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Chapter 9: Problem 12

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

Short Answer

Expert verified
To evaluate each vertex in a game tree, identify the players and their objectives. Traverse from terminal vertices up, evaluating parent vertices based on whether the player is aiming to maximize or minimize. Continue this process until the value of the root or game start state is reached, which is the value of the game.

Step by step solution

01

Identify the players and their objectives

In game theory, a tree involves players who make decisions in turns. The first player aims to maximize the value of the vertex and the second player aims to minimize. Identify who is aiming to maximize and who to minimize.
02

Traverse from terminal vertices

Begin at the terminal vertices - the leaves or ends of the tree. Work upwards from these given values.
03

Evaluate the parent vertices

The value of the parent vertex is evaluated by considering its child vertices. If the parent vertex is a maximizing player鈥檚 move, take the maximum value among the child vertices. If it鈥檚 a minimizing player鈥檚 move, take the minimum value.
04

Repeat the evaluation

Repeat the process for the next level of parent vertices, working upwards each time until reaching the root of the tree or the starting point of the game.
05

Value of the Game

Once the root is reached, the value of the game is the value of the root.

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Most popular questions from this chapter

Draw a graph having the given properties or explain why no such graph exists. Acyclic; four edges, six vertices

The minimum-queens problem asks for the minimum number of queens that can attack all of the squares of an \(n \times n\) board (i.e., the minimum number of queens such that each row, column, and diagonal contains at least one queen). Write a backtracking algorithm that determines whether \(k\) queens can attack all squares of an \(n \times n\) board.

An ordered tree is a tree in which the order of the children is taken into account. For example, the ordered trees are not isomorphic. Show that the number of nonisomorphic ordered trees with \(n\) edges is equal to \(C_{n},\) the \(n\) th Catalan number. Hint: Consider a preorder traversal of an ordered tree in which 1 means down and 0 means up.

Give an algorithm for constructing a full binary tree with \(n>1\) terminal vertices.

Report on the formulas for the number of nonisomorphic free trees and for the number of nonisomorphic rooted trees with \(n\) vertices
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