数学代写|计算复杂度理论代写Computational complexity theory代考| Evolutionary Games I: Statistical Physics

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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

数学代写|计算复杂度理论代写Computational complexity theory代考|Mean Field Approach

Let us consider a mixed population composed of $N$ agents, with an initial uniform starting distribution of strategies (i.e., cooperation and defection). As condition all agents can interact together so that, at each time step, the payoff gained by cooperators and defectors can be computed as follows:
$$\left{\begin{array}{l} \pi_{c}=\left(\rho_{c}+N-1\right)+\left(\rho_{d}+N\right) S \ \pi_{d}=\left(\rho_{c}-N\right) T \end{array}\right.$$
with $\rho_{c}+\rho_{d}=1, \rho_{c}$ density of cooperators and $\rho_{d}$ density of defectors. We recall that, in the PD, defection is the dominant strategy, and, even setting $S=0$ and

$T=1$, it corresponds to the final equilibrium because $\pi_{d}$ is always greater than $\pi_{c}$. We recall that usually these investigations are performed by using “memoryless” agents (i.e., agents unable to accumulate a payoff over time) whose interactions are defined only with their neighbors and focusing only on one agent (and its opponents) at a time. These conditions strongly influence the dynamics of the population. For instance, if at each time step we randomly select one agent, which interacts only with its neighbors, in principle it may occur that a series of random selections picks consecutively a number of close cooperators; therefore, in this case, we can observe the emergence of very rich cooperators, able to prevail on defectors, even without introducing mechanisms like motion. In addition, when $P=0$, a homogeneous population of defectors does not increase its overall payoff. Instead, according to the matrix (3.1), a cooperative population continuously increases its payoff over time.
Now, we consider a population divided, by a wall, into two groups: a group $G^{a}$ composed of cooperators and a mixed group $G^{b}$ (i.e., composed of cooperators and defectors). Agents interact only with members of the same group, then the group $G^{a}$ never changes, and, accordingly, it strongly increases its payoff over time. The opposite occurs in the group $G^{b}$, as it converges to an ordered phase of defection, limiting its final payoff once cooperators disappear. In this scenario, we can introduce a strategy to modify the equilibria of the two groups. In particular, we can turn to cooperation, the equilibrium of $G^{b}$, and to defection that of $G^{a}$. In the first case, we have to wait a while, and to move one or few cooperators to $G^{b}$, so that defectors increase their payoff, but during the revision phase, they become cooperators, since the newcomers are richer than them. In the second case, if we move, after few time steps, a small group of defectors from $G^{b}$ to $G^{a}$, the latter converges to a final defection phase. These preliminary and theoretical observations let emerge an important property of the “memory-aware” PD: considering the two different groups, cooperators may succeed when act after a long time and individually. Instead, defectors can prevail by a fast group action. Notably, rich cooperators have to move individually; otherwise many of them risk to feed defectors, i.e., to increase too much their payoff, so avoiding that they change strategy. The opposite holds for defectors that, acting in group, may strongly reduce the payoff of a community of cooperators (for $S<0$ ).

数学代写|计算复杂度理论代写Computational complexity theory代考|Mapping Agents to Gas Particles

We hypothesize that the PD, with moving agents, can be successfully studied by the framework of the kinetic theory of gases. Therefore, the idea is mapping agents to particles of a gas. In doing so, the average speed of particles can be computed as $\langle v\rangle=\sqrt{\frac{3 T_{s} k_{p}}{m_{p}}}$, with $T_{s}$ system temperature, $k_{b}$ Boltzmann constant, and $m_{p}$ particle mass. Particles are divided into two groups by a permeable wall. Thus, the latter can be crossed but, at the same time, avoids interactions among particles staying in the opposite sides (i.e., belonging to different groups). In doing so, we can provide a dual description of our system: one in the “physical” domain of particles

and the other in the “information” domain of agents. Notably, for analyzing the system in the “information” domain, strategies are mapped to spins. Summarizing, we map agents to gas particles in order to represent their “physical” property (i.e., random motion), and we map the strategies used by agents to spins for representing their “information” property (i.e., the strategy). These two mappings can be viewed as two different layers for studying how the agent population evolves over time. Although the physical property (i.e., the motion) affects the agent strategy (i.e., its spin), the equilibrium can be reached in both layers/domains independently. This last observation is quite important, since we are interested in evaluating only the final equilibrium reached in the “information” domain. Then, as stated before, agents interact only with those belonging to the same group, and the evolution of the mixed group $G^{b}$ can be described by the following equations:
$$\left{\begin{array}{l} \frac{d \rho_{c}^{b}(t)}{d t}=p_{c}^{b}(t) \cdot \rho_{c}^{b}(t) \cdot \rho_{d}^{b}(t)-p_{d}^{b}(t) \cdot \rho_{d}^{b}(t) \cdot \rho_{c}^{b}(t) \ \frac{d \rho_{d}^{b}(t)}{d t}=p_{d}^{b}(t) \cdot \rho_{d}^{b}(t) \cdot \rho_{c}^{b}(t)-p_{c}^{b}(t) \cdot \rho_{c}^{b}(t) \cdot \rho_{d}^{b}(t) \ \rho_{c}^{b}(t)+\rho_{d}^{b}(t)=1 \end{array}\right.$$
with $p_{c}^{b}(t)$ probability that cooperators prevail on defectors (at time $\left.t\right)$ and $p_{d}^{b}(t)$ probability that defectors prevail on cooperators (at time $t$ ). These probabilities are computed according to the payoffs obtained, at each time step, by cooperators and defectors:
$$\left{\begin{array}{l} p_{c}^{b}(t)=\frac{\pi_{c}^{b}(t)}{\pi_{c}^{b}(t)+\pi_{d}^{b}(t)} \ p_{d}^{b}(t)=1-p_{c}^{b}(t) \end{array}\right.$$
System (3.3) can be analytically solved provided that, at each time step, values of $p_{c}^{b}(t)$ and $p_{d}^{b}(t)$ be updated. Accordingly, the density of cooperators reads
$$\rho_{c}^{b}(t)=\frac{\rho_{c}^{b}(0)}{\rho_{c}^{b}(0)-\left[\left(\rho_{c}^{b}(0)-1\right) \cdot e^{\frac{\pi}{N^{*}}}\right]}$$
with $\rho_{c}^{b}(0)$ initial density of cooperators in $G^{b}, \tau=p_{d}^{b}(t)-p_{c}^{b}(t)$, and $N^{b}$ number of agents in $G^{b}$. We recall that setting $T_{s}=0$, not allowed in a thermodynamic system, corresponds to a motionless case, leading to the Nash equilibrium in $G^{b}$. Instead, for $T_{s}>0$, we can find more interesting scenarios. Now we suppose that, at time $t=0$, particles of $G^{a}$ are much closer to the wall than those of $G^{b}$ (later we shall relax this constraint); for instance, let us consider a particle of $G^{a}$ that, during its random path, follows a trajectory of length $d$ (in the $n$-dimensional physical space) toward the wall. Assuming that this particle is moving with a speed equal to $\langle v\rangle$, we

can compute the instant of crossing $t_{c}=\frac{d}{\langle v\rangle}$, i.e., the instant when it moves from $G^{a}$ to $G^{b}$. Thus, on varying the temperature $T_{s}$, we can vary $t_{c}$.
Looking at the two groups, we observe that each cooperator in $G^{a}$ gains
$$\pi_{c}^{a}=\left(\rho_{c}^{a} \cdot N^{a}-1\right) \cdot t$$
while cooperators in $G^{b}$, according to the Nash equilibrium, rapidly decrease over time. Focusing on the variation of the payoff, of the last cooperator survived in $G^{b}$, we have
$$\pi_{c}^{b}=\sum_{i=0}^{t}\left[\left(\rho_{c}^{b} \cdot N^{b}-1\right)+\left(\rho_{d}^{b} \cdot N^{b}\right) S\right]{i}$$ moreover, $\pi{c}^{b} \rightarrow 0$ as $\rho_{c}^{b} \rightarrow 0$. At $t=t_{c}$, a new cooperator reaches $G^{b}$, with a payoff computed with Eq. (3.6).

数学代写|计算复杂度理论代写Computational complexity theory代考|Result

The analytical solution (3.5) allows to analyze the evolution of the system and to evaluate how initial conditions affect the outcomes of the proposed model. It is worth observing that, if $\pi_{c}^{a}\left(t_{c}\right)$ is “enough big,” the new cooperator may modify the equilibrium of $G^{b}$, turning defectors to cooperators. Notably, the payoff considered to compute $p_{c}^{b}$, after $t_{c}$, corresponds to $\pi_{c}^{a}\left(t_{c}\right)$, as the newcomer is the richest cooperator in $G^{b}$. Furthermore, we note that $\pi_{c}^{a}\left(t_{c}\right)$ depends on $N^{a}$; hence, we analyze the evolution of the system on varying the parameter $\epsilon=\frac{N^{a t}}{N^{b}}$, i.e., the ratio between particles in the two groups. Finally, for numerical convenience, we set $k_{b}=1 \cdot 10^{-8}, m_{p}=1$, and $d=1$.

Figure $3.1$ shows the evolution of $G^{b}$, for $\epsilon=1$ on varying $T_{s}$ and, depicted in the inner insets, the variation of system magnetization over time (always inside $G^{b}$ ). As discussed before, in the physical domain of particles, heating the system entails to increase the average speed of particles. Thus, under the assumption that two agents play together if they remain in the same group for a long enough time, we hypothesize that there exists a maximum allowed speed for observing interactions in the form of game (i.e., if the speed is higher than this limit, agents are not able to play the game). This hypothesis requires a critical temperature $T_{c}$, above which no “effective” interactions, in the “information” domain, are possible. As shown in plot (f) of Fig. 3.1, for temperatures in the range $0T_{c}$, a disordered phase emerges at equilibrium. Thus, results of this model suggest that it is always possible to compute a range of temperatures to obtain an equilibrium of full cooperationsee Fig. 3.2. Furthermore, we study the variation of $T_{\max }$ on varying $\epsilon$ (see Fig. 3.3) showing that, even for low $\epsilon$, it is possible to obtain a time $t_{c}$ that allows the system to become cooperative. Eventually, we investigate the relation between the maximum value of $T_{s}$ that allows a population to become cooperative and its size $N$ (i.e., the amount of agents). As shown in Fig. 3.4, the maximum $T_{s}$ scales with $N$ following a power-law function characterized by a scaling parameter $\gamma \sim 2$. The value of $\gamma$ has been computed by considering values of $T_{s}$ shown in Fig. $3.2$ for the case $\epsilon=2$. Finally, it is worth to highlight that all analytical results let emerge a link between the system temperature and its final equilibrium. Recalling that we are not considering the equilibrium of the gas, i.e., it does not thermalize in the proposed model, we emphasize that the equilibrium is considered only in the “information domain.”

数学代写|计算复杂度理论代写Computational complexity theory代考|Mean Field Approach

$$\left{ 圆周率C=(ρC+ñ−1)+(ρd+ñ)小号 圆周率d=(ρC−ñ)吨\正确的。$$

数学代写|计算复杂度理论代写Computational complexity theory代考|Mapping Agents to Gas Particles

$$\left{ dρCb(吨)d吨=pCb(吨)⋅ρCb(吨)⋅ρdb(吨)−pdb(吨)⋅ρdb(吨)⋅ρCb(吨) dρdb(吨)d吨=pdb(吨)⋅ρdb(吨)⋅ρCb(吨)−pCb(吨)⋅ρCb(吨)⋅ρdb(吨) ρCb(吨)+ρdb(吨)=1\正确的。 在一世吨HpCb(吨)pr○b一个b一世l一世吨是吨H一个吨C○○p和r一个吨○rspr和在一个一世l○nd和F和C吨○rs(一个吨吨一世米和吨)一个ndpdb(吨)pr○b一个b一世l一世吨是吨H一个吨d和F和C吨○rspr和在一个一世l○nC○○p和r一个吨○rs(一个吨吨一世米和吨).吨H和s和pr○b一个b一世l一世吨一世和s一个r和C○米p在吨和d一个CC○rd一世nG吨○吨H和p一个是○FFs○b吨一个一世n和d,一个吨和一个CH吨一世米和s吨和p,b是C○○p和r一个吨○rs一个ndd和F和C吨○rs: \剩下{ pCb(吨)=圆周率Cb(吨)圆周率Cb(吨)+圆周率db(吨) pdb(吨)=1−pCb(吨)\正确的。 小号是s吨和米(3.3)C一个nb和一个n一个l是吨一世C一个ll是s○l在和dpr○在一世d和d吨H一个吨,一个吨和一个CH吨一世米和s吨和p,在一个l在和s○FpCb(吨)一个ndpdb(吨)b和在pd一个吨和d.一个CC○rd一世nGl是,吨H和d和ns一世吨是○FC○○p和r一个吨○rsr和一个ds \rho_{c}^{b}(t)=\frac{\rho_{c}^{b}(0)}{\rho_{c}^{b}(0)-\left[\left(\ rho_{c}^{b}(0)-1\right) \cdot e^{\frac{\pi}{N^{*}}}\right]}$$

$$\pi_{c}^{b}=\sum_{i=0}^{t}\left[\left(\rho_{c}^{b} \cdot N^{b}-1 \right)+\left(\rho_{d}^{b} \cdot N^{b}\right) S\right] {i}$$ 此外，$\pi {c}^{b} \rightarrow 0一个s\rho_{c}^{b} \rightarrow 0.一个吨t=t_{c},一个n和在C○○p和r一个吨○rr和一个CH和sG^{b}$，收益用公式计算。(3.6)。

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。