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

# 经济代写|博弈论代考Game theory代写|Evolutionary stability

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## 经济代写|博弈论代考Game theory代写|Evolutionary stability

For evolution to work, there must be some variation in a population. The shuffling of genes in sexual reproduction is one source of variation. Geographical migration and mutation are others. When can we expect a population to settle down in the face of such random variation? One approach is to look for an asymptotic attractor – a population of replicators that is stable in the face of any small perturbation.

The simplest possible model of a biological evolutionary process is called the replicator dynamics. Figure 14 on P. 50 shows how it works in a particular game when the players are drawn from two different populations that evolve separately. In this chapter, the corresponding diagrams are much simpler, because the focus will be on symmetric games in which the players are drawn from a single population.
The replicator dynamics assumes that the proportion of a population hosting a particular replicator increases at a rate proportional to two factors:

• The fraction of the population currently hosting the replicator.
• The difference between the current fitness of the replicator’s hosts and the average fitness of all the hosts in the population.

The first requirement simply recognizes that the rate of growth of a replicator is constrained by the number of parents available to transmit the replicator to the next generation. The second requirement recognizes that evolution can only take account of a replicator’s fitness relative to the fitness of the population as a whole.
If all the replicators to be considered are present when the replicator dynamics get started, then the system can only converge on a symmetric Nash equilibrium – if it converges on anything at all.
Evolutionarily stable strategies
The idea of an evolutionarily stable strategy or ESS begins with George Price, who submitted a 60-page essay on evolutionary mathematics to the journal Nature, which a more worldly author would have known publishes only short articles. Fortunately, his referee was John Maynard Smith. Together they wrote a paper which distilled Price’s essential wisdom into something readable. Maynard Smith went on to write Evolution and the Theory of Games, which put evolutionary game theory on the map. George Price eventually committed suicide, reportedly because he found it increasingly difficult to reconcile his fundamental contributions to evolutionary biology with his religious convictions.

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

Two birds drawn from the same species occasionally contest some valuable resource. The two replicators in the population make their hosts either passive or aggressive in such situations. A passive bird surrenders the entire resource to an aggressive bird. Two passive birds share the resource equally. Two aggressive birds fight.
Maynard Smith referred to passive birds as doves and aggressive birds as hawks, for which reason the game is called the Hawk-Dove Game – but don’t be misled into supposing that the birds are intended as representatives of different populations that evolve separately. The environment is intended to be entirely symmetric.
Prisoner’s Dilemma
If possession of the resource enhances a bird’s fitness by four utils and getting into a fight by only one util, then the Hawk-Dove Game reduces to the version of the Prisoner’s Dilemma shown in Figure 29. Recall that the only Nash equilibrium is for both players to use hawek (see Chapter 1). Since this strategy is strictly dominant, it is also an evolutionarily stable strategy.

The birds playing the Hawk-Dove Game are drawn from a single population, and so the replicator dynamics for the Prisoner’s Dilemma in Figure 29 on P. 126 are one-dimensional (rather than two-dimensional as in previous examples). The arrow shows that there is a unique asymptotic attractor in which the population consists entirely of hawks. If we were to perturb this population by throwing in a positive fraction of dovelike mutants, they would eventually all be eliminated. In fact, the basin of attraction consists of all population states other than that in which the whole population consists of doves. The appearance of even a tiny fraction of hawk mutants therefore dooms the doves to eventual extinction.

Group selection fallacy
The ardor with which game theorists deny the various fallacies claiming that cooperation is rational in the Prisoner’s Dilemma pales into insignificance when compared with the almost demonic ferocity with which evolutionary biologists denounce the group selection fallacy.
According to the group selection fallacy, evolution favours mutations that enhance the fitness of the species rather than the fitness of the mutated gene itself. A population playing dove in the Prisoner’s Dilemma would then be invulnerable to invasion by a mutant hawk because any fraction of hawks in the population would reduce the total fitness of the population. The fallacy lies in misplacing the relevant replicator at the level of the species. It is, after all, at the molecular level that replication takes place physically when the double helix divides. It is therefore right to confine attention to the unique ESS, which is hawk.
Charles Darwin knew nothing of modern genetics and so occasionally fell into a number of errors, of which the group selection fallacy was one. However, it is the biologist Vero Wynne-Edwards who is the luckless target of modern critics. He suggested, for example, that starlings gather in large numbers at nightfall in order to estimate their numbers with a view to controlling their population size. George Williams’s critique of his group selection arguments was very influential, leading to a literature explosion of which Dawkins’s Selfish Gene is just one example.

# 博弈论代写

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

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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。