二人零和连续对策上的判断与最优策略间的关系

引进了二人零和连续对策上的判断，判断块以及在判断下的最优策略等概念．研究了判断下的最优策略集与经典最优策略集之间的关系，得到了一些在理论上有趣而且有用的结果．     

矩阵对策的公平性研究

众所周知，零和二人有限对策也称为矩阵对策。设做一个矩阵对策的两个局中人都希望对策结果尽可能公平。当两个局中人使用对策解中的策略进行对策时，如果对策结果最公平，那么这个对策解称为最优的。本文证明了最优对策解集的一些性质，然后给出矩阵对策公平度的概念并证明了它的一些有趣的性质。     

模糊矩阵对策

本文考虑了三种类型的模糊矩阵对策问题，提出了最优解集和对策值对价值标准的连续依赖性，对策双方最优策略的可调和性等概念，得到了一些基本结果。模糊矩阵对策；价值标准；连续依赖性；可调和性     

半马尔可夫对策

本文考虑半马尔可夫随机对策.在一定条件下,我们证明随机对策有值函数,两个局中人相对于折扣报酬都有最优策略.     

一类带有感染年龄SIR模型最优化免疫策略的存在性

对于一个免疫策略来讲,付出(单位时间内接种疫苗的数量)和效果(再生数的大小)是两个重要概念.在给定的费用下找到带有最小再生数的策略和在给定的再生数下找到最小费用的策略是两个最优问题.对一个确定的免疫策略来说,人群中的易感群体和染病群体会趋于相对稳定的状态.当一种疾病侵袭已免疫人群时,用带有感染年龄的SIR模型去描述这类疾病的传播更为准确.因此,本文研究了一类带有感染年龄的SIR模型,得到了最优化策略的存在性.     

两个保险公司的最优合并时刻问题

Gerber和Shiu(2006)将两个公司的合并可行性问题与最优分红策略问题联系起来,给出公司合并会产生效益的条件,并提出一个更现实的问题:若选择合并,最优合并时刻是什么?本文分两步研究了这个问题.首先,利用混合奇异/二维最优停止理论,证明相应的验证定理,然后分两种情形讨论了这个问题.最终得到的最优策略为:情形1下不可能合并,两个公司分别采取最有利于自己的分红策略;情形2下,整个区域被分成3部分U_1、U_2和U_3,最优策略取决于两公司的盈余落在哪个区域.若落在区域U_1,最优策略同情形1;若落在区域U_3,两公司马上合并,合并后的公司采取它的最优分红策略;若落在区域U_2,两公司不分红并等待,直到其盈余过程到达U_1或U_3,然后,施行如上所述的合并和分红策略.     

约束折扣半马氏决策规划

本文研究约束折扣半马氏决策规划问题，即在一折扣期望费用约束下，使折扣期望报酬达最大的约束最优问题，假设状态集可数，行动集为紧的非空Ｂｏｒｅｌ集，本文给出了ｐ－约束最优策略的充要条件，证明了在适当的假设条件下必存在ｐ－约束最优策略。     

考虑消费者策略行为的全渠道两阶段广告投放与动态定价决策

在需求为导向的全渠道市场环境下，切实从消费者行为属性制定广告与价格营销策略是促进全渠道零售长足发展的关键。本文将消费者策略行为引入到全渠道零售中，构建零售商在两阶段均不投放广告、第一阶段正常期投放广告及第二阶段优惠期投放广告等三种广告决策模式下全渠道两阶段广告投放与动态定价模型，探讨全渠道零售商的最优响应策略，并对模型进行数值算例。研究发现：当消费者的策略性水平较低时，零售商选择在正常期(优惠期)投放广告策略下，正常期产品定价较高(较低),优惠期产品定价较低(较高);零售商在正常期或优惠期投放广告对正常期销量都是有利的；当消费者的策略性水平较低或广告的影响系数较低时，零售商选择在正常期投放广告能获得最优收益；当消费者的策略性水平偏高、广告的影响系数适中时，零售商在优惠期投放广告是最优的；当消费者的策略性水平及广告的影响系数都偏高时，零售商选择在两阶段均不投放广告是最优策略。     

报酬无界的平均准则马氏决策过程

本文对可数状态集、非空决策集、报酬无界的平均准则马氏决策过程，提出了一组新的条件，在此条件下存在（ε）最优平稳策略，且当最优不等式中的和有定义时最优不等式也成立。     

报酬无界的连续时间折扣马氏决策规划

本文讨论了报酬函数夫界，转移速率族一致有界，状态空间和行动集均可数的连续时间折扣马氏决策规划，文中引入了一为新的无界报酬函数，并在一新的马氏策略类中，证明了有界报酬下成立的所有结果。讨论了最优策略的结构，得到了该模型策略为最优的一个充要条件。     

Convex semi-infinite games

This paper introduces a generalization of semi-infinite games. The pure strategies for player I involve choosing one function from an infinite family of convex functions, while the set of mixed strategies for player II is a closed convex setC inR ⁿ. The minimax theorem applies under a condition which limits the directions of recession ofC. Player II always has optimal strategies. These are shown to exist for player I also if a certain infinite system verifies the property of Farkas-Minkowski. The paper also studies certain conditions that guarantee the finiteness of the value of the game and the existence of optimal pure strategies for player I.Many thanks are due to the referees for their detailed comments.     

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.     

Weakly Convex Sets and Their Properties

In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.     

模糊判断矩阵一致性逼近及排序方法

根据一致性模糊判断矩阵定义,提出了一种求取一致性判断矩阵及方案排序的新方法,该方法是通过建立一个线性目标规划模型来得到排序向量,并相应地得到逼近于决策偏好的一致性判断矩阵,最后给出了一个算例。     

Equilibrium payoffs of dynamic games

We give a characterization of the equilibrium payoffs of a dynamic game, which is a stochastic game where the transition function is either one or zero and players can only use pure actions in each stage. The characterization is in terms of convex combinations of connected stationary strategies; since stationary strategies are not always connected, the equilibrium set may not be convex. We show that subgame perfection may reduce the equilibrium set.     

极值问题与一类具有无限纯策略集的最优混合策略

证明了局部凸线性拓扑空间上实值连续泛函的极大极小定理与极大定理，并由此证明了一类具有无限纯策略集的二人对策中最优混合策略的存在性。该证明对经典情形也适用，而且比传统的证明要简洁。     

Optimal strategies for node selection games on oriented graphs: Skew matrices and symmetric games

In the node selection game Γ_D each of the two players simultaneously selects a node from the oriented graph D. If there is an arc between the selected nodes, then there is a payoff from the “dominated” player to the “dominating” player. We investigate the set of optimal strategies for the players in the node selection game Γ_D. We point out that a classical theorem from game theory relates the dimension of the polytope of optimal strategies for Γ_D to the nullity of certain skew submatrix of the payoff matrix for Γ_D. We show that if D is bipartite (with at least two nodes in each partite set), then an optimal strategy for the node selection game Γ_D is never unique. Our work also implies that if D is a tournament, then there is a unique optimal strategy for each player, a result obtained by Fisher and Ryan [Optimal strategies for a generalized “scissors, paper, and stone” game, Amer. Math. Monthly 99 (1992) 935–942] and independently by Laffond, Laslier, and Le Breton [The bipartisan set of a tournament game, Games Econom. Behav. 5 (1993) 182–201].     

A two-person zero-sum Markov game with a stopped set

We consider a two-person zero-sum Markov game with continuous time up to the time that the game process goes into a fixed subset of a countable state space, this subset is called a stopped set of the game. We show that such a game with a discount factor has optimal value function and both players will have their optimal stationary strategies. The same result is proved for the case of a nondiscounted Markov game under some additional conditions, that is a reward rate function is nonnegative and the first time τ (entrance time) of the game process going to the stopped set is finite with probability one (i.e., p(τ < ∞) = 1). It is remarkable that in the case of a nondiscounted Markov game, if the expectation of the entrance time is bounded, and the reward rate function need not be nonnegative, then the same result holds.