Posted on Categories:Game theory , 博弈论, 经济代写

# 经济代写|博弈论代考Game theory代写|The Model

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## 经济代写|博弈论代考Game theory代写|The Model

The building block of a repeated game, the game which is repeated, is called the staye game. Assume that the stage game is a finite I-player simultancous-move game with finite action spaces $A_i$ and stage-game payoff functions $g_i: A \rightarrow \mathbb{R}$, where $A=\times_{i \in}, A_i$. Let $\mathscr{A}_i$ be the space of probability distributions over $A_i$.

To define the repeated game, we must specify the playcrs’ strategy spaces and payoff functions. This section considers games in which the players obscrve the realized actions at the end of each period. Thus, let $a^t \equiv$ $\left(a_1^t \ldots, a_1^t\right)$ be the actions that are played in period $t$. Suppose that the game begins in period 0 , with the null history $h^0$. For $t \geq 1$, let $h^t=$ $\left(a^0, a^1, \ldots, a^{t-1}\right)$ be the realized choices of actions at all periods before $t$, and let $H^{\prime}=(A)^t$ be the space of all possible period-t histories.

Since all players observe $h^t$, a pure strategy $s_i$ for player $i$ in the repeated game is a sequence of maps $s_i^t$-one for each period $t$ that map possible period-t histories $h^t \in H^t$ to actions $a_i \in A_i$. (Remember that a strategy must specify play in all contingencies, even those that are not expected to occur.) A mixed (behavior) strategy $\sigma_i$ in the repeated game is a sequence of maps $\sigma_i^{\prime}$ from $H^{\prime}$ to mixed actions $\alpha_i \in \mathscr{X}i$. Note that a player’s strategy cannot depend on the past values of his opponents’ randomizing probabilities $\alpha{-i}$; it can depend only on the past values of $a_{{ }_i}$. Note also that each period of play begins a proper subgame. Moreover, since moves are simultaneous in the stage game, these are the only proper subgames, a fact that we will use in lesting for subgame perfection. ${ }^3$

This section considers infinitely repeated games; section 5.2 considers games with a fixed finite horizon. With a finite horizon, the set of subgameperfect equilibria is determined by backward-induction arguments that do not apply to the infinite-horizon model. The infinite-horizon case is a better description of situations where the players always think the game extends one more period with high probability; the finite-horizon model describes a situation where the terminal date is well and commonly foreseen. ${ }^4$

## 经济代写|博弈论代考Game theory代写|The Folk Theorem for Infinitely Repeated Games

The “folk thcorems” for repcated games assert that if the players are sufficiently patient then any fcasible, individually rational payoffs can be enforced by an equilibrium. Thus, in the limit of extreme patience, repeated play allows virtually any payoff to be an cquilibrium outcome.

To make this assertion precise, we must define “feasible” and “individuatly rational.” Define player is reservation utility or minmax value to be
$$r_i=\min {x_i}\left[\max {x_1} y_i\left(x_i, x_{-i}\right)\right] .$$
This is the lowest payoff player i’s opponents can hold him to by any choice of $x_i$, provided that player $i$ correctly foresees $x_{-i}$ and plays a best response 10it. I.et $m_{-i}^i$ be a strategy for player $i$ s opponents that attains the minimum in cquation 5.1. We call $m_{-i}^i$ the minmax profile against player $i$. Let $m_i^i$ be a strategy for player $i$ such that $g_i\left(m_i^i, m_{-i}^i\right)=v_i$.

To illustrate this definition, we compute the minmax values for the game in figure 5.1. To compute player 1 ‘s minmax value, we first compute his payoffs to $\mathrm{J}, \mathrm{M}$, and $\mathrm{D}$ as a function of the probability $q$ that player 2 assigns to $\mathrm{L}$; in the obvious notation, these payoffs are $v_{\mathrm{U}}(q)=-3 q+1$, $r_M(q)=3 q-2$. and $v_{\mathrm{D}}(q)=0$. Since player 1 can always attain a payoff of 0 by playing $D$, his minmax payoff is at lcast this large; the question is whether player 2 can hold player 1’s maximized payoff to 0 by some choice of $q$. Since $q$ does not enter into $v_D$, we can pick $q$ to minimize the maximum of $r_1$ and $r_M$, which occurs at the point where the two expressions are equal, i.e. $q=\frac{1}{2}$. Since $v_{\mathrm{L}}\left(\frac{1}{2}\right)=v_{\mathrm{M}}\left(\frac{1}{2}\right)=-\frac{1}{2}$, player 1 ‘s minmax value is the zero payoff he can achicve by playing D. (Note that $\max \left(v_{\mathrm{U}}(q), v_{\mathrm{M}}(q)\right) \leq 0$ for any $\left.q \in\left[\begin{array}{l}1 \ 3\end{array}-2\right] 3\right]$, so we can take player 2 ‘s minmax strategy against player $1, m_2$, to be any $q$ in this range.)

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|The Folk Theorem for Infinitely Repeated Games

$$r_i=\min {x_i}\left[\max {x_1} y_i\left(x_i, x_{-i}\right)\right] .$$

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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。