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Chapter 9: Problem 11

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

Short Answer

Expert verified
The value of each vertex in the game tree can be found by working backward from the terminal vertices towards the root, taking into account whether each decision point is used to maximize or minimize the outcome.

Step by step solution

01

Identify the Game Tree

Start by identifying the given game tree structure and the terminal vertices. The game tree is a graphical representation of a sequential game. Terminal vertices, found at the ends of the tree, are the final outcomes which have assigned values.
02

Work Backwards (Backward Induction)

This method involves starting from the end (terminal vertices) and moving backwards toward the start (root) of the game tree. In other words, determine the value of each preceding vertex based on the values of its subsequent vertices.
03

Determine the Value of Each Vertex

The value of each non-terminal vertex is determined by the player whose decision is represented at that vertex. If the player is trying to maximize, they will choose the decision that leads to the highest value. If they are trying to minimize, they will choose the decision that leads to the lowest value.
04

Complete for Entire Tree

Continue this process for all vertices, always choosing the decision leading to the highest or lowest value depending on the player's objective. Ultimately, the value at the root of the tree will represent the outcome of the optimal strategy for the game.

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