### 经济代写|博弈论代写Game Theory代考|ECON3503

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Convergence Stability

Any resident trait value $x^$ that is an equilibrium point under adaptive dynamics must satisfy $D\left(x^\right)=0$. Such a trait value is referred to as an evolutionarily singular strategy (Metz et al., 1996; Geritz et al., 1998). But what would happen if an initial resident trait is close to such an $x^$ ? Would it then evolve towards $x^$ or evolve away from this value? To analyse this we consider the derivative of $D(x)$ at $x=x^$. Suppose that $D^{\prime}\left(x^\right)<0$. Then for $x$ close to $x^$ we have $D(x)>0$ for $x$ and $D(x)<0$ for $x>x^$. Thus by the conditions in (4.6) $x$ will increase for $x$ and decrease for $x>x^$. Thus, providing the resident trait $x$ is close to $x^$, it will move closer to $x^$, and will converge on $x^$. We therefore refer to the trait $x^$ as being convergence stable if $$D^{\prime}\left(x^\right)<0 .$$ We might also refer to such an $x^*$ as an evolutionary attractor. Conversely, if $D^{\prime}\left(x^*\right)>0$, then any resident trait value close to (but not exactly equal to) $x^$ will evolve further away from $x^$. In this case we refer to $x^$ as an evolutionary repeller. Such a trait value cannot be reached by evolution. Figure $4.4$ illustrates these results. If $x^$ is a Nash equilibrium value of the trait, then $W\left(x^{\prime}, x^\right)$ has a maximum at $x^{\prime}=x^$. Thus if $x^$ lies in the interior of the range of possible trait values we must have $\frac{\partial W}{\partial x^{\prime}}\left(x^, x^\right)=0$; i.e. $D\left(x^\right)=0$. Thus $x^*$ is an equilibrium point under adaptive dynamics; i.e. it is an evolutionarily singular strategy. Since every ESS is also a Nash equilibrium, every internal ESS is also an evolutionarily singular strategy. The above analysis of the convergence stability of an ESS was originally developed by Eshel and Motro (1981) and Eshel (1983). Eshel and Motro (1981) refer to an ESS that is also convergence stable as a Continuously Stable Strategy (CSS). Some ESSs are also CSSs, others are not and are unattainable.

Figure $4.4$ shows the fitness derivative $D(x)$ at $x=0.5$ for each of the two formulations of the predator inspection game. In formulation I, $D(x)>0$ for $x<0.5$ and $D(x)<0$ for $x>0.5$, so that the trait $x=0.5$ is convergence stable and is hence a CSS. In contrast, in formulation II, $D(x)<0$ for $x<0.5$ and $D(x)>0$ for $x>0.5$, so that the trait $x=0.5$ is an evolutionary repeller and is unattainable. Figure $4.2$ illustrates why the two formulations result in the sign of $D(x)$ behaving differently. The two formulations differ in the dependence of the survival of the mutant on the mutant and resident distances travelled when these distances are similar. In formulation I an increase above $0.5$ in the distance travelled by the resident does not reduce the danger sufficiently for the mutant to also increase its distance by this much.

Eshel and Motro (1981) were the first to demonstrate that an ESS may not be convergence stable and so might not be attainable. In their model the trait under selection is the maximum risk that an individual should take in order to help kin. Since then unattainable ESSs have been demonstrated in a very wide variety of contexts. For example, Nowak (1990) shows that there can be an unattainable ESS in a variant of the repeated Prisoner’s Dilemma model.

## 经济代写|博弈论代写Game Theory代考|The Slope of the Best Response Function and Convergence Stability

The condition $D^{\prime}\left(x^*\right)<0$ for convergence stability in eq $(4.8)$ can be translated into a condition on the second derivatives of the payoff function $W$ (Box 4.2). This condition can then be used to show that if the slope of the best response function is less than 1 at an ESS, then the ESS is also convergence stable and is hence a CSS. Conversely if the slope exceeds 1 the ESS is unattainable (Box 4.2). In the predator inspection game the slope of the best response function at the ESS is less than 1 in Formulation I and greater than 1 for Formulation II (Fig. 4.3), in agreement with our previous findings for these two cases. We can also apply this result to other examples in this book. For example in the model of pairwise contests where individuals know their own fighting ability (Section 3.11), we can infer that the two ESSs shown in Fig. 3.9b are also convergence stable and are hence CSSs.

A pairwise invasibility plot (PIP) (van Tienderen and de Jong, 1986; Matsuda, 1995) shows, for each possible resident strategy, the range of mutant strategies that can invade. PIPs are useful graphical representations from which one can often ascertain whether a Nash equilibrium $x^$ is also an ESS at a glance. Figure $4.5$ illustrates these plots for the two cases of the predator inspection game that we have analysed. In both cases it can be seen from the figure that when the resident strategy is $x^=0.5$ any mutant different from $x^$ has lower fitness. Thus $x^=0.5$ is the unique best response

to itself and is hence an ESS. PIPs are, however, not so useful in cases where some mutants have equal fitness to residents when the resident strategy is $x^*$. This is because the ESS criterion then relies on the second-order condition (ES2)(ii). PIPs also can be used to infer whether $x^$ is convergence stable. In Fig. $4.5$ a if the resident trait is initially less than $0.5$ then those mutants that can invade have trait values greater than the resident trait, so that the resident trait will tend to increase over time. Conversely if the resident trait is initially greater than $0.5$ it will tend to decrease over time. The trait value $x^=0.5$ is therefore convergence stable. In contrast, in Fig. $4.5 \mathrm{~b}$ if the resident trait is initially less than $0.5$ those mutants that can invade have trait values less than the resident trait, so that the resident trait value will tend to decrease over time. Similarly, if the resident trait is initially greater than $0.5$ it will tend to increase over time. The trait value $x^*=0.5$ is therefore an evolutionary repeller in this case.

## 经济代写|博弈论代写Game Theory代考|Convergence Stability

Eshel 和 Motro (1981) 是第一个证明 ESS 可能不是收敛稳定的，因此可能无法实现。在他们的模型中，被选择的 特征是个人为了帮助亲属而应该承担的最大风险。从那时起，无法实现的 ESS已在各种环境中得到证明。例如， Nowak (1990) 表明，在重复囚徍困境模型的变体中可能存在无法实现的 ESS。

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