### 经济代写|博弈论代写Game Theory代考|ECON2112

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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|Game Trees with Perfect Information

This chapter considers game trees, the second main way for defining a noncooperative game in addition to the strategic form. In a game tree, players move sequentially and (in the case of perfect information studied in this chapter) are aware of the previous moves of the other players. In contrast, in a strategic-form game players move simultaneously. In this “dynamic” setting, a play means a specific run of the game given by a sequence of actions of the players.

A game tree is described by a tree whose nodes represent possible states of play. Some nodes are chance nodes where the next node is determined randomly according to a known probability distribution. A decision node belongs to a player who makes a move to determine the next node. Play ends at a terminal node or leaf of the game tree where each player receives a payoff.

Game trees can be solved by backward induction, where one finds optimal moves for each player, given that all future moves have already been determined. Backward induction starts with the decision nodes closest to the leaves, and assumes that a player always makes a payoff-maximizing move. Such a move is in general not unique, so that the players’ moves determined by backward induction may be interdependent (see Figure 4.4).

A strategy in a game tree is a derived concept. A strategy defines a move at every decision node of the player and therefore a plan of action for every possible state of play. Any strategy profile therefore defines a payoff to each player (which may be an expected payoff if there are chance moves), which defines the strategic form of the game.

Because every strategy is a combination of moves, the number of strategies in a game tree is very large. Reduced strategies reduce this number to some extent. In a reduced strategy, moves at decision nodes that cannot be reached due to an earlier own of the player are left unspecified.

Backward induction defines a strategy profile which is an equilibrium of the game (Theorem 4.6). This is called a subgame-perfect equilibrium (SPE) because it defines an equilibrium in every subgame. (In a game tree with perfect information,a subgame is just a subtree of the game. In games with imperfect information, treated in Chapter 10, an SPE can in general not be found by backward induction.)
In Section 4.7, we consider commitment games, which are two-player games where a given strategic-form game is changed to a game tree where one player moves first and the other player is informed about the first player’s move and can react to that move. It is assumed that the second player always chooses a best response, as in an SPE. This changes the game, typically to the advantage of the first mover. We study the commitment game for the Quality game and for the Cournot duopoly game of quantity competition from Section 3.6, where the commitment game is known as a Stackelberg leadership game.

Every game tree can be converted to strategic form. A general game in strategic form can only be represented by a tree with extra components that model imperfect information. Game trees with imperfect information will be treated in Chapter $10 .$

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

Chapter 3 is a prerequisite for this chapter. Chapter 1 is not, although combinatorial games are sequential games with perfect information as considered here, and therefore illustrative examples. We will recall the mathematical concepts of directed graphs and trees, and of the expected value of a random variable, as background material.
After studying this chapter, you should be able to

• interpret game trees with their moves and payoffs;
• create the strategies of a player as combinations of moves, and know how to count them;
• combine strategies into reduced strategies, which identify moves at decision nodes that are unreachable due to an earlier own move of the player, and understand where the corresponding “wildcard” symbol “*” is placed in the move list that represents the strategy;
• write down the strategic form of a game tree, with players’ payoffs in the correct place in each cell, and with computed expected payoffs if needed;
• explain backward induction and why it requires full (unreduced) strategies, which may not be unique;
• construct any equilibrium, even if not subgame-perfect, directly from the game tree, also for more than two players, as in Exercise 4.5;
• create the commitment game from a game in strategic form, and compare its subgame-perfect equilibria with the equilibria in the strategic form.

## 经济代写|博弈论代写Game Theory代考|Definition of Game Trees

Figure $4.1$ shows an example of a game tree. We always draw trees downwards, with the starting node, called the root, at the top. (Conventions on drawing game trees vary. Sometimes trees are drawn from the bottom upwards, sometimes from left to right, and sometimes with the root at the center with edges in any direction.)

The nodes of the tree denote states of play (which have been called “positions” in the combinatorial games considered in Chapter 1). Nodes are connected by lines, called edges. An edge from a node $u$ to a child node $v$ (where $v$ is drawn below $u$ ) indicates a possible move in the game. This may be a move of a “personal” player, for example move $X$ of player I in Figure 4.1. Then $u$ is also called a decision node. Alternatively, $u$ is a chance node, like the node $u$ that follows move $b$ of player II in Figure 4.1. We draw decision nodes as small filled circles and chance nodes as squares. After a chance node $u$, the next node $v$ is determined by a random choice made with the probability associated with the edge that leads from $u$ to $v$. In Figure $4.1$, these probabilities are $\frac{1}{3}$ for the left move and $\frac{2}{3}$ for the right move.
At a terminal node or leaf of the game tree, every player gets a payoff. In Figure $4.1$, leaves are not explicitly drawn, but the payoffs given instead, with the top payoff to player I and the bottom payoff to player II.

It does not matter how the tree is drawn, only how the nodes are connected by edges, as summarized in the background material on directed graphs and trees. The following is the formal definition of a game tree.

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

• 用他们的动作和回报来解释游戏树；
• 将玩家的策略创建为移动的组合，并知道如何计算它们；
• 将策略组合成简化策略，识别由于玩家之前自己的移动而无法到达的决策节点处的移动，并了解相应的“通配符”符号“*”在代表策略的移动列表中的位置；
• 写下博弈树的战略形式，在每个单元格的正确位置写下玩家的收益，并在需要时计算预期收益；
• 解释反向归纳以及为什么它需要完整（非简化）策略，这可能不是唯一的；
• 直接从博弈树构建任何均衡，即使不是子博弈完美的，也适用于两个以上的玩家，如练习 4.5 中所示；
• 从战略形式的博弈创建承诺博弈，并将其子博弈完美均衡与战略形式的均衡进行比较。

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