## 经济代写|博弈论代写Game Theory代考|Finitely Repeated Games

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。

## 经济代写|博弈论代写Game Theory代考|Finitely Repeated Games

These games represent the case of a fixed known horizon $T$. The strategy spaces at each $l=0,1, \ldots, T$ are as defined above; the utilities are usually taken to be the time average of the per-period payoffs. Allowing for a discount factor $\delta$ close to 1 will not change the conclusions we present.

The set of equilibria of a finitely repeated game can be very different from that of the corresponding infinitely repeated game, because the scheme of self-reinforcing rewards and punishments used in the folk theorem can unravel backward from the terminal date. The classic example of this is the repeated prisoner’s dilemma. As observed in chapter 4, with a fixed finite horizon “always defect” is the only subgame-perfect-equilibrium outcome. In fact, with a bit more work one can show this is the only Nash outcome:

Fix a Nash equilibrium $\sigma^$. Both players must cheat in the last period, $T$, for any history $h^T$ that has positive probability under $\sigma^$, since doing so increases their period- $T$ payoff and since there are no future periods in which they might be punished. Next, we claim that both players must defect in period $T-1$ for any history $h^{T-1}$ with positive probability: We have already established that both players will chcat in the last period along the equilibrium path, so in particular if player $i$ conforms to the equilibrium strategy in period $T-1$ his opponent will defect in the last period, and hence player $i$ has no incentive not to defect in period $T-1$. An induction argument completes the proof. This conclusion, though not pathological, relies on the fact that the static equilibrium gives the players exactly their minmax values, as the following theorem shows.

## 经济代写|博弈论代写Game Theory代考|Repeated Games with Long-Run and Short-Run Players

The first variant we will consider supposes that some of the players are long-run players, as in standard repeated games, while the roles corresponding to other “players” are filled by a sequence of short-run players, each of whom plays only once.
Example 5.1
Suppose that a single long-run firm faces a sequence of short-run consumers, each of whom plays only once but is informed of all previous play when choosing his actions. Each period, the consumer moves first, and chooses whether or not to purchase a good from the firm. If the consumer does not purchase, then both players receive a payoff of 0 . If the consumer decides to purchase, then the firm must decide whether to produce high or low quality. If it produces high quality, both players have a payoff of 1 ; if it produces low quality, the firm’s payoff is 2 and the consumer’s payoff is

This game is a simplified version of those considered by Dybvig and Spatt (1980), Klein and Lefler (1981), and Shapiro (1982). ${ }^{12}$ Simon (1951)

and Kreps (1986) use a similar game to analyze the employment relationship, and to argue that one reason for the existence of “firms” is precisely to provide a long-run player who can be induced to be trustworthy by the prospect of future rewards and punishments.

The following strategies are a subgame-perfect equilibrium of this game when the firm is sufficiently patient: The firm starts out producing high yuality cvery time a consumer purchases, and continues to do so as long as it has never produced low quality in the past. If ever the firm produces low quality, it produces low quality at every subsequent opportunity. The consumers start out purchasing the good from the firm, and continue to do so so long as the firm has never produced low quality. If ever the firm produces low quality, then no consumer ever purchases again. The consumer’s strategies are optimal because each consumer cares only about that period’s payoff, and thus should buy if and only if that period’s quality is expected to be high. The firm does incur a short-run cost by producing high quality. but when the firm is patient this cost is offset by the fear that producing low quality will drive away future consumers. Note that this equilibrium suggests why consumers may prefer to deal with a firm that is expected to remain in business for a while, as opposed to a “fly-by-night” firm for whom long-run considerations are unimportant.

# 博弈论代考

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|Preemption Games

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。

## 经济代写|博弈论代写Game Theory代考|Preemption Games

Preemption games are a rough opposite to the war of attrition, with $L(\hat{t})>F(\hat{t})$ for some range of times $\hat{t}$. Here the specification of the payoff to simultaneous stopping, $B(\cdot)$, is more important than in the war of attrition, as if $L$ exceeds $F$ we might expect both players to stop simultaneously. One example of a preemption game is the decision of whether and when to build a new plant or adopt a new innovation when the market is big enough to support only one such addition (Reinganum 1981a,b; Fudenberg and Tirole 1985). In this case $B(t)$ is often less than $F(t)$, as it cian be better to let an opponent have a monopoly than to incur duopoly losses.
()ne very stylized preemption game is “grab the dollar.” In this stationary game, time is discrete $(t=0,1, \ldots)$ and there is a dollar on the tablc, which either or both of the players can try to grab. If only one player grabs, that player receives 1 and the other 0 ; if both try to grab at once, the dollar is destroyed and both pay a fine of 1 ; if neither player grabs, the dollar remains on the table. The players use the common discount factor $\delta$, so that $l(t)=\delta^{\prime}, F(t)=0$, and $B(t)=-\delta^t$ for all $t$. Like the war of attrition, this game has asymmetric equilibria, where one player “wins” with probability 1. and also a symmetric mixed-strategy equilibrium, where each player grabs the dollar with probability $p^=\frac{1}{2}$ in cach period. (It is easy to check that this yields a symmetric equilibrium; to see that it is the only one, note that each player must be indifferent between stopping-i.e., grabbing-at date $t$, which yields payoff $\delta^{\prime}\left(\left(1-p^(t)\right)-p^(t)\right)$ if the other has not stopped before date $t$ and 0 otherwise, and never stopping, which yields payoff 0 , so that $p^(t)$ must equal $\frac{1}{2}$ for all $t$.) The payoffs in the symmetric cquilibrium are $(0,0)$, and the distribution over outcomes is that the probability that player 1 alone stops first at $t$, the probability that player 2 alone stops first at $t$, and the probability that both players stop simultaneously at $l$ are all equal to $\left({ }_4^1\right)^{t+1}$. Note that these probabilities are independent of the per-period discount factor, $\delta$, and thus of the period length, $\Lambda$, in contrast to the war of attrition, where the probabilities were proportional to the period length. This makes finding a continuous-time representation of this game more difficult.

## 经济代写|博弈论代写Game Theory代考|Iterated Conditional Dominance and the Rubinstein Bargaining Game

The last two sections presented several examples of infinite-horizon games with unique equilibria. The uniqueness arguments there can be strengthened, in that these games have a unique profile that satisfies a weaker concept than subgame perfection.

Definition 4.2 In a multi-stage game with observed actions, action $a_i^t$ is conditionally dominated at stage $t$ given history $h^t$ if, in the subgame beginning at $h^{\prime}$, every strategy for player $i$ that assigns positive probability to $a_i^{\prime}$ is strictly dominated. Iterated conditional dominance is the process that, at each round, deletes every conditionally dominated action in every subgame, given the opponents’ strategies that have survived the previous rounds.

It is easy to check that itcrated conditional dominance coincides with subgame perfection in finite games of perfect information. In these games it also coincides with Pearce’s (1984) extensive-form rationalizability. In general multi-stage games, any action ruled out by iterated conditional dominance is also ruled out by extensive-form rationalizability, but the cxact relationship between the two concepts has not been determined.
In a game of imperfect information, iterated conditional dominance can be weaker than subgame perfection, as it does not assume that players forecast that an equilibrium will occur in every subgame. To illustrate this point, consider a one-stage, simultaneous-move game. Then iterated conditional dominance coincides with iterated strict dominance, subgame perfection coincides with Nash equilibrium, and iterated strict dominance is in general weaker than Nash equilibrium.

# 博弈论代考

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|Iterated Strict Dominance and Nash Equilibrium

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。

## 经济代写|博弈论代写Game Theory代考|Iterated Strict Dominance and Nash Equilibrium

If the extensive form is finite, so is the corresponding strategic form, and the Nash existence theorem yields the existence of a mixed-strategy equilibrium. The notion of iterated strict dominance extends to extensiveform games as well; however, as we mentioned above, this concept turns out to have little force in most extensive forms. The point is that a player cannot strictly prefer one action over another at an information set that is not reached given his opponents’ play.

Consider figure 3.14. Here, player 2’s strategy $\mathrm{R}$ is not strictly dominated, as it is as good as $\mathrm{L}$ when player 1 plays $\mathrm{U}$. Morcover, this fact is not “pathological.” It obtains for all strategic forms whose payoffs are derived from an extensive form with the tree on the left-hand side of the figure. That is, for any assignment of payoffs to the terminal nodes of the tree, the payoffs to (U, L) and (U, R) must be the same, as both strategy profiles lead to the same terminal node. This shows that the set of strategic-form payoffs of a fixed game tree is of lower dimension than the set of all payoffs of the corresponding strategic form, so theorems based on generic strategic-form payoffs (see chapter 12) do not apply. In particular, there can be an even number of Nash equilibria for an open set of extensive-form payoffs. The game illustrated in figure 3.14 has two Nash equilibria, (U, R) and (D, L), and this number is not changed if the extensive-form payoffs are slightly perturbed. The one case where the odd-number theorem of chapter 12 applies is to a simultaneous-move game such as that of figure 3.4 ; in such a game, each terminal node corresponds to a unique strategy profile. Put differently: In simultaneous-move games, every strategy profile reaches every information set, and so no player’s strategy can involve a choice that is not implemented given his opponents’ play.

Recall that a game of perfect information has all its information sets as singletons, as in the games illustrated in figures 3.3 and 3.14 .

## 经济代写|博弈论代写Game Theory代考|Backward Induction and Subgame Perfection

As we have seen, the strategic form can be used to represent arbitrarily complex extensive-form games, with the strategies of the strategic form being complete contingent plans of action in the extensive form. Thus, the concept of Nash equilibrium can be applied to all games, not only to games where players choose their actions simultancously. However, many game theorists doubt that Nash equilibrium is the right solution concept for general games. In this section we will present a first look at “equilibrium refinements,” which are designed to separate the “reasonable” Nash equilibria from the “unreasonable” ones. In particular, we will discuss the ideas of backward induction and “subgame perfection.” Chapters 4, 5 and 13 apply these ideas to some classes of games of interest to economists.

Selten (1965) was the first to argue that in general extensive games some of the Nash equilibria are “more reasonable” than others. He began with the example illustrated here in figure 3.14. This is a finite game of perfect information, and the backward-induction solution (that is, the one obtained using Kuhn’s algorithm) is that player 2 should play $\mathrm{L}$ if his information set is reached, and so player 1 should play D. Inspection of the strategic form corresponding to this game shows that there is another Nash equilibrium, where player 1 plays $\mathrm{U}$ and player 2 plays $\mathrm{R}$. The profile $(\mathrm{U}, \mathrm{R})$ is a Nash equilibrium because, given that player 1 plays U, player 2’s information set is not reached, and player 2 loses nothing by playing R. But Selten argued, and we agree, that this equilibrium is suspect. After all, if player 2 ‘s information set is reached, then, as long as player 2 is convinced that his payoffs are as specified in the figure, player 2 should play $\mathbf{L}$. And if we were player 2 , this is how we would play. Moreover, if we were player 1 , we would expect player 2 to play $\mathbf{L}$, and so we would play $\mathrm{D}$.

In the now-familiar language, the equilibrium (U,R) is not “credible,” because it relies on an “empty threat” by player 2 to play $R$. The threat is “empty” because player 2 would never wish to carry it out.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Backward Induction and Subgame Perfection

Selten(1965)是第一个提出在一般广泛博弈中，某些纳什均衡比其他均衡“更合理”的人。他从图3.14所示的例子开始。这是一个完全信息的有限博弈，逆向归纳解(即使用库恩算法得到的解)是，如果达到参与人2的信息集，参与人2应该选择$\ mathm {L}$，因此参与人1应该选择d。检查该博弈对应的策略形式表明，存在另一个纳什均衡，其中参与人1选择$\ mathm {U}$，参与人2选择$\ mathm {R}$。配置文件$(\ mathm {U}， \ mathm {R})$是纳什均衡，因为假设参与人1选择U，参与人2的信息集没有达到，参与人2选择R不会损失任何东西。但塞尔滕认为，我们同意，这个均衡是可疑的。毕竟，如果达到参与人2的信息集，那么，只要参与人2确信他的收益如图中所示，参与人2就应该选择$\mathbf{L}$。如果我们是参与人2，我们会这样玩。此外，如果我们是参与人1，我们会期望参与人2玩$\mathbf{L}$，所以我们会玩$\mathbf{D}$。

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|Backward Induction and Subgame Perfection

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。

## 经济代写|博弈论代写Game Theory代考|Backward Induction and Subgame Perfection

In the Stackelberg game, it was easy to see how player 2 “ought” to play, because once $q_1$ was fixed player 2 faced a simple decision problem. This allowed us to solve for player 2 ‘s optimal second-stage choice for each $q_1$ and then work backward to find the optimal choice for player 1. This algorithm can be extended to other games where only one player moves at each stage. We say that a multi-stage game has perfect information if, for every stage $k$ and history $h^k$, exactly one player has a nontrivial choice set a choice set with more than one element and all the others have the one-element choice set “do nothing.” A simple example of such a game has player 1 moving in stages $0,2,4$, etc. and player 2 moving in stages $1,3,5$, and so on. More generally, some players could move several times in a row, and which player gets to move in stage $k$ could depend on the previous history. The key thing is that only one player moves at each stage $k$. Since we have assumed that each player knows the past choices of all rivals, this implies that the single player on move at $k$ is “perfectly informed” of all aspects of the game except those which will occur in the future.

Backward induction can be applied to any finite game of perfect information, where finite means that the number of stages is finite and the number of feasible actions at any stage is finite, too. The algorithm begins by determining the optimal choices in the final stage $K$ for each history $h^k$ that is, the action for the player on move, given history $h^\kappa$, that maximizes that player’s payoff conditional on $h^K$ being reached. (There may be more than one maximizing choice; in this case backward induction allows the player to choose any of the maximizers.) Then we work back to stage $K-1$, and determine the optimal action for the player on move there, given that the player on move at stage $K$ with history $h^K$ will play the action we determined previously. The algorithm proceeds to “roll back,” just as in solving decision problems, until the initial stage is reached. At this point we have constructed a strategy profile, and it is easy to verify that this maximizes that player’s payoff conditional on $h^{\mathrm{N}}$ being reached. (There may be more than one maximizing choice; in this case backward induction allows the player to choose any of the maximizers.) Then we work back to stage $K-1$, and determine the optimal action for the player on move there, given that the player on move at stage $K$ with history $h^K$ will play the action we determined previously. The algorithm proceeds to “roll back,” just as in solving decision problems, until the initial stage is reached. At this point we have constructed a strategy profile, and it is easy to verify that this profile is a Nash equilibrium. Moreover, it has the nice property that each player’s actions are optimal at every possible history.

The argument for the backward-induction solution in the two-stage Stackelberg game – that player 1 should be able to forecast player 2 ‘s second-stage play-strikes us as quite compelling. In a three-stage game,the argument is a bit more complex: The player on move at stage 0 must forecitst that the player on move at stage 1 will correctly forecast the play of the player on move at stage 2, which clearly is a more demanding hypothesis. And the arguments for backward induction in longer games require eorrespondingly more involved hypotheses. For this reason, backward-induction arguments may not be compelling in “long” games. For the moment, though, we will pass over the arguments against backward induction; section 3.6 discusses its limitations in more detail.

## 经济代写|博弈论代写Game Theory代考|The Value of Commitment and “Time Consistency”

One of the recurring themes in the analysis of dynamic games has been that in many situations players can benefit from the opportunity to make a binding commitment to play in a certain way. In a one-player game -i.e.. a decision problem – such commitments cannot be of value, as any payoff that a player could attain while playing according to the commitment could be attained by playing in exactly the same way without being committed to do so. With more than one player, though, commitments can be of value, since by committing himself to a given sequence of actions a player may be able to alter the play of his opponents. This “paradoxical” value of commitment is closely related to our observation in chapter 1 that a player can gain by reducing his action set or decreasing his payoff to some outcomes, provided that his opponents are aware of the change. Indeed, some forms of commitment can be represented in exactly this way.

The way to model the possibility of commitments (and related moves like “promises”) is to explicitly include them as actions the players can take. (Schelling (1960) was an early proponent of this view.) We have already seen one example of the value of commitment in our study of the Stackelberg game. which describes a situation where one firm (the “leader”) can commit itself to an output level that the follower is forced to take as given when making its own output decision. Under the typical assumption that each firm’s optimal reaction $r_i\left(q_j\right)$ is a decreasing function of its opponent’s output. the Stackelberg leader’s payoff is higher than in the “Cournot equilibrium” outcome where the two firms choose their output levels simultaneously.

In the Stackelberg example, commitment is achieved simply by moving earlier than the opponent. Although this corresponds to a different extensive form than the simultaneous moves of Cournot competition, the set of “physical actions” is in some sense the same. The search for a way to commit oneself can also lead to the use of actions that would not otherwise have been considered. Classic examples include a general burning his bridges behind him as a commitment not to retreat and Odysseus having himself lashed to the mast and ordering his sailors to plug their ears with wax as a commitment not to go to the Sirens” island. (Note that the natural way to model the Odysseus story is with two “players,” corresponding to ()dysseus before and Odysseus after he is exposed to the Sirens.) Both of these cases correspond to a “total commitment”: Once the bridge is burned, or Odysseus is lashed to the mast and the sailors’ ears are filled with wax, the cost of turning back or escaping from the mast is taken to be infinite. One can also consider partial commitments, which increase the cost of, c.g., turning back without making it infinite.

# 博弈论代考

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|An Application of Iterated Strict Dominance

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。

## 经济代写|博弈论代写Game Theory代考|An Application of Iterated Strict Dominance

We now make stronger assumptions on the (infinite-action) Cournot model introduced in example 1.3: Suppose that $u_i$ is strictly concave in $q_i\left(\hat{i}^2 u_i / \hat{i} q_i^2<0\right)$, that the cross-partial derivative is negative $\left(\hat{C}^2 u_i / \hat{c} q_i \hat{\theta} q_j<\right.$ 0 . which is the case if $p^{\prime}<0$ and $p^{\prime \prime} \leq 0$ ), and that the reaction curves $r_1$ and $r_2$ (which are continuous and downward-sloping from the previous two assumptions) intersect only once at a point $N$, at which $r_1$ is strictly steeper than $r_2$. This situation is depicted in figure 2.1. (Note that $N$ is stable, in the terminology introduced in subsection 1.2.5.)

Let $q_1^m$ and $q_2^m$ denote the monopoly outputs: $q_1^m=r_1(0)$ and $q_2^m=r_2(0)$. The first round of deletion of strictly dominated strategies yields $S_i^1=$ $\left[0, q_i^m\right]$. The second round of deletion yields $S_i^2=\left[r_i\left(q_j^m\right), q_i^m\right] \equiv\left[q_i^2, q_i^m\right]$, as indicated in figure 2.1. Consider, for instance, firm 2. Knowing that firm 1 won’t pick output greater than $q_i^m$, choosing output $q_2$ under $r_2\left(q_1^m\right) \equiv q_2^2$ is strictly dominated by playing $q_2^2$ by strict concavity of firm 2 s payoff in its own output. And similarly for firm 1. The third round of deletion yields $S_i^3=\left[q_i^2, r_i\left(q_j^2\right)\right] \equiv\left[q_i^2, q_i^3\right]$, and so on. More generally, itcrated deletion yields a sequence of shrinking intervals around the outputs $\left(q_1^, q_2^\right)$ corresponding to the intersection $N$ of the reaction curves. For $n-2 k+1$,
$$q_i^{2 k+1}=q_i^{2 k} \quad \text { and } \quad q_i^{2 k+1}=r_i\left(q_j^{2 k}\right) \text {; }$$
for $n=2 k$,
$$q_i^{2 k}=r_i\left(q_j^{2 k-1}\right) \quad \text { and } \quad q_i^{2 k}=\bar{q}_i^{2 k-1} \text {. }$$
A diflerence between this process and the case of finite strategy spaces is that the process of deletion does not stop after a finite number of steps. Nevertheless, the process does converge, because the sequences $q_i^n$ and $q_i^n$ both converge to $q_i^$, so that the process of iterated deletion of strictly dominated strategies yields $N$ as the unique “reasonable” prediction. (Let $q_i^s \equiv \lim q_i^n \leq q_i^$ and $\bar{q}_i^{\infty} \equiv \lim q_i^n \geq q_i^$. From the definition of $q_i^n$ and $\bar{q}_i^n$ and by continuity of the reaction curves, one has $\bar{q}_i^{\infty}=r_i\left(q_j^{\infty}\right)$ and $q_j^{\infty}=$ $r_j\left(q_i^x\right)$. Hence, $\bar{q}_i^x=r_i\left(r_j\left(\bar{q}_i^x\right)\right)$, which is possibie only if $\bar{q}_i^{\infty}=q_i^$; and similarly for $q_i^x$.)
We conclude that this Cournot game is solvable by iterated strict dominance. This need not be the case for other specifications of the payoff functions; see exercise 2.4 .

## 经济代写|博弈论代写Game Theory代考|Rationalizability

The concept of rationalizability was introduced independently by Bernheim (1984) and Pearce (1984), and was used by Aumann (1987) and by Brandenberger and Dekel (1987) in their papers on the “Bayesian approach” to the choice of strategies.

Like iterated strict dominance, rationalizability derives restrictions on play from the assumptions that the payoffs and the “rationality” of the players are common knowledge. The starting point of iterated strict dominance is the observation that a rational player will never play a strictly dominated strategy. The starting point of rationalizability is the complementary question: What are all the strategies that a rational player could play? The answer is that a rational player will use only those strategies that are best responses to some beliefs he might have about the strategies of his opponents. Or, to use the contrapositive, a player cannot reasonably play a strategy that is not a best response to some beliefs about his opponents’ strategies. Moreover, since the player knows his opponents’ payoffs, and knows they are rational, he should not have arbitrary beliefs about their strategies. He should expect his opponents to use only strategies that are best responses to some beliefs that they might have. And these opponents’ beliefs, in turn, should also not be arbitrary, which leads to an infinite regress. In the two-player case, the infinite regress has the form “I’m playing strategy $\sigma_1$ because I think player 2 is using $\sigma_2$, which is a reasonable belief because I would play it if I were player 2 and I thought player 1 was using $\sigma_1^{\prime}$, which is a reasonable thing for player 2 to expect because $\sigma_1^{\prime}$ is a best response to $\sigma_2^{\prime}, \ldots$,

Formally, rationalizability is defined by the following iterative process.
Definition 2.3 Set $\tilde{\Sigma}i^o \equiv \Sigma_i$, and for each $i$ recursively define \begin{aligned} & \tilde{\Sigma}_i^n=\left{\sigma_i \in \tilde{\Sigma}_i^{n-1} \mid \exists \sigma{-i} \in \underset{j \neq i}{\times} \text { convex hull }\left(\tilde{\Sigma}j^{n-1}\right)\right. \text { such that } \ & \left.u_i\left(\sigma_i, \sigma{-i}\right) \geq u_i\left(\sigma_i^{\prime}, \sigma_{-i}\right) \text { for all } \sigma_i^{\prime} \in \tilde{\Sigma}i^{n-1}\right} . \end{aligned} The rationalizable strategies for player $i$ are $R_i=\bigcap{n=0}^{\infty} \tilde{\Sigma}_i^n$.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|An Application of Iterated Strict Dominance

$$q_i^{2 k+1}=q_i^{2 k} \quad \text { and } \quad q_i^{2 k+1}=r_i\left(q_j^{2 k}\right) \text {; }$$

$$q_i^{2 k}=r_i\left(q_j^{2 k-1}\right) \quad \text { and } \quad q_i^{2 k}=\bar{q}_i^{2 k-1} \text {. }$$

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|Examples of Pure-Strategy Equilibria

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。

## 经济代写|博弈论代写Game Theory代考|Examples of Pure-Strategy Equilibria

A simple example of nonexistence is “matching pennies” (figure 1.8). Players 1 and 2 simultaneously announce heads $(\mathrm{H})$ or tails $(\mathrm{T})$. If the announcements match, then player 1 gains a util and player 2 loses a util. If the announcements differ, it is player 2 who wins the util and player 1 who loses. If the predicted outcome is that the announcements will match, then player 2 has an incentive to deviate, while playcr 1 would prefer to deviate from any prediction in which announcements do not match. The only “stable” situation is one in which cach player randomizes between his two pure strategics, assigning equal probability to each. To see this, note that if player 2 randomizes $\frac{1}{2}-\frac{1}{2}$ between $\mathrm{H}$ and $\mathrm{T}$, player 1’s payoff is $\frac{1}{2} \cdot 1+\frac{1}{2} \cdot(-1)=0$ when playing $\mathrm{H}$ and $\frac{1}{2} \cdot(-1)+\frac{1}{2} \cdot 1=0$ when playing $T$. In this case player 1 is completely indifferent between his possible choices and is willing to randomize himself.

This raises the question of why a player should bother to play a mixed strategy when he knows that any of the pure strategies in its support would do equally well. In matching pennies, if player 1 knows that player 2 will randomize between $\mathrm{H}$ and $\mathrm{T}$ with equal probabilities, player 1 has expected value 0 from all possible choices. As far as his payoff goes, he could just as well play “heads” with certainty, but if this is anticipated by player 2 the equilibrium disintegrates. Subsection 1.2.5 mentions one defense of mixed strategies, which is that it represents a large population of players who use different pure strategies. If we insist that there is only one “player 1,” though, this interpretation does not apply. Harsanyi (1973a) offered the alternative defense that the “mixing” should be interpreted as the result of small, unobservable variations in a player’s payoffs. Thus, in our example, sometimes player 1 might prefer matching on $T$ to matching on $\mathbf{H}$, and conversely. Then, for each value of his payoff, player 1 would play a pure strategy. This “purification” of mixed-strategy equilibria is discussed in chapter 6 .

## 经济代写|博弈论代写Game Theory代考|Multiple Nash Equilibria, Focal Points, and Pareto Optimality

Many games have several Nash equilibria. When this is the case, the assumption that a Nash equilibrium is played relies on there being some mechanism or process that leads all the players to expect the same equilibrium.

One well-known example of a game with multiple equilibria is the “battle of the sexes,” illustrated by figure 1.10a. The story that goes with the name “battle of the sexes” is that the two players wish to go to an event together, but disagree about whether to go to a football game or the ballet. Each player gets a utility of 2 if both go to his or her preferred event, a utility of 1 if both go to the other’s preferred event, and 0 if the two are unable to agree and stay home or go out individually. Figure $1.10 \mathrm{~b}$ displays a closely related game that goes by the names of “chicken” and “hawk-dove.” (Chapter 4 discusses a related dynamic game that is also called “chicken.”) One version of the story here is that the two players meet at a one-lane bridge and each must choose whether to cross or to wait for the other. If both play T (for “tough”), they crash in the middle of the bridge and get -1 each; if both play W (for “weak”), they wait and get 0 ; if one player chooses $\mathrm{T}$ and the other chooses $\mathrm{W}$, then the tough player crosses first, receiving 2, and the weak one receives 1 . In the bridge-crossing story, the term “chicken” is used in the colloquial sense of “coward.” (Evolutionary biologists call this game “hawk-dove,” because they interpret strategy T as “hawk-like” and strategy W as “dove-like.”)

Though the different payoff matrices in figures $1.10 \mathrm{a}$ and $1.10 \mathrm{~b}$ describe different sorts of situations, the two games are very similar. Each of them has three equilibria: two in pure strategies, with payoffs $(2,1)$ and $(1,2)$, and

Building on this result, one can compute the optimal contract, i.e., the $w$ that maximizes the principal’s expected payorI
$$r(1-x) \cdot w(1-x y)-h y=v(1-h / w)-w .$$
The optimal wage is thus $w-\sqrt{h}$ (assuming $\sqrt{h} v>g$ ). Note that the principal would be better off if he could “commit” to an inspection level. To see this, consider the different game in which the principal plays first and chooses a probability $y$ of inspection, and the agent, after vbserving $y$, chooses whether to shirk. For a given $w(>g)$, the principal can choose $y=4 / u+c$, where $\varepsilon$ is positive and arbitrarily small. The agent then works with probability 1, and the principal has (approximately) payoff
$$r-w \cdot h g(w>v(1-h ; w)-w \text {. }$$
Technically, commitment eliminates the constraint $x w \geq h$, (i.e., that it is ex post worthwhile to inspect). (It is crucial that the principal is committed to inspecting with probability $y$. If the “toss of the coin” determining inspection is not public, the principal has an ex post incentive not to inspect, as he knows that the agent works.) This reasoning will become familiar in chapter 3 . See chapters 5 and 10 for discussions of how repeated play might make the commitment credible whereas it would not be if the game was played only once.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Multiple Nash Equilibria, Focal Points, and Pareto Optimality

“两性之战”是具有多重平衡的游戏的一个著名例子，如图1.10a所示。与“性别之战”同名的故事是，两个玩家希望一起去参加一个活动，但不同意是去看足球比赛还是看芭蕾舞。如果双方都选择自己喜欢的事件，每个参与人的效用为2，如果双方都选择对方喜欢的事件，效用为1，如果双方不能达成一致，各自待在家里或外出，效用为0。图$1.10 \mathrm{~b}$显示了一个密切相关的游戏，名为“鸡”和“鹰鸽”。(游戏邦注:第4章讨论的是一个相关的动态游戏，也被称为“chicken”)故事的一个版本是，两个玩家在单车道的桥上相遇，每个人都必须选择是过去还是等待另一个。如果两人都选择T(代表“强硬”)，他们就会在桥的中间坠毁，各得-1分;如果两人都玩W(代表“弱”)，他们等待并得到0;如果一个玩家选择$\mathrm{T}$，另一个玩家选择$\mathrm{W}$，那么强壮的玩家先传中，得2分，弱的玩家得1分。在过桥的故事中，“chicken”这个词的口语意思是“懦夫”。(进化生物学家称这种博弈为“鹰鸽”，因为他们将策略T解释为“鹰式”，将策略W解释为“鸽式”。)

$$r(1-x) \cdot w(1-x y)-h y=v(1-h / w)-w .$$

$$r-w \cdot h g(w>v(1-h ; w)-w \text {. }$$

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|PROBABILITY

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。

## 经济代写|博弈论代写Game Theory代考|PROBABILITY

A coin toss produces a random event, where the outcome can be either “heads” or “tails.” For a “fair” coin, we assess that these outcomes are equally likely. Sometimes we use the phrase “fifty-fifty” to describe the prospects. In more formal terms, we would say that the probability of heads is $1 / 2$ and the probability of tails is $1 / 2$. Using the concept of probability, we have an organized and logical way of considering random events.

There are all sorts of situations that have random components. The weather is random, machines fail randomly, and sometimes people behave randomly. To study the random component, it is useful to consider a state space that describes all of the possible resolutions of the random forces. For example, if you are interested in a particular horse race, the state space may consist of all of the different orderings of the horses (ways in which they could finish in the race). If you are a poker player, the state space may comprise the different ways in which the cards can be dealt. If you like flipping coins, the state space is ${$ heads, tails $}$. Each of the states in the state space is assumed to describe an outcome that is mutually exclusive of what other states describe. Further, the states collectively exhaust all of the possibilities. In other words, one and only one state actually occurs.

To describe the relative likelihood of the different individual states, we can posit a probability distribution over the state space. For example, suppose the state space is ${A, B, C}$. Think of this as the possible outcomes of a horse race, where all you care about is which horse wins (horse $A$, horse $B$, or horse $C$ ). A probability distribution for this state space implies a function $p:{A, B, C} \rightarrow[0,1]$ from which we get three numbers, $p(A), p(B)$, and $p(C)$. The number $p(A)$ is the “probability that $A$ occurs,” $p(B)$ is the probability that $B$ occurs, and $p(C)$ is the probability that $C$ occurs. Each of these numbers is assumed to be between 0 and 1 (as you can see from the codomain designation of $p$ ) and the numbers sum to 1 . For example, if horse $A$ is twice as likely to win as are horses $B$ and $C$ individually and if $B$ and $C$ are equally likely to win, then $p(A)=1 / 2, p(B)=1 / 4$, and $p(C)=1 / 4$.

## 经济代写|博弈论代写Game Theory代考|DOMINANCE, BEST RESPONSE, AND CORRELATED CONJECTURES

To understand the formal relation between dominance and best response, you have to begin with the concept of correlated conjectures. Remember that a belief, or conjecture, of player $i$ is a probability distribution over the strategies played by the other players. In two-player games, this amounts to a probability distribution over the strategy adopted by player $j$ (player $i$ ‘s opponent). In games with more than two players, though, player $i$ ‘s belief is more complicated. It is a probability distribution over the strategy combinations (profiles) of player $i$ ‘s opponents. For example, consider a three-player game in which player 1 chooses between strategies A and B, player 2 chooses between M and $\mathrm{N}$, and player 3 chooses between $\mathrm{X}$ and $\mathrm{Y}$. The belief of player 1 represents his expectations about both player 2’s and player 3’s strategies. That is, player l’s conjecture is an element of $\Delta S_{-1}$. The belief is a probability distribution over
$${\mathrm{M}, \mathrm{N}} \times{\mathrm{X}, \mathrm{Y}}={(\mathrm{M}, \mathrm{X}),(\mathrm{M}, \mathrm{Y}),(N, X),(N, Y)} .$$
Let us explore the possible beliefs.
Suppose that player 1 thinks that with probability $1 / 2$ player 2 will select $M$, that with probability $1 / 2$ player 3 will choose $X$, and that his opponents’ actions are independent. The last property implies that the probability of any profile of the opponents’ strategies is the product of the individual probabilities. ${ }^1$ That is, player 1 thinks that $(\mathrm{M}, \mathrm{X})$ is his opponents’ strategy profile with probability $1 / 4,(\mathrm{M}, \mathrm{Y})$ occurs with probability $1 / 4$, and so on. We can represent this belief by the matrix in Figure B.1(a). The marginal distributions appear on the outside of the matrix, on the right for player 2 and below for player 3. The marginals are the probabilities of each strategy for these players individually. Note that the probability of each strategy profile (the number in a given cell) is the product of the marginal probabilities.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|DOMINANCE, BEST RESPONSE, AND CORRELATED CONJECTURES

$${\mathrm{M}, \mathrm{N}} \times{\mathrm{X}, \mathrm{Y}}={(\mathrm{M}, \mathrm{X}),(\mathrm{M}, \mathrm{Y}),(N, X),(N, Y)} .$$

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|CONSISTENCY OF BELIEFS

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。

## 经济代写|博弈论代写Game Theory代考|CONSISTENCY OF BELIEFS

In an equilibrium, player 2’s updated belief should be consistent with nature’s probability distribution and player l’s strategy. For example, as noted earlier, if player 2 knows that player 1 adopts strategy $\mathrm{N}^{\mathrm{F}} \mathrm{G}^{\mathrm{E}}$, then player 2’s updated belief should specify $q=0$; that is, conditional on receiving a gift, player 2 believes that player 1 is the enemy type. In general, consistency between nature’s probability distribution, player 1’s strategy, and player 2’s updated belief can be evaluated by using Bayes’ rule. Recall that Bayes’ rule was discussed in the context of information aggregation in the previous chapter. If you did not read about it there or if you could use a brief review, please read Appendix A.

The Bayes’ rule calculation is quite simple and intuitive. Here is the general form for the gift game in Figure 28.2. At player 2’s information set, his updated belief gives the relative likelihood that player 2 thinks his top and bottom nodes have been reached. Let $r^{\mathrm{F}}$ and $r^{\mathrm{E}}$ be the probabilities of arriving at player 2 ‘s top and bottom nodes, respectively. That is, $r^{\mathrm{F}}$ is the probability that nature selects $\mathrm{F}$ and then player 1 selects $\mathrm{G}^{\mathrm{F}}$. Likewise, $r^{\mathrm{E}}$ is the probability that nature selects $\mathrm{E}$ and then player 1 chooses $\mathrm{G}^{\mathrm{E}}$. As an example, suppose that $r^{\mathrm{F}}=1 / 8$ and $r^{\mathrm{E}}=1 / 16$. In this case, player 2 ‘s information set is reached with probability $1 / 8+1 / 16=3 / 16$, which is not a very likely event. But note that the top node is twice as likely as is the bottom node. Thus, conditional on player 2’s information set actually being reached, player 2 ought to believe that it is twice as likely that he is at the top node than at the bottom node. Because the probabilities must sum to 1 , this updated belief is represented by a probability of $2 / 3$ on the top node and $1 / 3$ on the bottom node.
In general, the relation between $r^{\mathrm{F}}, r^{\mathrm{E}}$, and $q$ is given by
$$q=\frac{r^{\mathrm{F}}}{r^{\mathrm{F}}+r^{\mathrm{E}}}$$

## 经济代写|博弈论代写Game Theory代考|EQUILIBRIUM DEFINITION

Perfect Bayesian equilibrium is a concept that incorporates sequential rationality and consistency of beliefs. In essence, a perfect Bayesian equilibrium is a coherent story that describes beliefs and behavior in a game. The beliefs must be consistent with the players’ strategy profile, and the strategy profile must specify rational behavior at all information sets, given the players’ beliefs. In more formal language:
Consider a strategy profile for the players, as well as beliefs over the nodes at all information sets. These are called a perfect Bayesian equilibrium (PBE) if: (1) each player’s strategy specifies optimal actions,given his beliefs and the strategies of the other players, and (2) the beliefs are consistent with Bayes’ rule wherever possible. ${ }^1$
Two additional terms are useful in categorizing the classes of potential equilibria. Specifically, we call an equilibrium separating if the types of a player behave differently. In contrast, in a pooling equilibrium, the types behave the same.
To determine the set of $\mathrm{PBE}$ for a game, you can use the following procedure.
Steps for calculating perfect Bayesian equilibria:

2. If possible, calculate updated beliefs ( $q$ in the example) by using Bayes’ rule. In the event that Bayes’ rule cannot be used, you must arbitrarily select an updated belief; here you will generally have to check different potential values for the updated belief with the next steps of the procedure.
3. Given the updated beliefs, calculate player 2’s optimal action.
4. Check whether player 1’s strategy is a best response to player 2’s strategy. If so, you have found a PBE.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|CONSISTENCY OF BELIEFS

$$q=\frac{r^{\mathrm{F}}}{r^{\mathrm{F}}+r^{\mathrm{E}}}$$

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|GOODWILL AND TRADING A REPUTATION

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。

## 经济代写|博弈论代写Game Theory代考|GOODWILL AND TRADING A REPUTATION

The word “trade” usually makes people think of the exchange of physical goods and services. But some less-tangible assets also are routinely traded. Reputation is one of them. Those who have studied accounting know that “goodwill” is a legitimate and often important item on the asset side of a firm’s balance sheet. Goodwill refers to the confidence that consumers have in the firm’s integrity, the belief that the firm will provide high-quality goods and services-in other words, the firm’s reputation. It is often said that a firm’s reputation is its greatest asset. Firms that have well-publicized failures (product recalls, for example) often lose customer confidence and, as a result, profits.

When a firm is bought or sold, its reputation is part of the deal. The current owners of a firm have an incentive to maintain the firm’s good reputation to the extent that it will attract a high price from prospective buyers. This incentive may outweigh short-term desires to take advantage of customers or to do other things that ultimately will injure the firm’s good name.

A game-theory model illustrates how reputation is traded. ${ }^5$ The following game-theoretic example is completely abstract-it is not a model of a firm per se-but it clearly demonstrates how reputation is traded. Consider the twoperiod repeated game analyzed at the beginning of Chapter 22; the stage game is reproduced in Figure 23.1. Here I add a new twist. Suppose there are three players, called player 1 , player $2^1$, and player $2^2$. In the first period, players 1 and $2^1$ play the stage game (with player $2^1$ playing the role of player 2 in the stage game). Then player $2^1$ retires, so he cannot play the stage game with player 1 again in period 2 . However, player $2^1$ holds the right to play in period 2 , even though he cannot exercise this right himself. Player $2^1$ can sell this right to player $2^2$, in which case players 1 and $2^2$ play the stage game in the second period.

## 有限元方法代写

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。

## 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。

## 经济代写|博弈论代写Game Theory代考|HOLD-UP EXAMPLE

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。

## 经济代写|博弈论代写Game Theory代考|HOLD-UP EXAMPLE

For an illustration of the hold-up problem, consider a setting in which two people-Joel Dean (JD) and Brynn-interact in the development of a new product. JD is a scientist who can, with effort, design a new device for treating a particular medical condition. He is the only person with a deep knowledge of both the medical condition and physics sufficient to develop the innovative design. But JD has neither the engineering expertise nor the resources that are needed to construct the device. Brynn is the CEO of an engineering company; she is capable of implementing the design and creating a marketable product. Thus, success relies on JD’s and Brynn’s combined contributions.

Suppose JD and Brynn interact over three dates as follows. At Date 1, JD decides how much to invest in the design of the medical device. His investmentin fact, the complete design specifications-is observed by Brynn. Then, at Date 2, JD and Brynn meet to negotiate a contract that sets conditions (a price) under which Brynn can produce and market the device at Date 3. Commercial production of the device will generate revenue for Brynn, and this revenue will be a function of JD’s initial investment level. The key issue is whether JD has the incentive to invest efficiently, given that he has to later negotiate with Brynn to obtain the fruits of his investment.

Here, a bit more formally, is a description of the sequence of events: At Date 1, JD selects between “high investment” (abbreviated H), “low investment” (L), and “no investment.” If he chooses not to invest, then the game ends, and both parties get a payoff of 0 . In contrast, if JD chooses L or H, then JD pays a personal investment cost, and the game continues at Date 2. JD’s cost of low investment is 1 , whereas his cost of high investment is 10. Assume that JD’s investment choice is observed by Brynn but is not verifiable to the court, so that the investment cannot directly influence a legal action.

At Date 2, JD and Brynn negotiate over contracted monetary transfer $p$, which is a transfer from Brynn to JD to be compelled by the external enforcer (the court) if and only if Brynn elects to produce at Date 3 . The default price is $p$, which represents the legal default rule in case $\mathrm{JD}$ and Brynn do not establish an agreement. Assume that the court always compels a transfer of 0 if Brynn selects $\mathrm{N}^4{ }^4$ Also assume that the players have equal bargaining weights, so $\pi_{\mathrm{JD}}=\pi_{\mathrm{B}}=1 / 2$. At Date 3 , Brynn chooses whether to “produce” $(\mathrm{P})$ or not $(\mathrm{N})$. If Brynn chooses to produce, then $p$ is the amount transfered from Brynn to JD; if Brynn chooses not to produce, then the transfer is 0 . Thus, Brynn’s choice of whether to produce is verifiable, and the contract simply prescribes the transfer as a function of this selection. The time line of the game is pictured in Figure 21.1; note that this is not the extensive-form diagram, which you can draw as an exercise.

## 经济代写|博弈论代写Game Theory代考|UP-FRONT CONTRACTING AND OPTION CONTRACTS

Key aspects of the hold-up story are that (1) investments are unverifiable, so the court cannot condition transfers directly on these actions, and (2) there is some barrier to the parties writing a comprehensive contract prior to choosing investments. In the example that I just presented, item (2) is represented by the assumption that JD and Brynn meet only after JD makes his investment decision. This assumption may be a stretch, for in many real settings the contracting parties can negotiate and form a contract before they are required to make significant investments and take other productive actions.

Let us consider, therefore, a version of the model in which JD and Brynn meet and form a contract at Date 0 , with interaction continuing in Dates 1-3 just as described in the previous section. Think of the contract at Date 0 as the “initial contract,” and think of any contracting at Date 2 as “renegotiation.” At Date 0, JD and Brynn jointly select the value of $p$ (the amount Brynn will have to pay JD if she produces at Date 3), and they also may specify an up-front transfer. Then $p$ becomes the default value for renegotiation at Date 2. If the players keep $p$ in place at Date 2 , then we say that renegotiation of the contract did not occur. Otherwise, the parties will have renegotiated their initial contract to alter the production-contingent transfer. Assume that the default decision for negotiation at Date 0 is $p=20$, the legal default rule assumed in the previous section. The time line of the game is pictured in Figure 21.2.

Hold up is an issue even though contracting occurs at Date 0. Here’s why. Because JD’s investment decision is unverifiable, there is no way for the contract to give JD a high-powered incentive to invest (as was achieved for Carina in the example at the end of Chapter 20; see pp. 263-265. Instead, the contract can only be used to motivate Brynn in her choice between $\mathrm{P}$ and $\mathrm{N}$ at Date 3 (an action that is verifiable). One hopes that Brynn’s action can be made contingent on JD’s investment in such a way as to motivate JD to invest. Unfortunately, renegotiation at Date 2 may interfere with the whole plan because it may undo something to which the players wanted to commit at Date 0 .

# 博弈论代考

## 有限元方法代写

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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。