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## 经济代写|博弈论代考Game theory代写|CONDITIONAL BELIEFS ABOUT TYPES

The gift game in Figure 28.1 illustrates the idea of a conditional belief. Recall that in this game, player 2 does not observe nature’s decision. Therefore, at the beginning of the game, player 2 knows only that player 1 is the friend type with probability $p$ and the enemy type with probability $1-p$. This belief $p$ is called player 2’s initial belief about player 1’s type. Keep in mind that this is a belief about player 1’s type, not a belief about player 1’s strategy (which is the sort of belief with which we were dealing in Parts I through III of this book).

Although player 2 does not observe nature’s decision, player 2 does observe whether player 1 decided to give a gift. Furthermore, player 2 might learn something about player 1’s type by observing player 1’s action. As a result, player 2 will have an updated belief about player 1’s type. For example, suppose that you are player 2 in the gift game and suppose that player 1 behaves according to strategy $\mathrm{N}^{\mathrm{F}} \mathrm{G}^{\mathrm{E}}$; thus, you expect only to receive a gift from the enemy type. What should you conclude, then, if player 1 actually gives you a gift? Given player 1’s strategy, you should conclude that player 1 is an enemy. In reference to Figure 28.1 , when your information set is reached, you believe that you are playing at the lower of the two nodes in the information set.

In general, player 2 has an updated belief about player 1’s type, conditional on arriving at player 2’s information set (that is, conditional on receiving a gift). Note that player 2’s updated belief about player l’s type can be put in terms of a probability distribution over the nodes in player 2’s information set. In Figure 28.1, this probability distribution is described by the numbers $q$ and $1-q$ that appear beside the nodes. Literally, $q$ is the probability that player 2 believes he is the top node when his information set is reached. Thus, $q$ is the probability that player 2 believes player 1 is the friend type, conditional on receiving a gift.

## 经济代写|博弈论代考Game theory代写|SEQUENTIAL RATIONALITY

Taking account of conditional beliefs allows us to evaluate rational behavior at all information sets, even those that may not be reached in equilibrium play. Consider, again, the gift game pictured in Figure 28.1. Regardless of player l’s strategy, player 2 will have some updated belief $q$ at his information set. This number has meaning even if player 2 believes that player 1 adopts the strategy $\mathrm{N}^{\mathrm{F}} \mathrm{N}^{\mathrm{E}}$ (where neither type gives a gift). In this case, $q$ represents player 2 ‘s belief about the type of player 1 when the “surprise” of a gift occurs. Given the belief $q$, we can determine player 2’s optimal action at his information set. You can readily confirm that action $\mathrm{A}$ is best for player 2, whatever is $q$. Thus, sequential rationality requires that player 2 select $A$.
For another example, consider the gift game pictured in Figure 28.2. This is the same game discussed in Chapter 24. Note that regardless of the probability $q$, player 2 receives a payoff of 0 if he selects $\mathrm{R}$ at his information set.
In contrast, if player 2 chooses $\mathrm{A}$, then he gets a payoff of 1 with probability $q$ (the probability that his decision is taken from the top node in his information set) and he gets a payoff of -1 with probability $1-q$. Player 2 ‘s expected payoff of selecting $A$ is therefore
$$q+(-1)(1-q)=2 q-1$$
Player 2 will select $\mathrm{A}$ if $q>1 / 2$, he will select $\mathrm{R}$ if $q<1 / 2$, and he will be indifferent between $\mathrm{A}$ and $\mathrm{R}$ if $q=1 / 2$.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|ENFORCING INTERNATIONAL TRADE AGREEMENTS

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

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## 经济代写|博弈论代考Game theory代写|JOINT DECISIONS

A simple way of inserting a “summary” negotiation component into a noncooperative game is to include joint-decision nodes in the game tree. ${ }^1 \mathrm{~A}$ jointdecision node is an abbreviated description of negotiation between players over some tangible objects, such as profit-sharing rules, monetary transfers, or whether to form a partnership. Thus, a joint-decision node represents a place in the game where players negotiate and establish a contract. We specify a joint decision when we do not want to create a full noncooperative model of the negotiation process and when we have a simple theory of how negotiation is resolved (by using, for example, the standard bargaining solution).

To represent joint decisions in a tree, we can employ the same devices currently used to specify individual decisions. We simply allow some decision nodes to be designated as joint-decision nodes. The joint-decision nodes are graphically represented by double circles to differentiate them from individual decision nodes. Furthermore, we label a joint-decision node with the set of players who are called on to make the joint decision. Branches represent the alternatives available to the players, as is the case with individual decision nodes. In addition, wherever there is a joint-decision node, we must designate one of the branches as the default decision, which is assumed to go into effect in the event that the players do not reach an agreement. ${ }^2$

A game with joint decisions is illustrated in Figure 20.1, which is a simple model of contracting between a supplier firm and a buyer firm. First, the firms jointly determine whether to contract and, if so, what damages $c$ to specify if the supplier (player 2) is caught providing a low-quality intermediate good. If they choose not to contract (which is the default decision), then the game ends and each receives nothing. If they contract, and firm 2 then provides a high-quality good, payoffs are 10 for the buyer and 5 for the supplier. By providing a lowquality good, firm 2 saves money. However, the low-quality good is useless to the buyer. (These ideas are captured by the numbers -6 and 10 .) But with probability $1 / 2$, the supplier is caught and damages are awarded by a court. (This is a payment of $c$ from the supplier to the buyer.)

## 经济代写|博弈论代考Game theory代写|NEGOTIATION EQUILIBRIUM

To analyze general games with joint decisions, we combine backward induction (more specifically, subgame perfection) with the standard bargaining solution; the former pins down behavior at individual decision nodes, whereas the latter identifies behavior at joint-decision nodes.

Given an extensive-form game with joint decisions, a specification of behavior at every information set is called a regime. This is simply a generalization of the “strategy” concept to include joint decisions. I use the following equilibrium definition:

A regime is called a negotiation equilibrium if its description of behavior at individual decision nodes is consistent with sequential rationality and its specification of joint decisions is consistent with the standard bargaining solution, for given bargaining weights.

This definition is not precise enough to be clear-cut in every game with joint decisions. In particular, we can run into two problems when trying to construct a negotiation equilibrium. First, we have to decide what is meant by “sequential rationality.” For example, we could use backward induction or subgame perfection. Second, how to apply the standard bargaining solution in some contexts may not be obvious, in particular where there is not transferable utility. I avoid these problems by focusing on games in which backward induction or subgame perfection can be easily employed (this is a wide class, by the way) and by assuming that players can transfer money whenever they negotiate. You can leave to interested hot-shot theorists the task of navigating the labyrinthine esoterica of more general application.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|A Note About Legal Default Rules

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

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## 经济代写|博弈论代考Game theory代写|Dynamic Monopoly

Retail firms frequently adjust prices. Sometimes price adjustments are due to changes in wholesale prices, but more often the adjustments have to do with customer demand. Consider the market for 3D television monitors. Not all consumers have the same interest in buying such a product; in other words, demand is differentiated. The “high-value” customers are people who either have a real need for the $3 \mathrm{D}$ technology or just love to buy the newest gadget on the market. “Low-value” customers would be interested in purchasing 3D monitors but only if these are priced to compete with standard displays.

Facing this heterogeneous consumer population-as is the case in most markets-retailers wish to extract a high price from the high-value customers and a lower price from the low-value customers. Although legal constraints and arbitrage force retailers to charge the same price to all people, a retailer can use time to its advantage. A common scheme is to initially set a high price that is intended to attract only the high-value customers. Then, after selling to the highvalue folks, the retailer will drop the price in the hopes of capturing the demand of low-value customers.

Superficially, this scheme appears to give the retailer a very high profit by allowing it to extract the surplus of trade from both the high- and the lowvalue customers. In other words, the retailer gets both types of customers to pay the most that they are willing to pay for the product. But the retailer does face some constraints. First, competition with other retailers may exert downward pressure on prices. Second, the retailer must realize that customers will behave strategically-in particular, high-value customers may delay their purchase of 3D monitors if they anticipate that the price will fall over time. In this section, I sketch a simple model that helps us explore the second strategic aspect. ${ }^3$ The model focuses on a monopoly setting (where a single firm is in the industry), and therefore it ignores the matter of competition between irms.

## 经济代写|博弈论代考Game theory代写|Price Guarantees as a Commitment to High Prices

As described earlier, under some conditions retail firms have an incentive to decrease prices over time to extract surplus from customers with different valuations. You might think that such a pricing policy is good for a firm-it helps the firm obtain profit from each type of customer. In fact, the incentive to decrease prices over time may actually have the opposite effect on profit. To make sense of my claim, let us continue with the dynamic monopoly example.
Consider again the optimal pricing scheme, C. Recall that when selecting the period 1 price $p_1$, Tony must consider the customers’ strategic calculations. In particular, all of the players know that if the low-type customers do not purchase and at least one of the high types does purchase a monitor in period 1 , then in period 2 Tony will have the incentive to select $p_2=200$. (You will recall that this is the “useful fact” noted on page 212.) Tony’s scheme C pricing policy is therefore a balancing act. On one hand, Tony wants to deal only with Hal and Hilbert in period 1, raising the price as high as they are willing to pay. On the other hand, once Hal and Hilbert have purchased monitors, Tony wants to decrease the price and sell two more units to Laurie and Lauren. Because Hal and Hilbert anticipate the second-period price, Tony can get them to pay only $\$ 1400$in period 1 . By comparison, consider how much Tony would make if he could somehow commit never to deal with Laurie and Lauren. For example, suppose that in period 1 Tony could make a legally enforceable promise not to sell monitors in period 2. Then suppose Tony chose a period 1 price of$p_1=1700$, or just below this amount. In this case, Hal and Hilbert would purchase monitors in period 1 . (By purchasing a monitor, a high type’s payoff would be$1700-p_1$; if he did not purchase a monitor, then the high type would get 0 , because no monitors would be sold in period 2.) Laurie and Lauren would not purchase at this high price. However, Tony’s payoff would be$\$3400$, which is greater than the payoff he would get without the ability to commit. Thus, commitment helps Tony.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Price Guarantees as a Commitment to High Prices

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Game theory , 博弈论, 经济代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 经济代写|博弈论代考Game theory代写|Dominance and Best Response Compared

There is a precise relation between dominance and best response, the latter of which underpins the theories of behavior to come. For a given game, let $U D_i$ be the set of strategies for player $i$ that are not strictly dominated. Let $B_i$ be the set of strategies for player $i$ that are best responses, over all of the possible beliefs of player $i$. Mathematically,
$$B_i=\left{s_i \mid \text { there is a belief } \theta_{-i} \in \Delta S_{-i} \text { such that } s_i \in B R_i\left(\theta_{-i}\right)\right} .$$
That is, if a strategy $s_i$ is a best response to some possible belief of player $i$, then $s_i$ is contained in $B_i$. As heretofore noted, the notion of best response will be of primary importance. Unfortunately, determining the set $B_i$ is sometimes a greater chore than determining $U D_i$. Fortunately, the two sets are closely related.
To build your intuition, examine the game in Figure 6.4. Note first that $R$ is dominated for player 2.9 ${ }^9$ Thus, $U D_2={L}$ in this game. Also note that strategy $\mathrm{R}$ can never be a best response for player 2, because $\mathrm{L}$ yields a strictly higher payoff regardless of what player 1 does. In other words, for any belief of player 2 about player 1’s behavior, player 2 ‘s only best response is to select $\mathrm{L}$. Therefore $B_2={L}$. Obviously, $B_2=U D_2$ and, for this game, dominance and best response yield the same conclusion about rational behavior for player 2 .

## 经济代写|博弈论代考Game theory代写|Weak dominance

Recall that a key aspect of the dominance definition is the strict inequality, so that a mixed strategy $\sigma_i$ is said to dominate a pure strategy $s_i$ if and only if $u_i\left(\sigma_i, s_{-i}\right)>u_i\left(s_i, s_{-i}\right)$ for all $s_{-i} \in S_{-i}$. One can also consider a version of dominance based on a weaker condition: We say that mixed strategy $\sigma_i$ weakly dominates pure strategy $s_i$ if $u_i\left(\sigma_i, s_{-i}\right) \geq u_i\left(s_i, s_{-i}\right)$ for all $s_{-i} \in S_{-i}$ and $u_i\left(\sigma_i, s_{-i}^{\prime}\right)>u_i\left(s_i, s_{-i}^{\prime}\right)$ for at least one strategy $s_{-i}^{\prime} \in S_{-i}$ of the other players. In other words, $\sigma_i$ performs at least as well as does strategy $s_i$, and it performs strictly better against at least one way in which the other players may play the game.
Figure 6.6 provides an illustration of weak dominance. In the game pictured, if player 1 were to select $Y$, then player 2’s strategy M delivers a strictly higher payoff than does $\mathrm{L}$. If player 1 selects $\mathrm{X}$, then strategies $\mathrm{L}$ and $\mathrm{M}$ yield the same payoff for player 2 . Thus, player 2 always weakly prefers $\mathrm{M}$ to $\mathrm{L}$, and she strictly prefers $\mathrm{M}$ in the event that player 1 picks $\mathrm{Y}$. This means that $\mathrm{M}$ weakly dominates $\mathrm{L}$.

In relation to best-response behavior, weak dominance embodies a form of cautiousness, as though the players cannot be too sure about each other’s strategies. In the example of Figure 6.6, player 2 might reasonably select $\mathrm{L}$ if she is certain that player 1 will choose $X$. On the other hand, if she entertains the slightest doubt-putting any small, strictly positive probability on $\mathrm{Y}$-then $\mathrm{M}$ is her only best response. The example suggests a general relation between weak dominance and best responses to “cautious” beliefs. To make this formal, for any game let $W U D_i$ be the set of strategies for player $i$ that are not weakly dominated. Call a belief $\theta_{-i}$ fully mixed if $\theta_{-i}\left(s_{-i}\right)>0$ for all $s_{-i} \in S_{-i}$. This simply means that $\theta_{-i}$ puts positive probability on every strategy profile of the other players. Then let $B_i^{c f}$ be the set of strategies for player $i$ that are best responses to fully mixed beliefs. In the superscript, $c$ denotes that correlated conjectures are allowed, and $f$ denotes that beliefs are fully mixed.
Result: For any finite game, $B_i^{c f}=W U D_i$ for each player $i=1,2, \ldots, n$.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Dominance and Best Response Compared

$$B_i=\left{s_i \mid \text { there is a belief } \theta_{-i} \in \Delta S_{-i} \text { such that } s_i \in B R_i\left(\theta_{-i}\right)\right} .$$

## 经济代写|博弈论代考Game theory代写|Weak dominance

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

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## 经济代写|博弈论代考Game theory代写|The Concept of Efficiency

The first strategic tension relates to the economic concept of efficiency, which is an important welfare criterion by which to judge behavior in a game. Suppose we wish to compare the outcomes induced by two strategy profiles, $s$ and $s^{\prime}$. We say that $s$ is more efficient than $s^{\prime}$ if all of the players prefer the outcome of $s$ to the outcome of $s^{\prime}$ and if the preference is strict for at least one player. In mathematical terms, $s$ is more efficient than $s^{\prime}$ if $u_i(s) \geq u_i\left(s^{\prime}\right)$ for each player $i$ and if the inequality is strict for at least one player. ${ }^6$

A strategy profile $s$ is called efficient if there is no other strategy profile that is more efficient; that is, there is no other strategy profile $s^{\prime}$ such that $u_i\left(s^{\prime}\right) \geq u_i(s)$ for every player $i$ and $u_j\left(s^{\prime}\right)>u_j(s)$ for some player $j$. The expression Pareto efficient is used to mean the same thing. Note that, in the prisoners’ dilemma, $(C, C)$ is more efficient than (D, D). Furthermore, (C, C), (C, D), and (D, C) are all efficient strategy profiles. In the game pictured in Figure 6.1(c), (D, R) is more efficient than both (M, L) and (U, R). In this game, (U, L) and (D, L) are the only efficient strategy profiles.

## 经济代写|博弈论代考Game theory代写|Best Response

It seems reasonable that rational people refrain from using dominated strategies. Dominance is a good descriptive and prescriptive concept. But it is just the beginning of our development of a theory of behavior. Indeed, in most games, players have more than one undominated strategy. Take, as an example, some of the simple games in Figure 3.4. Matching pennies, the battle of the sexes, and the coordination games have no dominated strategies, so one cannot predict how people should or will play these games on the basis of the dominance criterion. We must move on to explore the process by which players actually select their strategies, at least among those that are not dominated.

Rational folks think about the actions that the other players might take; that is, people form beliefs about one another’s behavior. In games, it is wise to form an opinion about the other players’ behavior before deciding your own strategy. For example, if you were to play the coordination game pictured in Figure 3.4 and you thought that the other player would definitely play strategy B, it would be prudent for you to play $\mathrm{B}$ as well. If you thought he would select $\mathrm{A}$, then you should follow suit. To maximize the payoff that you expect to obtain-which we assume is the mark of rational behavior-you should select the strategy that yields the greatest expected payoff against your belief. Such a strategy is called a best response (or best reply). Formally,
Suppose player $i$ has a belief $\theta_{-i} \in \Delta S_{-i}$ about the strategies played by the other players. Player $i$ ‘s strategy $s_i \in S_i$ is a best response if $u_i\left(s_i, \theta_{-i}\right) \geq u_i\left(s_i^{\prime}, \theta_{-i}\right)$ for every $s_i^{\prime} \in S_i$.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Best Response

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Game theory , 博弈论, 经济代写

## avatest™帮您通过考试

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## 经济代写|博弈论代考Game theory代写|Terminology and Notation for Strategies

It is now time to get acquainted with some standard notation. If you need help understanding the basic mathematical symbols and concepts used here, please consult Appendix A. We must be formal and precise going forward.

Given a game, we let $S_i$ denote the strategy space (also called the strategy set) of player $i$. That is, $S_i$ is a set comprising each of the possible strategies of player $i$ in the game. For the game shown in Figure 2.7(a), the strategy space of player 1 is $S_1={\mathrm{H}, \mathrm{L}}$, and the strategy space of player 2 is $S_2=\left{\mathrm{HH}^{\prime}, \mathrm{HL}^{\prime}, \mathrm{LH}^{\prime}, \mathrm{LL}^{\prime}\right}$. We use lowercase letters to denote single strategies (generic members of these sets). Thus, $s_i \in S_i$ is a strategy for player $i$ in the game. We could thus have $s_1=\mathrm{L}$ and $s_2=\mathrm{LH}^{\prime}$, for instance.

A strategy profile is a vector of strategies, one for each player. In other words, a strategy profile describes strategies for all of the players in the game. For example, suppose we are studying a game with $n$ players. A typical strategy profile then is a vector $s=\left(s_1, s_2, \ldots, s_n\right)$, where $s_i$ is the strategy of player $i$, for $i=1,2, \ldots, n$. Let $S$ denote the set of strategy profiles. Mathematically, we write $S=S_1 \times S_2 \times \ldots \times S_n$. Note that the symbol ” $\times$ ” denotes the Cartesian product. ${ }^1$

Given a single player $i$, we often need to speak of the strategies chosen by all of the other players in the game. As a matter of notation, it will be convenient to use the term $-i$ to refer to these players. Thus, $s_{-i}$ is a strategy profile for everyone except player $i$ :
$$s_{-i}=\left(s_1, s_2, \ldots, s_{i-1}, s_{i+1}, \ldots, s_n\right) .$$

knows only whether firm 1 is in or out of the market; firm 2 does not observe firm l’s competitive stance before taking its action. In this game, there is one information set for firm 1 (the initial node) and one for firm 2 . The strategy sets are $S_1={\mathrm{A}, \mathrm{P}, \mathrm{O}}$ and $S_2={\mathrm{A}, \mathrm{P}}$

## 经济代写|博弈论代考Game theory代写|The Normal Form

The extensive form is one straightforward way of representing a game. Another way of formally describing games is based on the idea of strategies. It is called the normal form (or strategic form) representation of a game. This alternative representation is more compact than the extensive form in some settings. As we develop concepts of rationality for games, you will notice the subtle differences between the two representations.

For any game in extensive form, we can describe the strategy spaces of the players. Furthermore, notice that each strategy profile fully describes how the game is played. That is, a strategy profile tells us exactly what path through the tree is followed and, equivalently, which terminal node is reached to end the game. Associated with each terminal node (which we may call an outcome) is a payoff vector for the players. Therefore, each strategy profile implies a specific payoff vector.

For each player $i$, we can define a function $u_i: S \rightarrow \mathbf{R}$ (a function whose domain is the set of strategy profiles and whose range is the real numbers) so that, for each strategy profile $s \in S$ that the players could choose, $u_i(s)$ is player i’s payoff in the game. This function $u_i$ is called player $i$ ‘s payoff function. As an example, take the game pictured in Figure 3.1(b). The set of strategy profiles in this game is
$$S={(\mathrm{OA}, \mathrm{O}),(\mathrm{OA}, \mathrm{I}),(\mathrm{OB}, \mathrm{O}),(\mathrm{OB}, \mathrm{I}),(\mathrm{IA}, \mathrm{O}),(\mathrm{IA}, \mathrm{I}),(\mathrm{IB}, \mathrm{O}),(\mathrm{IB}, \mathrm{I})}$$

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Terminology and Notation for Strategies

$$s_{-i}=\left(s_1, s_2, \ldots, s_{i-1}, s_{i+1}, \ldots, s_n\right) .$$

## 经济代写|博弈论代考Game theory代写|The Normal Form

$$S={(\mathrm{OA}, \mathrm{O}),(\mathrm{OA}, \mathrm{I}),(\mathrm{OB}, \mathrm{O}),(\mathrm{OB}, \mathrm{I}),(\mathrm{IA}, \mathrm{O}),(\mathrm{IA}, \mathrm{I}),(\mathrm{IB}, \mathrm{O}),(\mathrm{IB}, \mathrm{I})}$$

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Game theory , 博弈论, 经济代写

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## 经济代写|博弈论代考Game theory代写|Who should do how much housework?

Newspapers like to stoke the gender wars when short of copy.
Here is a typical quote: ‘Men pay lip service to equal rights in the home while letting women do three quarters of the household chores.’ Other things being equal, the fact that wives do more housework than husbands would indeed show that the balance of power within marriages is biased in favour of men, but are other things equal?
Alice and Bob are getting married. They have no interest in enjoying any of the benefits of marriage other than sharing the housework. In the modern style, they agree on a binding marriage contract that specifies how many hours a week of housework each will contribute. What deal does the Nash bargaining solution predict that they will reach?
In a toy version of the problem, Alice thinks a household should devote two hours a day to housework; Bob thinks one hour a day is adequate. Each player derives a benefit of 100 utils a week if at least the number of hours they think appropriate is worked; otherwise they see no benefit at all in any housework being done.

Neither Alice nor Bob likes doing housework. Alice loses 5 utils a week for each hour of housework that she does. Bob loses 10 utils per hour, because he dislikes doing housework more than Alice. In the status quo situation before the marriage, Alice therefore does 14 hours of housework a week from which she derives a utility of 30 utils; Bob does 7 hours of housework from which he also derives a utility of 30 utils.

The Coase theorem says that the bargaining outcome will be efficient, which means that Alice will get her way over the number of hours that the new household will spend on housework. To find the Nash bargaining solution, we need to find the extreme outcomes that just make the marriage worthwhile for both partners. One extreme arises when Alice does all the housework; she will then get 30 utils and Bob will get 100 utils. The other extreme arises when it is Bob who gets only 30 utils. He will then do one hour of housework a day. Alice must do the other hour of housework to make up the two hours a day she thinks necessary. Her utility will then be 65 utils.

Because the model has been fixed to make Alice and Bob risk neutral, the Nash bargaining solution is found by averaging the two extremes. So Alice will end up with 47.5 utils and Bob with 65 utils a week. To make this happen, Alice will have to work $10 \frac{1}{2}$ hours a week and Bob only $3 \frac{1}{2}$ hours a week.
The Nash bargaining solution therefore says that if Alice and Bob bargain on an equal basis, then Alice will get her way on the number of hours worked a week, but she will have to do three-quarters of the work. If it is indeed true that wives do three times as much housework as single women, then our toy model shows that it doesn’t necessarily follow that the balance of power within marriages is biased in favour of men. Who would do how much housework if all the factors left out of the toy model were taken into account? Even if I knew, I wouldn’t say!

## 经济代写|博弈论代考Game theory代写|Rubinstein’s bargaining model

In accordance with the Nash program, Nash defended his bargaining solution with a noncooperative bargaining model in which Alice and Bob each simultaneously commit themselves to take-it-or-leave-it demands. However, Schelling was later successful in casting doubt on the realism of attributing commitment power to the players in negotiation games.
For example, if Bob were able beat Alice to the draw when making an irrevocable commitment in Divide-the-Dollar, then he could scoop the pot by demanding 99 cents, leaving Alice with a choice between one penny or nothing. But how does Bob convince Alice that he is truly committed – that nothing she might do can make him revise his demand? Who believes someone who claims he is now making his ‘last and final offer’? Even prices posted on expensive items in fancy stores are seldom final. The seller will try to make you feel like a cheapskate for challenging the price, but folk wisdom is right for once. Everything is negotiable. Never take no for an answer.

It is genuinely hard to establish commitments. People sometimes make a career of building up a reputation for being stubborn or stupid for this purpose. Trade unionists occasionally succeed in committing themselves by voting for intransigent leaders. But outside such special circumstances, the vocabulary of commitment is usually just so much cheap talk. But if all threats must be credible, we have seen that we need to look at subgame-perfect equilibria.
So what happens when anything a player says has to be credible before the other player will believe it? This question led Ariel Rubinstein to make the most important of all contributions to the Nash program. In the most natural noncooperative model of bargaining, Alice and Bob alternate in making offers to each other until they reach agreement. If they are assumed to prefer making any particular deal now rather than later, then Rubinstein showed that the alternating-offers model has a unique subgame-perfect equilibrium.
My own contribution was to show that the unique subgame-perfect equilibrium outcome approximates an asymmetric version of the Nash bargaining solution when the time interval between successive offers becomes sufficiently small. In the symmetric version of the Nash bargaining solution, the ratio $N B / A N$ in Figure 33 is equal to one. In the asymmetric version $N B / A N$ equals the ratio of the rates at which Alice and Bob discount time.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Rubinstein’s bargaining model

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Game theory , 博弈论, 经济代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 经济代写|博弈论代考Game theory代写|Evolution of cooperation

We already know that cooperation can be sustained among animals that aren’t related by the mechanism that Bob Trivers called reciprocal altruism. A wonderful example is provided by the vampire bat (Desmodus rotundis).

Vampire bats roost together in caves during the day. At night they seek an animal from which to suck blood. Some $8 \%$ are unsuccessful, which is a big problem for bats, who need to feed every 60 hours or so. For this reason, the evolutionary pressure towards sharing is very strong. Gerald Wilkinson discovered that vampire bats share blood on a reciprocal basis with roostmates who aren’t always relatives. In brief, a bat is more likely to regurgitate blood for a begging roostmate, if the roostmate has shared blood with it in the past.
How does such cooperation get off the ground? Axelrod has muddied the waters by claiming to have shown that TIT-FOR-TAT is an ESS in the indefinitely repeated Prisoner’s Dilemma. Although Maynard Smith mistakenly endorsed the claim, it obviously isn’t true. A population of TIT-FOR-TATs can be invaded by the strategy that always plays dove. Such a mutant won’t displace TIT-FOR-TAT, but nor will it be expelled.

No pure strategy can be an ESS in the indefinitely repeated Prisoner’s Dilemma: a mutation that changes the strategy at an unreached subgame won’t even be detected, let alone driven out. The ESS concept needs to be widened to be useful in such a setting, so that whole sets of strategies through which a population may drift are regarded as evolutionarily stable aggregates. For example, the set $N$ in both Figure 14 and Figure 32 is a kind of aggregate asymptotic attractor within which the system is free to drift. There needn’t be a trajectory leading away from $N$, as in both these cases.

## 经济代写|博弈论代考Game theory代写|Hawk-Dove-Retaliator Game

The problem is already apparent in the Hawk-Dove-Retaliator Game with which Maynard Smith and Price originally explored the evolution of cooperation. A retaliator plays like a hawk against a hawk, and like a dove against a dove. The retaliate strategy is weakly dominated, and so the game has a symmetric Nash equilibrium in which retaliate is not played at all. As in the Hawk-Dove game, dove is played with probability $1 / 3$ and hawk with probability $2 / 3$. In the upper triangle of Figure 32 , this mixed equilibrium is marked with the letter $M$. There are also an infinity of Nash equilibria in which hawk is not played at all, marked in Figure 32 with the letter $N$. These require that retaliate is played with probability at least $3 / 5$.

The upper triangle shows the replicator dynamics for the Hawk-Dove-Retaliator Game. The shaded set is the basin of attraction for the set $N$. Maynard Smith and Price ignore this set because only $M$ is an ESS. However, if the system found its way into $N$, its only chance of escaping is if a new hawk mutation appears while it is close to $Q$. But this rare event might be delayed for a very long time. There have, in fact, been enormously long perods of stasis in the evolution of many species that might be attributed to this cause.
The lower triangle of Figure 32 shows the replicator dynamics for a modified version of the Hawk-Dove-Replicator Game in which a retaliator is realistically assumed to do a little better against a dove

and a little worse against a hawk. This game has three symmetric Nash equilibria. There is an analogue of the mixed equilibrium $M$ of the Hawk-Dove Game; a pure equilibrium $R$ in which only retaliate gets played; and an equilibrium $P$ in which all three strategies are played with positive probability. The equilibria $M$ and $R$ correspond to ESS strategies.
The basin of attraction of $R$ is shaded in Figure 32. Since this is a large set, we have a toy model in which it makes sense to apply the ESS concept, and which offers the beginnings of an explanation of the evolution of cooperation. Maynard Smith and Price expand the model by introducing a bullying type who displays like a retaliator but backs down when challenged. The bullies displace doves, but otherwise nothing much changes.

However, the most interesting application of the
Hawk-Dove-Replicator Game is to the case of local interaction. In real life, animals mostly play games with their geographical neighbours. Chance might therefore easily fix it so that a mutant retaliator becomes numerous in a small neighbourhood. The Hawk-Dove-Replicator Game then tells us that the other strategies will gradually be extinguished in that neighbourhood. But then the same will happen in overlapping neighbourhoods until the whole environment is taken over by retaliators.
This seems to me the most convincing toy explanation of the evolution of cooperation that is commonly offered.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Hawk-Dove-Retaliator Game

$R$的吸引力盆地在图32中被遮蔽。由于这是一个很大的集合，我们有一个玩具模型，在这个模型中应用ESS概念是有意义的，它为解释合作的进化提供了一个开端。梅纳德·史密斯和普赖斯扩展了这个模型，引入了一种恃强凌弱的类型，这种类型的人表现得像报复者，但在受到挑战时退缩。恃强凌弱者取代了鸽子，但除此之外没有什么变化。

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## 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。