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

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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|General Background

Evolutionary Game Theory (hereinafter EGT) represents the attempt to study the dynamics of a population, combining the principles of Game Theory with those of the Darwinian theory of evolution. At the first glance, these two fields have very few things in common. In very few words, Game Theory can be referred as an approach/framework aimed at finding the optimal strategy in a “competition,” as well as the way for finding a kind of equilibrium among the competitors. Instead, the Darwinian Theory deals with the evolution of life, considering both its competitive aspect and the idea of “transmission and optimization of information.” However, as later discussed, a closer look to these theories shows different points of connection. It is worth to remind that the early developments of the Game Theory, resulting from the works of John von Neumann and Oskar Morgenstern, were mainly focused on the modeling of the human behavior, with a clear reference to economic contexts. Later on, this area of Mathematics was strongly influenced by the Nobel laureate John Nash who introduced, among his relevant contributions to science, the so-called Nash equilibrium. The latter can be viewed as a particular state of a game, involving a number of “rational” agents that have to take an action without prior communications. On the other hand, the Darwinian theory of evolution, that constitutes one of the most important breakthroughs in Biology and more in general in science, after more than a century still constitutes a living theory, although yet debated in particular outside academia. His father, Charles Darwin, proposed it after a challenging journey around the world, which gave the opportunity to perform direct observations on a wild and unexplored Nature. So, after this experience, the young scientist envisioned a general theory for explaining natural evolution of life. As result, this fascinating combination of Game Theory and Darwinian Theory constitutes a powerful framework for modeling several scenarios, spanning from social systems to biological phenomena, and representing specific mechanisms (e.g., reproduction, imitation), interaction patterns, and behaviors. Therefore, a number of “complex phenomena” can be characterized by EGT models. Then, in full agreement with the conclusions reported in the famous Anderson’s work “More is different,” this modern and interdisciplinary area can be actually classified as one of the components of the field of complexity. For instance, emergent behaviors that can be referred as a kind of phase transition can be observed studying an agent population whose local interactions are simple, e.g., based on a game like the Prisoner’s Dilemma. Notably, from a state of disorder one can observe an evolution toward an ordered equilibrium, and the latter can be even opposite to the expected one (i.e., the Nash equilibrium). The last observation deserves particular attention, since one may experience some confusion. Notably, while the mathematical framework of Game Theory has very solid roots and deep basis, direct observations on the real world suggest that the Nash equilibrium is not always respected. For example, cooperation is a common phenomenon and takes different forms also in Nature, like mutualism. Here, one of the goals of EGT is to find out the motivations, and the mechanisms, that can lead to the phenomena we observe and that cannot be predicted using only the Game Theory. At the same time, also the Darwinian Theory takes profit by its combination with Game Theory. For instance, the altruistic behaviors detected among animals cannot be explained referring only to the Darwin’s Theory. So, in this case as well as in others, the contribution of the Game Theory allows to cover this important lack. A fundamental key point of EGT is given by the adaptive behavior introduced in the dynamics of a population. Notably, this behavior is driven by a “rational mindset,” obtained by implementing agents that take actions for optimizing their own gain. The latter, usually defined as “payoff”, in EGT is equivalent to the concept of “fitness.” This correspondence (i.e., payoff-fitness) is quite important since it represents one of the most relevant connections between the Game Theory and the Darwinian Theory, where the best individual is not the strongest but the fittest one or, in general, that who better adapts herself/himself to new environments. Thus, the equivalence payoff-fitness has a deep meaning that allows to use the “rationality” of Game Theory and the “evolutionary mechanisms” of the Darwinian Theory. In particular, as discussed later with more detail, “rationality” and “evolutionary mechanisms” are implemented in the process defined “strategy revision phase.” The latter allows agents to revise their strategy, e.g., imitating their neighbors, and to support the replication of successful strategies. Accordingly, this process is fundamental for the actual evolution of a population and allows to introduce different mechanisms that can influence the way the system moves toward an equilibrium. Finally, even if EGT models can be based on different kinds of games, in this book we refer to the twostrategy games, i.e., games characterized by only two strategies. Even if this choice can be perceived as too limiting for modeling real scenarios, in our view it offers two main advantages. First, it allows to introduce in a very easy way the field of EGT; second, it facilitates the connection with models well-known in Statistical Physics. We can now go on with the next sections of this chapter, reporting some important concepts useful for proceeding through the presentation of more interesting models (i.e., Chap. 3).

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Nash Equilibrium

Developed by John Forbes Nash Jr., the Nash equilibrium constitutes a milestone in Game Theory. For the sake of simplicity, let us consider two agents playing a generic game that are not allowed to communicate before to take an action (i.e., a move). Here, the Nash equilibrium represents the particular situation where neither of them has something to gain from being the only one to change strategy. The lack of opportunities in doing prior agreements, i.e., preliminary communications, has deep consequences. First of all, if both agents want to increase their gain, some form of “blind coordination” is required, which in turn entails that reaching the maximum gain might need to take a very risky strategy. In the next section, introducing real games like the Prisoner’s Dilemma, the “risky” side of some strategies will become more clear. Beyond its general description, the Nash equilibrium has obviously a formal mathematical definition. Let us consider a game described by a set $S$ of strategy profiles and a payoff function $f$, i.e., $(S, f)$. This game might involve $n$ players, so that each one adopts a strategy. Here, a strategy profile $S$ is defined as $S=S_{1} \times S_{2} \times \ldots \times S_{n}$, and the payoff function computes that gain for a specific $x \in S$ as $f(x)=\left(f_{1}(x), f_{2}(x), \ldots, f_{n}(x)\right)$. Then, a set of strategies $x^{} \in S$ is a Nash equilibrium if $\forall_{i}, x_{i} \in S_{i}: f_{i}\left(x_{i}^{}, x_{-i}^{}\right) \geq f_{i}\left(x_{i}, x_{-i}^{}\right)$. Summarizing, reaching the Nash equilibrium can be viewed as a tradeoff, which ensures the potentiality of a good payoff while reducing possible risks. For this reason, if we consider a population whose agents interact according to a dilemma game, the expected equilibrium should correspond to the Nash equilibrium, i.e., all agents take the most convenient strategy, following a selfish behavior. However, in doing so, our population, as a whole, risks to be unable to produce a common wellness. The latter, in this context, has a very general meaning, as for instance might refer to people paying taxes for receiving public services or to people who vaccinate themselves for avoiding viral spreading, and so on. Now, beyond the need to identify strategies for the common wellness, a challenging point in EGT is the understanding of the mechanisms that lead toward non-expected equilibria, in particular because, often, the real world shows the emergence of equilibria different from the Nash equilibrium.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|The Prisoner’s Dilemma

The Prisoner’s Dilemma (PD hereinafter) is one of the most famous games who shows why rational approaches might lead people to avoid cooperation. Its general dynamics can be told using different stories; however, here, we report the classical one who gave the name to this game, i.e., the story of two criminals captured by police, and undergone to a questioning. Notably, after being captured, our criminals are separated to avoid potential agreements. In addition, in order to ensure at least one guilty, they are offered a bargain: each criminal has two options, i.e., betray (i.e., to defect) or remaining silent (i.e., to cooperate). Thus, despite what that the common sense might suggest, in this case being a cooperator indicates to collaborate with the “partner,” not with the police. So, if the two criminals cooperate, they are sentenced to 1 year in prison, while if both betray the conviction is 2 years in prison. Instead, in the third case, i.e., one betrays and the other remains silent, the cooperator is sentenced to 5 years in prison, whereas the defector is set free. Recalling that they cannot communicate before to take an action (i.e., remaining silent or betray), it appears quite clear the motivation that leads our criminals to defect. The Nash equilibrium of this game is “defection,” entailing criminals will spend 2 years in prison. At the same time, if both cooperate they can save 1 year, so that the highest “common” benefit can be achieved only by a coordinate action of cooperation, although very risky. Anyway, beyond this story or similar ones, the PD can be characterized by the following payoff matrix:
$$\begin{array}{cc} C & D \ C & \left(\begin{array}{cc} 1 & S \ T & 0 \end{array}\right) \end{array}$$
with $T$ representing the Temptation, i.e., the payoff an agent gains defecting when the other cooperates, and $S$ representing the Sucker’s payoff, i.e., the gain achieved by a cooperator when its opponent defects. In principle, the general structure of the payoff matrix (3.1) can be adapted also for describing other games (e.g., the HawkDove game). Notably, on varying the range of $T$ and $S$, one can refer to different scenarios. In our case, i.e., in the PD, the values of $T$ and $S$ are $1 \leq T \leq 2$ and $-1 \leq S \leq 0$. Accordingly, we can draw the $T-S$ plane and analyze the outcomes of the game from point to point, among the possible combinations $(T, S)$. For instance, low values of $T$ entail defectors receive a small payoff when they meet cooperators, while high value of $S$ entail cooperators have small losses when meet defectors.

CD C(1小号 吨0)

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