对抗仿真中防空导弹微分对策模型研究

针对防空导弹与目标的空间对抗,建立了防空导弹微分对策空间对抗模型,并推算得到了最优策略,运用自适应神经网络评判方法对模型进行求解,通过给定条件下的仿真证明了该算法的有效性,显示了该模型可应用于防空导弹与目标的空间对抗问题。     

One-person games of economic survival

This paper discusses the one-person economic survival game model with a discrete probability distribution for the contribution to surplus variable. The general game model and strategies in these games are examined, and necessary conditions which an undominated, stationary strategy must satisfy are obtained. For a special class of these games a mathematical formulation of the value of the game is given, and examples and theorems which relate to undominated strategies in this class are presented. This paper, in some sense, is a sequel to a portion of a paper by Shubik and Thompson [7] which appeared in this journal.     

A search game taking account of attributes of searching resources

This article deals with a two‐person zero‐sum game called a search allocation game (SAG), in which a searcher and a target participate as players. The searcher distributes his searching resources in a search space to detect the target. The effect of resources lasts a certain period of time and extends to some areas at a distance from the resources' dropped points. On the other hand, the target moves around in the search space to evade the searcher. In the history of search games, there has been little research covering the durability and reachability of searching resources. This article proposes two linear programming formulations to solve the SAG with durable and reachable resources, and at the same time provide an optimal strategy of distributing searching resources for the searcher and an optimal moving strategy for the target. Using examples, we will analyze the influences of two attributes of resources on optimal strategies. © 2007 Wiley Periodicals, Inc. Naval Research Logistics 2008     

基于梯度迭代法的一类追逃对抗模型研究

针对一类追逃对抗问题,基于微分对策理论,建立了三维空间中的追逃对抗模型,进而得到了最优性条件和最优策略.运用梯度迭代法给出了模型的数值解,并做了实例分析.分析结果显示,该模型可以很好地应用于潜艇对潜艇、潜艇对水面舰艇等实际追逃对抗问题.     

A saddle-point theorem for a class of infinite games

In this paper, the existence of a saddle point for two-person zero-sum infinite games of a special type is proved. The games have continuous bilinear payoff functions and strategy sets which are convex, noncompact subsets of an infinite-dimensional vector space. The closures of the strategy sets are, however, compact. The payoff functions satisfy conditions which allow the use of dominance arguments to show that points in the closure of a strategy set are dominated by or are strategically equivalent to points in the strategy set itself. Combining the dominance arguments with a well-known existence theorem produces the main result of the paper. The class of games treated is an extension of a class studied by J. D. Matheson, who obtained explicit solutions for the saddle points by using necessary conditions.     

A family of games on 1,∞)2 with payoff a function of y/x

A two-parameter class of games on 1,∞)² is studied. The games may be regarded as analogs of Silverman games, having continuous payoff function in place of a step function of y/x. This change is motivated by a desire to move toward a model for competitive situations where the penalty for overspending increases with the amount of overspending. There are some similarities to games with bell-shaped kernel. For most of the region considered in the plane of the two parameters there are solutions of finite type, which are obtained explicitly. There are, however, pockets in this plane where no optimal strategies have been found and possibly where none of finite type exist.     

Solving partially observable agent‐intruder games with an application to border security problems

In this paper, we introduce partially observable agent‐intruder games (POAIGs). These games model dynamic search games on graphs between security forces (an agent) and an intruder given possible (border) entry points and high value assets that require protection. The agent faces situations with dynamically changing, partially observable information about the state of the intruder and vice versa. The agent may place sensors at selected locations, while the intruder may recruit partners to observe the agent's movement. We formulate the problem as a two‐person zero‐sum game, and develop efficient algorithms to compute each player's optimal strategy. The solution to the game will help the agent choose sensor locations and design patrol routes that can handle imperfect information. First, we prove the existence of ?‐optimal strategies for POAIGs with an infinite time horizon. Second, we introduce a Bayesian approximation algorithm to identify these ?‐optimal strategies using belief functions that incorporate the imperfect information that becomes available during the game. For the solutions of large POAIGs with a finite time horizon, we use a solution method common to extensive form games, namely, the sequence form representation. To illustrate the POAIGs, we present several examples and numerical results.     

Differential-game examination of optimal time-sequential fire-support strategies

Optimal time-sequential fire-support strategies are studied through a two-person zero-sum deterministic differential game with closed-loop (or feedback) strategies. Lanchester-type equations of warfare are used in this work. In addition to the max-min principle, the theory of singular extremals is required to solve this prescribed-duration combat problem. The combat is between two heterogeneous forces, each composed of infantry and a supporting weapon system (artillery). In contrast to previous work reported in the literature, the attrition structure of the problem at hand leads to force-level-dependent optimal fire-support strategies with the attacker's optimal fire-support strategy requiring him to sometimes split his artillery fire between enemy infantry and artillery (counterbattery fire). A solution phenomnon not previously encountered in Lanchester-type differential games is that the adjoint variables may be discontinuous across a manifold of discontinuity for both players' strategies. This makes the synthesis of optimal strategies particularly difficult. Numerical examples are given.     

On nash subsets of bimatrix games

This work considers a class of bimatrix games to which some well-known structure theorems of 0-sum matrix games can be made to generalize. It is additionally shown how to construct such games and how to generate the equilibrium points defining a given game as a member of that class.     

Modified fictitious play

We describe a modification of Brown's fictitious play method for solving matrix (zero-sum two-person) games and apply it to both symmetric and general games. If the original game is not symmetric, the basic idea is to transform the given matrix game into an equivalent symmetric game (a game with a skew-symmetric matrix) and use the solution properties of symmetric games (the game value is zero and both players have the same optimal strategies). The fictitious play method is then applied to the enlarged skew-symmetric matrix with a modification that calls for the periodic restarting of the process. At restart, both players' strategies are made equal based on the following considerations: Select the maximizing or minimizing player's strategy that has a game value closest to zero. We show for both symmetric and general games, and for problems of varying sizes, that the modified fictitious play (MFP) procedure approximates the value of the game and optimal strategies in a greatly reduced number of iterations and in less computational time when compared to Brown's regular fictitious play (RFP) method. For example, for a randomly generated 50% dense skew-symmetric 100 × 100 matrix (symmetric game), with coefficients |a_ij| ≤ 100, it took RFP 2,652,227 iterations to reach a gap of 0.03118 between the lower and upper bounds for the game value in 70.71 s, whereas it took MFP 50,000 iterations to reach a gap of 0.03116 in 1.70 s. Improved results were also obtained for general games in which the MFP solves a much larger equivalent symmetric game. © 1996 John Wiley & Sons, Inc.     

Finding saddle points on polyhedra: Solving certain continuous minimax problems

This article reviews procedures for computing saddle points of certain continuous concave-convex functions defined on polyhedra and investigates how certain parameters and payoff functions influence equilibrium solutions. The discussion centers on two widely studied applications: missile defense and market-share attraction games. In both settings, each player allocates a limited resource, called effort, among a finite number of alternatives. Equilibrium solutions to these two-person games are particularly easy to compute under a proportional effectiveness hypothesis, either in closed form or in a finite number of steps. One of the more interesting qualitative properties we establish is the identification of conditions under which the maximizing player can ignore the values of the alternatives in determining allocation decisions. © 1996 John Wiley & Sons, Inc.     

The timing of third-party intervention in social conflict

This paper analyzes how the equilibrium outcome of social conflict between factions is strategically altered by third-party intervention. We consider an intervening third party that commits financial support to one of two contending factions for reducing its cost in conflict. Within the framework of three-player sequential-move games, we investigate the questions as follows. What is the optimal intervention intensity in terms of the third party’s financial support? Is there a first-mover advantage in conflict when there is third-party intervention? Fighting against all odds, will the unsupported faction have a chance to prevail when its opponent receives third-party support? What is the optimal timing of third-party intervention? The analysis in the paper has implications for the conditions under which the strategic intervention of a third party may or may not break a conflict between factions.     

A game theoretic approach to a two firm bidding problem

In this paper a model is developed for determining optimal strategies for two competing firms which are about to submit sealed tender bids on K contracts. A contract calls for the winning firm to supply a specific amount of a commodity at the bid price. By the same token, the production of that commodity involves various amounts of N different resources which each firm possesses in limited quantities. It is assumed that the same two firms bid on each contract and that each wants to determine a bidding strategy which will maximize its profits subject to the constraint that the firm must be able to produce the amount of products required to meet the contracts it wins. This bidding model is formulated as a sequence of bimatrix games coupled together by N resource constraints. Since the firms' strategy spaces are intertwined, the usual quadratic programming methods cannot be used to determine equilibrium strategies. In lieu of this a number of theorems are given which partially characterize such strategies. For the single resource problem techniques are developed for determining equilibrium strategies. In the multiple resource problem similar methods yield subequilibrium strategies or strategies that are equilibrium from at least one firm's point of view.     

Designing and pricing menus of extended warranty contracts

Extended warranties provide “piece of mind” to a consumer in that product failures which occur after the base warranty expires are rectified at little or no cost. They also provide an additional source of revenue for manufacturers or third‐party providers, such as retailers or insurance providers, and help cultivate consumer loyalty. In this article, we analyze a number of extended warranty contracts which differ in design, including restrictions on deferrals and renewals. With the use of dynamic programming, we compute the optimal strategy for a consumer with perfect information and determine the optimal pricing policy for the provider given the consumer's risk characterization. We also provide insight into when different contracts should be issued. Finally, we illustrate how profits can be dramatically increased by offering menus of warranty contracts, as opposed to stand alone contracts, with the use of integer programming. Surprisingly, risk‐taking consumers provide the greatest benefit to offering menus. These insights can help a company develop a comprehensive warranty planning strategy for given products or product lines. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Capacity investment decisions under risk aversion

This article studies the optimal capacity investment problem for a risk‐averse decision maker. The capacity can be either purchased or salvaged, whereas both involve a fixed cost and a proportional cost/revenue. We incorporate risk preference and use a consumption model to capture the decision maker's risk sensitivity in a multiperiod capacity investment model. We show that, in each period, capacity and consumption decisions can be separately determined. In addition, we characterize the structure of the optimal capacity strategy. When the parameters are stationary, we present certain conditions under which the optimal capacity strategy could be easily characterized by a static two‐sided (s, S) policy, whereby, the capacity is determined only at the beginning of period one, and held constant during the entire planning horizon. It is purchased up to B when the initial capacity is below b, salvaged down to Σ when it is above σ, and remains constant otherwise. Numerical tests are presented to investigate the impact of demand volatility on the optimal capacity strategy. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 218–235, 2016     

Newsvendor problems with sequentially revealed demand information

This article analyzes a capacity/inventory planning problem with a one‐time uncertain demand. There is a long procurement leadtime, but as some partial demand information is revealed, the firm is allowed to cancel some of the original capacity reservation at a certain fee or sell off some inventory at a lower price. The problem can be viewed as a generalization of the classic newsvendor problem and can be found in many applications. One key observation of the analysis is that the dynamic programming formulation of the problem is closely related to a recursion that arises in the study of a far more complex system, a series inventory system with stochastic demand over an infinite horizon. Using this equivalence, we characterize the optimal policy and assess the value of the additional demand information. We also extend the analysis to a richer model of information. Here, demand is driven by an underlying Markov process, representing economic conditions, weather, market competition, and other environmental factors. Interestingly, under this more general model, the connection to the series inventory system is different. © 2012 Wiley Periodicals, Inc. Naval Research Logistics 2012     

New results on a stochastic duel game with each force consisting of heterogeneous units

Two forces engage in a duel, with each force initially consisting of several heterogeneous units. Each unit can be assigned to fire at any opposing unit, but the kill rate depends on the assignment. As the duel proceeds, each force—knowing which units are still alive in real time—decides dynamically how to assign its fire, in order to maximize the probability of wiping out the opposing force before getting wiped out. It has been shown in the literature that an optimal pure strategy exists for this two‐person zero‐sum game, but computing the optimal strategy remained cumbersome because of the game's huge payoff matrix. This article gives an iterative algorithm to compute the optimal strategy without having to enumerate the entire payoff matrix, and offers some insights into the special case, where one force has only one unit. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 56–65, 2014     

Optimal nonmyopic gambling strategy for the generalized Kelly criterion

We consider the optimal wagers to be made by a gambler who starts with a given initial wealth. The gambler faces a sequence of two-outcome games, i.e., “win” vs. “lose,” and wishes to maximize the expected value of his terminal utility. It has been shown by Kelly, Bellman, and others that if the terminal utility is of the form log x, where x is the terminal wealth, then the optimal policy is myopic, i.e., the optimal wager is always to bet a constant fraction of the wealth provided that the probability of winning exceeds the probability of losing. In this paper we provide a critique of the simple logarithmic assumption for the utility of terminal wealth and solve the problem with a more general utility function. We show that in the general case, the optimal policy is not myopic, and we provide analytic expressions for optimal wager decisions in terms of the problem parameters. We also provide conditions under which the optimal policy reduces to the simple myopic case. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 639–654, 1997     

Optimal search for a Markovian target

This article is concerned with an optimal search method for detecting a randomly moving target whose dynamics are described by a stochastic differential equation. The key notions are formulating the problem as one of optimal control and establishing the searcher's strategy by finding the control signal minimizing the probability that the searcher fails to detect the target. The search equation and the search function are derived, and sufficient conditions are given for the existence of an optimal search control. Finally, in order to circumvent difficulties arising in the realization of the optimal search algorithm, a successive approximation is presented with simulation studies.     

Optimal cutoff strategies in capacity expansion problems

Capacity expansion refers to the process of adding facilities or manpower to meet increasing demand. Typical capacity expansion decisions are characterized by uncertain demand forecasts and uncertainty in the eventual cost of expansion projects. This article models capacity expansion within the framework of piecewise deterministic Markov processes and investigates the problem of controlling investment in a succession of same type projects in order to meet increasing demand with minimum cost. In particular, we investigate the optimality of a class of investment strategies called cutoff strategies. These strategies have the property that there exists some undercapacity level M such that the strategy invests at the maximum available rate at all levels above M and does not invest at any level below M. Cutoff strategies are appealing because they are straightforward to implement. We determine conditions on the undercapacity penalty function that ensure the existence of optimal cutoff strategies when the cost of completing a project is exponentially distributed. A by-product of the proof is an algorithm for determining the optimal strategy and its cost. © 1995 John Wiley & Sons, Inc.