连续对策上的计策问题

限定一个连续对策不是平凡地无意义（例如对某个局中人绝对有利等），我们提出了连续对策上的计策的基本概念。最后得到结论，如果局中人1使用经典对策，那么他的赢得期望必不是赢得函数的最大值。如果局中人1使用计策成功（即使得局中人2中计），那么局中人1必取得赢得函数的最大值，局中人2也有对偶的结果。     

矩阵计策的支撑解系

令[aij]n×n是二人零和对策的支付矩阵.局中人1可用其"计策"得到最大支付a＝max{aij|1≤i≤n,1≤j≤n},然而,一个开放问题是如何找到全体计策解,本文首先引进计策解系的一种特殊类型--支撑解系.然后研究支撑解系的特征、性质、代数结构.最后给出寻找全体基本支撑解系的一个算法.     

基于风险偏好与满意度的区间值合作对策

研究区间Shapley值一般是以超可加区间值合作对策或凸区间值合作对策为前提,但这限制了区间Shapley值的适用范围。本文以区间数的接受指标及局中人对风险的偏好水平为基础,提出了局中人满意度的概念,并利用满意度对区间值合作对策进行了探讨。通过计算区间值合作对策的局中人与联盟对其区间Shapley值的满意度,来判断区间Shapley值是否被局中人或联盟接受,形成的联盟是否稳定,拓展了区间值合作对策Shapley值的适用范围。同时,得到了当区间值合作对策满足一定条件时满意度的一些性质。     

三步矩阵对策上的无中生有计及其实例

基于一个历史实例及假定:①三步矩阵对策中赢得矩阵都不变,②每步都是局中人1先行动,③对于每步对策,局中人2观测不到对手究竟使用了何策略;但局中人1可以观测到对手所用的策略,建立了三步矩阵对策上的无中生有计(《三十六计》中的第七计)的对策模型.研究了当局中人2中计,半识破和完全识破对手的无中生有计时的赢得和所用的策略的情况.并用上述实例对模型作了说明.     

关于零和二人有限计策问题(英文)

令［ａｉｊ］ｍ ×ｎ是二人零和对策的支付矩阵．本文研究局中人１ 用“计策”获取最大支付ａ＝ ｍ ａｘ｛ａｉｊ｜１≤ｉ≤ｍ ,１≤ｊ≤ｎ｝的问题     

一种扩张n人正规对策上的计策问题

1°本文扩充了经典 n人正规对策的结构及平衡局势的概念 ;2°引进最优结盟及最优局势集以及求最优结盟方案的算法 ;3°在施计论中给出计策的结构和性质 ;4°在破计论中 ,给出将计就计的模型及其值得度的概念 ;5°讨论了某结盟的计策有恰被一人识破的可能时最好以多大概率选用其各个隐蔽策略     

多重因素约束下的网格检查对策问题研究

基于物品数量及每列容量等限制因素,构造局中人的可行策略集合;考虑隐藏成本,处罚规则与检查成功概率等因素,构造相应的支付函数,建立多重因素约束下的网格检查对策模型.根据矩阵对策性质,将对策论问题转化为非线性整数规划问题,利用H(o|¨)lder不等式获得实数条件下的规划问题的解,然后转化为整数解,得到特定条件下的模型的对策值及局中人的最优混合策略.最后,给出一个实例,说明上述模型的实用性及方法的有效性.     

带策略约束的区间数双矩阵博弈的双线性规划求解方法

传统区间数双矩阵博弈理论研究局中人支付值为区间数的策略选择问题,但没有考虑局中人策略选择可能受到各种约束.创建一种求解局中人策略选择受约束且支付值为区间数的双矩阵博弈（简称带策略约束的区间数双矩阵博弈）的简单、有效的双线性规划求解方法.首先,将局中人的博弈支付看作支付值区间中数值的函数.通过证明这种函数具有单调性,据此利用支付值区间的上、下界,构造了一对辅助双线性规划模型,可分别用于显式地计算任意带策略约束的区间数双矩阵博弈中局中人区间数博弈支付的上、下界及其相应的最优策略.最后,利用考虑策略约束条件下企业和政府针对发展低碳经济策略问题的算例,通过比较其与不考虑策略约束情形下的结果,说明了提出的模型和方法的有效性、优越性及可应用性.     

带策略约束的区间数双矩阵博弈的双线性规划求解方法

传统区间数双矩阵博弈理论研究局中人支付值为区间数的策略选择问题,但没有考虑局中人策略选择可能受到各种约束.创建一种求解局中人策略选择受约束且支付值为区间数的双矩阵博弈(简称带策略约束的区间数双矩阵博弈)的简单、有效的双线性规划求解方法.首先,将局中人的博弈支付看作支付值区间中数值的函数.通过证明这种函数具有单调性,据此利用支付值区间的上、下界,构造了一对辅助双线性规划模型,可分别用于显式地计算任意带策略约束的区间数双矩阵博弈中局中人区间数博弈支付的上、下界及其相应的最优策略.最后,利用考虑策略约束条件下企业和政府针对发展低碳经济策略问题的算例,通过比较其与不考虑策略约束情形下的结果,说明了提出的模型和方法的有效性、优越性及可应用性.     

连续对策的最大熵策略密度对策解

对于正方形[0,2]×[0,2]上的连续对策,将局中人的非纯策略(概率分布函数)的导数称为这个局中人的策略密度(概率密度函数).建立了这种连续对策的最大熵理论.主要证明了当每个局中人都没有最优纯策略时,具有最大熵的最优策略密度集合的非空紧凸性,研究了最优策略密度的最大熵,给出一类带有最大熵的连续对策.     

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.     

Zero-sum constrained stochastic games with independent state processes

We consider a zero-sum stochastic game with side constraints for both players with a special structure. There are two independent controlled Markov chains, one for each player. The transition probabilities of the chain associated with a player as well as the related side constraints depend only on the actions of the corresponding player; the side constraints also depend on the player’s controlled chain. The global cost that player 1 wishes to minimize and that player 2 wishes to maximize, depend however on the actions and Markov chains of both players. We obtain a linear programming formulations that allows to compute the value and saddle point policies for this problem. We illustrate the theoretical results through a zero-sum stochastic game in wireless networks in which each player has power constraints     

Single-suit two-person card play II. Dominance

This paper studies a two-person constant sum perfect information game, the End Play Game, arising from an abstraction of end play in bridge. This game was described by Emanuel Lasker who called it whistette. The game uses a deck of cards consisting of a single totally ordered suit of 2n cards. The deck is divided into two hands A and B of n cards each, held by players Left and Right, and one player is designated as having the lead. 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 the value of the game to each player is the number of tricks he takes. The strategy of this game seems to be quite complicated, despite its simple appearance. This paper studies partial orderings on hands. One partial order recognizes regularities in the value function that persist when extra cards are added to hands. A pair of hands (A ^* , B ^* ) dominates a pair of hands (A, B) for Left, if for any set of extra cards (C ₁, C ₂) added to the deck such that A B (which equals A ^* B ^* ) is a block of consecutive cards in the expanded deck A B {C ₁ , C ₂} the value of (A C ₁, B C ₂) to Left always is at least as much as the value to Left of (A ^* C ₁, B ^* C ₂) both when Left has the lead in both games and when Right has the lead in both games. The main result is that ({4, 1}, {3, 2}) dominates ({3, 2}, {4, 1}). Note that with just four cards the hands {4, 1} and {3, 2} are of identical value — they both take one trick independent of the lead or how the hands are played. The dominance result shows that {4, 1} is preferable to {3, 2} when other cards are present. We show that the dominance relation gives a partial order that is not a total order on hands of 3 or more cards. We also study the total point count ordering, which gives a rough estimate for the value of a hand. We derive upper and lower bounds for the value of a hand with given point count.     

带有分布时滞带菌者的疾病模型行波解

对于带有分布时滞带菌者的疾病模型,当分布时滞核是—般的г分布时滞核时,通过线性链技巧和几何奇异扰动理论,本文证明带菌者的疾病模型行波解存在性.     

A BSDE approach to stochastic differential games with incomplete information

We consider a two-player zero-sum stochastic differential game in which one of the players has a private information on the game. Both players observe each other, so that the non-informed player can try to guess his missing information. Our aim is to quantify the amount of information the informed player has to reveal in order to play optimally: to do so, we show that the value function of this zero-sum game can be rewritten as a minimization problem over some martingale measures with a payoff given by the solution of a backward stochastic differential equation.     

Right preconditioned MINRES for singular systems†

We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.     

A Characterization of Nash Equilibrium for the Games with Random Payoffs

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.