is controlled to prevent buffer overflow. We use a risk-sensitive escape time criterion, which in comparison to the ordinary escape time criteria heavily penalizes exits which occur on short time intervals. A limit as the buffer sizes tend to infinity is considered. In [2] we showed that, for a large class of networks, the limit of the normalized cost agrees with the value function of a differential game. In this game, one player controls the service discipline (who to serve and whether to serve), and the other player chooses arrival and service rates in the network. The game's value is characterized in [2] as the unique solution to a Hamilton–Jacobi–Bellman Partial Differential Equation (PDE). In the current paper we apply this general theory to the important case of a network of queues in tandem. Our main results are: (i) the construction of an explicit solution to the corresponding PDE, and (ii) drawing out the implications for optimal risk-sensitive and robust regulation of the network. In particular, the following general principle can be extracted. To avoid buffer overflow there is a natural competition between two tendencies. One may choose to serve a particular queue, since that will help prevent its own buffer from overflowing, or one may prefer to stop service, with the goal of preventing overflow of buffers further down the line. The solution to the PDE indicates the optimal choice between these two, specifying the parts of the state space where each queue must be served (so as not to lose optimality), and where it can idle. Referring to those queues which must be served as bottlenecks, one can use the solution to the PDE to explicitly calculate the bottleneck queues as a function of the system's state, in terms of a simple set of equations.     

不动点理论与里卡提方程两个正周期解的存在性

利用压缩映射原理,得到里卡提方程一个正周期解的存在性;利用变量变换方法,将里卡提方程转化为伯努利方程.根据伯努利方程的周期解和变量变换,得到里卡提方程的另一个周期解.并讨论了两个正周期解的稳定性,一个周期解在某个区间上是吸引的,另一个周期解在R上是不稳定的.     

Linear differential games with delayed and noisy information

This paper deals with a linear-quadratic-Gaussian zero-sum game in which one player has delayed and noisy information and the other has perfect information. Assuming that the player with perfect information can deduce his opponent's state estimate, the optimal closed-loop control laws are derived. Then, it is shown that the separation theorem is satisfied for the player with imperfect information and his optimal state estimate is given by a delay-differential equation.     

Importance Sampling,Large Deviations,and Differential Games

A standard heuristic in the area of importance sampling is that the changes of measure used to prove large deviation lower bounds give good performance when used for importance sampling. Recent work, however, has shown that a naive implementation of the heuristic is incorrect in many situations. The present paper considers the simple setting of sums of independent and identically distributed (iid) random variables, and shows that under mild conditions asymptotically optimal changes of measure can be found if one considers dynamic importance sampling schemes. For such schemes, the change of measure applied to an individual summand can depend on the historical empirical mean of the simulation (i.e. the system state). In analyzing the asymptotic performance of importance sampling, we show that the value function of a differential game characterizes the optimal performance, with player A corresponding to the choice of change of measure and player B arising due to a large deviations analysis. From this perspective, the traditional implementation of the heuristic is shown to correspond to player A choosing a control that is fixed, regardless of the system state or player B's choice of control. This leads to an "open loop" control for player A, which is suboptimal except in special cases. Our final contribution is a method, based on the Isaacs equation associated with the differential game, for constructing dynamic schemes. Numerical examples are presented to illustrate the results.     

Backward induction algorithm for a class of closed-loop Stackelberg games

In the paper a new deterministic continuum-strategy two-player discrete-time dynamic Stackelberg game is proposed with fixed finite time duration and closed-loop information structure. The considered payoff functions can be widely used in different applications (mainly in conflicts of consuming a limited resource, where one player, called leader, is a superior authority choosing strategy first, and another player, called follower, chooses after).     

Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

The Value of Information Structures in Zero-sum Games with Lack of Information on One Side

Two players are engaged in a zero-sum game with lack of information on one side, in which player 1 (the informed player) receives some stochastic signal about the state of nature. I consider the value of the game as a function of player 1’s information structure, and study the properties of this function. It turns out that these properties reflect the fact that in zero sum situation the value of information for each player is positive.     

Epistemic characterizations of iterated deletion of inferior strategy profiles in preference-based type spaces

Bonanno (Logics and the foundations of game and decision theory, Amsterdam University Press, Amsterdam, 2008) provides an epistemic characterization for the solution concept of iterated deletion of inferior strategy profiles (IDIP) by embedding strategic-form games with ordinal payoffs in non-probabilistic epistemic models which are built on Kripke frames. In this paper, we will follow the event-based approach to epistemic game theory and supplement strategic games with type space models, where each type is associated with a preference relation on the state space. In such a framework, IDIP can be characterized by the conditions that at least one player has correct beliefs about the state of the world and that there is common belief that every player is rational, has correct beliefs about the state of the world and has strictly monotone preferences. Moreover, we shall compare the epistemic motivations for IDIP and its mixed strategy variant known as strong rationalizability (SR). Presuppose the above conditions. Whenever there is also common belief that players’ preferences are representable by some expected utility function IDIP still applies. But if there is common belief that players’ preferences are representable by some expected payoff function, then SR results.     

Single-suit two-person card play

We consider a two-person constant sum perfect information game, which we call theEnd Play Game, which arises from an abstraction of simple end play positions in card games of the whist family, including bridge. This game was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called itwhistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two handsA andB ofn cards each, held by players Left and Right, and one player is designated as having thelead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a “trick” and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and thevalue of the game to each player is the number of tricks he takes. Despite its simple appearance, this game is quite complicated, and finding an optimal strategy seems difficult. This paper derives basic properties of the game, gives some criteria under which one hand is guaranteed to be better than another, and determines the optimal strategies and value functions for the game in several special cases.     

Constructive methods in the nonlinear problems of control theory

We consider nonlinear control problems of transformation of a substance from one state into another given some boundary data. Our goal is to recover the second order differential operators with three time-independent coefficients. Some constructive analytical methods of study are proposed that use, in particular, the Burman-Lagrange inversion formulas for analytic functions. Some formulas are given for solving the control problems in a few particular cases of the initialboundary data. Alongside the theoretical study, we carry out, on the basis of a logical system of symbolic computation, the practical implementation of the algorithms and programs for automatic generation of the formulas that give the solutions of control problems.     

On the theory of three-person differential games

A differential game of three players with dynamics described by linear differential equations under geometric constraints on the control parameters is considered. Sufficient conditions are obtained for the existence of the first player’s strategy guaranteeing that the trajectory of the game reaches a given target set for any admissible control of the second player and avoids the terminal set of the third player. An algorithm of constructing the first player’s strategy guaranteeing the game’s termination in finite time is suggested. A solution of a model example is given.     

Bidders and dispenser: manipulative hypergames in a multinational context

The object of this paper is to present an introduction to the basic ideas of Hypergame Analysis, and to illustrate these by building some models of a particular type of situation.Hypergame Analysis is an extension of the Game-theoretic framework, the purpose of which is to enable one to model situations in which the various parties are not well-informed of each other's preferences and strategies. we take as a basic structure not a single game, but a linked set of ‘perceived’ games: this, in essence, is what constitutes a Hypergame. Misperceptions may arise accidentally or be deliberately induced. Thus, a player may be acting ‘rationally’ relative to the game he perceives, but this game itself may have been ‘set up’ to suit the interests of some other party.The technique is used to explore situations in which several parties (the ‘bidders’) negotiate competitively with another (the ‘dispenser’) who is able to accept whichever bidder's offer is most advantageous to him. In particular, the ability of the dispenser to ‘play off’ one bidder against another is examined. This is related to an account of the siting of new plant by a Multi-National Corporation. Some general implications are suggested: especially, it is argued that to have a reasonable chance of producing adequate forecasts in such difficult situations, modelling techniques must embody at least this degree of conceptual complexity.     

Stronger players win more balanced knockout tournaments

A player is said to be stronger than another player if he has a better chance of beating the other player than vice versa and his chance of beating any third player is at least as good as that of the other player. Recently, Israel gave an example which shows that a stronger player can have a smaller probability of winning a knockout tournament than a weaker one when players are randomly assigned to starting positions. In this paper we prove that this anomaly cannot happen if the tournament plan is a balanced one.The work by this author was done while consulting at Bell Laboratories.     

一种估算马尔柯夫状态转移概率矩阵的新方法

在现有文献研究的基础上,对马尔柯夫状态转移概率矩阵估算方法又作了进一步研究,根据马尔柯夫状态转移概率矩阵的性质和特点,提出了一种新的估算方法.方法首先构造了一个以相对误差绝对值之和最小为目标,以某一状态转移到其他状态的概率之和等于1以及状态转移概率不小于零为约束条件的优化模型.在此基础上,通过变量替换,将该模型转化为线性规划模型.由于线性规划模型不仅能够求得解析解,而且有现成的求解软件,因此不但便于问题求解,而且更加方便、可靠.最后进行了示例计算,验证了给出的马尔柯夫状态转移概率矩阵优化算法的可行性和正确性.