1. Min-Max Trees Based on slides by: Rob Powers Ian Gent Yishay Mansour

2. Two Players Games One Search Tree for both Players –Even layers – Max Player move –Odd Layers – Min Player move The state evaluated according to heuristic function.

3. MinMax search strategy Generate the whole game tree. (Or up to a constant depth) Evaluate Terminal states (Leafs) propagate Min-Max values up from leafs Search for MAX best next move, so that no matter what MIN does MAX will be better off For branching factor b and depth search d the complexity is O(b d )

4. MinMax first Example

7. 1

8. 1-2

9. 1-2 1

10. Cuting Off Search We want to prune the tree: stop exploring subtrees with values that will not influence the final MinMax root decision In the worst case, no pruning. –The complexity is O(b d ). –In practice, O(b d/2 ), with branching factor of b 1/2 instead of b.

11. Alpha Beta First Example

12. Alpha and Beta values At a Max node we will store an alpha value –the alpha value is lower bound on the exact minimax score –the true value might be   –if we know Min can choose moves with score <  then Min will never choose to let Max go to a node where the score will be  or more At a Min node,  value is similar but opposite Alpha-Beta search uses these values to cut search

13. Alpha Beta in Action Why can we cut off search? Beta = 1 < alpha = 2 where the alpha value is at an ancestor node At the ancestor node, Max had a choice to get a score of at least 2 (maybe more) Max is not going to move right to let Min guarantee a score of 1 (maybe less)

14. Alpha and Beta values Max node has  value –the alpha value is lower bound on the exact minimax score –with best play Max can guarantee scoring at least  Min node has  value –the beta value is upper bound on the exact minimax score –with best play Min can guarantee scoring no more than  At Max node, if an ancestor Min node has  <  –Min’s best play must never let Max move to this node therefore this node is irrelevant –if  = , Min can do as well without letting Max get here so again we need not continue

15. Alpha-Beta Pruning Rule Two key points: –alpha values can never decrease –beta values can never increase Search can be discontinued at a node if: – Max node: the alpha value is  the beta of any Min ancestor this is beta cutoff –Min node: the beta value is  the alpha of any Max ancestor this is alpha cutoff

16. #2b Left->Right  = - ,  = + 

17. #2b Left->Right  = - ,  = + 

18. #2b Left->Right  = - ,  = + 

19. #2b Left->Right  = - ,  = +   = - ,  = 4  = - ,  = + 

20. #2b Left->Right  = - ,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4

21. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4

22. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = + 

23. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = + 

24. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = + 

25. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = +   = 8,  = +   = 4,  = + 

26. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = +   = 8,  = +   = 4,  = +  = 8,  = + 

27. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = +   = 8,  = +   = 4,  = +  = 8,  = + 

28. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = +   = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = + 

29. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = +   = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +  (Alpha pruning)

30. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +  (Alpha pruning)

31. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8 (Alpha pruning)

32. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8 (Alpha pruning)

33. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8 (Alpha pruning)

34. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)

35. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)

36. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8

37. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 5,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8

38. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 5,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 8

39. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 5,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 8

40. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 5,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8

41. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 5,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6

42. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 5,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6

43. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 6,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6

44. #2b Left->Right  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 6  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 6,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6

45. #2b Left->Right  = 6,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 6  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 6,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6

46. #2b Left->Right  = 6,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 6  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 6,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6  = 6,  = + 

47. #2b Left->Right  = 7,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 6  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 6,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8  = 5,  = 6  =5,  =8  =5,  =6  = 6,  = + 

48. Beta Pruning  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 4,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8 9

49. Beta Pruning  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 9,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8 9

50. Beta Pruning  = 4,  = +   = - ,  = 4  = - ,  = +  = - ,  = 4  = 4,  = 8  = 8,  = +   = 4,  = +  = 8,  = 6  = 8,  = +   = 9,  = 8  = 4,  = 3  = 4,  = 8 (Alpha pruning)  = 4,  = 8 9 (Beta pruning)

51. #2b Right->Left  = - ,  = + 

52. #2b Right->Left  = - ,  = + 

53. #2b Right->Left  = 7,  = +   = - ,  = + 

54. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = + 

55. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = + 

56. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = + 

57. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = + 

58. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 12  =7,  =+ 

59. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 12  =7,  =+ 

60. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+ 

61. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)

62. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = + 

63. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = + 

64. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = + 

65. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = + 

66. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+ 

67. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)

68. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)

69. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)

70. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)

71. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)  = 7,  = + 

72. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)  = 7,  = + 

73. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)  = 7,  = 7  = 7,  = + 

74. #2b Right->Left  = 7,  = +   = - ,  = +  = 7,  = 6  = 7,  = +   = 7,  = 6  =7,  =+  (Alpha pruning)  = 7,  = +  = 7,  = -1  =7,  =+  (Alpha pruning)  = 7,  = 7  = 7,  = +  (Alpha pruning)

75. - 2a