# 经济代写|博弈论代写Game Theory代考|ECON3301

#### Doug I. Jones

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## 经济代写|博弈论代写Game Theory代考|Taking Risks in a Finitely Repeated Prisoner’s Dilemma Game

McNamara et al. (2004) consider a game in which two individuals play a sequence of rounds of the Prisoner’s Dilemma game (Section 3.2) with one another. If in any round either player defects then the interaction ends and no more rounds are played. The idea is that individuals go off to seek more cooperative partners after a defection. This search phase is not explicitly incorporated into this model, although it is explicitly considered in a related model in Section 7.10. The interaction ends after the $N$ th round if it has not already done so. Here $N$ is known to both players. A strategy for an individual specifies the number of rounds in which to cooperate before defecting. Each partner attempts to maximize its total payoff over the rounds that are played. For illustrative purposes we assume that each round has payoffs given by Table 7.1.
Note that if the last (i.e. $N$ th) round is played both players should defect. They will then each receive a payoff of 1 for this round. Now consider the $(N-1)$ th round. If a partner defects there will be no further rounds, so it is best to defect since $1>0$. If a partner cooperates then cooperation will give a reward of 2 from the current round and the individual will go on to get a reward of 1 from the $N$ th round. This is less than the payoff of 5 from defection. Thus it is best to defect whatever the action of the partner. A similar consideration applies to the partner, so both players should defect and the game will end with both players receiving 1 . We can iterate backwards to earlier rounds in this way, deducing that both players should defect in the first round. This is the only Nash equilibrium for the game. Furthermore, since it is the unique best response to itself, it is an ESS.

Models in which two players play a number of rounds of the Prisoner’s Dilemma against one another have become a testbed for the evolution of cooperation. These models assume that in any round there is always the possibility of at least another round, otherwise the type of backward induction argument presented above will rule out any cooperation. McNamara et al. (2004) incorporated a maximum on the number of round specifically because they wished to show that cooperation could still evolve when variation is maintained even though the standard arguments precluded it. Figure $7.3$ illustrates the crucial role of variation. In each case illustrated the mean of the strategy values in the resident population is 5 . The cases differ in the range of strategies that are present. When there is no variation, so that all residents have strategy 5, the best response of a mutant to the resident population is to cooperate for 1 rounds; i.e. defect before partner does. As the amount of variation about the mean of 5 increases it becomes best to cooperate for more rounds. To understand why this occurs suppose that the 9 strategies $1,2, \ldots, 9$ are present in equal proportions. If 4 rounds of cooperation have passed, then partner’s strategy is equally likely to be $4,5,6,7,8$, or 9 . Thus the probability that partner will defect on the next round is only $\frac{1}{6}$. This makes it worthwhile to take a chance and cooperate for at least one

## 经济代写|博弈论代写Game Theory代考|Males Signal Parental Ability, Females Respond with Clutch Size

Suppose that a female and male bird have just paired. The female has to decide whether to lay one egg or two. Her best decision depends on the ability of the male to provision the nest. This ability, $q$, is either low $(L)$ or high $(H)$. All eggs survive unless the female lays two eggs when $q=L$. In this case $b(\leq 2)$ eggs survive, and furthermore the female has to work harder to help the male, reducing her future reproductive success by $K$. The payoffs to the female for each of the four combinations of clutch size and male ability are given in Table 7.2. We assume that $b-K<1$ so that the best action of the female is to lay one egg when $q=L$ and to lay two eggs when $q=H$.
It would be advantageous for the female to have information on the ability of the male. If ability is positively correlated with size then she can gain information from the male’s size. Assuming that she can observe size, this cue is not something the male can control by his behaviour. A cue that is not under the control of the signaller (the male in this case) is referred to as an index. Female frogs prefer large mates. Since larger males can produce lower-frequency croaks, the deepness of the croak of a male is a cue of his size that cannot be faked and is an index (Gerhardt, 1994).

From now on we assume that there are no indices; the female bird has no cue as to the ability of her partner. Instead, the male transmits one of two signal $s_1$ or $s_2$ to the female. For example, he might bring her a small amount of food (signal $s_1$ ) or a large amount of food (signal $s_2$ ). We assume that producing signal $s_1$ is cost free, whereas the signal $s_2$ costs the male $c_L$ if he has low ability and $c_H$ if he has high ability. The payoff to a male is the number of surviving offspring minus any signalling cost. We examine circumstances in which there is a signalling equilibrium at which the male’s signal is an honest indicator of his ability. That is, we seek a Nash equilibrium at which males signal $s_1$ when having low ability and $s_2$ when having high ability, and females lay one egg when the signal is $s_1$ and lay two eggs when the signal is $s_2$. We describe two circumstances in which this is a Nash equilibrium.

# 博弈论代考

## 经济代写|博弈论代写博弈论代考|在有限重复的囚徒困境博弈中冒险

McNamara et al.(2004)考虑了一个游戏，在这个游戏中，两个人彼此玩了一系列的囚徒困境游戏(章节3.2)。如果在任何一轮中有任何一个玩家叛变，那么互动就会结束，不再进行任何一轮游戏。其理念是，在叛变后，个体会去寻找更多的合作伙伴。这个搜索阶段没有显式地合并到这个模型中，尽管在第7.10节中的一个相关模型中显式地考虑了它。如果交互还没有完成，那么交互在$N$第一轮之后结束。在这里，玩家都知道$N$。针对个人的策略规定了叛变前合作的回合数。每个伙伴都试图在玩的回合中最大化自己的总收益。为了便于说明，我们假设每一轮的收益如表7.1所示。

## 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。

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