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Jarrar.lecture notes.aai.2011s.ch6.games

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Games Chapter 6 Dr. Mustafa Jarrar University of Birzeit [email_address] www.jarrar.info Lecture Notes, Advanced Artificial Intelligence (SCOM7341) University of Birzeit 2 nd Semester, 2011 Advanced Artificial Intelligence (SCOM7341)
Can you plan ahead with these games
Game Tree (2-player, deterministic, turns) Calculated by utility function, depends on the game. Last state, game is over
Two-Person Perfect Information Deterministic Game Your Moves My Moves My Moves Your Moves Your Moves My Moves My Moves My Moves ,[object Object],[object Object],[object Object],[object Object]
Computer Games ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Minimax ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Minimax Tree Max Min Max Min
Minimax Tree Evaluation ,[object Object],[object Object],[object Object],[object Object]
Minimax Tree Max Min Max Min 100 -24 -8 -14 -73 -100 -5 -70 -12 -3 70 4 12 -3 21 28 23 ,[object Object],[object Object]
Minimax Tree Evaluation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Minimax Tree -24 -8 -14 -73 -100 -5 -70 -12 -3 70 4 12 -3 21 28 23 Max Min Max Min ,[object Object]
Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -3 70 4 12 -3 21 28 23 28 -3 12 70 -3 -73 -14 -8 Max Min Max Min ,[object Object]
Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -4 70 4 12 -3 21 28 23 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 Max Min Max Min
Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -4 70 4 12 -3 21 28 23 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 -3 Max Min Max Min
Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -4 70 4 12 -3 21 28 23 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 -3 Max Min Max Min The best next move for Max
MiniMax Example-2 ,[object Object],4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 Max Max Min Min
MiniMax Example-2 ,[object Object],4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 Max Max Min Min
MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 Max Max Min Min
MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 Max Max Min Min
MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 5 Max Max Min Min
MiniMax Example-2 ,[object Object],Max Max Min Min 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 5
Properties of minimax ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Pruning the Minimax Tree ,[object Object],[object Object],[object Object],[object Object]
 Cut Example 100 21 -3 12 70 -4 -73 -14 -3 -4 -3 -73 Max Max Min
 Cut Example ,[object Object],100 21 -3 12 -70 -4 -73 -14 Max Max Min
 Cut Example ,[object Object],[object Object],100 21 -3 12 -70 -4 -73 -14 21 Max Max Min
 Cut Example ,[object Object],[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 Max Max Min
 Cut Example ,[object Object],[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 12 Max Max Min
 Cut Example ,[object Object],[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 -70 Max Max Min
 Cut Example ,[object Object],[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 -70 Max Max Min
 Cut Example ,[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 -70 -73 Max Max Min
 Cut Example ,[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 -70 -73 Max Max Min
 Cut Example ,[object Object],100 21 -3 12 -70 -4 -73 -14 -3 -3 -70 -73 Max Max Min
 Cuts ,[object Object],[object Object]
 Cut Example ,[object Object],100 21 -3 12 70 -4 73 -14 Min Min Max 21 21 70 73
Alpha-Beta Example 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[-∞, +∞] [-∞, +∞] ,[object Object],[object Object],Max Min
Alpha-Beta Example 2 Max Min [-∞, 7] [-∞, +∞] ,[object Object],[object Object],7
Alpha-Beta Example 2 Max Min [-∞, 6] [-∞, +∞] ,[object Object],[object Object],7 6
Alpha-Beta Example 2 Max Min 5 [5, +∞] ,[object Object],[object Object],7 6 5 ,[object Object],[object Object],[object Object]
Alpha-Beta Example 2 Max Min [-∞, 5] [5, +∞] ,[object Object],[object Object],7 6 5 ,[object Object],[object Object],[object Object],[-∞,3] 3
Alpha-Beta Example 2 Max Min [-∞, 5] [5, +∞] ,[object Object],[object Object],7 6 5 ,[object Object],[object Object],[-∞,3] 3
Alpha-Beta Example 2 Max Min [-∞, 5] [5, +∞] ,[object Object],[object Object],7 6 5 ,[object Object],[object Object],[-∞,3] 3 [-∞,6] 6
Alpha-Beta Example 2 Max Min [-∞, 5] [5, +∞] ,[object Object],[object Object],7 6 5 ,[object Object],[object Object],[-∞,3] 3 [-∞,5] 6 5
Alpha-Beta Example 2 Max Min [-∞, 5] 5 ,[object Object],[object Object],7 6 5 ,[object Object],[object Object],[object Object],[-∞,3] 3 [-∞,5] 6 5
Exercise max min max min
Exercise (Solution) max min max min 10 9 14 2 4 10 14 4 10 4 10
 -  Pruning ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
 -  Pruning ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Pruning at MAX node ,[object Object],[object Object],[object Object]
Pruning at MIN node ,[object Object],[object Object],[object Object]
Idea of  -  Pruning ,[object Object],[object Object],100 21 -3 12 70 -4 21 21 70
Why is it called α-β? ,[object Object],[object Object],[object Object],[object Object]
Imperfect Decisions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Utility Evaluation Function ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Utility Evaluation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Utility Evaluation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Evaluation Functions ,[object Object],[object Object],[object Object]
Evaluation Functions for Ordering ,[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object]
Example: Tic-Tac-Toe (evaluation function) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Tic-Tac-Toe 1-Ply E(s12) 8 - 6 = 2 E(s13) 8 - 5 = 3 E(s14) 8 - 6 = 2 E(s15) 8 - 4 = 4 E(s16) 8 - 6 = 2 E(s17) 8 - 5 = 3 E(s18) 8 - 6 = 2 E(s19) 8 - 5 = 3 X X X X X X X X X E(s11) 8 - 5 = 3 E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4
Tic-Tac-Toe 2-Ply E(s2:16) 5 - 6 = -1 E(s2:15) 5 -6 = -1 E(s28) 5 - 5 = 0 E(s27) 6 - 5 = 1 E(s2:48) 5 - 4 = 1 E(s2:47) 6 - 4 = 2 E(s2:13) 5 - 6 = -1 E(s2:9) 5 - 6 = -1 E(s2:10) 5 -6 = -1 E(s2:11) 5 - 6 = -1 E(s2:12) 5 - 6 = -1 E(s2:14) 5 - 6 = -1 E(s25) 6 - 5 = 1 E(s21) 6 - 5 = 1 E(s22) 5 - 5 = 0 E(s23) 6 - 5 = 1 E(s24) 4 - 5 = -1 E(s26) 5 - 5 = 0 E(s1:6) 8 - 6 = 2 E(s1:7) 8 - 5 = 3 E(s1:8) 8 - 6 = 2 E(s1:9) 8 - 5 = 3 E(s1:5) 8 - 4 = 4 E(s1:3) 8 - 5 = 3 E(s1:2) 8 - 6 = 2 E(s1:1) 8 - 5 = 3 E(s2:45) 6 - 4 = 2 X X X X X X X X X E(s1:4) 8 - 6 = 2 X O X O X O E(s2:41) 5 - 4 = 1 E(s2:42) 6 - 4 = 2 E(s2:43) 5 - 4 = 1 E(s2:44) 6 - 4 = 2 E(s2:46) 5 - 4 = 1 O X O X O X O X X O X O X O X O X X O X O O X X O X O X O X O X O X O X O X O X O O E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4
Checkers Case Study 31 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23
Checkers MiniMax Example 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 27 16 -> 11 31 -> 27 31 -> 24 22 -> 13 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -8 -8 1 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23
Checkers Alpha-Beta Example 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 27 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23 ,[object Object],[object Object],MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1
Checkers Alpha-Beta Example 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 27 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23 ,[object Object],[object Object],MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1
Checkers Alpha-Beta Example 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1  cutoff : no need to examine further branches MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1
Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1  cutoff : no need to examine further branches MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 13 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  0 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  -4  cutoff : no need to examine further branches MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
Checkers Alpha-Beta Example 22 -> 31 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  -8 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1

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Jarrar.lecture notes.aai.2011s.ch6.games

  • 1. Games Chapter 6 Dr. Mustafa Jarrar University of Birzeit [email_address] www.jarrar.info Lecture Notes, Advanced Artificial Intelligence (SCOM7341) University of Birzeit 2 nd Semester, 2011 Advanced Artificial Intelligence (SCOM7341)
  • 2. Can you plan ahead with these games
  • 3. Game Tree (2-player, deterministic, turns) Calculated by utility function, depends on the game. Last state, game is over
  • 4.
  • 5.
  • 6.
  • 7. Minimax Tree Max Min Max Min
  • 8.
  • 9.
  • 10.
  • 11.
  • 12.
  • 13. Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -4 70 4 12 -3 21 28 23 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 Max Min Max Min
  • 14. Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -4 70 4 12 -3 21 28 23 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 -3 Max Min Max Min
  • 15. Minimax Tree 100 -24 -8 -14 -73 -100 -5 -70 -12 -4 70 4 12 -3 21 28 23 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 -3 Max Min Max Min The best next move for Max
  • 16.
  • 17.
  • 18. MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 Max Max Min Min
  • 19. MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 Max Max Min Min
  • 20. MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 5 Max Max Min Min
  • 21.
  • 22.
  • 23.
  • 24. Cut Example 100 21 -3 12 70 -4 -73 -14 -3 -4 -3 -73 Max Max Min
  • 25.
  • 26.
  • 27.
  • 28.
  • 29.
  • 30.
  • 31.
  • 32.
  • 33.
  • 34.
  • 35.
  • 36.
  • 37.
  • 38.
  • 39.
  • 40.
  • 41.
  • 42.
  • 43.
  • 44.
  • 45. Exercise max min max min
  • 46. Exercise (Solution) max min max min 10 9 14 2 4 10 14 4 10 4 10
  • 47.
  • 48.
  • 49.
  • 50.
  • 51.
  • 52.
  • 53.
  • 54.
  • 55.
  • 56.
  • 57.
  • 58.
  • 59.
  • 60.
  • 61. Tic-Tac-Toe 1-Ply E(s12) 8 - 6 = 2 E(s13) 8 - 5 = 3 E(s14) 8 - 6 = 2 E(s15) 8 - 4 = 4 E(s16) 8 - 6 = 2 E(s17) 8 - 5 = 3 E(s18) 8 - 6 = 2 E(s19) 8 - 5 = 3 X X X X X X X X X E(s11) 8 - 5 = 3 E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4
  • 62. Tic-Tac-Toe 2-Ply E(s2:16) 5 - 6 = -1 E(s2:15) 5 -6 = -1 E(s28) 5 - 5 = 0 E(s27) 6 - 5 = 1 E(s2:48) 5 - 4 = 1 E(s2:47) 6 - 4 = 2 E(s2:13) 5 - 6 = -1 E(s2:9) 5 - 6 = -1 E(s2:10) 5 -6 = -1 E(s2:11) 5 - 6 = -1 E(s2:12) 5 - 6 = -1 E(s2:14) 5 - 6 = -1 E(s25) 6 - 5 = 1 E(s21) 6 - 5 = 1 E(s22) 5 - 5 = 0 E(s23) 6 - 5 = 1 E(s24) 4 - 5 = -1 E(s26) 5 - 5 = 0 E(s1:6) 8 - 6 = 2 E(s1:7) 8 - 5 = 3 E(s1:8) 8 - 6 = 2 E(s1:9) 8 - 5 = 3 E(s1:5) 8 - 4 = 4 E(s1:3) 8 - 5 = 3 E(s1:2) 8 - 6 = 2 E(s1:1) 8 - 5 = 3 E(s2:45) 6 - 4 = 2 X X X X X X X X X E(s1:4) 8 - 6 = 2 X O X O X O E(s2:41) 5 - 4 = 1 E(s2:42) 6 - 4 = 2 E(s2:43) 5 - 4 = 1 E(s2:44) 6 - 4 = 2 E(s2:46) 5 - 4 = 1 O X O X O X O X X O X O X O X O X X O X O O X X O X O X O X O X O X O X O X O X O O E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4
  • 63.
  • 64. Checkers MiniMax Example 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 27 16 -> 11 31 -> 27 31 -> 24 22 -> 13 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -8 -8 1 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23
  • 65.
  • 66.
  • 67. Checkers Alpha-Beta Example 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1  cutoff : no need to examine further branches MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1
  • 68. Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
  • 69. Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1  cutoff : no need to examine further branches MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
  • 70. Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  1 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
  • 71. Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 13 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  0 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
  • 72. Checkers Alpha-Beta Example 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 22 -> 31 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  -4  cutoff : no need to examine further branches MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1
  • 73. Checkers Alpha-Beta Example 22 -> 31 1 0 -4 -8 20 -> 16 21 -> 17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22 -> 25 22 -> 26 23 -> 26 23 -> 27 21 -> 14 16 -> 11 31 -> 27 16 -> 11 31 -> 27 31 -> 22 16 -> 11 31 -> 27 31 -> 24 22 -> 18 23 -> 30 23 -> 32 20 -> 16 31 -> 27 31 -> 26 21 -> 17 20 -> 16 21 -> 17 20 -> 16 20 -> 16 21 -> 17 31 1 2 3 4 8 6 5 9 10 11 12 16 14 13 17 18 19 20 24 22 21 25 26 27 28 32 30 29 7 15 23  1  -8 MAX MAX MIN 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1