# 经济代写|博弈论代写Game Theory代考|ECON40010

## 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 $\mathrm{I}$ 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 noule $u$, the next nekle $n$ 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|Backward Induction

Which moves should the players choose in a game tree? “Optimal” play should maximize a player’s payoff. This can be decided irrespective of other players’ actions when the player is the last player to move. In the game in Figure 4.3(a), player II maximizes her payoff by move $r$. Going backward in time, player I makes his move $T$ or $B$ at the root of the game tree, where he will receive either 1 or 2 , assuming the described future behavior of player II. Consequently, he will choose $B$, and play ends with payoff 2 to each player. These chosen moves are shown in Figure 4.3(b) with arrows on the edges; similar to the boxes that we put around best-response payoffs in a strategic-form game, this is additional information to help the analysis of the game and not part of the game specification.

This process is called backward induction: Starting with the decision nodes closest to the leaves, a player’s move is chosen that maximizes that player’s payoff at the node. In general, a move is chosen in this way for each decision node provided all subsequent moves have already been decided. Eventually, this will determine a move for every decision node, and hence for the entire game.The move selected by backward induction is not necessarily unique, if there is more than one move that gives maximal payoff to the player. This may influence the choice of moves earlier in the tree. In the game in Figure 4.4, backward induction chooses either move $b$ or move $c$ for player II, both of which give her payoff 5 (which is an expected payoff for move $b$ in the original game in Figure 4.1) that exceeds her payoff 4 for move $a$. At the rightmost node, player I chooses $Q$. This determines the preceding move $d$ by player II which gives her the higher payoff 3 as opposed to 2 (via move Q). In turn, this means that player I, when choosing between $X, Y$, or $Z$ at the root of the game tree, will get payoff 2 for $Y$ and payoff 1 for $\mathrm{Z}$. The payoff when he chooses $X$ depends on the choice of player II: if that is $b$, then player I gets 2, and he can choose either $X$ or $Y$, both of which give him maximal payoff 2. These choices are shown with white arrows in Figure 4.4. If player II chooses $c$, however, then the payoff to player $\mathrm{I}$ is 4 when choosing $X$, so this is the unique optimal move.

To summarize, the possible combinations of moves that can arise in Figure $4.1$ by backward induction are, by listing the moves for each player: $(X Q, b d),(X Q, c d)$, and $(Y Q, b d)$. Note that $Q$ and $d$ are always chosen, but that $Y$ can only be chosen in combination with the move $b$ by player II.

We do not try to argue that player II “should not” choose $b$ because of the “risk” that player I might choose $Y$. This would be a “refinement” of backward induction that we do not pursue.

The moves determined by backward induction are therefore, in general, not unique, and possibly interdependent.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Backward Induction

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: