### 经济代写|博弈论代写Game Theory代考|ECON2070

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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|Bayesian Games

Bayesian games are games where the players have incomplete information about the game settings. In this section, we introduce Harsanyi’s approach to incomplete information (Harsanyi 1967, 1968), which uses random variables, known as “types,” to capture uncertainties in the game.

The setting of incomplete information refers to where at least one player does not know the payoff of at least one of the other players. It differs from the complete information setting where the set of players $\mathcal{N}$, the action space for each player $\mathcal{A}{i}$, and the payoff function for each player $u{i}$ are completely known to all the players. The set of players $\mathcal{N}$ and the action spaces $\mathcal{A}_{i}, i \in \mathcal{N}$ are considered as common knowledge in the games of incomplete information.

Harsanyi has introduced an approach to capture a game with incomplete information using the notion of types of players. The type of player can be considered as an aggregation of her private information. Thus, the player’s type is only known to herself. The type is also called an epistemic type since it can involve the player’s beliefs of different levels about the other players’ payoffs. By considering the assignments of types as prior moves of the fictitious player, Nature, we can transform a game with incomplete information into a game with imperfect information about Nature’s moves, and use standard techniques to analyze equilibrium behaviors.

To define Bayesian games formally, we let $\Theta=\left(\Theta_{1}, \ldots, \Theta_{N}\right)$ denote the type spaces of players in $\mathcal{N}$. Associate the type space with a prior distribution $p: \Theta \rightarrow[0,1]$, which is common knowledge. The payoffs of the players are defined by functions of their types and the actions of all the players as $u_{i}: \mathcal{A} \times \Theta \rightarrow \mathbb{R}$. Then, a Bayesian game is defined as the tuple $(\mathcal{N}, \mathcal{A}, \Theta, p, u)$, where $u=$ $\left(u_{1}, \ldots, u_{N}\right)$ and $\mathcal{A}=\left(\mathcal{A}{1}, \ldots, \mathcal{A}{N}\right)$. A pure strategy of player $i$ is defined as a mapping $s_{i}: \Theta_{i} \rightarrow \mathcal{A}_{i}$. Mixed strategies extend pure strategies to probability distributions on the action set of the player. Note that the strategies and payoffs are mappings of the types.

A Bayesian game can be naturally transformed into an NFG by enumerating all the pure strategies, each of which specifies what action the player should take for each of the possible types. For example, in a two-player game, each row or column of the payoff matrix represents a vector of $\left|\Theta_{i}\right|$ actions specifying a pure strategy of the player.

## 经济代写|博弈论代写Game Theory代考|Applications in Cybersecurity

Bayesian games are useful to model many cybersecurity applications since they often involve unknown or unobservable information, as we have seen in man-in-the-middle attacks in remotely controlled robotic systems (Xu and Zhu 2015), spoofing attacks (Zhang and Zhu 2017), compliance control (Casey et al. 2015), deception over social networks (Mohammadi et al. 2016), and denial of service (Pawlick and Zhu 2017). The attackers often know more about the defenders than what defenders know about the attackers. The information asymmetry naturally creates an attacker’s advantage. Cyber deception is an essential mechanism (Pawlick et al. 2018, 2019) to reverse the information asymmetry and create additional uncertainties to deter the attackers. In addition to the later chapters in this book, readers can also refer to Pawlick et al. (2019) for a recent survey of game-theoretic methods for defensive deception and Zhang et al. (2020) for a summary of Bayesian game frameworks for cyber deception. The Bayesian games can also be extended to dynamic ones to model multistage deception. Interested readers can see Huang and Zhu (2019) for more details.
Another important application of Bayesian games is mechanism design. According to the fundamental revelation principle (Myerson 1989, 1981), an equivalent revenue-maximizing mechanism can be designed under the scenario where players reveal their private types. Mechanism design has been applied to understanding security as a service (Chen and Zhu 2016, 2017), and pricing in IoT networks (Farooq and Zhu 2018).

## 经济代写|博弈论代写Game Theory代考|Stochastic Games

Stochastic games are a class of games where the players interact for multiple rounds in different games. This section first presents the basic concepts of stochastic games along with their solution concepts. We will discuss the Markov perfect equilibria and the applications of stochastic games in cybersecurity.

The attribute that makes stochastic games is the concept of “states.” At each state, players interact in a state-dependent game whose outcome determines the state and the associated game in the next round. Let $Q$ denote a finite state space. Each state has an associated strategic game. The state transition is captured by the kernel function $P: \mathcal{Q} \times \mathcal{A} \times \mathcal{Q} \rightarrow[0,1]$, with $P\left(q_{1}, a, q_{2}\right)$ being the probability of transitioning to state $q_{2}$ from state $q_{1}$ when the players play action profile ( $a_{1}, a_{2}$ ). Since the game played at each stage is determined by its state, the payoff functions also depend on the state. They are defined as $r=\left(r_{1}, \ldots, r_{N}\right)$ and $r_{i}: \mathcal{Q} \times \mathcal{A} \rightarrow \mathbb{R}$. That is, each player gets a payoff in each state and the payoff is dependent on the current state and the joint actions of the players. A stochastic game is thus defined as the tuple $(Q, \mathcal{N}, \mathcal{A}, P, r)$. We assume the game at each stage is finite. It is clear that repeated games are special cases of stochastic games when the state space is a singleton.
The record of multiround interactions up to round $t$, consisting of actions and states, is called history denoted by $h_{l}=\left(q^{0}, a^{0}, \ldots, a^{l-1}, q^{t}\right)$, where the superscripts denote the rounds or stages. Let $s_{i}\left(h_{t}, a_{i}\right)$ be a behavioral strategy of player $i$, which indicates the probability of playing $a_{i}$ under history $h_{l}$. Let $q_{t}$ and $q_{l}^{\prime}$ denote the final states of histories $h_{t}$ and $h_{t}^{\prime}$, respectively. The behavioral strategy is a Markov strategy if, for each $t, s_{i}\left(h_{t}, a_{i}\right)=s_{i}\left(h_{t}^{\prime}, a_{i}\right)$ whenever $q_{t}=q_{i}^{\prime}$. In other words, the dependency on the history is only through the last state. It is analogous to the Markov property of stochastic processes. A Markov strategy is stationary if $s_{i}\left(h_{l_{1}}, a_{i}\right)=s_{i}\left(h_{l_{2}}^{\prime}, a_{i}\right)$ whenever $q_{l_{1}}=q_{l_{2}}^{\prime}$.
Stochastic games can also be viewed as a generalization of Markov decision processes (MDPs) to multiagent scenarios. Stochastic games are also referred to as Markov games. Here we briefly introduce the MDP formulation and explain how stochastic games generalize it. An MDP is defined by the set of states, the set of actions, the state transition probability and the reward or payoff function. Let $q^{0}$ be the initial state of the MDP and $p\left(q^{t+1} \mid q^{t}, a^{t}\right)$ be probability of transitioning from state $q^{t}$ to state $q^{l+1}$ under action $a^{t}$. The objective of a single-player MDP under discounted payoff aims to find a control policy $u$ that maps from the states to actions to maximize the following infinite-horizon payoffs.
$$\max {u} \sum{t=t_{0}}^{\infty} \beta^{t} \mathbb{E}\left[r^{t}\left(q^{t}, u\left(q^{t}\right)\right)\right]$$
Here, $r^{t}$ is the scalar payoff under state $q^{t}$ and action $u\left(q^{t}\right)$ at stage $t$. Dynamic programming can be used to find the optimal solutions to $(2.19)$. Let $v^{}\left(q^{0}\right)$ denote the optimal cost-to-go, which is the optimal value of the problem $(2.19)$ given an initial state $q^{0}$. The Bellman equations state that the optimal cost-to-go at the stage $t$ is the optimized current reward plus the discounted expected future reward at the next stage. More formally, the Bellman equations are given by $$v^{}\left(q^{l}\right)=\max {a} r^{l}\left(q^{l}, a\right)+\beta \sum{q^{t+1} \in Q} p\left(q^{t+1} \mid q^{l}, a\right) v\left(q^{t+1}\right)$$
The optimal payoff given an initial state can be further formulated as an LP problem by transforming $(2.20)$ into a set of constraints as:
$$\begin{array}{ll} & \min {v} v(q) \ \text { s.t. } & v\left(q^{l}\right) \geq r^{l}\left(q^{t}, a\right)+\beta \sum{q^{t+1} \in O} p\left(q^{l+1} \mid q^{t}, a\right) v\left(q^{l+1}\right), \forall a, \forall q^{t} . \end{array}$$

## 经济代写|博弈论代写Game Theory代考|Bayesian Games

Harsanyi 引入了一种使用玩家类型的概念来捕获信息不完整的游戏的方法。玩家的类型可以被认为是她的私人信息的聚合。因此，玩家的类型只有她自己知道。该类型也称为认知类型，因为它可以涉及玩家对其他玩家收益的不同级别的信念。通过将类型分配视为虚构玩家 Nature 的先行动作，我们可以将信息不完整的博弈转化为 Nature 动作信息不完整的博弈，并使用标准技术分析均衡行为。

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