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# 数学代写|运筹学代写OPERATIONS RESEARCH代考|KMA255 The Dice Game Pig

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## 数学代写|运筹学代写Operations Research代考|The Dice Game Pig

A popular dice game in the United States is Pig. Two persons take turns. The player whose turn it is rolls one die at a time but may roll the die several times during his turn. If he rolls a 1 , his turn is over; otherwise, the player must choose whether to roll the die again or stop. If the player’s turn is over because he rolls a 1 , all points scored that turn are lost, and his total score remains what it was at the beginning of the turn. If the player decides to end his turn without rolling a 1 , the total number of points scored during the turn is added to the player’s total score. The winner is the first player to score 100 or more points. A coin toss determines which player begins.

We only consider the one-player version with goal to minimize the expected number of turns needed to reach the 100 points. ${ }^1$ We say that the state is $(i, 0)$ if it is the beginning of a turn and $i$ more points are needed to reach the goal of $G=100$ points (so the current total score is $100-i$ points). When a turn is in progress, the state of the system is defined by $s=(i, k)$ with $i$ the number of points still needed at the beginning of the current turn and $k(\geq 2)$ the number of points scored in the current turn without rolling a 1 . In every state $(i, k)$, there are two decisions, “stop” and “roll,” where, of course, the decision “roll” is the only possible one in state $(i, 0)$. The value function $V(s)$ is defined by
$V(s)=$ minimum expected value of the number of new turns to still play to reach 100 points from state $s$.

## 数学代写|运筹学代写Operations Research代考|Optimal Stopping and the One-Stage-Look-Ahead Rule

In Section 5.8 .2 we already considered an instance of an optimal stopping problem. In this section the optimal stopping problem in stochastic dynamic optimization is put in a more general context and the optimality of the appealing one-stage-lookahead rule is discussed. Let us consider a random process that is observed at time points $t=0,1,2, \ldots$ to be in one of the states of a finite or countably infinite set $I$. In each state there are no more than two possible actions: $a=0$ (stop) and $a=1$ (continue). When in state $i$ and choosing the stopping action $a=0$, a terminal reward $R(i)$ is received. Suppose that there is a termination state that is entered upon choosing the stopping action. Once this state is entered the system stays in that state and no further costs or rewards are made thereafter. When in state $i$ and choosing action $a=1$, a continuation cost $c(i)$ is incurred and the process moves on to the next state according to the transition probabilities $p_{i j}$ for $i, j \in I$. The following assumption is made:
(A) $\sup _{i \in I} R(i)<\infty$ and $c(i) \geq 0$ for all $i \in I$.
(B) There is a nonempty set $S_0$ consisting of the states in which stopping is mandatory and having the property that the process will reach the set $S_0$ within a finite expected time when always the continuation action $a=1$ is chosen in the states $i \notin S_0$, whatever the starting state is.

## 数学代写|运筹学代写OPERATIONS RESEARCH代 考|THE DICE GAME PIG

$V(s)=$ 从状态达到 100 点的新回合数的最小期望值 $s$.

## 数学代写|运筹学代写OPERATIONS RESEARCH代 考|OPTIMAL STOPPING AND THE ONE-STAGELOOK-AHEAD RULE

$A \sup _{i \in I} R(i)<\infty$ 和c $(i) \geq 0$ 对全部 $i$
$\notin S_0$ ，无论起始状态是什么

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。