### 经济代写|博弈论代写Game Theory代考|ECON3050

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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|Games in Strategic Form

In this chapter we start with the systematic development of non-cooperative game theory. Its most basic model is the game in strategic form, the topic of this chapter. The available actions of each player, called strategies, are assumed as given. The players choose their strategies simultaneously and independently, and receive individual payoffs that represent their preferences for strategy profiles (combinations of strategies).

For two players, a game in strategic form is a table where one player chooses a row and the other player a column, with two payoffs in each cell of the table. We present a number of standard games such as the Prisoner’s Dilemma, the Quality game, Chicken, the Battle of the Sexes, and Matching Pennies.

The central concept of equilibrium is a profile of strategies that are mutual best responses. A game may have one, several, or no equilibria. As shown in Chapter 6 , allowing for mixed (randomized) strategies will ensure that every finite game has an equilibrium, as shown by Nash (1951). An equilibrium without randomization as considered in the present chapter is also known as a pure Nash equilibrium.
A strategy dominates another strategy if the player strictly prefers it for any fixed strategies of the other players. Dominated strategies are never played in equilibrium and can therefore be eliminated from the game. If iterated elimination of dominated strategies results in a unique strategy profile, the game is called dominance solvable. We illustrate this with the “Cournot duopoly” of quantity competition. (In Section $4.7$ in Chapter 4, we will change this game to a commitment game, known as Stackelberg leadership.)

A strategy weakly dominates another strategy if the player weakly prefers it for any fixed strategies of the other players, and in at least one case strictly prefers it. Eliminating a weakly dominated strategy does not introduce new equilibria, but may lose equilibria, which reduces the understanding of the game. Unless one is interested in finding just one equilibrium of the game, one should therefore not eliminate weakly dominated strategies, nor iterate that process.The final Section $3.8$ shows that symmetric $N$-player games with two strategies per player always have an equilibrium.

## 经济代写|博弈论代写Game Theory代考|Prerequisites and Learning Outcomes

This is a first core chapter of the book. The previous Chapter 2 has introduced the concepts of strategies and equilibrium for the special congestion games and is therefore useful but not a prerequisite. We deal mostly with finite sets. You should be thoroughly familiar with Cartesian products $S_{1} \times \cdots \times S_{N}$ of sets $S_{1}, \ldots, S_{N}$. The Cournot game on intervals is analyzed with basic calculus.
After studying this chapter, you should

• know the components of a game in strategic form: strategies, strategy profiles, and payoffs;
• be able to write down and correctly interpret tables for two-player games;
• understand the concept of an equilibrium, use this term correctly (and its plural “equilibria”), and know how it relates to strategies, partial strategy profiles, and best responses;
• be familiar with common $2 \times 2$ games such as the Prisoner’s Dilemma or the Stag Hunt, and understand how they differ;
• know the difference between dominance and weak dominance and why dominated strategies can be eliminated in a complete equilibrium analysis but weakly dominated strategies cannot;
• know the Cournot quantity game, also when strategy sets are real intervals;
• understand symmetry in games with two players and more than two players.

## 经济代写|博弈论代写Game Theory代考|Games in Strategic Form

A game in strategic form is the fundamental model of non-cooperative game theory. The game has $N$ players, $N \geq 1$, and each player $i=1, \ldots, N$ has a nonempty set $S_{i}$ of strategies. If each player $i$ chooses a strategy $s_{i}$ from $S_{i}$, the resulting $N$-tuple $s=\left(s_{1}, \ldots, s_{n}\right)$ is called a strategy profile. The game is specified by assigning to each strategy profile $s$ a real-valued payoff $u_{i}(s)$ to each player $i$.

The payoffs represent each player’s preference. For two strategy profiles $s$ and $\hat{s}$, player $i$ strictly prefers $s$ to $\hat{s}$ if $u_{i}(s)>u_{i}(\hat{s})$, and is indifferent between $s$ and $\hat{s}$ if $u_{i}(s)=u_{i}(\hat{s})$. If $u_{i}(s) \geq u_{i}(\hat{s})$ then player $i$ weakly prefers $s$ to $\hat{s}$. Player $i$ is only interested in maximizing his own payoff, and not interested in the payoffs to other players (other than in trying to anticipate their actions); any “social” concern a player has about a particular outcome has to be (and could be) built into his own payoff. All players know the available strategies and payoffs of the other players, and know that they know them, etc. (that is, the game is “common knowledge”).
The game is played as follows: The players choose their strategies simultaneously (without knowing what the other players choose), and receive their respective payoffs for the resulting strategy profile. They cannot enter into any binding agreements about what they should play (which is why the game is called “noncooperative”). Furthermore, the game is assumed to be played only once, and therefore also called a one-shot game. Playing the same game (or a varying game) many times leads to the much more advanced theory of repeated games.

Much of this book is concerned with two-player games. The players are typically named 1 and 2 or I and II. Then the strategic form is conveniently represented by a table. The rows of the table represent the strategies of player I (also called the row player), and the columns represent the strategies of player II (the column player). A strategy profile is a strategy pair, that is, a row and a column, with a corresponding cell of the table that contains two payoffs, one for player I and the other for player II.

If $m$ and $n$ are positive integers, then an $m \times n$ game is a two-player game in strategic form with $m$ strategies for player I (the rows of the table) and $n$ strategies for player II (the columns of the table).

Many interesting game-theoretic observations apply already to $2 \times 2$ games. Figure 3.1(a) shows the famous Prisoner’s Dilemma game. Each player has two strategies, called $C$ and $D$, which stand for “cooperate” and “defect”. The payoffs are as follows: If both players choose $C$, then both get payoff 2 . If both players choose $D$, then both get payoff 1 . If the two players choose different strategies, then the player who chooses $C$ gets payoff 0 and the player who chooses $D$ gets the highest possible payoff 3 .

## 经济代写|博弈论代写Game Theory代考|Prerequisites and Learning Outcomes

• 以战略形式了解游戏的组成部分：战略、战略概况和收益；
• 能够写下并正确解释两人游戏的表格；
• 理解均衡的概念，正确使用该术语（及其复数“均衡”），并了解它与策略、部分策略配置文件和最佳响应的关系；
• 熟悉常见2×2诸如囚徒困境或猎鹿之类的游戏，并了解它们的不同之处；
• 知道支配和弱支配之间的区别，以及为什么支配策略可以在完全均衡分析中消除而弱支配策略不能；
• 知道古诺数量博弈，当策略集是真实区间时也是如此；
• 了解有两个玩家和两个以上玩家的游戏中的对称性。

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