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# GSP 480 Week 4 Homework and iLab

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Overview

For this week you will be asked to work AI algorithms on paper rather than coding them to access your understanding. The problems are specifically designed to test your understanding so you should make sure you complete all readings and have a good understanding of the weekly topics before completing the homework. Please ask in the discussions if you have any questions.

Part 1 –Minimax

Imagine a game where the ending states are awarded points according to figure 5.2 from your textbook on page 164. Figure 5.2 indicates a tree where the idea is to maximize your point value based on your opponent’s next move. Please answer the following questions:

1)Assuming both opponents play optimally what is the maximum value that the first opponent (max) can achieve from any move he makes?

2)Assume that opponent 2 (min) has had a few too many drinks and there is a % chance that he may make a suboptimal move. What is the best point value outcome opponent 1 (max) can hope for?

3)Let’s assume both opponents play 100 games from this same exact point in the game pictured in figure 5.2. Opponent 2 (min) has exactly a 10% chance of picking the worst possible move (but you don’t know when that might happen). The rest of the time he plays optimally. What is the move opponent max should make to maximize his 100 game total score?

4)Continuing the scenario from question 3 -- What if the % chance of min picking the worst move is x%. What is the minimum value of x such that MAX should alter his choice of moves from assuming both opponents play optimally? Show your calculation or explain how you came to this conclusion.

Part 2 –Alpha-Beta pruning

Again refer to figure 5.2 from page 164 of your text. Assume that MAX now wishes to minimize his score and MIN wishes to maximize his. Go through a step by step process of how the alpha-beta pruning algorithm works on this new situation. Show each step of the algorithm similar to how it is presented in figure 5.5 on page 168. At each step show the current intervals for nodes A,B,C and D. Make sure you clearly indicate which nodes are pruned and not considered for evaluation

Questions 2 and 3 no longer refer to figure 5.2 and instead are asking questions about a general search tree so you should give your answers in terms of the depth of the tree and/or branching factors of the tree.

1)In the optimal case, how many nodes can be pruned using alpha-beta pruning? Give your answer in terms of the branching factor of the tree and depth.

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