连续对策之判断下的最优策略集

本文引进连续对策上的判断块、判断准确、判断下的最优策略集等概念，得到了如下几个主要结果：1.判断下的最优策略集是一个局部凸空间的非空有界闭凸集；2.两个判断下的最优策略集相等的充要条件是这两个判断位于同一判断块中；3.若局中人判断准确，则在一次性对策下不论他使用此判断下的那一个最优策略（不论是纯的还是混合的），都可无风险地取得最优赢得。     

矩阵对策的两个注记

设(x*,y*)是以A=[aij]m×n为赢得矩阵G的对策解,则当局中人1,2各自独立地使用其最优策略x*=(x*1,x*2,…,xmn),y*=(y*1,y*2,…,y*n)时,局中人1的赢得期望为对策值v*=x*Ay*T.若局中人双方使用使得方差D(x*,y*)=∑∑(aij-v*)2x*iy*j达最小的对策解(x*,y*),则其赢得靠近v*的概率达到最大.以O记使方差达到最小的对策解的集合.若O满足(x(1),y(1)),(x(2),y(2))∈O蕴涵(x(1),y(2)),(x(2),y(1))∈O,则说O是可换的.本文首先证明了若矩阵对策G有纯解,则O是可换的.然后证明了如果限定局中人1在其混合扩充策略集的一个非空紧凸子集X中选取策略,那么存在X的一个非空紧子集O(X),它是有限个非空互不相交紧凸集之并,使得只要局中人1使用O(X)中的策略,那么在最坏的情况下可以取得最好的赢得.     

一类多步矩阵对策上的计策问题

以有序树为工具,研究了可以描述连环计,诱敌深入等多步矩阵对策上的一类计策模型.在不考虑信息环境的封闭对策系统中,及局中人对每一步矩阵对策的赢得矩阵,两个局中人的策略集合以及局中人的理性等的了解都是局中人的共同知识的假定下,提出了局中人的最优计策链及将计就计等概念,研究了局中人中计和识破计策的固有概率,讨论了局中人在什么情况下最好主动用计,在什么情况下最好从动用计以及求解最优计策等问题.     

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.     

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。     

网格状有向图上的部分合作对策

本文通过在有向图上每个状态结点处定义合作函数,运用Berge C的关于图匕对策中策略的概念,在网格状有向图上考察部分合作动态对策.局中人在对策进程中将采取部分合作而不是完全合作,部分合作的主要特征是每个局中人的行为是合作行动与单独行动的组合.本文合作函数的设定允许局中人加入某个联盟之后再脱离该联盟,同时给出了有向图上部分合作对策的值、最优路径的算法及示例.     

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。     

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.     

一类特殊凸对策的求解方法

针对支付函数对每个自变量都是严格凸函数的一类特殊凸对策问题，提出了求解局中人双方最优策略的一种简单方法。     

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.     

On maximum tests for normal distributions

Lety be a normally distributed random vector with known regular covariance matrix and letA, B be disjoint closed convex sets inR ⁿ . To be tested is the zero-hypothesisE(y)εA against the alternative hypothesisE(y) ε B at a level of significanceα. Taking the set of admissible tests as one strategy set, the set of probability densities corresponding toB as the other strategy set and the power function of the test problem as the pay-off function this game has an equilibrium point. Thus there is a test, in particular a Neyman-Pearson test, which is simultaneously a maximin and a minimax test. The optimal test is uniquely determined, except on sets with measure zero. Finally the case of non-convexA, B is briefly considered.     

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.     

Representation of finite action large games

A large game can be formalized as a probability distribution on the set of players' characteristics or as a function from a measure space of players to the set of players' characteristics. Given a game as a probability distribution on the set of players' characteristics, a representation of that game is a function from a set of players to the set of players' characteristics which induces the same distribution. It is shown that if the playoffs are continuous and there are only finite number of actions, then the set of Nash equilibria of any representation of a game induces essentially all the Cournot-Nash equilibrium distributions of the given game.     

Combinatorial analysis of Tetris-like games

We study some combinatorial properties of Tetris-like games by using Schützenberger methodology and probability generating functions. We prove that every Tetris-like game is equivalent to a finite state automaton and propose a straight foward algorithm to transform a Tetris-like game into its corresponding automaton. In this way, we can study the average number of pieces inserted during a game and the average score as a function of the player's ability and the pieces extrusion.     

Maximum entropy interpretation of autoregressive spectral densities

A new proof is given of the maximum entropy characterization of autoregressive spectral densities as models for the spectral density of a stationary time series. The new proof is presented in parallel with a proof of the maximum entropy characterization of exponential models for probability densities. Concepts of entropy, cross-entropy and information divergence are defined for probability densities and for spectral densities.     

博弈参与人的偏好对最优反应的影响分析

在现实社会中，每个人都有自己的偏好，因此在博弈过程中，参与人的不同偏好在其选择策略时起着不同的作用。本文运用多目标决策方法研究了总需求不确定情况下具有风险偏好的企业决策者关于一种产品价格博弈模型和均衡，并进一步讨论了参与人具有相同偏好和不同偏好情况下风险厌恶程度、价格对需求影响程度等参数对参与人最优反应的影响。     

Probability of Winning at Tennis I. Theory and Data

The probability of winning a game, a set, and a match in tennis are computed, based on each player's probability of winning a point on serve, which we assume are independent identically distributed (iid) random variables. Both two out of three and three out of five set matches are considered, allowing a 13-point tiebreaker in each set, if necessary. As a by-product of these formulas, we give an explicit proof that the probability of winning a set, and hence a match, is independent of which player serves first. Then, the probability of each player winning a 128-player tournament is calculated. Data from the 2002 U.S. Open and Wimbledon tournaments are used both to validate the theory as well as to show how predictions can be made regarding the ultimate tournament champion. We finish with a brief discussion of evidence for non-iid effects in tennis, and indicate how one could extend the current theory to incorporate such features.     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.