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3- Game Tree_solution

# 3- Game Tree_solution - CS188 Spring 2011 Section 3 Game...

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CS188 Spring 2011 Section 3: Game Trees 1 Warm-Up: Column-Row You have a 3x3 matrix of values like the one below. In a somewhat boring game, player A first selects a row, and then player B selects a column. The game is over after these two moves and the outcome of the game is the value in the square that lies in the intersection of the chosen row and column. 1 8 6 7 3 4 5 9 2 For the following questions, assume that player A wants to maximize the final number that is selected. For each question state which action the player takes and justify your decision in one sentence. (a) What is player A’s move if player B is trying to minimize the final number? Draw out the corresponding game tree. A should pick the middle row, resulting in the final outcome of 3. (b) What is player A’s move if player B is moving randomly? Draw out the corresponding game tree, labeling non-leaf nodes with their expected value assuming B moves randomly and A maximizes expected value. A should pick the bottom row, resulting in an expected value of 5.33. (c) What is player A’s move if player B shares A’s value function (i.e. wants to maximize the final value)? Draw out the corresponding game tree. A should pick the bottom row, resulting in a final outcome of 9. 1

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2 Min-Max Search In this problem, we will explore adversarial search. Consider the zero-sum game tree shown below. Trapezoids that point up, such as at the root, represent choices for the player seeking to maximize; trapezoids that point down represent choices for the minimizer. Outcome values for the maximizing player are listed for each leaf node. It is your move, and you seek to maximize the expected value of the game. (a) Assuming both opponents act optimally, carry out the min-max search algorithm. Write the value of each node inside the corresponding trapzoid. What move should you make now? How much is the game worth to you?
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