1. PENGANTAR INTELIJENSIA BUATAN (64A614)
Kuliah 5 : Game Playing
Fakultas Teknik Jurusan Informatika
Universitas Surabaya

2. Outline
Game Playing
Game Trees
Minimax
Alpha-Beta Pruning

3. Games
Some games:
Ball games
Card games
Board games
Computer games
...

4. Human Game Playing

5. Garry Kasparov and Deep Blue, 1997
Computer Game Playing
Garry Kasparov and Deep Blue, 1997

6. Game-Playing Agent
sensors
actuators
agent
environment ?
Environment

7. Type of games

8. Typical Case 2-person game Players alternate moves
Zero-sum: one player’s loss is the other’s gain
Perfect information: both players have access to complete information about the state of the game. No information is hidden from either player.

9. Typical Case No chance (e.g., using dice) involved
Examples: Tic-Tac-Toe, Checkers, Chess, Go, Nim, Othello
Not : Bridge, Solitaire, Backgammon, ...

10. Two-player games A game formulated as a search problem:
Initial state: board position and turn
Successor function(Action) : definition of legal moves
Terminal state: conditions for when game is over
Utility function: a numeric value that describes the outcome of the game. E.g., -1, 0, 1 for loss, draw, win. (payoff function)

11. The Minimax Algorithm
Basic idea : Choose move with highest minimax value = Best achievable payoff against best play.
Algorithm :
Generate game tree completely.
Determine utility of each terminal state.
Propagate the utility values upward in the tree by applying MIN and MAX operators on the nodes in the current level.
At the root node use minimax decision to select the move with the max (of the min) utility value.
Steps 2 and 3 in the algorithm assume that the opponent will play perfectly.

12. Tic-Tac-Toe

14. Generate Game Tree

15. Generate Game Tree x x x x

16. Generate Game Tree x x o x o x o

17. Generate Game Tree x
1 ply
1 move x o x o x o

18. A subtree win lose draw x o x o x o x o x x o x o x o x o x o x o o o

19. What is a good move? win lose draw x o x o x o x o x x o x o x o x o x

20. Minimax Minimize opponent’s chance Maximize your chance 3 8 12 4 6 145
Minimize opponent’s chance
Maximize your chance

21. Minimax Minimize opponent’s chance Maximize your chance 3 2 8 12 4 614 5
MIN
Minimize opponent’s chance
Maximize your chance

22. Minimax Minimize opponent’s chance Maximize your chance 3 2 8 12 4 614 5
MAX
MIN
Minimize opponent’s chance
Maximize your chance

23. Minimax Minimize opponent’s chance Maximize your chance 3 2 8 12 4 614 5
MAX
MIN
Minimize opponent’s chance
Maximize your chance

24. Minimax = Maximum of the minimum
1st ply
2nd ply

25. MAX A B C MIN D E F G MAX 1 1 -3 4 1 2 -3 4 -5 1 -7 2 -3 -8
Select this move B C
MIN 1 -3 D E F G
MAX 4 1 2 -3 4 -5 1 -7 2 -3 -8
= terminal position
= agent
= opponent

26. MiniMax search on Tic-Tac-Toe
Evaluation function Eval(n) for A
infinity if n is a win state for A (Max)
-- infinity if n is a win state for B (Min)
(# of 3-moves for A) -- (# of 3-moves for B) a 3-move is an open row, column, diagonal

27. MiniMax search on Tic-Tac-Toe
A is X
Eval(s) = 6 - 4

28. Tic-Tac-Toe MiniMax search, d=2

29. Tic-Tac-Toe MiniMax search, d=4

30. Tic-Tac-Toe MiniMax search, d=6

31. Partial Game Tree for Tic-Tac-Toe
f(n) = +1 if the position is a win for X.
f(n) = -1 if the position is a win for O.
f(n) = 0 if the position is a draw.

32. 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

33. Minimax Exercise 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

34. Alpha-Beta Prunning
A disadvantage of MinMax search is that it has to expand every node in the subtree to depth m. Can we make a correct MinMax decision without looking at every node?
We want to prune the tree: stop exploring subtrees for which their value will not influence the final MinMax root decision
Observation: all nodes whose values are greater (smaller) that the current minimum (maximum) need not be explored !

35. Do We Have To Do All That Work ?
MAX
MIN 3 12 8

36. Do We Have To Do All That Work ?3
MAX 3
MIN 3 12 8

37. Do We Have To Do All That Work ?3
MAX 3 2
MIN 3 12 8 2
Since 2 is smaller than 3, then there is no need for
further search

38. Do We Have To Do All That Work ?3
MAX 3 X 2
MIN 3 12 8 14 5 2

39. Alpha-Beta Pruning Generally applied optimization on Mini-max.
Instead of:
first creating the entire tree (up to depth-level)
then doing all propagation
Interleave the generation of the tree and the propagation of values.
Point:
some of the obtained values in the tree will provide information that other (non-generated) parts are redundant and do not need to be generated.

40. Alpha-Beta Pruning
MIN
MAX 2 2 1 2 =2 1 5

41. - The (temporary) values at MAX-nodes are ALPHA-values
- The (temporary) values at MIN-nodes are BETA-values
Alpha-value
MIN
MAX 2 2 5 =2 2 1 1
Beta-value

42. Alpha-Beta pruning Example

43. Alpha-Beta pruning Example

44. Alpha-Beta pruning Example

45. Alpha-Beta pruning Example

46. Alpha-Beta pruning Example

47. Properties of Alpha-Beta pruning
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

48. Alpha-Beta pruning Example
 6
MAX
MIN 6

49. Alpha-Beta pruning Example
 6
MAX
MIN 6
 2 6 12 8 2

50. Alpha-Beta pruning Example
 6
MAX
MIN 6
 2
 5 6 12 8 2 5

51. Alpha-Beta pruning Example
 6
MAX
Selected move
MIN 6
 2
 5 6 12 8 2 5

52. A B C MAX D E F G MIN H I J K L M MAX 6 6 2 beta cutoff 6 >=8
alpha cutoff 2 H I J K L M
MAX 6 5 8 2 1
= agent
= opponent

53. Alpha-Beta pruning Example8 7 3 9 1 6 2 4 5
 4 16
 5 31 39
= 5
MAX 6
 8
 5 23 15
= 4 30
= 5
 3 38
MIN 33
 1 2
 8 10
 2 18
 1 25
 3 35
 2 12
 4 20
 3 5
= 8 8
 9 27
 9 29
 6 37
= 3 14
= 4 22
= 5
MAX 1 3 4 7 9 11 13 17 19 21 24 26 28 32 34 36
11 static evaluations saved !!

54. Example of a perfectly ordered tree
MAX
MIN 21

55. Exercise Minimax Alpha Beta Pruning A B C D E F G H I J K L M N O P QV W Y X 2 3 8 5 7 6 1 4 10
Max
Min