### 数学代写|计算复杂度理论代写Computational complexity theory代考|The Public Goods Game

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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|The Public Goods Game

The Public Goods Game (PGG hereinafter) considers a set of individuals that have to secretly decide if to contribute to the wellness of their own community by offering a token. Like for the PD, cooperators are those that aim to the “common” wellness, while defectors are those that follow a selfish behavior. In addition, being the choice “secret,” prior communications are avoided also in this game. The token, or coin, provided by cooperators represents a very general form of contribution. For instance, in an economical context, a coin can be a kind of tax; in online platforms can be the sharing of knowledge (e.g., in forums, blogs, etc.). Thus, the contribution actually refers to an effort made by an individual for improving the services of her/his society. Then, the total amount of coins is enhanced by a numerical parameter, named synergy factor, that promotes collaborative efforts, and its final value is equally divided among all individuals, no matter their action. Therefore, defectors, i.e., those whose contribution is null (or smaller than the average value), can be considered as free riders. At the same time, since both defectors and cooperators receive an equal fraction of the total pot (i.e., the enhanced summation of coins), the most rational (and convenient) strategy is defection. In addition, the latter constitutes the Nash equilibrium of the PGG. According to the described dynamics, and in a more formal way, we can defined the payoff received by cooperators (i.e., $\pi^{c}$ ) and by defectors (i.e., $\pi^{d}$ ):
$$\left{\begin{array}{l} \pi^{c}=r \frac{N^{c}}{G}-c \ \pi^{d}=r \frac{N^{c}}{G} \end{array}\right.$$
where $N^{c}$ indicates the number of cooperators among the $G$ agents involved in the game, $r$ indicates the synergy factor, and $c$ represents the agents’ contribution. Without loss of generality, usually $c$ is set to 1 . It is worth to highlight that the value of $G$ strongly depends on agent topology, i.e., the way they interact. For instance, when they are arranged in a square lattice, $G$ is equal to 5 . This last point will be clarified in the next chapters, where some practical cases are presented. Finally, we deem interesting to emphasize that like for the PD, the “common wellness” requires a “blind” coordinate effort, otherwise following the Nash equilibrium we cannot observe the improvement of a society (no matter of what the contribution and the payoff represent).

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Evolutionary Game Theory

After the very brief introduction to some concepts of Game Theory (the reader interested in knowing more is invited to consult the references, as well as to read one the many books on this field), we can start to move toward the modeling of evolutionary games. First of all, we want to pay attention to the understanding of mechanisms that lead to cooperation in dilemma games, with particular emphasis for those dilemmas whose Nash equilibrium is defection. The underlying motivation is born from observations on the real world where, fortunately, we can find clear examples of cooperation. In addition, now we move from the local level before discussed, i.e., the dynamics of a single game, to the global level of an agent population. Notably, here, agent interactions take the form of a game, and, being the system adaptive, we can study the evolution of strategies over time. This approach allows to obtain a thermodynamic view of our population and, at the same time, to study the local mechanisms that lead toward a particular equilibrium (or steady state), i.e., a particular distribution of strategies. As result, being particularly interested in defection-based games (i.e., games whose Nash equilibrium is defection), we pay a special attention for those mechanisms/conditions that allow to reach a state of full cooperation. At this point, one might begin to understand why Statistical Physics can constitute the optimal framework for analyzing the dynamics of EGT models. Notably, as we will see later, agent populations playing evolutionary games show critical behaviors, e.g., order-disorder phase transitions (well known in Statistical Physics). For this reason, Chap. 2 is devoted to summarize some mathematical methods and tools for studying these phenomena, as the Ising model. Giving a quick look to the literature, we can find several works focused on the connections between EGT and Physics, as the early works of Hauert and Szabo, or the more recent works of Perc, Szolnoki, and their colleagues. Actually, even considering the classical Game Theory, we can find physicists interested in defining a link with Physics, as shown in some works of Galam. As before mentioned, an agent population whose interactions are based on simple games like the PD constitutes an adaptive system. Due to its relevance, this point deserves attention. Notably, being adaptive means that some forms of adaptation/evolution can be detected in the system. In our case, the evolution refers to the strategies adopted by the agents and, in most agent based models, the mechanism responsible for this evolution is a process usually defined “strategy revision phase.” The latter allows agents to change their strategy according to a particular rule, where usually “rationality” constitutes the main ingredient. In addition, further approaches can be used for modeling the dynamics of evolutionary games. For instance, without considering physical agents, a famous class of analytical methods is the “replicator dynamics.” The latter, proposed by Taylor and Jonker, uses differential equations. This approach, better discussed in Chap. 2, is based on the following conditions: given a strategy $i$, used with a frequency $x_{i}$ (in a population), the frequency rate reads with fi expected payoff associated to the strategy i and average payoff.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Strategy Revision Phase

Let us now go back to the process before introducing the evolution of an agent population, i.e., the “strategy revision phase.” The latter can be implemented according to different methods, usually related to the analysis of the payoff of the involved agent. In addition, further methods can consider different behaviors, as conformity (see Chap. 4), and pure imitation (see Chap. 3). In general, methods based on the payoff analysis can be divided in the following categories:

• Comparison
• Self-evaluation
• Imitation
The first one, i.e., the payoff comparison, is often implemented as a stochastic rule by a Fermi-like function. The latter allows to compute the probability an agent $y$ takes the strategy of an agent $x$ and reads
$$W\left(s_{y} \leftarrow s_{x}\right)=\left(1+\exp \left[\frac{\pi_{y}-\pi_{x}}{K_{y}}\right]\right)^{-1}$$
where $\pi_{x}$ and $\pi_{y}$ correspond to the payoffs of two agents, and $s_{x}$ and $s_{y}$ indicate their strategy. $K_{y}>0$ is an agent-dependent parameter whose role will be described in the Chap. 3. The Fermi-like function actually is adopted in a wide number of contexts and applications. A fast inspection to its shape-see Fig. 1.1 clarifies why it can be efficiently used for implementing stochastic and rational processes. Notably, its “stochastic” behavior comes from the opportunity to use it as a weighted distribution, where even inconvenient choices can be performed (e.g., imitating a a poorer agent, even if with a very low probability), while its “rationality” is represented by the temperature $K$ (or $K_{y}$ if referred to a specific agent). The second part of Chap. 3 focuses on a complete analysis on the role of the temperature (indicated also as “noise”) in the PGG. Then, the second category in the list, i.e., self-evaluation methods, entails agents decide to change their strategy whether the current payoff is smaller than the previous one. This approach can be viewed as a kind of evaluation on the own performance and entails that agents have some memory (we recall that usually the agent payoff is reset after each iteration). Last, methods based on imitative mechanisms (considering the payoff as reference) usually lead agents to imitate a richer opponent in their neighborhood. However, it is also possible (as shown in the application presented in the Chap. 4) to provide agents with behaviors not related to the payoff (e.g., conformity).

## 数学代写|计算复杂度理论代写Computational complexity theory代考|The Public Goods Game

$$\左{ 圆周率C=rñCG−C 圆周率d=rñCG\正确的。$$

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Strategy Revision Phase

• 比较
• 自我评估
• 模仿
第一个，即收益比较，通常通过类费米函数实现为随机规则。后者允许计算代理的概率是采取代理策略X并阅读
在(s是←sX)=(1+经验⁡[圆周率是−圆周率Xķ是])−1
在哪里圆周率X和圆周率是对应于两个代理的收益，并且sX和s是表明他们的策略。ķ是>0是一个依赖于代理的参数，其作用将在第 1 章中描述。3. 类费米函数实际上在广泛的上下文和应用中被采用。对其形状的快速检查（见图 1.1）阐明了为什么它可以有效地用于实施随机和合理的过程。值得注意的是，它的“随机”行为来自于将其用作加权分布的机会，即使是不方便的选择也可以执行（例如，模仿较差的代理，即使概率非常低），而它的“合理性”被表示为由温度ķ（或者ķ是如果提到特定的代理）。章的第二部分。图 3 重点对温度（也表示为“噪声”）在 PGG 中的作用进行了全面分析。然后，列表中的第二类，即自我评估方法，需要代理人决定改变他们的策略，无论当前的收益是否小于前一个。这种方法可以看作是对自身性能的一种评估，并且需要代理有一些记忆（我们记得通常代理支付在每次迭代后都会重置）。最后，基于模仿机制（将收益作为参考）的方法通常会导致代理人模仿他们附近更富有的对手。然而，也有可能（如第 4 章中介绍的应用程序所示）为代理人提供与收益无关的行为（例如，从众）。

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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。