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|ECON3050 Definition of Game Trees

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## 经济代写|博弈论代考Game theory代写|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 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|博弈论代考Game theory代写|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.

# 博弈论代写

## 经济代写|博弈论代考博弈论代写|逆向归纳法

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## MATLAB代写

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