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Game Trees – The Minimax Method

Game Trees – The Minimax Method. Anthony Brown COT 4810 January 25, 2008. Minimax. Mini mize your max imum losses Works best on: Complete information games Two Person games Zero-Sum games Opponent is logical. Minimax History. Money Money Money John Von Neumann Chess Chess Chess

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Game Trees – The Minimax Method

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  1. Game Trees – The Minimax Method Anthony Brown COT 4810 January 25, 2008

  2. Minimax Minimize your maximum losses Works best on: Complete information games Two Person games Zero-Sum games Opponent is logical

  3. Minimax History Money Money Money John Von Neumann Chess Chess Chess Claude Shannon

  4. Minimax Generate all nodes in a game tree Score each leaf node Score each MIN node with min(children) Score each MAX node with max(children) MAX (Player) MIN MAX

  5. Minimax – Leaves evaluated MAX MIN MAX $ 3 0 9 5 0 2 4 0 3 6 1

  6. Minimax – Max(children) MAX MIN MAX 3 $ 9 5 2 4 6 1 3 0 9 5 0 2 4 0 3 6 1

  7. Minimax – Min(children) MAX MIN 3 5 2 1 MAX 3 $ 9 5 2 4 6 1 3 0 9 5 0 2 4 0 3 6 1

  8. Minimax – final Max(Children) MAX 5 MIN 3 5 2 1 MAX 3 $ 9 5 2 4 6 1 3 0 9 5 0 2 4 0 3 6 1

  9. Prisoner’s Dilemma – non-zero sum MAX Keep Quiet Rat out Prisoner B MIN Keep Quiet Rat out Prisoner A Keep Quiet Rat out Prisoner A A B 6 months/ 6 months 10 years/ Go free Go free/ 10 years 5 years/ 5 years

  10. Minimax pseudo code If N is a leaf return eval(N) If this level is a MAX return max(children) If this level is a MIN return min(children)

  11. OPTIMIZATIONS Variable cutoff levels Alpha Beta Pruning Heuristics – Check power moves first Transposition Tables

  12. Alpha beta pruning Post-order traversal of the tree Alpha – recorded as the largest value scanned by a MAX node Beta – recorded as the smallest value scanned by a MIN node Prune when Alpha rises above Beta Or Prune when Beta falls below Alpha Max ? Min A D Max 7 B 4 E 9 C

  13. Large Alpha Beta Tree Max 5 Min 2 5 4 Max 7 2 5 8 4 9 Min 3 7 2 5 2 8 4 9 1 2 4 3 8 7 9 2 5 6 2 8 4 9 1 2 6

  14. Alpha Beta Efficiency Pruning can reduce the number of leaf nodes scanned by the square root of non-pruned Minimax trees. Best Case: O(bd/2) b = # of children d = # of recursions (turns)

  15. Transposition Tables Used when many moves can reach the same position Positions are stored in a hash table for speed Takes up much more memory

  16. Homework Questions Find the path to the best position using the Minimax algorithm on this tree. How could Minimax fail in the real world? Why is tic-tac-toe solvable and chess unsolvable with Minimax? 1 2 3 4 0 3 1 0 1 7 9 7 5 8 3 1

  17. Bibliography Dewdney, A. K. The New Turing Omnibus. New York: Henry Holt, 1989. 36-41. Shah, Rajiv Bakulesh "Minimax", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 10 January 2007. (accessed TODAY) Available from: http://www.nist.gov/dads/HTML/minimax.html

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