homework4

# homework4 - nodes intermediate and final values for , , and...

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Nau: Game Theory 1 Updated 10/5/11 Problem 4.1. Here is the game tree for the Sharing Game, and a table showing several different strategy profiles. For each strategy profile, tell whether or not it is a subgame-perfect equilibrium. d

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Nau: Game Theory 2 Updated 10/5/11 1,0 0,2 3,1 2,4 5,3 4,6 6,5 1 2 1 2 1 2 R R R R R R L L L L L L 4,6 5,3 1,0 0,2 3,1 2,4 Homework 4.2. At right is a game tree for a version of the Centipede game. Suppose each player looks only at his/her own payoff values, and runs the minimax algorithm on those values. (a) What value will each player compute for each node? (b) Do the above values equal the player’s maximin values, minimax values, or both? (c) If the answer to (b) is “both”, then can you give an example of a perfect-information non-zero- sum game in which the maximin and minimax values differ?
Nau: Game Theory 3 Updated 10/5/11 Homework 4.3. Above is a game tree for a perfect-information zero-sum game. Run the alpha-beta algorithm on this game tree. Next to each node, write all of the

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Unformatted text preview: nodes intermediate and final values for , , and v . v : Nau: Game Theory 4 Updated 10/5/11 v : 1 1 1 1 1 1 1 1 Homework 4.4. At each node x of the tree shown below, let v ( x ) be x s minimax value. Suppose Max uses an evaluation function e ( x ) that returns v ( x ) with probability 0.9, and v ( x ) with probability 0.1. At the root node, what is Maxs probability of choosing the best move if Max searches (a) to depth 1? (b) to depth 2? (c) to depth 3? Nau: Game Theory 5 Updated 10/5/11 Homework 4.5. Let x and y be P-game boards of size 2 2 (hence it is Mins move at both x and y). Suppose that e(x) = e(y) = 1/2, where e is the evaluation function that counts the fraction of 1 squares in the board. (a) What is the probability that x is a forced win for Max? Explain why. (b) Suppose Max is trying to choose whether to move to x or to y. What is the probability that Max is at a critical node?...
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## homework4 - nodes intermediate and final values for , , and...

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