标签： ECON6025

经济代写|博弈论代写Game Theory代考|ECON90022

Posted on 2023年3月15日2023年3月15日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

博弈论是对理性主体之间战略互动的数学模型的研究。它在社会科学的所有领域，以及逻辑学、系统科学和计算机科学中都有应用。最初，它针对的是两人的零和博弈，其中每个参与者的收益或损失都与其他参与者的收益或损失完全平衡。在21世纪，博弈论适用于广泛的行为关系；它现在是人类、动物以及计算机的逻辑决策科学的一个总称。

statistics-lab™ 为您的留学生涯保驾护航在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|A Theory of Hypergames on Graphs

A game arena is a transition system with labels assigned to the states. It captures different configurations of the network and the actions that the attacker and the defender may use to change the current configuration. A configuration of system is a set of state variables that jointly define the current state of the system. For instance, a state variable may be a collection of the current host compromised by the attacker, IP addresses of different hosts over the network, an enumeration of services running over each host, or a list of users currently accessing the hosts with their privileges (root, user, and none). Suppose that there are $n$ state variables and we denote the $i$ th state variable as $X_i$, then the domain of a state space can be given by $S=X_1 \times X_2 \times \cdots \times X_n$. Given this notion of state, we formally define a game arena as follows:

Definition 6.1 (Arena) A turn-based, deterministic game arena between two players P1 (defender, pronoun “he”) and P2 (attacker, pronoun “she”) is a tuple
$$
G=\langle S, A, T, \mathcal{A} \mathcal{P}, L\rangle,
$$
whose components are defined as follows:

$S=S_1 \cup S_2$ is a finite set of states partitioned into two sets $S_1$ and $S_2$. At a state in $S_1$, P1 chooses an action and at a state in $S_2, \mathrm{P} 2$ selects an action.
$A=A_1 \cup A_2$ is the set of actions. $A_1$ (resp., $A_2$ ) is the set of actions for P1 (resp., P2);
$T:\left(S_1 \times A_1\right) \cup\left(S_2 \times A_2\right) \rightarrow S$ is a deterministic transition function that maps a state-action pair to the next state.
$\mathcal{A P}$ is the set of atomic propositions.
$L: S \rightarrow 2^{\mathcal{A P}}$ is the labeling function that maps each state $s \in S$ to a set $L(s) \subseteq \mathcal{A} \mathcal{P}$ of atomic propositions that evaluate to true at that state.

经济代写|博弈论代写Game Theory代考|Specifying the Security Properties in LTL

We consider qualitative formal specifications for defender and attacker objectives. Different from quantitative utility functions in terms of costs, qualitative logic formulas capture hard security constraints that the network defense system must satisfy.

The defender has two types of goals, namely (i) operational objectives, such as the services should eventually be available to the legitimate users, and (ii) defense objectives, such as the attacker should never be able to compromise servers with sensitive information. However, the intention of attacker is often unknown. Thus, we consider the worst-case scenario where the attacker’s objective is to violate the security goal of the defender.
We choose to express the security goal of the defender using LTL (Manna and Pnueli 1992). LTL allows us to express the security properties of system with respect to time. We shall now present the formal syntax and semantics of LTL and then discuss several examples.

Let $\mathcal{A P}$ be a set of atomic propositions. Linear Temporal Logic (LTL) has the following syntax,
$$
\varphi:=\mathrm{T}|\perp| p|\varphi| \neg \varphi\left|\varphi_1 \wedge \varphi_2\right| \bigcirc \varphi \mid \varphi_1 \mathrm{U} \varphi_2
$$
where

$T, \perp$ represent universally true and false, respectively.
$p \in \mathcal{A} \mathcal{P}$ is an atomic proposition.
$O$ is a temporal operator called the “next” operator (see semantics below).
$\mathrm{U}$ is a temporal operator called the “until” operator (see semantics below).
Let $\Sigma:=2^{\mathcal{A} P}$ be the finite alphabet. Given a word $w \in \Sigma^\omega$, let $w[i]$ be the $i$ th element in the word and $w[i \ldots]$ be the subsequence of $w$ starting from the ith element. For example, $w=a b c, w[0]=a$, and $w[1 \ldots]=b c$. Formally, we have the following definition of the semantics:
$w \vDash p$ if $p \in w[0]$;
$w \vDash \neg p$ if $p \notin w[0]$
$w \vDash \varphi_1 \wedge \varphi_2$ if $w \vDash \varphi_1$ and $w \vDash \varphi_2$.
$w \vDash \bigcirc \varphi$ if $w[1 \ldots] \vDash \varphi$.
$w \vDash \varphi \cup \psi$ if $\exists i \geq 0, w[i \ldots] \vDash \psi$ and $\forall 0 \leq j<i, w[j \ldots] \vDash \varphi$.

博弈论代考

经济代写|博弈论代写Game Theory代考|A Theory of Hypergames on Graphs

游戏竞技场是一个转换系统，标签分配给状态。它捕获网络的不同配置以及攻击者和防御者可能用来更改当前配置的操作。系统配置是一组共同定义系统当前状态的状态变量。例如，状态变量可能是当前被攻击者攻陷的主机的集合、网络上不同主机的 IP 地址、在每个主机上运行的服务的枚举，或者当前使用其权限访问主机的用户列表（root、用户和无）。假设有 $n$ 状态变量，我们表示 $i$ 第状态变量为 $X_i$ ，那么状态空间的域可以由下式给出 $S=X_1 \times X_2 \times \cdots \times X_n$. 鉴于这种状态概念，我们正式定义一个游戏竞技场如下:
定义 6.1 (竞技场) 两个玩家 P1 (防御者，代词“他”) 和 P2（攻击者，代词“她”) 之间的回合制确定性游戏竞技场是一个元组
$$
G=\langle S, A, T, \mathcal{A} \mathcal{P}, L\rangle
$$
其组件定义如下:

$S=S_1 \cup S_2$ 是分成两组的有限状态集 $S_1$ 和 $S_2$. 在一个状态 $S_1$ ，P1 选择一个动作并且处于一个状态 $S_2, \mathrm{P} 2$ 选择一个动作。
$A=A_1 \cup A_2$ 是动作集。 $A_1$ (分别， $A_2$ ) 是 P1 (resp., P2) 的动作集；
$T:\left(S_1 \times A_1\right) \cup\left(S_2 \times A_2\right) \rightarrow S$ 是将状态-动作对映射到下一个状态的确定性转换函数。
$\mathcal{A} \mathcal{P}$ 是原子命题的集合。
$L: S \rightarrow 2^{\mathcal{A P}}$ 是映射每个状态的标签函数 $s \in S$ 一组 $L(s) \subseteq \mathcal{A} \mathcal{P}$ 在该状态下评估为真的原子命题。

经济代写|博弈论代写Game Theory代考|Specifying the Security Properties in LTL

我们考虑防御者和攻击者目标的定性正式规范。与成本方面的定量效用函数不同，定性逻辑公式捕获网络防御系统必须满足的硬安全约束。
防御者有两种类型的目标，即 (i) 操作目标，例如服务最终应该对合法用户可用，以及 (ii) 防御目标，例如攻击者永远无法破坏具有敏感信息的服务器。然而，攻击者的意图往往是末知的。因此，我们考虑最坏的情况，即攻击者的目标是破坏防御者的安全目标。
我们选择使用 LTL (Manna and Pnueli 1992) 来表达防御者的安全目标。LTL 允许我们表达系统相对于时间的安全属性。我们现在将介绍 LTL 的形式语法和语义，然后讨论几个例子。
让 $\mathcal{A} \mathcal{P}$ 是一组原子命题。线性时间逻辑 (LTL) 具有以下语法，
$$
\varphi:=\mathrm{T}|\perp| p|\varphi| \neg \varphi\left|\varphi_1 \wedge \varphi_2\right| \bigcirc \varphi \mid \varphi_1 \mathrm{U} \varphi_2
$$
在哪里

$T, \perp$ 分别代表普遍正确和普遍错误。
$p \in \mathcal{A} \mathcal{P}$ 是一个原子命题。
$O$ 是一个称为“下一个”运算符的时间运算符（参见下面的语义）。
U是一个称为”until”运算符的时间运算符 (参见下面的语义)。
让 $\Sigma:=2^{\mathcal{A P}}$ 是有限的字母表。给了一个字 $w \in \Sigma^\omega$ ，让 $w[i]$ 成为 $i$ 单词中的第 th 个元素和 $w[i \ldots]$ 是的后续 $w$ 从第 $\mathrm{i}$ 个元素开始。例如， $w=a b c, w[0]=a$ ，和 $w[1 \ldots]=b c$. 形式上，我们有以下语义定义:
$w \models p$ 如果 $p \in w[0]$;
$w \models \neg p$ 如果 $p \notin w[0]$
$w \models \varphi_1 \wedge \varphi_2$ 如果 $w \models \varphi_1$ 和 $w \models \varphi_2$.
$w \models \bigcirc \varphi$ 如果 $w[1 \ldots] \models \varphi$.
$w \models \varphi \cup \psi$ 如果 $\exists i \geq 0, w[i \ldots] \models \psi$ 和 $\forall 0 \leq j<i, w[j \ldots] \models \varphi$.

经济代写|博弈论代写Game Theory代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON40010

Posted on 2023年3月9日2023年3月9日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|Solution Concepts

Since EFGs can be converted into NFGs, the solution concepts introduced in Section 2.3 .1 still apply. Also, there are other solution concepts specifically for EFGs. We briefly introduce some of them at a high level.

One solution concept is subgame perfect equilibrium (or subgame perfect Nash equilibrium, SPE in short). It is a refinement of NE. This means the set of SPE is a subset of NE. A strategy profile is an SPE equilibrium if, in every subgame of the original game, the strategies remain an NE. A subgame is defined as the partial tree consisting of a node and all its successors, with the requirement that the root node of the partial tree is the only node in its information set. This means if the players play a small game that is part of the original game, they still have no incentive to deviate from their current strategy. A subgame can be a subgame of another subgame rooted at its ancestor node. A finite EFG with perfect recall always has an SPE. To find an SPE in a finite game with perfect information, one can use backward induction as all the information sets are singletons. First, consider the smallest subgames rooted at a parent node of a terminal node. After solving these subgames by determining the action to maximize the acting player’s expected utility, one can solve a slightly larger subgame whose subgames have already been solved, assuming the player will play according to the smaller subgames’ solution. This process continues until the original game is solved. The resulting strategy is an SPE. For games of imperfect or incomplete information, backward induction cannot be applied as it will require reasoning about the information sets with more than one node. The SPE concept can also be defined through the one-shot deviation principle: a strategy profile is an SPE if and only if no player can gain any utility by deviating from their strategy for just one decision point and then reverting back to their strategy in any subgame.

In the example game in Figure 2.1, if we remove the dashed box, the game becomes one with perfect information, and it has two subgames in addition to the original game, with the root nodes being nodes 2 and 3 respectively. In this new game, Player 1 choosing $L$ and Player 2 choosing $r$ at both nodes 2 and 3 is an NE but not an SPE, because, in the subgame rooted at node 3, Player 2 choosing $\mathrm{r}$ is not the best action. In contrast, Player 1 choosing $\mathrm{R}$ and Player 2 choosing $\mathrm{r}$ at node 2 , 1 at node 3 is an SPE.

Another solution concept is sequential equilibrium (Kreps and Wilson 1982), which is a further refinement of SPE. A sequential equilibrium consists of not only the players’ strategies but also the players’ beliefs for each information set. The belief describes a probability distribution on the nodes in the information set since the acting player cannot distinguish them when they play the game. Unlike SPE which only considers the subgames rooted from a node that is the only element in its information set, the sequential equilibrium requires that the players are sequentially rational and takes the best action in terms of expected utility concerning the belief at every information set even if it is not a singleton.

经济代写|博弈论代写Game Theory代考|Stackelberg Game

Consider a two-player NFG with finite actions. The two players choose an action in their action set simultaneously without knowing the other player’s strategy ahead of time. In an NE of the game, each player’s strategy is a best response to the other player’s strategy. But what if one player’s strategy is always known to other players ahead of time? This kind of role asymmetry is depicted in a Stackelberg game.

In a Stackelberg game, one player is the leader and chooses her strategy first, and the other players are followers who observe the leader’s strategy and then choose their strategies. The leader has to commit to the strategy she chooses, i.e. she cannot announce a strategy to the followers and play a different strategy when the game is actually played. She can commit to either a pure strategy or a mixed strategy. If she commits to a mixed strategy, the followers can observe her mixed strategy but not the realization of sampled action from this mixed strategy. A Stackelberg game can be described in the same way as before, with a tuple $(\mathcal{N}, \mathcal{A}, u)$ for games in normal form and $(\mathcal{N}, \mathcal{A}, \mathcal{H}, \mathcal{Z}, \chi, \rho, \sigma, u)$ for games in extensive form, with the only difference that one of the players (usually Player 1 ) is assumed to be the leader.

At first sight, one may think that the followers have an advantage as they have more information than the leader when they choose their strategies. However, it is not the case. Consider the Football vs. Concert game again. If the row player is the leader and commits to choosing the action $\mathrm{F}$, a rational column player will choose action $\mathrm{F}$ to ensure a positive payoff. Thus, the row player guarantees herself a utility of 2, the highest utility she can get in this game, through committing to playing pure strategy $\mathrm{F}$. In the real world, it is common to see similar scenarios. When a couple is deciding what to do during the weekend, if one of them has a stronger opinion and insists on going to the activity he or she prefers, they will likely choose that activity in the end. In this example game, the leader ensures a utility that equals the best possible utility she can get in an NE. In some other games, the leader can get a utility higher than any NE in the game by committing to a good strategy. In the example game in Table 2.5 (Up, Right) is the only NE, with the row player’s utility being 4 . However, if the row player is the leader and commits to a uniform random strategy, the column player will choose Left as a best response, yielding an expected utility of 5 for the row player. This higher expected utility for the row players shows the power of commitment.

博弈论代考

经济代写|博弈论代写Game Theory代考|Solution Concepts

由于 EFG 可以转换为 NFG，因此第 2.3.1 节中介绍的解决方案概念仍然适用。此外，还有其他专门针对 EFG 的解决方案概念。我们简要介绍了其中的一些。

一种解决方案概念是子博弈完美均衡（或子博弈完美纳什均衡，简称 SPE）。它是NE的改进。这意味着 SPE 的集合是 NE 的子集。如果在原始博弈的每个子博弈中，策略仍然是 NE，则策略配置文件是 SPE 均衡。子博弈定义为由一个节点及其所有后继节点组成的部分树，要求部分树的根节点是其信息集中的唯一节点。这意味着如果玩家玩的是原游戏的一部分的小游戏，他们仍然没有动力偏离他们当前的策略。一个子博弈可以是另一个以其祖先节点为根的子博弈的子博弈。具有完美召回率的有限 EFG 始终具有 SPE。要在具有完美信息的有限游戏中找到 SPE，可以使用反向归纳法，因为所有信息集都是单例。首先，考虑以终端节点的父节点为根的最小子博弈。在通过确定使行动玩家的预期效用最大化的行动来解决这些子博弈之后，可以解决一个稍大的子博弈，其子博弈已经被解决，假设玩家将根据较小的子博弈的解决方案进行游戏。这个过程一直持续到原始游戏被解决为止。由此产生的策略是 SPE。对于信息不完善或不完整的博弈，不能应用反向归纳，因为它需要对具有多个节点的信息集进行推理。SPE的概念也可以通过单次偏差原则来定义：

在图 2.1 的示例游戏中，如果我们去掉虚线框，则游戏变成一个完全信息游戏，除了原游戏之外还有两个子游戏，根节点分别为节点 2 和 3。在这个新游戏中，玩家 1 选择大号和玩家 2 选择r在节点 2 和 3 上都是 NE 但不是 SPE，因为在以节点 3 为根的子博弈中，玩家 2 选择r不是最好的行动。相反，玩家 1 选择R和玩家 2 选择r在节点 2 处，节点 3 处的 1 是一个 SPE。

另一个解决方案概念是序贯平衡（Kreps 和 Wilson 1982），它是 SPE 的进一步改进。序贯均衡不仅包括参与者的策略，还包括参与者对每个信息集的信念。信念描述了信息集中节点的概率分布，因为代理玩家在玩游戏时无法区分它们。与 SPE 只考虑以节点为根的子博弈不同，该节点是其信息集中的唯一元素，序贯均衡要求参与者是序贯理性的，并且根据与每个信息集的信念有关的预期效用采取最佳行动，即使它不是单身人士。

经济代写|博弈论代写Game Theory代考|Stackelberg Game

考虑具有有限动作的两人 NFG。两个玩家同时在他们的行动集中选择一个行动，而事先不知道对方的策略。在一个 NE 游戏中，每个玩家的策略是对其他玩家策略的最佳反应。但是如果一个玩家的策略总是被其他玩家提前知道呢？Stackelberg 博弈描述了这种角色不对称。

在 Stackelberg 博弈中，一个玩家是领导者并首先选择她的策略，而其他玩家是观察领导者策略然后选择自己策略的追随者。领导者必须对她选择的策略做出承诺，即她不能向追随者宣布策略并在实际进行游戏时采取不同的策略。她可以致力于纯策略或混合策略。如果她承诺采用混合策略，则追随者可以观察到她的混合策略，但看不到从该混合策略中采样的动作的实现。Stackelberg 游戏可以用与之前相同的方式描述，用一个元组(否,A,在)对于正常形式的游戏和(否,A,H,和,H,r,p,在)对于广泛形式的游戏，唯一的区别是其中一名玩家（通常是玩家 1 ）被假定为领导者。

乍一看，人们可能会认为追随者具有优势，因为他们在选择策略时比领导者拥有更多的信息。然而，事实并非如此。再次考虑足球与音乐会比赛。如果行玩家是领导者并承诺选择行动F, 一个理性的列玩家会选择行动F以确保获得积极的回报。因此，划船玩家通过致力于玩纯策略来保证自己的效用为 2，这是她在该游戏中可以获得的最高效用F. 在现实世界中，类似的场景很常见。当一对夫妇在决定周末做什么时，如果其中一方有更强烈的意见并坚持参加他或她喜欢的活动，他们很可能最终会选择那个活动。在这个示例游戏中，领导者确保效用等于她在 NE 中可以获得的最佳效用。在其他一些游戏中，领导者可以通过承诺一个好的策略来获得比游戏中任何 NE 更高的效用。在表 2.5 的示例游戏中（上，右）是唯一的 NE，行玩家的效用为 4 。但是，如果行玩家是领导者并采用统一的随机策略，则列玩家将选择左作为最佳响应，从而为行玩家产生 5 的预期效用。划船运动员的这种更高的预期效用显示了承诺的力量。

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非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON6025

Posted on 2023年3月9日2023年3月9日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|Normal-Form Games

In this section, we introduce normal-form games (NFGs) and formally introduce the solution concepts we mentioned in Section 2.2. Most of the notations and definitions are adapted from Leyton-Brown and Shoham 2008. The normal form is the most basic form of games to represent players’ interactions and strategy space. An NFG captures all possible combinations of actions or strategies for the players and their corresponding payoffs in a matrix, or multiple matrices for more than two players. A player can choose either a pure strategy that deterministically selects a single strategy or play a mixed strategy that specifies a probability distribution over the pure strategies. The goal for all players is to maximize their expected utility. Formally, a finite, $N$-person NFG is described by a tuple $(\mathcal{N}, \mathcal{A}, u)$, where:

$\mathcal{N}={1, \ldots, N}$ is a finite set of $N$ players, indexed by $i$.
$A=A_1 \times \cdots \times A_N$ is a set of jnint actions of the players, where $\mathcal{A}_i$ is a finite set of actions available to player $i . a=\left(a_1, \ldots, a_N\right) \in \mathcal{A}$ is called an action profile with $a_i \in \mathcal{A}_i$.
$u=\left(u_1, \ldots, u_N\right)$ where $u_i: \mathcal{A} \mapsto \mathbb{R}$ is a utility (or payoff) function for player $i$. It maps an action profile $a$ to a real value. An important characteristic of a game is that player $i$ ‘s utility depends on not only his own action $a_i$ but also the actions taken by other players; thus, the utility function is defined over the space of $\mathcal{A}$ instead of $\mathcal{A}i$. A player can choose a mixed strategy. We use $S_i=\Delta^{\left|\mathcal{A}_l\right|}$ to denote the set of mixed strategies for player $i$, which is the probability simplex with dimension $\left|\mathcal{A}_i\right|$. Similarly, $S=S_1 \times \cdots \times S_N$ is the set of joint strategies and $s=\left(s_1, \ldots, s_N\right) \in S$ is called a strategy profile. The support of a mixed strategy is defined as the set of actions that are chosen with a nonzero probability. An action of player $i$ is a pure strategy and can be represented by a probability distribution with support size 1 (of value 1 in one dimension and 0 in all other dimensions). The utility function can be extended to mixed strategies by using expected utility. That is, if we use $s_l\left(a_i\right)$ to represent the probability of choosing action $a_i$ in strategy $s_i$, the expected utility for player $i$ given strategy profile $s$ is $u_i(s)=\sum{a \in \mathcal{A}} u_i(a) \prod_{l^{\prime}=1}^n s_{l^{\prime}}\left(a_{l^{\prime}}\right)$. A game is zero-sum if the utilities of all the players always sum up to zero, i.e. $\sum_l u_i(s)=0, \forall s$ and is nonzero-sum or general-sum otherwise.
Many classic games can be represented in normal form. Table 2.3 shows the game Prisoner’s Dilemma (PD). Each player can choose between two actions, Cooperate (C) and Defect (D). If they both choose $\mathrm{C}$, they both suffer a small loss of -1 . If they both choose $\mathrm{D}$, they both suffer a big loss of -2 . However, if one chooses $\mathrm{C}$ and the other chooses $\mathrm{D}$, the one who chooses $\mathrm{C}$ suffers a huge loss while the other one does not suffer any loss. If the row player chooses a mixed strategy of playing C with probability 0.4 and D 0.6 , while the column player chooses the uniform random strategy, then the row player’s expected utility is $-1.4=(-1) \cdot 0.4 \cdot 0.5+(-3) \cdot 0.4 \cdot 0.5+(-2) \cdot 0.6 \cdot 0.5$. Now we provide the formal definition of best response. We use $-i$ to denote all players but $i$.

经济代写|博弈论代写Game Theory代考|Extensive-Form Games

Extensive-form games (EFGs) represent the sequential interaction of players using a rooted game tree. Figure 2.1 shows a simple example game tree. Each node in the tree belongs to one of the players and corresponds to a decision point for that player. Outgoing edges from a node represent actions that the corresponding player can take. The game starts from the root node, with the player corresponding to the root node taking an action first. The chosen action brings the game to the child node, and the corresponding player at the child node takes an action. The game continues until it reaches a leaf node (also called a terminal node), i.e. each leaf node in the game tree is a possible end state of the game. Each leaf node is associated with a tuple of utilities or payoffs that the players will receive when the game ends in that state. In the example in Figure 2.1, there are three nodes. Node 1 belongs to Player 1 (P1), and nodes 2 and 3 belong to Player 2 (P2). Player 1 first chooses between action L and R, and then Player 2 chooses between action 1 and r. Player 1 ‘s highest utility is achieved when Player 1 chooses L, and Player 2 chooses 1 .

There is sometimes a special fictitious player called Chance (or Nature), who takes an action according to a predefined probability distribution. This player represents the stochasticity in many problems. For example, in the game of Poker, each player gets a few cards that are randomly dealt. This can be represented by having a Chance player taking an action of dealing cards. Unlike the other real players, the Chance player does not rationally choose an action to maximize his utility since he does not have a utility function. In the special case where Nature only takes an action at the very beginning of the game, i.e. the root of the game tree, the game is essentially a Bayesian game, as we will detail in Section 2.7.

In a perfect information game, every player can perfectly observe the action taken by players in the previous decision points. For example, when two players are playing the classic board game of Go or Tic-Tac-Toe, each player can observe the other players’ previous moves before she decides her move. However, it is not the case in many other problems. An EFG can also capture imperfect information, i.e. a game where players are sometimes uncertain about the actions taken by other players and thus do not know which node they are at exactly when they take actions. The set of nodes belonging to each player is partitioned into several information sets. The nodes in the same information set cannot be distinguished by the player that owns those nodes. In other words, the player knows that she is at one of the nodes that belong to the same information set, but does not know which one exactly. For example, in a game of Poker where each player has private cards, a player cannot distinguish between certain nodes that only differ in the other players’ private cards. Nodes in the same information set must have the same set of actions since otherwise, a player can distinguish them by checking the action set. It is possible that an information set only contains one node, i.e. a singleton. If all information sets are singletons, the game is a perfect information game. The strategy of a player specifies what action to take at each information set. In the example game in Figure 2.1, the dashed box indicates that nodes 2 and 3 are in the same information set, and Player 2 cannot distinguish between them. Thus, nodes 2 and 3 have the same action set. This information set effectively makes the example game a simultaneous game as Player 2 has no information about Player 1s previous actions when he makes a move.

博弈论代考

经济代写|博弈论代写Game Theory代考|Normal-Form Games

在本节中，我们介绍了规范形式的博変 (NFG) 并正式介绍了我们在 2.2 节中提到的解决方案概念。大多数符号和定义改编自 Leyton-Brown 和 Shoham 2008。范式是博恋中表示玩家交互和策略空间的最基本形式。NFG 捕获参与者的所有可能的行动或策略组合以及他们在矩阵中的相应收益，或者捕获两个以上参与者的多个矩阵。玩家可以选择确定性地选择单一策略的纯策略，也可以选择指定纯策略概率分布的混合策略。所有参与者的目标都是最大化他们的预期效用。形式上，一个有限的， $N$-person NFG 由元组描述 $(\mathcal{N}, \mathcal{A}, u)$ ，在哪里:

$\mathcal{N}=1, \ldots, N$ 是一个有限集 $N$ 球员，索引i.
$A=A_1 \times \cdots \times A_N$ 是玩家的一组 jnint 动作，其中 $\mathcal{A}_i$ 是玩家可用的一组有限动作 i. $a=\left(a_1, \ldots, a_N\right) \in \mathcal{A}$ 被称为动作配置文件 $a_i \in \mathcal{A}_i$.
$u=\left(u_1, \ldots, u_N\right)$ 在哪里 $u_i: \mathcal{A} \mapsto \mathbb{R}$ 是玩家的效用 (或收益) 函数 $i$. 它映射了一个动作配置文件 $a$ 到一个真正的价值。游戏的一个重要特征是玩家 $i$ 的效用不仅取决于他自己的行动 $a_i$ 还有其他玩家采取的行动；因此，效用函数定义在 $\mathcal{A}$ 代替 $\mathcal{A} i$. 玩家可以选择混合策略。我们用 $S_i=\Delta^{\left|\mathcal{A}l\right|}$ 表示玩家的混合策略集 $i$ ，这是具有维度的概率单纯形 $\left|\mathcal{A}_i\right|$. 相似地， $S=S_1 \times \cdots \times S_N$ 是一组联合策略，并且 $s=\left(s_1, \ldots, s_N\right) \in S$ 称为策略配置文件。混合策略的支持被定义为以非零概率选择的一组动作。玩家的一个动作 $i$ 是一种纯策略，可以用支持大小为 1 的概率分布表示（在一个维度上值为 1 ，在所有其他维度上值为 0 ) 。效用函数可以通过使用期望效用扩展到混合策略。也就是说，如果我们使用 $s_l\left(a_i\right)$ 表示选择动作的概率 $a_i$ 在战略上 $s_i$ ，玩家的预期效用 $i$ 给定的战略概兄 $s$ 是 $u_i(s)=\sum a \in \mathcal{A} u_i(a) \prod{l^{\prime}=1}^n s_{l^{\prime}}\left(a_{l^{\prime}}\right)$. 如果所有玩家的效用总和为零，则游戏是零和游戏，即 $\sum_l u_i(s)=0, \forall s$ 否则是非零和或一般和。
许多经典游戏都可以用正常形式表示。表 2.3 显示了囚徒困境 (PD) 游戏。每个玩家都可以在合作 (C) 和背叛 (D) 两种行动之间进行选择。如果他们都选择C，他们都遭受 -1 的小损失。如果他们都选择 $\mathrm{D}$ ，他们都遭受了 -2 的巨大损失。然而，如果选择C另一个选择D，选择的人C遭受巨大损失，而另一个则没有遭受任何损失。如果行玩家选择以概率 0.4 和 D 0.6 玩 C 的混合策略，而列玩家选择均匀随机策略，则行玩家的期望效用为
$-1.4=(-1) \cdot 0.4 \cdot 0.5+(-3) \cdot 0.4 \cdot 0.5+(-2) \cdot 0.6 \cdot 0.5$. 现在我们提供最佳响应的正式定义。我们用 $-i$ 表示所有玩家，但 $i$.

经济代写|博弈论代写Game Theory代考|Extensive-Form Games

广泛形式的博弈 (EFG) 表示玩家使用有根博弈树的顺序交互。图 2.1 显示了一个简单的博弈树示例。树中的每个节点都属于其中一名玩家，并对应于该玩家的一个决策点。来自节点的出边表示相应玩家可以采取的动作。游戏从根节点开始，根节点对应的玩家先行动。选择的动作将游戏带到子节点，子节点的相应玩家采取动作。游戏继续进行，直到到达一个叶节点（也称为终端节点），即游戏树中的每个叶节点都是游戏的一个可能结束状态。每个叶节点都与一个效用或收益元组相关联，当游戏在该状态下结束时，玩家将收到这些效用或收益。在图 2.1 的示例中，有三个节点。节点 1 属于玩家 1 (P1)，节点 2 和 3 属于玩家 2 (P2)。玩家 1 首先在动作 L 和 R 之间进行选择，然后玩家 2 在动作 1 和 r 之间进行选择。当玩家 1 选择 L 而玩家 2 选择 1 时，玩家 1 的效用最高。

有时有一个特殊的虚构玩家，称为机会（或自然），他根据预定义的概率分布采取行动。这个玩家代表了很多问题中的随机性。例如，在扑克游戏中，每个玩家都会得到几张随机发的牌。这可以通过让 Chance 玩家采取发牌动作来表示。与其他真实玩家不同，机会玩家不会理性地选择一个行动来最大化他的效用，因为他没有效用函数。在自然只在博弈开始时采取行动的特殊情况下，即博弈树的根，博弈本质上是贝叶斯博弈，我们将在 2.7 节中详细介绍。

在一个完美的信息博弈中，每个玩家都可以完美地观察到玩家在之前的决策点采取的行动。例如，当两个玩家正在玩围棋或井字棋等经典棋盘游戏时，每个玩家在决定自己的走法之前都可以观察其他玩家之前的走法。然而，在许多其他问题中并非如此。EFG 还可以捕获不完全信息，即玩家有时不确定其他玩家采取的行动的游戏，因此不知道他们在采取行动时确切处于哪个节点。属于每个玩家的节点集被分成几个信息集。同一信息集中的节点不能被拥有这些节点的玩家区分。换句话说，玩家知道她在属于同一个信息集的节点之一，但不知道具体是哪一个。例如，在每个玩家都有私人牌的扑克游戏中，玩家无法区分仅在其他玩家的私人牌上不同的某些节点。同一信息集中的节点必须具有相同的动作集，否则玩家可以通过检查动作集来区分它们。一个信息集有可能只包含一个节点，即单例。如果所有的信息集都是单例的，那么这个博弈就是一个完美的信息博弈。玩家的策略指定了在每个信息集上采取什么行动。在图2.1的示例游戏中，虚线框表示节点2和节点3在同一个信息集中，玩家2无法区分它们。因此，节点 2 和 3 具有相同的动作集。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECOS3012

Posted on 2022年12月29日2022年12月29日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|STRATEGIC GAMES

Prisoner’s Dilemma is a canonical example of a strategic game because, as we will see shortly, it typifies many scenarios that confront decision makers. Further, being a simple scenario, it can be used to illustrate many of the fundamental concepts of game theory, and it also clearly demonstrates a fundamental dilemma in our (human) decision-making processes.

We model this scenario as a strategic game in which the two suspects, each confined in a separate interrogation room, are the players. We will often refer to our two players in strategic games as Rose and Colin. (This convention helps later to emphasize the distinction between row and column players and was popularized by Phil Straffin in his book Game Theory and Strategy [110].) They each have two strategies available to them which we name Quiet and Confess. Table $3.1$ lists each of the strategy profiles in the form (Rose, Colin) and the resulting outcome.

We assume that each suspect is primarily concerned about their own sentence and wants to minimize it. Table $3.2$ provides payoffs (a common synonym for utilities) for each player. Here we use the utility function 6 minus the number of years in prison; this is consistent with the player preferences. Based on our assumptions, these payoffs are ordinal. For these payoffs to also be vNM, we would need to assume that the suspects are risk neutral in the number of years to be served in prison.

Tables $3.1$ and $3.2$ complete the construction of the model by identifying the strategies, outcomes, and payoffs. We will refer to this model of the Prisoner’s Dilemma scenario as the Prisoner’s Dilemma strategic game.

We are now ready to look for a solution that maximizes the payoffs to the players. By observing that $5>3$, we see that Confess is the best response strategy for Rose if she knows that Colin will choose Quiet. Further, we can observe that Confess is also a best response for Rose if she knows Colin will choose Confess. We formalize the definition of a best response strategy below.

经济代写|博弈论代写Game Theory代考|FINALJEOPARDY

As we observed in Section 3.1, the phrase “Prisoner’s Dilemma” has been used to describe many real-world scenarios; however, not all of these scenarios actually fit the mathematical definition. This can only be revealed by constructing and analyzing a model of the scenario.

We examine a situation in Jeopardy! which fans have identified as a Prisoner’s Dilemma. In the Final Jeopardy round, each contestant makes a wager as to whether they can answer a specific question correctly. When making the wager, contestants know the category of the question, but not the question itself, and the amount of money each of the other contestants has available. Each player’s wager can be between 0 and their current winnings. Depending on whether the contestant answers the question correctly, they win or lose the amount of money wagered. The contestant with the most money after this final round of play wins the game. The winner keeps all of their winnings, and the other two contestants lose essentially all of their money. If there is a tie at the end of the round, a simple, essentially random, tie-breaker rule is applied to identify the winner.

The so-called Prisoner’s Dilemma situation occurs when two contestants are tied for the lead, and the third contestant has less than half of the money of either of the first two contestants. For simplicity we will assume that it is contestants 1 and 2 who are tied with the most money.

In this situation, aficionados of Jeopardy! often refer to “Jeek’s Rule,” which asserts that while they could wager any amount up to their current winnings, contestants 1 and 2 should either wager nothing or everything. We discuss the reasonableness of this rule and then make it an assumption when we define our strategic game.

Let $E$ ‘ be the amount of money contestants 1 and ‘ 2 have each won at the time Final Jeopardy begins. Let $w_i$ denote the wager of contestant $i$ and suppose that contestant 1 ‘s wager satisfies $0<w_1<E$. There are four cases to consider:

Case 1: Both contestants answer the question correctly. In this case, if $w_1<w_2$, contestant 1 regrets not wagering $E$ in order to win. If $w_1 \geq w_2$, then contestant 1 regrets not wagering $E$ to maximize their winnings.

Case 2: Contestant 1 answers the question correctly and contestant 2 does not. Here contestant 1 regrets not wagering $E$ in order to maximize their winnings.

Case 3 : Contestant 1 answers the question incorrectly and contestant 2 answers correctly. Then contestant 1 is indifferent about their bet unless $w_2=0$, in which case they regret not wagering $w_1=0$.

Case 4: Both contestants answer the question incorrectly. Here, if $w_1 \geq w_2$, contestant 1 regrets not wagering $w_1=0$ in order to win. If $w_1<w_2$, then contestant 1 regrets not wagering $w_1=0$ to maximize their winnings.

博弈论代考

经济代写|博弈论代写Game Theory代考|STRATEGIC GAMES

囚徒困境是战略游戏的一个典型例子，因为正如我们很快就会看到的，它代表了决策者面临的许多场景。此外，作为一个简单的场景，它可以用来说明博弈论的许多基本概念，它也清楚地展示了我们（人类）决策过程中的一个基本困境。

我们将此场景建模为一个战略游戏，其中两名嫌疑人是玩家，每个人都被关在一个单独的审讯室里。我们经常将战略游戏中的两名球员称为罗斯和科林。（此约定后来有助于强调行和列玩家之间的区别，并由 Phil Straffin 在他的书博弈论和策略 [110] 中推广。）他们每个人都有两种可用的策略，我们将其命名为 Quiet 和 Confess。桌子3.1列出表格中的每个策略配置文件 (Rose, Colin) 和结果结果。

我们假设每个嫌疑人主要关心的是他们自己的刑期，并希望将其减至最少。桌子3.2为每个玩家提供收益（公用事业的常见同义词）。这里我们使用效用函数 6 减去入狱年数；这符合玩家的喜好。根据我们的假设，这些收益是有序的。为了让这些回报也成为 vNM，我们需要假设嫌疑人在监狱服刑的年数上是风险中性的。

表3.1和3.2通过确定策略、结果和收益来完成模型的构建。我们将这种囚徒困境场景模型称为囚徒困境战略博弈。

我们现在准备寻找一种解决方案，使玩家的收益最大化。通过观察5>3，我们看到如果 Rose 知道 Colin 会选择 Quiet，Confess 是她最好的应对策略。此外，我们可以观察到如果 Rose 知道 Colin 会选择 Confess，那么 Confess 也是她的最佳回应。我们在下面正式定义最佳响应策略。

经济代写|博弈论代写Game Theory代考|FINALJEOPARDY

正如我们在第 $3.1$ 节中观察到的， “图困境”一词已被用来描述许多现实世界的场景；然而，并非所有这些场景实际上都符合数学定义。这只能通过构建和分析场景模型来揭示。
我们检查处于危险中的情况！粉丝将其确定为囚徒困境。在 Final Jeopardy 回合中，每位参赛者都会就他们是否能正确回答特定问题下注。下注时，参塞者知道问题的类别，但不知道问题本身，也不知道其他参塞者每个人可用的金额。每个玩家的赌注可以介于 0 和他们当前的奖金之间。根据参赛者是否正确回答问题，他们赢或输下注的金额。最后一轮比赛结束后，钱最多的参寒者将赢得比䗙。获胜者保留了所有的奖金，而另外两名参寒者基本上输掉了所有的钱。如果在回合结束时出现平局，一个简单的，基本上是随机的，
当两名参塞者并列领先，而第三名参塞者的钱少于前两名参塞者中任何一方的一半时，就会出现所谓的囚徒困境情况。为简单起见，我们假设参塞者 1 和 2 的奖金最多。
在这种情况下，危险的爱好者！通常提到”Jeek 规则”，该规则断言，虽然他们可以下注任何金额，但不超过他们当前的奖金，但参寒者 1 和 2 要么什么都不下，要么全下。我们讨论这条规则的合理性，然后在定义我们的战略游戏时将其作为假设。
让 $E^{\prime}$ 是参赛者 1 和 2 在 Final Jeopardy 开始时各自赢得的金额。让 $w_i$ 表示参赛者的赌注 $i$ 并假设参赛者 1 的赌注满足 $0<w_1<E$. 有四种情况需要考虑:
案例 1: 两位参寨者都正确回答了问题。在这种情况下，如果 $w_1<w_2$ ，选手1后悔没有下注 $E$ 为了赢。如果 $w_1 \geq w_2$ ，那么选手 1 后悔没有下注 $E$ 以最大化他们的奖金。
案例 2: 参寒者 1 正确回答了问题，而参赛者 2 没有。这里选手 1 后悔没有下注 $E$ 为了最大化他们的奖金。，在这种情况下，他们后悔没有下注 $w_1=0$.
案例 4: 两位参寒者都答错了问题。在这里，如果 $w_1 \geq w_2$ ，选手 1 后悔没有下注 $w_1=0$ 为了赢。如果 $w_1<w_2$ ，那么选手 1 后悔没有下注 $w_1=0$ 以最大化他们的奖金。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON6025

Posted on 2022年12月29日2022年12月29日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|CONSTRUCTING UTILITIES

We have already suggested how we might construct a utility function to model a player’s choices when there are a finite number of outcomes. Ordinal preferences can be revealed by asking the player to choose among all outcomes and assign those outcomes the highest utility, asking the player to choose among all outcomes not previously chosen and assign those outcomes the second highest utility, and so forth. vNM preferences can be obtained by asking the player to name the highest and lowest ranked outcomes $o_h$ and $o_l$, assign utilities of $u\left(o_h\right)=1$ and $u\left(o_l\right)=0$ to these outcomes, and then for each remaining outcome $o$ determine a probability $p$ for which the player would be willing to choose either the outcome $o$ or the lottery $(1-p) o_l+p o_h$ and assign $u(o)=p$.

In this section, we examine four specific scenarios to illustrate a variety of ways utility functions may be created.

To model Self-Interest and Other-Interest, we simplify our scenario to examine the monthly salaries of the job offers for each spouse. Suppose Scarlett and Regis receive $\$ x$ thousand and $\$ y$ thousand, respectively; we will denote this by $(x, y)$. Consider the following four possible outcomes: $(7,0),(6,6),(5,7)$, and $(1,6)$. If Scarlett is exclusively self-interested, she would rank order these outcomes in the given order. If Scarlett is primarily interested in Regis receiving money and only secondarily interested in receiving money for herself, Scarlett would rank order the outcomes $(5,7),(6,6),(1,6)$, and $(7,0)$. If Scarlett had a mixture of self-interest, other-interest, and a desire for equity, she might rank order the outcomes $(6,6),(5,7),(7,0)$, and $(1,6)$.

In fact, this last rank order would be obtained if Scarlett considered $\$ 1,000$ given to Regis to be worth the same to her as her receiving $\$ 500$, suggesting the utility function $u(x, y)=x+0.5 y$. Of course, this is only an ordinal utility function unless, at minimum, Scarlett is indifferent between the outcome $(7,0)$ with utility $u(7,0)=7$ and the lottery $L=0.6(6,6)+0.4(1,6)$ with utility
$$
u(L)=0.6 u(6,6)+0.4 u(1,6)=0.6(9)+0.4(4)=7 .
$$
This example demonstrates how we can incorporate both self-interest and altruistic interests into a player’s utility function. Therefore, maximizing a utility function does not necessarily imply selfishness, but rather achieving the most preferred outcome based on the player’s interests.

经济代写|博弈论代写Game Theory代考|DETERMINING RISK

In the duopoly scenario, and in most other economic models, the utility of an outcome is cquivalent to some dollar value associated with the outcome. While we can see how dollar values might capture the intensity of a player’s preferences, dollar values are not necessarily vNM utilities. For example, receiving $\$ 11.00$ instead of $\$ 10.00$ means significantly more than receiving $\$ 1001.00$ instead of $\$ 1000.00$ to most people. To explore this difference, we consider the relationship between the expected utility of a lottery, as given by the Expected Utility Hypothesis, and the utility of the expected value of the lottery.

Consider the following raffle: For $\$ 25$, you can purchase a $\frac{1}{400}$ chance for a $\$ 10,000$ college scholarship. We can represent this lottery with our usual notation
$$
\left.\frac{399}{400}(\text { losing } \$ 25)+\frac{1}{400} \text { (winning } \$ 9,975\right) \text {, }
$$ but since the outcomes are numerical, we can calculate the expected monetary value of the raffle as
$$
\frac{399}{400}(-\$ 25)+\frac{1}{400}(\$ 9,975)=\$ 0 .
$$
The expected monetary value of entering or not entering the raffle is the same, however, entering the raffle involves a small chance of a large gain offset by a large chance of a small loss, while not entering the raffle involves no chance of a gain or a loss. Entering the raffle involves risk while not entering the raffle does not.

Most parents of college students are willing to enter the raffle, but many college students themselves are not. For the college parents,
$$
\left.\left.u\left(\frac{399}{400} \text { (losing } \$ 25\right)+\frac{1}{400} \text { (winning } \$ 9,975\right)\right)>u(\$ 0),
$$
but for the students themselves,
$$
\left.u\left(\frac{399}{400}(\text { losing } \$ 25)+\frac{1}{400} \text { (winning } \$ 9,975\right)\right)<u(\$ 0) .
$$
For the parents, the utility of the lottery is greater than the utility of the expected value, making them risk loving in this scenario. On the other hand, the students are risk adverse since the utility of the lottery is less than the utility of the expected value. This principle holds in general, as we describe in the following definition.

博弈论代考

经济代写|博弈论代写Game Theory代考|CONSTRUCTING UTILITIES

我们已经提出了如何构建效用函数来模拟玩家在结果数量有限时的选择。序数偏好可以通过要求玩家在所有结果中进行选择并为这些结果分配最高效用，要求玩家在所有先前末选择的结果中进行选择并为这些结果分配第二高效用等来揭示。可以通过要求玩家说出排名最高和最低的结果来获得 vNM 偏好 $o_h$ 和 $o_l$ ，分配实用程序 $u\left(o_h\right)=1$ 和 $u\left(o_l\right)=0$ 这些结果，然后是每个剩余的结果 $o$ 确定一个概率 $p$ 玩家愿意选择的结果 $o$ 或彩票 $(1-p) o_l+p o_h$ 并分配 $u(o)=p$.
在本节中，我们将研究四个特定场景来说明可以创建效用函数的各种方式。
为了对自利和其他利益建模，我们简化了我们的场景，以检查每个配偶的工作机会的月薪。假设思嘉和瑞吉斯收到 $\$ x$ 千和 $\$ y$ 千，分别；我们将用 $(x, y)$. 考虑以下四种可能的结果: $(7,0),(6,6),(5,7)$ ，和 $(1,6)$. 如果斯嘉丽完全是自利的，她会按照给定的顺序对这些结果进行排序。如果 Scarlett 主要对 Regis 收钱感兴趣，仅次于为自己收钱，Scarlett 将对结果进行排序 $(5,7),(6,6),(1,6)$ ，和 $(7,0)$. 如果思嘉混合了自身利益、他人利益和对公平的渴望，她可能会对结果进行排序 $(6,6),(5,7),(7,0)$ ，和 $(1,6)$
事实上，如果斯嘉丽考虑，就会获得最后的排名顺序 $\$ 1,000$ 给予 Regis 的价值与她收到的价值相同 $\$ 500$ ，提示效用函数 $u(x, y)=x+0.5 y$. 当然，这只是一个有序的效用函数，除非至少 Scarlett 对结果无动于衷 $(7,0)$ 实用 $u(7,0)=7$ 和彩票 $L=0.6(6,6)+0.4(1,6)$ 实用
$$
u(L)=0.6 u(6,6)+0.4 u(1,6)=0.6(9)+0.4(4)=7 .
$$
这个例子展示了我们如何将自利和利他利益结合到玩家的效用函数中。因此，最大化效用函数并不一定意味着自私，而是基于参与者的利益实现最偏好的结果。

经济代写|博弈论代写Game Theory代考|DETERMINING RISK

在双头垄断的情况下，以及在大多数其他经济模型中，结果的效用等同于与结果相关的一些美元价值。虽然我们可以看到美元价值如何捕捉玩家偏好的强度，但美元价值不一定是 VNM 效用。例如，接收 $\$ 11.00$ 代替 $\$ 10.00$ 意味着远远超过接收 $\$ 1001.00$ 代替 $\$ 1000.00$ 对大多数人来说。为了探索这种差异，我们考虑了预期效用假设给出的彩票预期效用与彩票预期价值的效用之间的关系。
考虑以下抽奖活动：对于 $\$ 25$ ，你可以购买一个 $\frac{1}{400}$ 一个机会 $\$ 10,000$ 大学奖学金。我们可以用我们通常的符号来表示这张彩票
$$
\left.\left.\frac{399}{400} \text { ( losing } \$ 25\right)+\frac{1}{400} \text { (winning } \$ 9,975\right) \text {, }
$$
但由于结果是数字的，我们可以计算抽奖的预期货币价值
$$
\frac{399}{400}(-\$ 25)+\frac{1}{400}(\$ 9,975)=\$ 0 .
$$
参加或不参加抽奖的预期货币价值是相同的，但是，参加抽奖有小概率的大收益被大概率的小损失所抵消，而不参加抽奖则没有获得收益的机会或亏损。参加抽奖有风险，不参加抽奖则没有风险。
大多数大学生家长都愿意参加抽奖，但很多大学生自己却不愿意。对于大学生家长来说，
$$
u\left(\frac{399}{400}(\text { losing } \$ 25)+\frac{1}{400}(\text { winning } \$ 9,975)\right)>u(\$ 0)
$$
但对于学生自己来说，
$$
u\left(\frac{399}{400}(\text { losing } \$ 25)+\frac{1}{400}(\text { winning } \$ 9,975)\right)<u(\$ 0) .
$$
对于父母来说，彩票的效用大于期望值的效用，使他们在这种情况下冒险去爱。另一方面，学生是风险厌恶的，因为彩票的效用小于期望值的效用。正如我们在下面的定义中所描述的那样，这个原则通常是成立的。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON2112

Posted on 2022年12月29日2022年12月29日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|ORDINAL UTILITIES

In this scenario, there are four possible outcomes: Scarlett leaves with vanilla ice cream, which we will label $V$, leaves with chocolate ice cream, labeled $C$, leaves with strawberry ice cream, labeled $S$, or leaves with no ice cream, labeled $N$. Our outcome set is then ${V, C, S, N}$. The rules dictate that Scarlett may choose any outcome that is available.
To model Scarlett’s preferences and build a utility function, we consider the choices she would make when certain outcomes are available. Supposing that all potential outcomes are available, Scarlett would choose $V$; if $C$ were not an available outcome, Scarlett would choose $S$; if only $V$ and $N$ were available outcomes, Scarlett would flip a coin to determine whether to select $V$ or $N$; and if only $C$ and $S$ were available outcomes, Scarlett would be unable to make a choice. While this behavior would be possible, most would find it bizarre: Scarlett’s choice of $V$ when presented with the outcome set ${V, C, S, N}$ suggests she prefers $V$ to $S$, but Scarlett’s choice of $S$ when presented the outcome subset ${V, S, N}$ suggests she prefers $S$ to $V$. To avoid such absurd possibilities, we will assume that individual behavior is governed by self-consistent internal preferences over the outcomes, which is reflected in the mathematical definition of ordinal preferences below.

Definition 2.1.1. A player $i$ is said to have ordinal preferences among outcomes if there exists a utility function $u_i$ from the set $O$ of outcomes into the real numbers, $\mathbb{R}$, such that whenever presented with a subset $O^{\prime} \subseteq O$ of outcomes, player $i$ chooses any of the outcomes that maximize $u_i$ over all outcomes $o \in O^{\prime}$.

To ensure that ordinal preferences align with the players real-world choice behavior, we note that whenever player $i$ prefers outcome $o_j$ over outcome $o_k$, we should have $u_i\left(o_j\right)>$ $u_i\left(o_k\right)$, and when player $i$ is indifferent between $o_j$ and $o_k$ we should have $u_i\left(o_j\right)=u_i\left(o_k\right)$. We now ask under what conditions Scarlett’s outcome choice behavior can be modeled by ordinal preferences and describe three reasonable properties for a self-consistent set of choices. Since it is usually easier for a player to choose between two rather than among many outcomes, these properties will focus on pairwise choices.

First, we want Scarlett’s pairwise choices to be complete, meaning that whenever she is presented with a pair of outcomes, Scarlett is able to make a choice. Equivalently, for each pair of outcomes, $A$ and $B$, exactly one of the following conditions holds: (a) Scarlett chooses $A$ over $B$, (b) Scarlett chooses $B$ over $A$, or (c) Scarlett is willing to flip a coin to determine which outcome to choose (in this case we will often say Scarlett chooses either $A$ or $B$ ). Hence, this condition excludes the following option as a possibility: (d) Scarlett chooses neither $A$ nor $B$. (When we want the rules to allow a player to choose none of the options, we must include that as an outcome, as we did in the Ice Cream Parlor scenario.) For (a), we will assign utilities so that $u(A)>u(B)$. Likewise for (b) we will assign utilities so that $u(B)>u(A)$. Finally for (c), since Scarlett is willing to flip a coin to determine the outcome, we assume she is indifferent between $A$ and $B$ and assign $u(A)=u(B)$.

经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

A significant limitation of ordinal preferences and their associated utility functions is that they cannot describe the intensity of a player’s preference for a particular outcome. That is, they cannot capture the difference between Scarlett preferring vanilla ice cream over chocolate ice cream and Scarlett so strongly preferring vanilla that she would pay for it rather than have a free serving of chocolate. Notice how we have once again translated our intuitive sense of internal preference intensity into something that is observable (a real-world choice) so we can create a utility function based on these choices. While asking players to choose among outcomes that include the receipt or payment of money would be one observable way to determine intensity of preferences, we will take an approach that does not rely on the availability of money.

We begin by introducing a new, probability-based outcome called a lottery. Suppose that when a second customer, Regis, enters the ice cream parlor, he encounters a college student conducting a taste test involving different flavors of ice cream. The college student offers Regis the choice of either a sample of chocolate ice cream (his second-most favorite) or an unknown sample that is either vanilla (his favorite) or strawberry (his least favorite). The second option in this example is a simple lottery.

Definition 2.2.1. Given a set of outcomes, $O$, a simple lottery is a probability distribution over this set. When $O=\left{o_1, o_2, \ldots, o_m\right}$, a finite set, a simple lottery can be denoted by $p_1 o_1+p_2 o_2+\ldots+p_m o_m$ where $p_i$ is the probability of outcome $o_i$. A compound lottery is a probability distribution over other lotteries. Because an outcome $o \in O$ can be written as the simple lottery $1 o$, we see that $O \subset \mathcal{L}$, the set of all (simple and compound) lotteries.
Tó reveál thè strength of a player’s préference for one outcomé over another, wè must examine not nnly chnires hetween single nutromes hut chnices hetween single nutcomes and lotteries. Suppose Regis prefers vanilla $V$ over chocolate $C$ and prefers $C$ over strawberry $S$. The choice Regis makes between $C$ and the lottery $0.5 S+0.5 V$ tells us about the strength of his preference for $V$ over $C$ and for $C$ over $S$. If he would choose either of the two possibilities, then the strength of Regis’s preference for $C$ is exactly halfway between $S$ and $V$. If Regis were to choose $C$ over $0.5 S+0.5 \mathrm{~V}$, it reveals that his preference intensity for $C$ is closer to $V$ than to $S$. If Regis were willing to choose either $C$ or the lottery $0.1 S+0.9 \mathrm{~V}$, it would reveal that his preference for $V$ over $C$ is very small and his preference for $C$ over $S$ is relatively large. However, if he were instead willing to choose $C$ or the lottery $0.9 S+0.1 \mathrm{~V}$, it would reveal a strong preference for $V$ over $C$ and that his preferences for $C$ over $S$ is small. When a player is willing to choose either of two lotteries, this reveals the player is indifferent between these choices. This motivates the following generalization of the utility function concept.

博弈论代考

经济代写|博弈论代写Game Theory代考|ORDINAL UTILITIES

在这种情况下，有四种可能的结果: Scarlett 带着香草冰淇淋离开，我们将其标记为 $V$ ，带有巧克力冰淇淋的叶子，贴有标签 $C$ ，叶子和草莓冰淇淋，贴有标签 $S$ ，或没有冰淇淋的叶子，贴有标签 $N$. 我们的结果集是 $V, C, S, N$. 规则规定斯嘉丽可以选择任何可用的结果。
为了对 Scarlett 的偏好进行建模并构建效用函数，我们考虑了当某些结果可用时她会做出的选择。假设所有可能的结果都是可用的，斯嘉丽会选择 $V$; 如果 $C$ 没有可用的结果，斯嘉丽会选择 $S$; 要是 $V$ 和 $N$ 如果有可用的结果，斯嘉丽会掷硬币来决定是否选择 $V$ 要么 $N$; 如果只有 $C$ 和 $S$ 如果有可用的结果，斯嘉丽将无法做出选择。虽然这种行为是可能的，但大多数人会觉得很奇怪：斯嘉丽选择 $V$ 当出现结果集时 $V, C, S, N$ 表明她更喜欢 $V$ 到 $S$, 但思嘉的选择 $S$ 当呈现结果子集时 $V, S, N$ 表明她更喜欢 $S$ 到 $V$. 为了避免这种荒谬的可能性，我们将假设个人行为受对结果的自洽内部偏好的支配，这反映在下面的序数偏好的数学定义中。
定义 2.1.1。一个玩家 $i$ 如果存在效用函数，则据说在结果之间具有顺序偏好 $u_i$ 从集合 $O$ 结果转化为实数， $\mathbb{R}$ ，这样每当出现一个子集 $O^{\prime} \subseteq O$ 结果，玩家 $i$ 选择任何最大化的结果 $u_i$ 在所有结果上 $o \in O^{\prime}$.
为了确保序数偏好与玩家在现实世界中的选择行为一致，我们注意到每当玩家 $i$ 更喜欢结果 $o_j$ 超过结果 $o_k$ ，我们本应该 $u_i\left(o_j\right)>u_i\left(o_k\right)$ ，当玩家 $i$ 无所谓 $o_j$ 和 $o_k$ 我们本应该 $u_i\left(o_j\right)=u_i\left(o_k\right)$. 我们现在询问在什么条件下 Scarlett 的结果选择行为可以通过序数偏好来建模，并描述自洽选择集的三个合理属性。由于玩家通常更容易在两个结果之间做出选择，而不是在许多结果中做出选择，因此这些属性将侧重于成对选择。
首先，我们㳍望 Scarlett 的成对选择是完整的，这意味着无论何时向她呈现一对结果，Scarlett 都能够做出选择。等价地，对于每对结果， $A$ 和 $B$, 怙好满足以下条件之一：(a) Scarlett 选择 $A$ 超过 $B$ ， (b) 思嘉选择 $B$ 超过 $A$ ，或者 (c) Scarlett 愿意郑硬币来决定选择哪个结果（在这种情况下，我们通常会说 Scarlett 选择 $A$ 要么 $B$ ). 因此，这个条件排除了以下选项的可能性： (d) Scarlett 既不选择 $A$ 也不 $B$. (当我们㣇望规则允许玩家不选择任何选项时，我们必须将其作为结果包括在内，就像我们在冰淇淋店场景中所做的那样。）对于 (a)，我们将分配实用程序，以便 $u(A)>u(B)$. 同样对于 (b) 我们将分配实用程序，以便 $u(B)>u(A)$. 最后对于 (c)，由于 Scarlett 愿意掷硬币来决定结果，我们假设她对两者无动于衷 $A$ 和 $B$ 并分配 $u(A)=u(B)$.

经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

序数偏好及其相关效用函数的一个重要限制是它们无法描述玩家对特定结果的偏好强度。也就是说，他们无法区分 Scarlett 更喜欢香草冰淇淋而不是巧克力冰淇淋，以及 Scarlett 如此强烈地喜欢香草以至于她愿意为它付钱而不是免费享用巧克力之间的区别。请注意我们如何再次将我们对内部偏好强度的直觉转化为可观察的东西 (现实世界的选择)，因此我们可以根据这些选择创建效用函数。虽然要求玩家在包括收款或付款在内的结果中进行选择是确定偏好强度的一种可观察方法，但我们将采用一种不依赖于金钱可用性的方法。
我们首先介绍一种新的、基于概率的结果，称为彩票。假设当第二位顾客 Regis 进入冰淇淋店时，他遇到一名大学生正在对不同口味的冰淇淋进行味觉测试。这位大学生让瑞吉斯选择巧克力冰淇淋样品（他第二喜欢) 或末知样品，香草 (他最喜欢) 或草莓（他最不喜欢) 。此示例中的第二个选项是简单的彩票。
定义 2.2.1。给定一组结果， $O$ ，一个简单的彩票是这个集合的概率分布。什么时候
O=\left{0_1，o_2, Vdots, o_m\right } } \text { ，一个有限集，一个简单的彩票可以表示为 }
$p_1 o_1+p_2 o_2+\ldots+p_m o_m$ 在哪里 $p_i$ 是结果的概率 $o_i$. 复合彩票是其他彩票的概率分布。因为一个结果 $o \in O$ 可以写成简单的彩票 $l o$ ，我们看到 $O \subset \mathcal{L}$ ，所有（简单和复合）彩票的集合。
为了揭示玩家对一个结果的偏好程度优于另一个结果，我们不仅要检查单个结果之间的关系，还要检查单个结果和彩票之间的关系。假设 Regis 更喜欢香草味 $V$ 巧克力 $C$ 并且喜欢 $C$ 在草莓上 $S$. 瑞吉斯的选择 $C$ 和彩票 $0.5 S+0.5 V$ 告诉我们他偏爱的强度 $V$ 超过 $C$ 并为 $C$ 超过 $S$. 如果他选择这两种可能性中的任何一种，那么 Regis 偏好的强度 $C$ 恰好在中间 $S$ 和 $V$. 如果瑞吉斯选择 $C$ 超过 $0.5 S+0.5 \mathrm{~V}$ ，它揭示了他对 $C$ 更接近 $V$ 比到 $S$. 如果瑞吉斯愿意选择 $C$ 或彩票 $0.1 S+0.9 \mathrm{~V}$ ，这将揭示他对 $V$ 超过 $C$ 很小，他偏爱 $C$ 超过 $S$ 比较大。但是，如果他愿意选择 $C$ 或彩票 $0.9 S+0.1 \mathrm{~V}$ ，它会显示出对 $V$ 超过 $C$ 他的偏好 $C$ 超过 $S$ 是小。当玩家愿意选择两个彩票中的任何一个时，这表明玩家对这些选择无动于衷。这激发了效用函数概念的以下概括。

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金融工程代写

非参数统计代写

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经济代写|博弈论代写Game Theory代考|ECON3050

Posted on 2022年12月12日2022年12月12日 by statistics-lab

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我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|Backward induction, Kuhn’s Theorem

Let $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ be an extensive form game with perfect information. Recall that $A(\varnothing)$ is the set of allowed initial actions for player $i=P(\varnothing)$. For each $b \in A(\varnothing)$, let $s^{G(b)}$ be some strategy profile for the subgame $G(b)$. Given some $a \in A(\varnothing)$, we denote by $s^a$ the strategy profile for $G$ in which player $i=P(\varnothing)$ chooses the initial action $a$, and for each action $b \in A(\varnothing)$ the subgame $G(b)$ is played according to $s^{G(b)}$. That is, $s_i^a(\varnothing)=a$ and for every player $j, b \in A(\varnothing)$ and $b h \in H \backslash Z, s_j^a(b h)=s_j^{G(b)}(h)$.
Lemma 2.11 (Backward Induction). Let $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ be a finite extensive form game with perfect information. Assume that for each $b \in A(\varnothing)$ the subgame $G(b)$ has a subgame perfect equilibrium $s^{G(b)}$. Let $i=P(\varnothing)$ and let a be the $>_i$-maximizer over $A(\varnothing)$ of $o_a\left(s^{G(a)}\right)$. Then $s^a$ is a subgame perfect equilibrium of $G$.

Proof. By the one deviation principle, we only need to check that $s^a$ does not have deviations that differ at a single history. So let $s$ differ from $s^a$ at a single history $h$.

If $h$ is the empty history then $s=s^{G(b)}$ for $b=s_i(\varnothing)$. In this case $o\left(s^a\right)>_i o(s)=o_b\left(s^{G(b)}\right)$, by the definition of $a$ as the maximizer of $o_a\left(s^{G(a)}\right)$.

Otherwise, $h$ is equal to $b h^{\prime}$ for some $b \in A(\varnothing)$ and $h^{\prime} \in H_b$, and $o(s)=o_b(s)$. But since $s^a$ is a subgame perfect equilibrium when restricted to $G(b)$ there are no profitable deviations, and the proof is complete.
Kuhn [22] proved the following theorem.
Theorem $2.12$ (Kuhn, 1953). Every finite extensive form game with perfect information has a subgame perfect equilibrium.

Given a game $G$ with allowed histories $H$, denote by $\ell(G)$ the maximal length of any history in $H$.

Proof of Theorem 2.12. We prove the claim by induction on $\ell(G)$. For $\ell(G)=0$ the claim is immediate, since the trivial strategy profile is an equilibrium, and there are no proper subgames. Assume we have proved the claim for all games $G$ with $\ell(G)<n$.

Let $\ell(G)=n$, and denote $i=P(\varnothing)$. For each $b \in A(\varnothing)$, let $s^{G(b)}$ be some subgame perfect equilibrium of $G(b)$. These exist by our inductive assumption, as $\ell(G(b))<n$.

Let $a^* \in A(\varnothing)$ be a $\leq_i$-maximizer of $o\left(s^{a^}\right)$. Then by the Backward Induction Lemma $s^{a^}$ is a subgame perfect equilibrium of $G$, and our proof is concluded.

经济代写|博弈论代写Game Theory代考|Classical examples

Extensive form game with perfect information. Let $G=\left(N, A, H, P,\left{u_i\right}_{i \in N}\right)$ be an extensive form game with perfect information, where, instead of the usual outcomes and preferences, each player has a utility function $u_i: Z \rightarrow \mathbb{R}$ that assigns her a utility at each terminal node. Let $G^{\prime}$ be the strategic form game given by $G^{\prime}=\left(N^{\prime},\left{S_i\right}_{i \in N},\left{u_i\right}_{i \in \mathbb{N}}\right)$, where
$-N^{\prime}=N$.
$S_i$ is the set of $G$-strategies of player $i$.
For every $s \in S, u_i(s)$ is the utility player $i$ gets in $G$ at the terminal node at which the game arrive when players play the strategy profile $s$.

We have thus done nothing more than having written the same game in a different form. Note, however, that not every game in strategic form can be written as an extensive form game with perfect information.

Exercise 3.1. Show that $s \in S$ is a Nash equilibrium of $G$ iff it is a Nash equilibrium of $G^{\prime}$.

Note that a disadvantage of the strategic form is that there is no natural way to define subgames or subgame perfect equilibria.

Matching pennies. In this game, and in the next few, there will be two players: a row player (R) and a column player (C). We will represent the game as a payoff matrix, showing for each strategy profile $s=\left(s_R, s_C\right)$ the payoffs $u_R(s), u_C(s)$ of the row player and the column player.

博弈论代考

经济代写|博弈论代写Game Theory代考|Backward induction, Kuhn’s Theorem

是玩家允许的初始动作集 $i=P(\varnothing)$. 对于每个 $b \in A(\varnothing)$ ，让 $s^{G(b)}$ 是子博亦的一些策略配置文件 $G(b)$. 给定一些 $a \in A(\varnothing)$ ，我们用 $s^a$ 的战略概况 $G$ 在哪个播放器中 $i=P(\varnothing)$ 选择初始动作 $a$ ，并且对于每个动作 $b \in A(\varnothing)$ 子博亦 $G(b)$ 根据播放 $s^{G(b)}$. 那是， $s_i^a(\varnothing)=a$ 并且对于每一位玩家 $j, b \in A(\varnothing)$ 和 $b h \in H \backslash Z, s_j^a(b h)=s_j^{G(b)}(h)$.
引理 $2.11$ (逆向归纳法) 。让 $G=| l$ eft(N, $A, H, O, 0, P$, left{\leq_ilright__{i lin $N} \backslash$ ight) 是一个具有完美信息的有限扩展形式博亦。假设对于每个 $b \in A(\varnothing)$ 子博亦 $G(b)$ 有一个子博亦完美均衡 $s^{G(b)}$. 让 $i=P(\varnothing)$ 让a 成为 $>_i$-最大化器结束 $A(\varnothing)$ 的 $o_a\left(s^{G(a)}\right)$. 然后 $s^a$ 是子博孪的完美均衡 $G$.
证明。由一偏差原则，我们只需要检查 $s^a$ 没有在单一历史上不同的偏差。所以让 $s$ 与…..不同 $s^a$ 在单一的历史 $h$.
如果 $h$ 那么就是空的历史 $s=s^{G(b)}$ 为了 $b=s_i(\varnothing)$. 在这种情况下 $o\left(s^a\right)>_i o(s)=o_b\left(s^{G(b)}\right)$, 根据定义 $a$ 作为最大化者 $o_a\left(s^{G(a)}\right)$.
否则， $h$ 等于 $b h^{\prime}$ 对于一些 $b \in A(\varnothing)$ 和 $h^{\prime} \in H_b$ ，和 $o(s)=o_b(s)$. 但是由于 $s^a$ 是一个子博栾完美均衡，当限制为 $G(b)$ 没有有利可图的偏差，并且证明是完整的。 Kuhn [22] 证明了以下定理。
定理 $2.12$ (库恩， 1953 年)。每个具有完美信息的有限扩展形式博孪都有一个子博栾完美均衡。
给定一个游戏 $G$ 有允许的历史 $H$ ，表示为 $\ell(G)$ 任何历史的最大长度 $H$.
定理 $2.12$ 的证明。我们通过归纳证明了这一主张 $\ell(G)$. 为了 $\ell(G)=0$ 索赔是直接的，因为平凡的策略配置文件是均衡的，并且没有适当的子博亦。假设我们已经证明了所有游戏的要求 $G$ 和 $\ell(G)<n$.
让 $\ell(G)=n$ ，并表示 $i=P(\varnothing)$. 对于每个 $b \in A(\varnothing)$ ，让 $s^{G(b)}$ 是某个子博亦的完美均衡 $G(b)$. 这些根据我们的归纳假设存在，如 $\ell(G(b))<n$.
.ThenbytheBackwardInductionLemmas $\left{\left{a^{\wedge}\right}\right.$ isasubgameperfectequilibriumof $\mathrm{G} \$$ ，我们的证明就结束了。

经济代写|博弈论代写Game Theory代考|Classical examples

具有完美信息的广泛形式的博亦，其中每个玩家都有一个效用函数，而不是通常的结果和偏好 $u_i: Z \rightarrow \mathbb{R}$ 在每个终端节点为她分配一个实用程序。让 $G^{\prime}$ 是给出的战略形式游戏里
$$
-N^{\prime}=N \text {. }
$$

$S_i$ 是一组 $G$-玩家的策略 $i$.
对于每一个 $s \in S, u_i(s)$ 是效用播放器 $i$ 进入 $G$ 在玩家玩策略配置文件时游戏到达的终端节点 $s$.
因此，我们所做的只不过是用不同的形式编写了相同的游戏。但是请注意，并非所有战略形式的博亦都可以写成具有完美信息的扩展形式博亦。
练习 3.1。显示 $s \in S$ 是一个纳什均衡 $G$ 当且仅当它是一个纳什均衡 $G^{\prime}$.
请注意，策略形式的一个缺点是没有自然的方式来定义子博亦或子博恋完美均衡。
匹配的便士。在这个游戏中，以及接下来的几个游戏中，将有两个玩家：一个行玩家 (R) 和一个纵列玩家 (C)。我们将游戏表示为收益矩阵，显示每个策略配置文件 $s=\left(s_R, s_C\right)$ 收益 $u_R(s), u_C(s)$ 行播放器和列播放器。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON3503

Posted on 2022年12月12日2022年12月12日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|The ultimatum game

In the ultimatum game player 1 makes an offer $a \in{0,1,2,3,4}$ to player 2. Player 2 either accepts or rejects. If player 2 accepts then she receives $a$ dollars and player 1 receives $4-a$ dollars. If 2 rejects then both get nothing. This is how this game can be written in extensive form:

$N={1,2}$.
$A={0,1,2,3,4, a, r}$.
$Z={0 a, 1 a, 2 a, 3 a, 4 a, 0 r, 1 r, 2 r, 3 r, 4 r}$.

$O={(0,0),(0,4),(1,3),(2,2),(3,1),(4,0)}$. Each pair corresponds to what players 1 receives and what player 2 receives.
For $b \in{0,1,2,3,4}, o(b a)=(4-b, b)$ and $o(b r)=(0,0)$.
$P(\varnothing)=1, P(0)=P(1)=P(2)=P(3)=P(4)=2$.
For $a_1, b_1, a_2, b_2 \in{0,1,2,3,4},\left(a_1, a_2\right) \leq_1\left(b_1, b_2\right)$ iff $a_1 \leq b_1$, and $\left(a_1, a_2\right) \leq_2\left(b_1, b_2\right)$ iff $a_2 \leq b_2$.

A strategy for player 1 is just a choice among ${0,1,2,3,4}$. A strategy for player 2 is a map from ${0,1,2,3,4}$ to ${a, r}$ : player 2 ‘s strategy describes whether or not she accepts or rejects any given offer.

Remark 2.3. A common mistake is to think that a strategy of player 2 is just to choose among ${a, r}$. But actually a strategy is a complete contingency plan, where an action is chosen for every possible history in which the player has to move.

经济代写|博弈论代写Game Theory代考|Subgames and subgame perfect equilibria

A subgame of a game $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ is a game that starts after a given finite history $h \in H$. Formally, the subgame $G(h)$ associated with $h=\left(h_1, \ldots, h_n\right) \in H$ is $G(h)=$ $\left(N, A, H_h, O, o_h, P_h,\left{\leq_i\right}_{i \in N}\right)$, where
$$
H_h=\left{\left(a_1, a_2, \ldots\right):\left(h_1, \ldots, h_n, a_1, a_2, \ldots\right) \in H\right}
$$
and
$$
o_h\left(h^{\prime}\right)=o\left(h h^{\prime}\right) \quad P_h\left(h^{\prime}\right)=P\left(h h^{\prime}\right) .
$$
A strategy $s$ of $G$ can likewise used to define a strategy $s_h$ of $G(h)$. We will drop the $h$ subscripts whenever this does not create (too much) confusion.

A subgame perfect equilibrium of $G$ is a strategy profile $s^$ such that for every subgame $G(h)$ it holds that $s^$ (more precisely, its restriction to $H_h$ ) is a Nash equilibrium of $G(h)$. We will prove Kuhn’s Theorem, which states that every finite extensive form game with perfect information has a subgame perfect equilibrium. We will then show that Zermelo’s Theorem follows from Kuhn’s.

As an example, consider the following Cold War game played between the USA and the USSR. First, the USSR decides whether or not to station missiles in Cuba. If it does not, the game ends with utility 0 for all. If it does, the USA has to decide if to do nothing, in which case the utility is 1 for the USSR and $-1$ for the USA, or to start a nuclear war, in which case the utility is $-1,000,000$ for all.

Exercise 2.7. Find two equilibria for this game, one of which is subgame perfect, and one which is not.

Exercise 2.8. Find two equilibria of the ultimatum game, one of which is subgame perfect, and one which is not.

An important property of finite horizon games is the one deviation property. Before introducing it we make the following definition.

Let $s$ be a strategy profile. We say that $s_i^{\prime}$ is a profitable deviation from $s$ for player $i$ at history $h$ if $s_i^{\prime}$ is a strategy for $G$ such that
$$
o_h\left(s_{-i}, s_i^{\prime}\right)>_i o_h(s) .
$$
Note that a strategy profile has no profitable deviations if and only if it is a subgame perfect equilibrium.

博弈论代考

经济代写|博弈论代写Game Theory代考|The ultimatum game

在最后通牒中，玩家 1 提出要约 $a \in 0,1,2,3,4$ 给玩家 2。玩家 2 接受或拒绝。如果玩家 2 接受，那么她会收到 $a$ 美元和玩家 1 收到 $4-a$ 美元。如果 2 拒绝，则两者都一无所获。这就是这个游戏可以用扩展形式编写的方式:

$N=1,2$.
$A=0,1,2,3,4, a, r$.
$Z=0 a, 1 a, 2 a, 3 a, 4 a, 0 r, 1 r, 2 r, 3 r, 4 r$.
$O=(0,0),(0,4),(1,3),(2,2),(3,1),(4,0)$. 每对对应于玩家 1 收到什么和玩家 2 收到什么。
为了 $b \in 0,1,2,3,4, o(b a)=(4-b, b)$ 和 $o(b r)=(0,0)$.
$P(\varnothing)=1, P(0)=P(1)=P(2)=P(3)=P(4)=2$.
为了 $a_1, b_1, a_2, b_2 \in 0,1,2,3,4,\left(a_1, a_2\right) \leq_1\left(b_1, b_2\right)$ 当且仅当 $a_1 \leq b_1$ ，和 $\left(a_1, a_2\right) \leq_2\left(b_1, b_2\right)$ 当且仅当 $a_2 \leq b_2$.
玩家 1 的策略只是一个选择 $0,1,2,3,4$. 玩家 2 的策略是来自 $0,1,2,3,4$ 至 $a, r:$ 玩家 2 的策略描述了她是接受还是拒绝任何给定的提议。
备注 2.3。一个常见的错误是认为参与者 2 的策略只是在其中进行选择 $a, r$. 但实际上，策略是一个完整的应急计划，其中为玩家必须移动的每一个可能的历史选择一个动作。

经济代写|博弈论代写Game Theory代考|Subgames and subgame perfect equilibria

$h \in H$. 形式上，子博亦 $G(h)$ 有关联 $h=\left(h_1, \ldots, h_n\right) \in H$ 是 $G(h)=$ left(N, A, H_h, O, o_h, P_h,Veft{ㄴleq_ilright}_{i in N}Yright), 在哪里
和
$$
o_h\left(h^{\prime}\right)=o\left(h h^{\prime}\right) \quad P_h\left(h^{\prime}\right)=P\left(h h^{\prime}\right) .
$$
一个策略 $s$ 的 $G$ 同样可以用来定义一个策略 $s_h$ 的 $G(h)$. 我们将放弃 $h$ 下标只要这不会造成 (太多) 混乱。
的子博娈完美均衡 $G$ 是策略概况 $\$ \mathrm{~S}^{\wedge}$ suchthat foreverysubgame克 (小时) itholdsthat ${ }^{\wedge}$ (moreprecisely, itsrestrictionto $\mathrm{H} \mathrm{h})$ isaNashequilibriumof $\mathrm{G}(\mathrm{h}) \$$ 。我们将证明库恩定理，该定理指出每个具有完美信息的有限扩展形式博孪都有一个子博栾完美均衡。然后我们将证明策梅洛定理是从库恩定理推导出来的。
例如，考虑以下美国和苏联之间的冷战博孪。首先，苏联决定是否在古巴部署导弹。如果不是，则游戏以所有人的效用 0 结束。如果是，美国必须决定是否不采取任何行动，在这种情况下，苏联的效用为 1 ，而 $-1$ 对于美国，或者发动核战争，在这种情况下，效用是 $-1,000,000$ 对所有人。
练习 2.7。为这个游戏找到两个均衡，一个是完美的子博栾，一个不是。
练习2.8。找到最后通牒博亦的两个均衡，一个是完美的子博亦，一个不是。
有限时域博亦的一个重要属性是单偏差属性。在介绍它之前，我们先作如下定义。
让 $s$ 成为战略概况。我们说 $s_i^{\prime}$ 是一个有利可图的偏离 $s$ 对于玩家 $i$ 在历史上 $h$ 如果 $s_i^{\prime}$ 是一个策略 $G$ 这样
$$
o_h\left(s_{-i}, s_i^{\prime}\right)>_i o_h(s) .
$$
请注意，当且仅当它是子博娈完美均衡时，策略配置文件没有盈利偏差。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON90022

Posted on 2022年12月12日2022年12月12日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|The Sweet Fifteen Game

We assume the students are familiar with chess, but the details of the game will, in fact, not be important. We will choose the following (non-standard) rules for the ending of chess: the game ends either by the capturing of a king, in which case the capturing side wins and the other loses, or else in a draw, which happens when there a player has no legal moves, or more than 100 turns have elapsed.
As such, this games has the following features:

There are two players, white and black.
There are (at most) 100 times periods.
In each time period one of the players chooses an action. This action is observed by the other player.
The sequence of actions taken by the players so far determines what actions the active player is allowed to take.

Every sequence of alternating actions eventually ends with either a draw, or one of the players winning.

We say that white can force a victory if, for any moves that black chooses, white can choose moves that will end in its victory. Zermelo showed in 1913 [34] that in the game of chess, as described above, one of the following three holds:

White can force a victory.
Black can force a victory.
Both white and black can force a draw.
We will prove this later.
We can represent the game of chess mathematically as follows.
Let $N={W, B}$ be the set of players.
Let $A$ be the set of actions (or moves) that any of the players can potentially take at any stage of the game. E.g., “rook from A1 to B3” is an action.
A history $h=\left(a_1, \ldots, a_n\right)$ is a finite sequence of elements of $A$; we denote the empty sequence by $\varnothing$. We say that $h$ is legal if each move $a_i$ in $h$ is allowed in the game of chess, given the previous moves $\left(a_1, \ldots, a_{i-1}\right)$, and if the length of $h$ is at most 100. We denote by $H$ the set of legal histories, including $\varnothing$.
Let $Z \subset H$ be the set of terminal histories; these are histories of game plays that have ended (as described above, by capturing or a draw). Formally, $Z$ is the set of histories in $H$ that are not prefixes of other histories in $H$.
Let $O={W, B, D}$ be the set of outcomes of the game (corresponding to $W$ wins, $B$ wins, and draw). Let $o: Z \rightarrow O$ be a function that assigns to each terminal history the appropriate outcome.

经济代写|博弈论代写Game Theory代考|Definition of finite extensive form games

In general, an extensive form game (with perfect information) $G$ is a tuple $G=\left(N, A, H, O, o, P, \subseteq_i\right.$ }$_{i \in N}$ ) where

$N$ is a finite set of players.
$A$ is a finite set of actions.
$H$ is a finite set of allowed histories. This is a set of sequences of elements of $A$ such that if $h \in H$ then every prefix of $h$ is also in $H$.
$Z$ is the set of sequences in $H$ that are not subsequences of others in $H$. Note that we can specify $H$ by specifying $Z ; H$ is the set of subsequences of sequences in $Z$.
$O$ is a finite set of outcomes.
$o$ is a function from $Z$ to $O$.
$P$ is a function from $H \backslash Z$ to $N$.
For each player $i \in N, \leq_i$ is a preference relation over $O$. (I.e., a complete, transitive and reflexive binary relation). So for outcomes $x, y \in O$ we write $x \prec_1 y$ if player 1 strictly prefers $y$ to $x$. This means that given the choice, player 1 would rather have $y$ than $x$. We write $x \leq_1 y$ if player 1 is either indifferent between $x$ and $y$ or strictly prefers $y$.
We denote by $A(h)$ the actions available to player $P(h)$ after history $h$ :
$$
A(h)={a \in A: h a \in H} .
$$
Strategies are defined as for chess. A strategy profile $s=\left{s_i\right}_{i \in N}$ constitutes a strategy for each player. We can, as for chess, define $h(s)$ and $o(s)$ as the history and outcome associated with a strategy profile.

博弈论代考

经济代写|博弈论代写Game Theory代考|The Sweet Fifteen Game

我们假设学生熟悉国际象棋，但实际上游戏的细节并不重要。我们将为国际象棋的结束选择以下（非标准）规则：游戏通过捕获国王结束，在这种情况下，捕获一方获胜，另一方失败，或者以平局结束，这发生在玩家没有合法的动作，或者已经超过 100 回合。
因此，这款游戏具有以下特点：

有两个玩家，白色和黑色。
有（最多）100 个周期。
在每个时间段中，其中一位玩家选择一个动作。这个动作被其他玩家观察到。
到目前为止，玩家采取的行动顺序决定了允许活跃玩家采取什么行动。
每个交替动作序列最终都以平局或其中一名玩家获胜而告终。

我们说，如果对于黑方选择的任何着法，白方可以选择将以其胜利告终的着法，则白方可以取得胜利。策梅洛在 1913 年 [34] 表明，在国际象棋游戏中，如上所述，以下三个之一成立：

白方可以迫使胜利。
黑色可以迫使胜利。
白色和黑色都可以强制平局。
我们稍后会证明这一点。
我们可以用数学方法表示象棋游戏如下。
让否=在,乙成为球员的集合。
让一个是任何玩家在游戏的任何阶段都可能采取的一组动作（或移动）。例如，“车从 A1 到 B3”是一个动作。
历史H=(一个1,…,一个n)是元素的有限序列一个; 我们将空序列表示为∅. 我们说H如果每一步都是合法的一个一世在H考虑到之前的动作，在国际象棋游戏中是允许的(一个1,…,一个一世−1), 如果长度H最多为 100。我们用H法律历史集，包括∅.
让从⊂H是终端历史的集合；这些是已经结束的游戏历史（如上所述，通过捕获或平局）。正式地，从是历史集合H不是其他历史的前缀H.
让欧=在,乙,丁是游戏的结果集（对应于在赢，乙赢，平局）。让欧:从→欧是一个函数，它为每个终端历史分配适当的结果。

经济代写|博弈论代写Game Theory代考|Definition of finite extensive form games

一般来说，广泛形式的博亦 (具有完美信息) $G$ 是一个元组 $G=\left(N, A, H, O, o, P, \subseteq_i\right}_{i \in N}$ ) 在哪里

$N$ 是一组有限的参与者。
$A$ 是一组有限的动作。
$H$ 是一组有限的允许历史。这是一组元素序列 $A$ 这样如果 $h \in H$ 那么每个前缀 $h$ 也在 $H$.
$Z$ 是序列的集合 $H$ 不是其他人的子序列 $H$. 请注意，我们可以指定 $H$ 通过指定 $Z ; H$ 是序列的子序列集 $Z$.
$O$ 是一组有限的结果。
$o$ 是一个函数 $Z$ 至 $O$.
$P$ 是一个函数 $H \backslash Z$ 至 $N$.
对于每个玩家 $i \in N, \leq_i$ 是一个偏好关系 $O$. (即完整的、传递的和自反的二元关系) 。所以对于结果 $x, y \in O$ 我们写 $x \prec_1 y$ 如果玩家 1 严格偏好 $y$ 至 $x$. 这意味着如果有选择，参与者 1 更愿意 $y$ 比 $x$. 我们写 $x \leq_1 y$ 如果玩家 1 对两者都无动于市 $x$ 和 $y$ 或严格偏好 $y$. 我们用 $A(h)$ 玩家可用的动作 $P(h)$ 历史之后 $h$ :
$$
A(h)=a \in A: h a \in H .
$$
策略定义为国际象棋。战略概况 $\mathrm{s}=\backslash l$ eft{s_ilright $}_{-}{i$ in $\mathrm{N}}$ 构成每个玩家的策略。对于国际象棋，我们可以定义 $h(s)$ 和 $o(s)$ 作为与战略概况相关的历史和结果。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON40010

Posted on 2022年11月29日2022年11月29日 by statistics-lab

如果你也在怎样代写博弈论Game Theory这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的博弈论Game Theory及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

经济代写|博弈论代写Game Theory代考|Nim, Poker Nim, and the Mex Rule

In this section, we prove a sweeping statement (Theorem 1.14): Every impartial game is equivalent to some Nim heap. Note that this means a single Nim heap, even if the game itself is rather complicated, for example a general Nim position, which is a game sum of several Nim heaps. This can proved without any reference to playing Nim optimally. As we will see at the end of this section with the help of Figure 1.2, the so-called mex rule that underlies Theorem $1.14$ can be used to discover the role of powers of two for playing sums of Nim heaps.

We use the following notation for Nim heaps. If $G$ is a single Nim heap with $n$ tokens, $n \geq 0$, then we denote this game by $* n$. This game is completely specified by its $n$ options, and they are defined recursively as follows:
options of $* n: \quad * 0, * 1, * 2, \ldots, (n-1)$. Note that $ 0$ is the empty heap with no tokens, that is, $* 0=0$; we will normally continue to just write 0 .

We can use (1.18) as the definition of $* n$. For example, the game $* 4$ is defined by its options $* 0, * 1, * 2, * 3$. It is very important to include $* 0$ in that list of options, because it means that $* 4$ has a winning move. Condition (1.18) is a recursive definition of the game $* n$, because its options are also defined by reference to such games $* k$, for numbers $k$ smaller than $n$. This game fulfills the ending condition because the heap gets successively smaller in any sequence of moves.

A general Nim position is a game sum of several Nim heaps. Earlier we had written such a position by just listing the sizes of the Nim heaps, such as $1,2,3$ in (1.1). The fancy way to write this is now $* 1+* 2+* 3$, a sum of games.

The game of Poker Nim is a variation of Nim. Suppose that each player is given, at the beginning of the game, some extra “reserve” tokens. Like Nim, the game is played with heaps of tokens. In a move, a player can choose, as in ordinary Nim, a heap and remove some tokens, which he can add to his reserve tokens. A second, new kind of move is to add some of the player’s reserve tokens to some heap (or even to create an entire new heap with these tokens). These two kinds of moves are the only ones allowed.

经济代写|博弈论代写Game Theory代考|Sums of Nim Heaps

In this section, we derive how to compute the Nim value for a general Nim position, which is a sum of different Nim heaps. This will be the Nim sum that we have defined using the binary representation, now cast in the language of game sums and equivalent games, and without assuming the binary representation.

For example, we know that $* 1+* 2+* 3 \equiv 0$, so by Lemma $1.12, * 1+* 2$ is equivalent to $* 3$. In general, however, the sizes of the Nim heaps cannot simply be added to obtain the equivalent Nim heap, because $* 2+* 3$ is also equivalent to $* 1$, and $* 1+* 3$ is equivalent to $* 2$.

If $* k \equiv * n+* m$, then we call $k$ the $\operatorname{Nim}$ sum of $n$ and $m$, written $k=n \oplus m$. The following theorem states that the Nim sum of distinct powers of two is their arithmetic sum. For example, $1=2^0$ and $2=2^1$, so $1 \oplus 2=1+2=3$.

Theorem 1.15. Let $n \geq 1$, and $n=2^a+2^b+2^c+\cdots$, where $a>b>c>\cdots \geq 0$. Then
$$

n \equiv \left(2^a\right)+\left(2^b\right)+*\left(2^c\right)+\cdots \text {. }
$$
We first discuss the implications of this theorem, and then prove it. The expression $n=2^a+2^b+2^c+\cdots$ is an arithmetic sum of distinct powers of two. Any $n$ is uniquely given as such a sum. It amounts to the binary representation of $n$, which, if $n<2^{a+1}$, gives $n$ as the sum of all powers $2^a, 2^{a-1}, 2^{a-2}, \ldots, 2^0$ where each power of two is multiplied with 0 or 1 , the binary digit for the respective position. For example,
$$
9=8+1=1 \cdot 2^3+0 \cdot 2^2+0 \cdot 2^1+1 \cdot 2^0,
$$
so that 9 in decimal is written as 1001 in binary. Theorem $1.15$ uses only the distinct powers of two $2^a, 2^b, 2^c, \ldots$ that correspond to the digits 1 in the binary representation of $n$.

博弈论代考

经济代写|博弈论代写Game Theory代考|Nim, Poker Nim, and the Mex Rule

在本节中，我们证明了一个笼统的陈述（定理 1.14)：每个公正的游戏都等同于一些 Nim 堆。请注意，这意味着单个 $\mathrm{Nim}$ 堆，即使游戏本身相当复杂，例如一般的 Nim 位置，它是几个 Nim 堆的游戏总和。这可以在没有任何参考以最佳方式玩 $\operatorname{Nim}$ 的情况下证明。正如我们将在本节末尾借助图 $1.2$ 看到的，所谓的 mex 规则是定理的基础1.14可用于发现 Nim 堆求和的 2 次幂的作用。
我们对 $\operatorname{Nim}$ 堆使用以下表示法。如果 $G$ 是单个 $\operatorname{Nim}$ 堆 $n$ 代币， $n \geq 0$ ，那么我们将这个游戏表示为 $* n$. 这个游戏完全由它指定 $n$
选项，它们递归定义如下: $* n: * 0, * 1, * 2, \ldots,(n-1)$. 注意 0 是没有标记的空堆，即 $* 0=0$; 我们通常会继续只写 0 。
我们可以用 (1.18) 作为定义 $* n$. 例如，游戏 $* 4$ 由其选项定义 $* 0, * 1, * 2, * 3$. 包括在内非常重要 $* 0$ 在该选项列表中，因为这意味着 $* 4$ 有一个获胜的举动。条件 (1.18) 是游戏的递归定义 $* n$ ，因为它的选项也是通过参考此类游戏来定义的 $* k$ ，对于数字 $k$ 小于 $n$. 这个游戏满足结束条件，因为堆在任何移动序列中都逐渐变小。
一般的 $\operatorname{Nim}$ 位置是几个 $\operatorname{Nim}$ 堆的游戏总和。早些时候我们通过列出 $\operatorname{Nim}$ 堆的大小来编写这样的位置，例如 $1,2,3$ 在 (1.1) 中。现在写这个的好方法 $* 1+* 2+* 3$ ，游戏总和。
Poker Nim 游戏是 Nim 的变体。假设在游戏开始时给每个玩家一些额外的“保留”标记。与 Nim 一样，该游戏是用大量代币玩的。在移动中，玩家可以像在普通 Nim 中一样选择一个堆并移除一些令牌，他可以将这些令牌添加到他的储备令牌中。第二种新的移动方式是将玩家的一些储备令牌添加到一些堆中（或者甚至用这些令牌创建一个全新的堆）。这两种动作是唯一允许的。

经济代写|博弈论代写Game Theory代考|Sums of Nim Heaps

在本节中，我们将推导如何计算一般 Nim 位置的 $\mathrm{Nim}$ 值，该位置是不同 $\mathrm{Nim}$ 堆的总和。这将是我们使用二进制表示定义的 Nim 和，现在用游戏总和和等效游戏的语言进行转换，并且不假设二进制表示。
例如，我们知道 $* 1+* 2+* 3 \equiv 0$, 所以由引理 $1.12, * 1+* 2$ 相当于 $* 3$. 然而，一般来说， $\operatorname{Nim}$ 堆的大小不能简单地相加以获得等效的 $\operatorname{Nim}$ 堆，因为 $* 2+* 3$ 也相当于 $* 1$ ，和 $* 1+* 3$ 相当于 $* 2$.
如果 $* k \equiv * n+* m$ ，然后我们称 $k$ 这 $\mathrm{Nim}$ 的总和 $n$ 和 $m$, 写 $k=n \oplus m$. 下面的定理表明，两个不同幕的 $\operatorname{Nim}$ 和是它们的算术和。例如， $1=2^0$ 和 $2=2^1$ ，所以 $1 \oplus 2=1+2=3$.
定理 1.15。让 $n \geq 1$ ，和 $n=2^a+2^b+2^c+\cdots$ ，在哪里 $a>b>c>\cdots \geq 0$. 然后 $\$ \$$
Wefirstdiscusstheimplicationsofthistheorem, andthenproveit. Theexpression $\$ n=2$
$9=8+1=1 \backslash c$ dot $2^{\wedge} 3+0 \backslash c$ dot $2^{\wedge} 2+0 \backslash c$ dot $2^{\wedge} 1+1 \backslash c$ dot $2^{\wedge} 0$,
$\$ \$$
这样十进制的9写成二进制的 1001。定理 $1.15$ 只使用两个不同的权力 $2^a, 2^b, 2^c, \ldots$ 对应于二进制表示中的数字 $1 n$.

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