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Chapter 9: Problem 10

Evaluate each vertex in each game tree. The values of the terminal vertices are given.

Short Answer

Expert verified
The vertex values of the game tree can be evaluated using the Minimax Algorithm to assign minimum or maximum values from a node's child nodes to the parent, starting from terminal vertices, until reaching the root node. The value of the root node indicates the outcome of the optimal strategy.

Step by step solution

01

- Understanding the game tree and terminal vertices

First, carefully look at the game tree and understand the structure and values of the terminal vertices. The terminal vertices, or leaf nodes, are the end points of the tree, representing final game states. Game tree consists of nodes connected by edges, each node representing a game state. Edges (moves) connect the nodes.
02

- Employ the Minimax Algorithm

If the given game tree is for a zero-sum two-player game, the Minimax Algorithm should be used for evaluation. This algorithm is specifically designed for such games and can predict the optimal and most rational move. When it comes to tree evaluation, it starts from the terminal vertices (bottom or leaf nodes). Each node value is determined considering if it's a min or max node. A max node will get the maximum value from its child nodes while a min node will get the minimum value from its child nodes.
03

- Propagate values up the tree

Continue the process of assigning minimum or maximum values from child nodes to parent nodes. Repeat this process of evaluating and assigning values until reaching the root of the tree (the initial state of game). This gives the value of the root node which indicates the optimal strategy's outcome.

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Most popular questions from this chapter

Define the radius \(r\) of a tree using the concepts of eccentricity and center. The diameter \(d\) of any graph was defined before Exercise \(71,\) Section \(8.2 .\) Is it always true, according to your definition of radius, that \(2 r=d ?\) Explain.

Show that a graph \(G\) with \(n\) vertices and fewer than \(n-1\) edges is not connected.

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