1. Pengantar Kecerdasan Buatan
GAME PLAYING

2. Two-player games
The object of a search is to find a path from the starting position to a goal position
In a puzzle-type problem, you (the searcher) get to choose every move
In a two-player competitive game, you alternate moves with the other player
The other player doesn’t want to reach your goal
Your search technique must be very different

3. Permainan yang melibatkan 2 orang berarti melibatkan juga unpredictable opponent
Search space game besar & biasanya ada batas waktu

4. Types of Games
battleship
Kriegspiel
Not Considered: Physical games like tennis, croquet, ice hockey, etc.
(but see “robot soccer”

5. Game Trees
This is an example of a partial game tree for the game tic-tac-toe.
Even for this simple game, the game tree is very large.

6. Minimax Minimax is a method used to evaluate game trees.
A static evaluator is applied to leaf nodes, and values are passed back up the tree to determine the best score the computer can obtain against a rational opponent.

7. Game Trees

8. Minimax – Animated Example3 6
The computer can obtain 6 by choosing the right hand edge from the first node.
Max 5 1 3 6 2 7 6
Min 5 3 1 3 7
Max 6 5

9. Properties of minimax Complete? Yes (if tree is finite). Optimal?
Can it be beaten by an opponent playing sub-optimally?
No. (Why not?)
Time complexity?
O(bm)
Space complexity?
O(bm) (depth-first search, generate all actions at once)
O(m) (backtracking search, generate actions one at a time)

10. Searching Game Trees
Exhaustively searching a game tree is not usually a good idea.
Even for a game as simple as tic-tac-toe there are over 350,000 (9!) nodes in the complete game tree. [9! If computer goes first, 8! If computer goes second]
For chess :
b ≈ 35 (approximate average branching factor)
d ≈ 100 (depth of game tree for “typical” game)
bd ≈ ≈ nodes!!
exact solution completely infeasible
It is usually impossible to develop the whole search tree.

11. Applying MiniMax to tic-tac-toe
The static heuristic evaluation function

12. Backup Values

14. −∞ means MIN wins.
+∞ means MAX wins.

15. Alpha-beta Pruning
A method that can often cut off a half the game tree.
Based on the idea that if a move is clearly bad, there is no need to follow the consequences of it.
alpha – highest value we have found so far
beta – lowest value we have found so far

16. Another Alpha-Beta Example
Do DF-search until first leaf
Range of possible values
[-∞,+∞]
[-∞, +∞]

17. Alpha-Beta Example (continued)
[-∞,+∞]
[-∞,3]

18. Alpha-Beta Example (continued)
[-∞,+∞]
[-∞,3]

19. Alpha-Beta Example (continued)
[3,+∞]
[3,3]

20. Alpha-Beta Example (continued)
[3,+∞]
This node is
worse for MAX
[3,3]
[-∞,2]

21. Alpha-Beta Example (continued),
[3,14]
[3,3]
[-∞,2]
[-∞,14]

22. Alpha-Beta Example (continued),
[3,5]
[3,3]
[−∞,2]
[-∞,5]

23. Alpha-Beta Example (continued)
[3,3]
[3,3]
[−∞,2]
[2,2]

24. Alpha-Beta Example (continued)
[3,3]
[3,3]
[-∞,2]
[2,2]

25. Alpha-beta Algorithm Depth first search
only considers nodes along a single path from root at any time
a = highest-value choice found at any choice point of path for MAX
(initially, a = −infinity)
b = lowest-value choice found at any choice point of path for MIN
(initially,  = +infinity)
Pass current values of a and b down to child nodes during search.
Update values of a and b during search:
MAX updates  at MAX nodes
MIN updates  at MIN nodes
Prune remaining branches at a node when a ≥ b

26. When to Prune Prune whenever  ≥ .
Prune below a Max node whose alpha value becomes greater than or equal to the beta value of its ancestors.
Max nodes update alpha based on children’s returned values.
Prune below a Min node whose beta value becomes less than or equal to the alpha value of its ancestors.
Min nodes update beta based on children’s returned values.

27. Alpha-Beta Example Revisited
Do DF-search until first leaf
, , initial values
=−
 =+
, , passed to kids
=−
 =+

28. Alpha-Beta Example (continued)
=−
 =+
=−
 =3
MIN updates , based on kids

29. Alpha-Beta Example (continued)
=−
 =+
=−
 =3
MIN updates , based on kids.
No change.

30. Alpha-Beta Example (continued)
MAX updates , based on kids.
=3
 =+
3 is returned
as node value.

31. Alpha-Beta Example (continued)
=3
 =+
, , passed to kids
=3
 =+

32. Alpha-Beta Example (continued)
=3
 =+
MIN updates ,
based on kids.
=3
 =2

33. Alpha-Beta Example (continued)
=3
 =+
=3
 =2
 ≥ ,
so prune.

34. Alpha-Beta Example (continued)
MAX updates , based on kids.
No change.
=3
 =+
2 is returned
as node value.

35. Alpha-Beta Example (continued)
=3
 =+ ,
, , passed to kids
=3
 =+

36. Alpha-Beta Example (continued)
=3
 =+ ,
MIN updates ,
based on kids.
=3
 =14

37. Alpha-Beta Example (continued)
=3
 =+ ,
MIN updates ,
based on kids.
=3
 =5

38. Alpha-Beta Example (continued)
=3
 =+
2 is returned
as node value. 2

39. Alpha beta pruning example

40. Example
-which nodes can be pruned? 6 5 3 4 1 2 7 8

41. Answer to Example -which nodes can be pruned? Max Min Max 6 5 3 4 1 27 8
Answer: NONE! Because the most favorable nodes for both are explored last (i.e., in the diagram, are on the right-hand side).

42. Second Example (the exact mirror image of the first example)
-which nodes can be pruned? 4 3 6 5 8 7 2 1

43. Answer to Second Example (the exact mirror image of the first example)
-which nodes can be pruned?
Max
Min
Max 4 3 6 5 8 7 2 1
Answer: LOTS! Because the most favorable nodes for both are explored first (i.e., in the diagram, are on the left-hand side).

44. Checker Example
Black to move,
White = “Min”,
Black = “Max”

45. Chess Evaluation Functions
Alan Turing’s f(n)=(sum of your piece values)- (sum of opponent’s piece values)
More complex: weighted sum of positional features:
Deep Blue has > 8000 features
(IBM Computer vs. Gary Kasparov)
Pawn
1.0
Knight
3.0
Bishop
3.25
Rook
5.0
Queen
9.0
Pieces values for a simple Turing-style evaluation function

46. Min Max another example

48. Alpha Beta Another Example