1. Game playing

2. Imperfect Information
Type of games
Deterministic
Chance
Imperfect Information
Backgammon
Bridge
Perfect Information
Chess
Here we consider Games with Perfect Information

3. Example - Tic-Tac-Toe O X O X O X -10 O X O X O X O X 10 -10 10 O X O
-10 O X O X O X O X 10
-10 10 O X O X O X O X O X O X 10 -1 10

4. Minimax search
MAX node
MIN node
value computed
by minimax
f value

5. Minimax search

6. Minimax algorithm

7. Properties of minimax Complete? Yes (if tree is finite)
Optimal? Yes (against an optimal opponent)
Time complexity? O(bm)
Space complexity? O(bm) (depth-first exploration)
For chess, b ≈ 35, m ≈100 for "reasonable" games  exact solution completely infeasible

8. Alpha-beta pruning
We can improve on the performance of the minimax algorithm through alpha-beta pruning.
We don’t need to compute the value at this node.
No matter what it is it can’t effect the value of the root node. 2 1 2 7 1 ?

9. Alpha-beta pruning

10. Alpha-beta pruning

11. Alpha-beta pruning

12. Why is it called α-β?
Define β similarly for min

13. The Alpha-Beta Procedure
There are two rules for terminating search:
Search can be stopped below any MIN node having a beta value less than or equal to the alpha value of any of its MAX ancestors.
Search can be stopped below any MAX node having an alpha value greater than or equal to the beta value of any of its MIN ancestors.

14. The α-β algorithm

15. The α-β algorithm

16. Properties of α-β Pruning does not affect final result
Good move ordering improves effectiveness of pruning
With "perfect ordering," time complexity = O(bm/2)
 doubles depth of search

17. The Alpha-Beta Procedure
Example:
max
min
max
min

18. The Alpha-Beta Procedure
Example:
max
min
max
 = 4
min 4

19. The Alpha-Beta Procedure
Example:
max
min
max
 = 4
min 4 5

20. The Alpha-Beta Procedure
Example:
max
min
 = 3
max
 = 3
min 4 5 3

21. The Alpha-Beta Procedure
Example:
max
min
 = 3
max
 = 3
 = 1
min 4 5 3 1

22. The Alpha-Beta Procedure
Example:
max
 = 3
min
 = 3
max
 = 3
 = 1
 = 8
min 4 5 3 1 8

23. The Alpha-Beta Procedure
Example:
max
 = 3
min
 = 3
max
 = 3
 = 1
 = 6
min 4 5 3 1 8 6

24. The Alpha-Beta Procedure
Example:
max
 = 3
min
 = 3
 = 6
max
 = 3
 = 1
 = 6
min 4 5 3 1 8 6 7

25. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
min
 = 3
 = 6
max
 = 3
 = 1
 = 6
min 4 5 3 1 8 6 7

26. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
min
 = 3
 = 6
max
 = 3
 = 1
 = 6
 = 2
min 4 5 3 1 8 6 7 2

27. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
min
 = 3
 = 6
 = 3
max
 = 3
 = 1
 = 6
 = 2
min 4 5 3 1 8 6 7 2

28. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
min
 = 3
 = 6
 = 3
max
 = 3
 = 1
 = 6
 = 2
 = 5
min 4 5 3 1 8 6 7 2 5

29. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
min
 = 3
 = 6
 = 3
max
 = 3
 = 1
 = 6
 = 2
 = 4
min 4 5 3 1 8 6 7 2 5 4

30. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
 = 4
min
 = 3
 = 6
 = 4
max
 = 3
 = 1
 = 6
 = 2
 = 4
min 4 5 3 1 8 6 7 2 5 4 4

31. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
 = 4
min
 = 3
 = 6
 = 4
max
 = 3
 = 1
 = 6
 = 2
 = 4
 = 6
min 4 5 3 1 8 6 7 2 5 4 4 6

32. The Alpha-Beta Procedure
Example:
 = 3
max
 = 3
 = 4
min
 = 3
 = 6
 = 4
max
 = 3
 = 1
 = 6
 = 2
 = 4
 = 6
min 4 5 3 1 8 6 7 2 5 4 4 6 7

33. The Alpha-Beta Procedure
Example:
 = 4
max
 = 3
 = 4
min
 = 3
 = 6
 = 4
 = 6
max
 = 3
 = 1
 = 6
 = 2
 = 4
 = 6
min 4 5 3 1 8 6 7 2 5 4 4 6 7 7

34. The Alpha-Beta Procedure
Example:
 = 4
Done!
max
 = 3
 = 4
min
 = 3
 = 6
 = 4
 = 6
max
 = 3
 = 1
 = 6
 = 2
 = 4
 = 6
min 4 5 3 1 8 6 7 2 5 4 4 6 7 7

35. Games that include chance
Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-16) and (5-11,11-16)

36. Games that include chance
chance nodes
Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-16) and (5-11,11-16)
[1,1], [6,6] chance 1/36, all other chance 1/18

37. Games that include chance
[1,1], [6,6] chance 1/36, all other chance 1/18
Cannot calculate definite minimax value, only expected value

38. Expected minimax value
EXPECTED-MINIMAX-VALUE(n)=
UTILITY(n) If n is a terminal
maxs  successors(n) MINIMAX-VALUE(s) If n is a max node
mins  successors(n) MINIMAX-VALUE(s) If n is a max node
s  successors(n) P(s) . EXPECTEDMINIMAX(s) If n is a chance node
These equations can be backed-up recursively all the way to the root of the game tree.

39. Position evaluation with chance nodes
Left, A1 wins
Right A2 wins
Outcome of evaluation function may change when values are scaled differently.
Behavior is preserved only by a positive linear transformation of EVAL.

40. Alpha-beta pruning in games with chance
We can improve on the performance of the minimax algorithm for games with chance through alpha-beta pruning.
We don’t need to compute the value at these nodes. 3 4
- 4
0.5 ?
- 1 5
 = 1
 2.5
 = 0.5
Suppose -5  e(n)  5
= 3 * (- 1) * 0.5 = 1.0